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The differential production model with quasi-fixed inputs

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The differential production model with quasi-fixed inputs A panel data approach to U.S. banking
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THE DIFFERENTIAL PRODUCTION MODEL WITH QUASI-FIXED INPUTS:
A PANEL DATA APPROACH TO U.S. BANKING

















By

GRIGORIOS T. LIVANIS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2004
































Copyright 2004

by

Grigorios T. Livanis
































This dissertation is dedicated to my parents, Theodosios and Konstantina; my brothers,
Harilaos and Ioannis; and the love of my life, Maria Chatzidaki, who made this happen.














ACKNOWLEDGMENTS

First and foremost, I would like to express my deep gratitude and sincere

appreciation to my advisor, Dr. Charles B. Moss, for his outstanding guidance,

encouragement, and advice during my graduate studies and the development of this

dissertation. He has always been a source of motivation and inspiration. I would like

especially to acknowledge Dr. Elias Dinopoulos for the endless discussions, advice, and

encouragement during the research process that contributed to the quicker completion of

this dissertation. Sincere appreciation is also extended to the other members of my

committee Dr. James Seale, Dr. Timothy Taylor and Dr. Mark Flannery for their

guidance, and constructive criticisms that led to improvements in this dissertation.

I would like to express my immeasurable gratitude to my parents, Theodosios and

Konstantina Livanis; and my brothers, Harilaos and loannis Livanis, for their continuous

love and moral support, despite the distance. I especially thank my parents, who taught

me that I could achieve anything that I committed myself to fully. In the last years of my

studies I was privileged to have my brother, loannis, studying at the same University. His

humor and support made those years more enjoyable.

Finally, I would like to express my deepest love and gratitude to my partner in life,

Maria Chatzidaki, for all of her love, support and sacrifice. Without her by my side, I

would not have reached my goals successfully. Words cannot express how thankful I am

to be sharing my life with someone so loving, patient, and thoughtful.















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ................................................................................................. iv

LIST O F TA BLES ............................................................................................................ vii

ABSTRACT............................................................ viii

CHAPTER

I INTRODUCTION AND OBJECTIVES ................................................................. 1

1. 1 Introduction....................................... .......................................................... 1
1.2 O objectives .............................................................................................. ........ 4
1.3 O overview ............... ........................................................................................ 5

2 M ETH O D O LO G Y .................................................................................................. 7

2.1 Introduction................. .................................................................................. 7
2.2 The Case of Multiple Quasi-Fixed Inputs .......................................................8
2.3 Cost M inim ization ....................................................................................... 11
2.3.1 Returns to Scale and Elasticities of Variable Cost............................12
2.3.2 Factor and Product Shares .............................................................17
2.3.3 Marginal Shares of Variable Inputs ..................................................20
2.3.4 Input Demand Equations................................................................21
2.3.5 Comparative Statics in Demand.....................................................29
2.4 Conditions for Profit Maximization ................................... ...................33
2.4.1 O utput Supply ..................................................................... .......... 35
2.4.2 Comparative Statics in Supply......................................................41
2.5 Rational Random Behavior in the Differential Model ...................................43
2.6 Comparison to the Original LT Model ........................................................46

3 PARAMETERIZATION AND ALTERNATIVE SPECIFICATION .......................48

3.1 Input Demand Parameterization ..................................................................48
3.1.1 The Case of Multiple Quasi-Fixed Inputs...........................................50
3.1.2 The Case of One Quasi-Fixed Input ...........................................55
3.2 Output Supply Parameterization..................................................................56
3.3 Alternative Specification for the Cost-Based System ....................................57
3.4 Capacity Utilization and Quasi-Fixity ......................................................... 61









4 ESTIMATION METHODS.....................................................................................64

4.1 Choice of Estimation Method......................................................................64
4.2 Fixed Effects and Pooled Model ..................................................................73
4.3 Random Effects ...........................................................................................84

5 APPLICATION TO U.S. BANKING INDUSTRY ..............................................89

5.1 Introduction......................... ........................................................................ 89
5.2 The US Banking Industry in the 90s............................................................91
5.3 Brief Literature Review ...............................................................................92
5.4 D ata D description .......................................................................................... 96
5.5 Em pirical M odel ........................................................................................ 102
5.6 Em pirical R esults.........................................................................................109

6 SUMMARY AND CONCLUSIONS ..................................................................122

APPLNDIX ANALYTICAL GRADIENT VECTOR ..............................................128

A.1 Gradient Vector for Section 4.1 ................................................................128
A.2 Gradient Vector for Section 4.2................................................................ 129

LIST OF REFERENCES ............................................................................................. 130

BIOGRAPHICAL SKETCH ......................................................................................... 137















LIST OF TABLES


Table page

5-1 Financial indicators for the U.S. banking industry, 1990-2000............................92

5-2 Definition of variables and descriptive statistics (mean
and standard deviation) .................................................................................... 100

5-3 Parameter estimates and standard errors for the differential model,
1990-2000 .................................................. ................................................1......

5-4 Parameter estimates and standard errors for the translog, 1990-2000.................114

5-5 C oncavity test..................................................................................................... 116

5-6 Allen-Uzawa elasticities of substitution........................................................... 118

5-7 Economies of scale for the mean size U.S. bank, 1990-2000 ......................... 121














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE DIFFERENTIAL PRODUCTION MODEL WITH QUASI-FIXED INPUTS:
A PANEL DATA APPROACH TO U.S. BANKING

By

Grigorios T. Livanis

August 2004

Chair: Charles B. Moss
Major Department: Food and Resource Economics

This study assesses the empirical and policy implications of using the differential

approach in opposition to dual specifications for the decisions of the multiproduct firm.

In applied production analysis, the dual specifications of the firm's technology usually

fail to satisfy the theoretical properties of the cost or profit function. If the validity of

those properties is not examined, then empirical results should be interpreted with

caution. On the other hand, the differential production model of the multiproduct firm has

rarely been tested empirically, since it was first developed by Laitinen and Theil in 1978.

The novelty of this study is that it generalizes the differential production model for

the multiproduct firm to account for quasi-fixed inputs in production; and to account for

production technologies that are not output homogeneous, as assumed in the original

model. Another objective of this study was to provide alternative parameterizations of the

differential model, to account for variable coefficients over time. For this reason a

supermodel was developed that contains different specifications that can be tested by








simple parameter restrictions. Further, maximum likelihood estimators were provided for

the case of panel data in the differential model. The contribution of these estimators to the

econometrics' literature was the consideration of nonlinear symmetry constraints for the

differential model under balanced and unbalanced panel data designs.

The extended differential production model was applied to the U.S. banking

industry for the period 1990-2000. To assess the empirical results of the differential

model (and to provide a direct comparison with a dual specification), a translog cost

function was applied to the same dataset. Results indicated that the differential model is

consistent with economic theory, while the translog specification failed to satisfy the

concavity property of the cost function for each year in the sample. Concerning the Allen

elasticity of substitution both models found similar results. One disadvantage of the

differential model was the assumption of perfect competition, which resulted in total

revenue over total cost being the measure of scale economies.













CHAPTER 1
INTRODUCTION AND OBJECTIVES


1.1 Introduction

This study extends the multiproduct differential production model, developed by

Laitinen and Theil (1978), to incorporate quasi-fixed inputs and applies this formulation

to the U.S. banking industry. The differential approach differs from the dual

specifications of cost and profit functions that have become the cornerstone of the

literature in applied production analysis. Specifically, in the differential approach there is

no particular specification of the firm's true technology, and thus it can describe different

technologies without being exact for any particular form. The differential approach

entails differentiation of the first-order conditions in a cost or profit optimization

problem, to attain the input-demand and output-supply equations, respectively.

In contrast, the dual approach involves specifying a flexible functional form for the

cost or profit function, to describe the firm's technology, which yields a system of

equations to be estimated (e.g., a translog cost function with respective input shares).

Thus, it can be considered as an approximation in the space of the variables (quantities

and prices), while the differential approach is an approximation in parameter space. The

disadvantage of the dual approach is that usually different functional forms lead to

different results for the same dataset, as Howard and Shumway (1989) indicated, often

failing to satisfy parameter restrictions. Especially, concavity restrictions tend to be

nonlinear and more difficult to impose (Diewert and Wales 1987); and as a result, few






2

empirical studies examine the concavity of their results in detail (exceptions are

Featherstone and Moss 1994, Salvanes and Tjotta 1998, see also Shumway, 1995 for a

recent survey of studies testing various parameter restrictions).

Numerous models have been developed for analyzing consumer demand based on

the differential approach (Rotterdam, AIDS, CBS, NBR). Further, as demonstrated by

Barten (1993), Lee et al. (1994) and Brown et al. (1994) a number of competing systems

can be generated from alternative parameterizations of the differential system of demand

that was originally introduced by Theil (1965, 1976, 1980). Thus the form of consumer

demand can be selected through simple parameter restrictions.

In applied production analysis, a similar differential input-demand system was

developed by Theil (1977) and Laitinen and Theil (LT, 1978). The Theil (1977) model

concerns one-output transformation technologies, while the LT model extends to the

multiproduct case. However, neither model (especially not the LT model) has been used

much in empirical analysis because of their complexity. Exceptions include Rossi (1984),

who extended the LT model to account for fixed inputs. However, he assumed that the

production function was separable into variable and quasi-fixed inputs. Davis (1997)

provided an application of the Theil (1977) model; while Fousekis and Pantzios (1999)

generalized the Theil (1977) parameterization by including Rotterdam-type, CBS-type,

and NBR-type effects. Recently, Washington and Kilmer (2000, 2002) applied the LT

model in international agricultural trade. However, they assumed input-output

separability and independence, which transformed the model into a single output model.

Our study extends the LT model to account for quasi-fixed inputs that are not

separable from the variable inputs in the firm's technology. The model nests the Rossi








(1984) model, and a testable hypothesis is this separability. Further, in order to generalize

the LT model, the output homogeneous assumption for the transformation technology is

relaxed, and a comparison to the LT model is provided. Testable hypotheses were input

independence, output independence, and input-output separability, as in the LT model. In

the empirical section, the usual parameter restrictions of homogeneity and symmetry of

the cost or profit function are imposed; and the concavity of the cost function in input

prices and the convexity of the profit function in output prices were tested. Going one

step farther, alternative specifications of the extended model were provided, forming the

base for a test for quasi-fixity, based on a simple Hausman specification test

(Schankerman and Nadiri 1986) or on direct test of the coefficients of the estimated

model.

The proposed model was applied to the U.S. banking industry, giving specific

attention to the concavity property of the cost function. The banking industry was

selected because, most probably, the assumption of perfect competition in both input and

output markets of a specific bank will hold; and because of the availability of data. The

proposed model was compared with a standard transcendental logarithmic (translog)

specification with quasi-fixed inputs, which is the most common specification applied to

banking data. Comparison of the two models centers on whether the concavity property

of the cost function is rejected. Contributions in the field of production analysis often

check whether concavity is fulfilled by the estimated parameters of the cost function.

Since the seminal papers of Lau (1978) and Diewert and Wales (1987), concavity is often

directly imposed either locally or globally on the parameters. More recently, Ryan and

Wales (1998, 2000) and Mochini (1999) discussed further techniques to impose








concavity. Symmetry and homogeneity properties of the cost function can be regarded as

technical properties, since they are a result of the continuity property and the definition of

the cost function, respectively. On the other hand, concavity is the first property with true

economic context, since it is a result of the optimization process. For instance, Koebel

(2002) showed that a priori imposition of concavity may lead to estimation biases, when

aggregation across goods is considered. Further, a radical failure in concavity may in fact

be attributable to an inappropriate specification of the functional form.

Finally, traditional measures of efficiency (such as economies of scale) were

provided for the differential model and other measures of substitutability or

complementarity in the input and output sectors of the banks, such as Allen elasticities of

substitution. These measures were compared with those of the translog cost specification.

1.2 Objectives

Specific objectives of our study can be summarized in the following:

1. To mathematically derive the LT differential production model of multiproduct
firms under the assumption of quasi-fixed inputs and use a more general production
technology that is not output homogeneous as in the LT model.

2. To provide alternative parameterizations of the extended LT model, especially for
the cost-based system (input-demand system of equations). This will be useful for
deriving a new test for asset quasi-fixity.

3. To provide alternative econometric procedures using Maximum Likelihood
estimators for balanced and unbalanced panel data, for estimating the extended LT
model.

4. To apply the extended LT multiproduct model to the U.S. banking industry and to
econometrically estimate the system of derived-demand and output-supply
equations using the econometric methods developed in this study.

5. To compare the results of the extended LT model with those of a flexible functional
form specification, such as the translog. Specific attention was given to the
concavity property of the cost function in input prices. The two models were also
compared in terms of Allen elasticities of substitution and degree of economies of
scale.








1.3 Overview

Chapter 2 provides the mathematical derivation of the basic model used in this

analysis. It borrows heavily from the derivation techniques as presented by Laitinen

(1980), but differs in terms of the added generalizations of a non-homogeneous, in the

output vector, production technology; and of the existence of quasi-fixed inputs. Also, the

extended model was compared with the original LT model, showing that the assumption

of output-homogeneous production technology affects only the input-demand system and

does not need to be imposed.

The basic parameterization of the extended model, closely following Laitinen

(1980), is provided in Chapter 3. The novelty in this is the parameterization of the

coefficients of the quasi-fixed inputs in the input-demand system, and the development of

a "supermodel" for the cost-based system of equations. Specifically, the coefficients of

the quasi-fixed inputs are a function of the respective shadow price of the quasi-fixed

input. To parameterize those coefficients, the procedure of Morrison-Paul and

MacDonald (2000) was used, whereby shadow prices are decomposed to their ex-ante

market rental prices plus a deviation term. The "supermodel" for the cost-based system,

developed in this chapter, accommodates for a new test for asset quasi-fixity and different

assumptions on the estimated coefficients through simple parameter tests.

Chapter 4 concerns the econometrics of the differential approach. Section 4.1

presents the econometric issues related to the differential model, and the two step

Maximum Likelihood procedure, provided by Laitinen (1980). Since this procedure does

not conform to the data used in the empirical analysis, Maximum Likelihood estimators

were developed for time-specific, fixed-effects, and individual-specific, random-effects

panel data based on previous studies by Magnus (1982) and Biorn (2004). These








procedures are useful for systems of equations with balanced or unbalanced panel data

designs with nonlinear restrictions on the parameters.

Chapter 5 covers the empirical part of the present study. The time-specific, fixed-

effects econometric method, presented in Chapter 4 is adapted for estimating the

extended LT model and the translog specification for the banking industry. Then the

results of both models are compared in terms of rejection (or not) of concavity, and

elasticity measures.

Finally, Chapter 6 provides a summary, conclusions of the present study, and

presents unresolved issues for future research.














CHAPTER 2
METHODOLOGY


2.1 Introduction

The Laitinen-Theil (LT, 1978) model extends previous studies by Hicks (1946) and

Sakai (1974), to explicitly account for input-output separability, input independence,

homotheticity and non-jointness of production. It concerns long-run behavior of risk-

neutral multiproduct firms under competitive circumstances. Moreover, it is generally

applicable, since it does not require specific assumptions, such as input-output

separability or constant elasticities of scale or substitution. Before the LT model, Pfouts

(1961, 1964, and 1973) had extended the Hicks' model to account for fixed inputs, but it

was a special case since he assumed input-output separability and output independence.

In the empirical literature, the LT model has hardly been applied. To my knowledge, only

Rossi (1984) extended the LT model to account for fixed inputs (but he assumed

separability between variable and fixed inputs) and applied the model in Italian farms.

Washington and Kilmer (2000, 2002) were two other studies that used the LT model in

international trade of agricultural products. However, by assuming input and output

independence and input-output separability, the model became a single output model

(Theil 1977).

The advantage of the LT model is that it avoids the use of a functional form for the

dual specification (either cost or profit functions). That is, it does not specify a functional

form for the true technology of the firm. However, the parameterization of the model








provided by Laitinen (1980) implies constant price effects, and implies that the change in

the cost share of the i'h input due to the change in r'h product is also constant. Therefore,

there is a need for parameterization allowing for variable output and price effects.

Fousekis and Pantzios (1999) provided such a general model but for the single output

firm.

This chapter provides the general methodology and derivation of the short-run

system of input-demand and output-supply equations for a multiproduct firm, under

perfect competition in both markets of the firm. The model used was developed by LT,

but it was transformed to account for a more general transformation technology that does

not impose any restrictions on the returns to scale of the firm; nor imposes any restriction

on homogeneity, homotheticity, input-output separability, or any other separability

assumptions. These assumptions could be tested through parameter restrictions of the

model. Further, the LT model was extended to account for quasi-fixed inputs. Apart from

Clements (1978) and Rossi (1984), who used a transformation technology separable in

the fixed inputs, there is no other attempt to specify or extend and test a more general

model.

2.2 The Case of Multiple Quasi-Fixed Inputs

Let the production technology of a multiproduct, multifactor (MP-MF) individual

firm be represented by a transformation function:

T(x,y,z) = 0 (2-1)

where y e R" denotes a vector of variable outputs, x e R" a set of variable inputs and

z e Rk a set of quasi-fixed inputs (inputs that are difficult to adjust). Strictly positive








prices of outputs and inputs are denoted by p E R", and w E R" respectively. This

transformation technology satisfies certain regularity conditions (Lau 1972):

* The domain of T (x, y, z) is a convex set containing the origin.
* T(x, y. z) is convex and closed in {y, x, z}, in the nonnegative orthant R .
* T (x, y, z) is continuous and twice differentiable in y, x and z.
* T(x, y,z) is strictly increasing in y and strictly decreasing in x

Mittelhammer et al. (1981) showed that a single-equation multiproduct, multifactor

in an implicit form production function, is not as general as it was thought to be. The

production function shown by Equation 2-1 restricts each output to depend on all inputs,

and other outputs that appear as arguments in the implicit form. Further, they showed that

it cannot represent separability in the form of two independent functional constraints,

such as T(.) = g (.) + g2 (.), on the arguments of T(x, y, z). In such cases, the gradient

vector of T(x, y, z) is zero, which further implies that the Kuhn-Tucker conditions do not

hold. Therefore, our study did not examine separability of that form; and instead left it for

future research.

Assume that a MP-MF firm minimizes variable costs of producing the vector of

outputs y, conditional on the vector of quasi-fixed inputs z and fixed prices w for the

variable inputs. This short-run or restricted cost function can be denoted as

VC = VC (y, w; z), and it is assumed that it satisfies the following properties (Chambers

1988):

* VC (y, w; z) is monotonically non-decreasing, homogeneous of degree one and
concave in w.

* VC (y, w; z) is non decreasing and convex in y.









* VC(y,w; z) is non increasing and convex in z.

* VC(y, w; z) is twice continuously differentiable on (w, y; z).

Applying Shephard's lemma on the restricted cost function, the conditional factor

aVC
demands are then obtained as x, = = VC (y, w; z). If v denotes the vector of ex-
0w,

ante market rental prices of the quasi-fixed inputs, then the short-run total cost of

producing the vector y is given by SC = VC(y, w; z)+ v -z'. The long-run cost function

C(w,y) of the multiproduct firm is then obtained by minimizing short-run total cost with

respect to quasi-fixed inputs, while holding the variable inputs and the level of output at

the observed cost-minimizing levels. That is,

C(w,y) minSC min (VC(y, w;z)+ v z')

The first-order condition of this minimization problem implies that

aSC OVC(y,w;z) +
= +v=0
az az*

where z* denotes the static equilibrium levels of z. This condition can be written as

QVC(y,w;z')
-- (Y. = v. which states that a necessary condition for a firm to be in long-run
9z"

equilibrium is that the shadow prices of the quasi-fixed inputs be equal to the observed

ex-ante market rental prices v (Samuelson 1953). Therefore, the shadow price of a quasi-

fixed input is defined as the potential reduction in expenditures on other variable inputs

that can be achieved by using an additional unit of the input under consideration, while

maintaining the level of outputs. Further, Berndt and Fuss (1989) showed that when this

condition holds, temporary and full-equilibrium demand levels for the quasi-fixed inputs









are equal. The same result holds for the short-run and long-run marginal cost and

demands for variable inputs of the multiproduct firm.

2.3 Cost Minimization

For the multiproduct-multifactor firm, let y, be the r'h product (r = 1,...,m) to

which corresponds a price p,. Let x, be the ith factor of production (i = 1,...,n) whose

price is denoted by w, and zk be the k"' quasi-fixed factor of production (k = 1,...,1)

with an ex-ante market rental price denoted by vk. Assume a production function in an

implicit form that is not separable into the quasi-fixed inputs, as in Rossi (1984), nor is it

negatively linearly homogeneous in the output vector as in LT (1978); and assume that it

satisfies the properties mentioned in Section 2.2. Thus, it can be written as

T(x, y, z) = 0 (2-2)

Then in the short-run, the firm's objective is to minimize variable cost ( VC)

subject to its transformation technology, by varying the input quantities for given output

and input prices, and for given quasi-fixed input levels. Thus, the problem that the firm

faces is


min VC(w,x)- w,x,:T(x,y,z)=0 (2-3)
S=1

n
The Lagrangean of the above problem can be written as L = w,x, -AT(x,y,z) and the


first-order conditions needed to attain a minimum are given by the following equations:

9L 9T(.)
aL w x, T() =0 (2-4)
SIn x, In x,

6L
T(x,y, z)=O 0 (2-5)








In this formulation, A > 0 is implied by the positivity of x, and the assumption that the

marginal physical product of each input is positive (aT(.) / a In x, > 0). Further, Equations

2-4 and 2-5 are assumed to yield unique positive values for x, and A; and Equation 2-4

is a vector of n x 1.

The second-order conditions are given by the following equations:

a2L 82T
-= 8, w,x, A (2-6)
SIn x, n xn, x In x, In x,


where 3 is a Kronecker delta. That is 3 =0, i = j


a2L T a 'L
and = 0
2a In x, l 1nx, 9A

The solution of the minimization problem described in Equation 2-3 gives the

conditional or compensated short-run demands of the inputs as a function of all input

prices, output quantities, and quasi-fixed inputs. That is, x"s = xsR (w, y, z) and

As = A" (w, y, z), where xsR denotes the vector of inputs and As"R is the Lagrangean

multiplier. To obtain a minimum cost in the short-run, it is sufficient that the matrix of

the second order derivatives that has a size n x n (Equation 2-6), is symmetric and

positive definite. The minimum short-run cost is then given by

VC(w,y,z) = w,x,(w,y, Z) (2-7)


2.3.1 Returns to Scale and Elasticities of Variable Cost

Consider first the total differential of T(x, y, z) = 0 in natural logarithmic form:

n 9T "' aT ', aT
S-- dlnx, + dlny, d-lnzk = (2-8)
,= a n x, =, a 1n yr k = alnzk








The degree of returns to scale (RTS) is defined as the proportional increase in all outputs,

resulting from a proportional increase in all inputs, variable and quasi-fixed. Letting this

be the case, and defining x" = [x, z], then d In x, = d In x, = d In zk = d In z, and

d In y, = d In y, can each be put before its summation sign. Then we have

dln T +dlnz T z dln y, T
-cInx, = -lnzk = alnyr

This can also be written as

d nT 'x T "' TFy
Inx + +dlny, =0
,1 ln x, k1 a nzk r= In y,

Therefore.



R dlInyr y a In, x, k=1 anzk
RTS nx' (2-9)
d In x1 T
r=1 a In yr

Notice that this relationship for the returns to scale is the same as the relationship derived

by Caves et al. (1981).

The marginal cost of the r"' output can be found by taking the derivative of the

optimum variable cost function (Equation 2-7) with respect to output Y :

OVC Ox, VC In x,
= I = _Y f, (2-10)
Yr, y,. Yr, In y,

where f = is the variable cost share of input i and the last expression has been
VC

derived from the second by multiplying the second term by (yVCI/y,VC) and noting


that 0 In x, = I x,. Also, notice that the above equation can be written as
x,









SnVClnx, (2-11)
8 In y, lny,

Next, differentiating the optimum transformation technology T(x, y, z) = 0 with respect

to In yr, holding input prices, other outputs and quasi-fixed inputs constant, we get

aT 81n x BT
T ax, +0= 0 (2-12)
,= In x, In y, In y,

aT wxI
However, by using the first-order condition W= w' and by multiplying the first
SIn x, A

Y VC
term by y Equation 2-12 becomes
yVC

y -VC wx, ln x, aT w,x,
+ = 0, where f
Ayr y, VC ln y, a In y, VC

Using now Equation 2-10 the above expression can be written as

y, 9VC aT
Y + = 0 (2-13)
SaY, a In y,.

If we sum Equation 2-13 over r then we get

8VC
I 3VC '" aT r In y
S-0 or A =T l (2-14)
SZl Y ,=, Q1n y,. ST
A a In yaT
n anyr


Letting = then from Equation 2-14 we have that
VC

8 In VC
In yr
7 alnY (2-15)
VC n
Y a In y,








The elasticity of variable cost with respect to proportionate output changes, holding

quasi-fixed inputs constant, is obtained by substituting in Equation 2-15 the expression

for the lagrangean multiplier (A) from the first-order condition (Equation 2-4). That is,

fT
we substitute A = VC/ in Equation 2-15 to obtain
SIn x,

ST
8 ln VC 1y
lnVC_ Olnyr (2-16)

alny In xy
llnx,

To find the elasticity of variable cost with respect to proportionate quasi-fixed input

changes, we follow similar analysis as above, holding output constant. Therefore taking

the derivative of the optimum variable cost function with respect to a quasi-fixed input

we obtain

QVC x VC 8 Inx
ac W Ox--- = VC ln (2-17)
zk zk k IOlnZk

QVC
Notice that = w, denotes the shadow price of the quasi-fixed input. Also, from the
azk

analysis in Section 2.1, in order for the firm to be in long-run equilibrium, it has to be the

8VC
case that -z = vk where v, is the ex-ante market rental price of the quasi-fixed input.
azk

Further, Equation 2-17 can also be transformed into the following expression

8 In VC alnx,
= -- f (2-18)
SIn zk In zk

Now, taking the derivative of the optimum production technology T(x, y, z) = 0 with

respect to In zk, holding input prices, other quasi-fixed inputs, and outputs constant, we

get









a T Inx, aT
Sn, nz+--
, lnx, ln z, 8d In z,


Again, using the first-order condition, i
9 In x,


= w,, multiplying the first term of the
2


VC
above equation by z and using Equation 2-17 we obtain the following relationship
z VC


zk VC OT
+ z nz
2 az In zA


Summing this equation over k. we obtain the second interpretation for 2:

Ia("-
3VC
SaT
k lnzk
k alnz,

However. Equation 2-21 must be equal to Equation 2-14 implying the following

relationship


SVC


IaT
k kan z


alnyr


(2-20)


(2-21)









(2-22)


I-' 8lny


Solving for the elasticity of cost with respect to proportionate quasi-fixed input change

from the above equation, we obtain


al n FC
ln nz,


aT
S9lnz, yClnVC
S9T In yr
a In y,


(2-23)


which can also be written as


(2-19)









OT
z 1n z
e = V, k (2-24)

,r In y,

or equivalently, Equation 2-23 (through the use of Equation 2-16) can be written as

aT
8 In VC zk
C- VC k (2-25)
S n zk
In x,

Finally, taking into consideration Equations 2-16 and 2-25, the degree of returns to scale

(RTS) in terms of derivatives of the variable cost function (Equation 2-9) can be written

as

O T + OT 'nlnVC
,= anx, k= anzk a In z
RTS k (2-26)
S9T 8o In VC
r=1 In yr In y,

2.3.2 Factor and Product Shares

We have already defined the variable cost share of input i as

W X,
f 'x (2-27)
VC

Taking the total differential of Equation 2-27 we have

df = fd In w, + fd In x, fd In VC (2-28)

Summing Equation 2-28 over i and noting that = 1 and so d f = 0, we have
I I

d n VC = fd In w, + fd In x, (2-29)
1 /

or in a more compact form

d In VC = d in W + d In X (2-30)






18

where d n W = fd In w,, d n X = fd In x, are the Divisia indexes of variable input
I I

prices and variable input quantities, respectively (Divisia input price index and Divisia

input volume index).

Then considering Equation 2-14 for A, define as in Laitinen and Theil (1978)


gR Y, a aVC/aInyr I yT (2-31)
A yr VCl / Ca1ny, liny,


as the share of the r'' product in total variable marginal cost multiplied by a
a Iny,

Notice that if we had assumed negatively linear-homogeneous production function in the

aT
output vector, which implies that I -1, as in LT, then g, would be just the
in y,

share of the r'h product in total variable marginal cost. It is the case though that at the

point of the firm's optimum (from Equation 2-13):

aT
gr =- n (2-32)
a In y,.


Noting that gr, = y we can define the share of the r' product in total
r r In y,

variable marginal cost as

g, 9MVC / In y, .
V--=VcT/aln with rs=1.m


These shares are necessarily positive and have unit sum over r. Further, we can define


the Divisia volume index of outputs as d In Y = g d In y .

Similarly, considering Equation 2-21 for define
Similarly, considering Equation 2-21 for A, define









zk OVC aVC/alnzk T (2-)T
Pk (- I3)
t A ~ )zk J6VC/Olnzk nk (2-33)
k

as the share of the k"' quasi-fixed input shadow value in total shadow value of the quasi-

___T FuthT s u
fixed inputs, multiplied by Further, substituting for its equivalent
k, In Zk k lz

form from Equation 2-22 we obtain the ratio of the k'h quasi-fixed input shadow value in

AT
the variable marginal cost of m outputs, multiplied by ---O
a In y,

OVC//1nz T
Pk = v_ k9 C -In 2 IC OT (2-34)
a VC / n y, a in y,
I'

Using now Equation 2-31, the above equation transforms to

8VC / In z,
Pk VCalnz r=l,..., m (2-35)
SVC / ln y,.

Also, at the point of the firm's optimum (from Equations 2-33 and 2-20), it holds

9T
k aln (2-36)
a In z,


As in the case of outputs, note that / = -- Therefore, we can define the
k k alnzk

share of the k'1 quasi-fixed input shadow value in total shadow value of the quasi-fixed

inputs as

pk c / 8 n z, .
k aVC/alnzk with k,e =1,...,
2u, I VC / Inz


which are positive and have unit sum over k. Further, as in the case of outputs, the


Divisia volume index of quasi-fixed inputs is defined as d In Z = p / e dlnzk .
k e








2.3.3 Marginal Shares of Variable Inputs

Like in LT model, define the share of i'~ variable input in the marginal cost of the

r"h product as

a(w,x,) / Soy,
= V lyr (2-37)
VCl/gy,


Then multiply Equation 2-37 by -' and sum over r to get



'j g. or cVC/ lny, a(wx,)/yr
,= g, 0, VC/alny, OVC/dlny,


The above equation can be written as

a(w,x,)/aln y,
V, = C / (2-38)
r

Equation 2-38 defines the share of the i'" input in variable marginal cost of outputs.

Finally, as Laitinen and Theil mentioned, summation of ,', or 0, over i gives always

unity, but need not be non-negative.

In a similar fashion define the share of i'h variable input in the shadow price of

quasi-fixed input zk as

k =(w,x,)/ (2-39)
a VC / Ozk


Then multiply Equation 2-39 by and sum over k to get the share of the i'h
Lue


variable input in variable marginal cost of m outputs:









SPk k =_ a VC/lnzk a(w,x,)/zk
k P"Z. k aVC/alnz, a VC / z,
e G

which can be simplified to

Sa(w,x,) / a In k
k (2-40)
", _OVC / lnz,
k

As in the case of the outputs summation of 'k, over i is always unity but need not be

non-negative.

2.3.4 Input Demand Equations

The first step is to write the first-order conditions as identities and then to

differentiate them with respect to their arguments. That is, with respect to each output y,,

input prices w,, and quasi-fixed input quantity zk, in order to determine how the

optimum changes in response to changes in these given variables. Therefore, the first-

order conditions as identities are

aT(x(;w,y,z),y,z)
w,x, (w, y, z)- 2(w, y, z) -0 (2-41)
xlnx,

T(x(w. y, z). y, z) 0 (2-42)

Totally differentiating Equation 2-41 with respect to In Yr, In w and In zk, it gives the

following relationships, respectively

8Inx, 1nA T 02T T dInx, 2T
w, x,- A- 2 / 0 (2-43)
SIln y,. c1ny,. O lnx, I1nx x, In xa Dlny, a nx, In yr

SOlnx, 2 T aln2 xT 6 lnx(,
,, w,x, + w, x, '- A A -n=0 (2-44)
\alnw c In x, a In w, = ,1 In x-, In x, 8 In wJ









alnx, a1nA2 T n 82T alnx, ( )2
wUx -A A-A 0 (2-45)
0,lnzk alnzk, lnx, =, lnx,Olnx Olnzk In x,a In zk

Notice that Equation 2-43 represents n distinct equations, equal to the number of inputs.

However, if we consider all the outputs we are going to have nx m distinct equations.

Similar arguments can be used to show that Equations 2-44 and 2-45 represent nx n and

nxl (k = 1,...,I) distinct equations, respectively.

Then totally differentiating Equation 2-42 with respect to In y,, In w,, and In zk we have,

respectively


= lnx, ln y, a nyr


0T-- 0 (2-47)
8, In x, 8 In w

aT 81nx 8T
T+ 0 (2-48)
,=a 1nx, 8l nzk, lnzk

Since we differentiate with respect to each output, input price and quasi-fixed input level,

Equations 2-46 to 2-48 are vectors of dimension m x 1, n x 1 and l x 1, respectively.

The next steps for the derivation of the input-demand system consist of the

following

* Divide Equations 2-43 to 2-45 by variable cost (VC), use the definition of the cost

shares ', = f,, and use from the first-order conditions the relationship
VC
9T
A = wx, .
1 Inx,


* Multiply Equations 2-46 to 2-48 by and use the following relationships
VC
aT 9T A
g, = /k = and = y,.
a In y, 8 In/p, VC








These transformations give the following relationships


aIn2z
ln zk,


2A -l
VC 8 1n


c In A A
~'anw, VC


a2T a nx x 2
x, lnx, l n yr VC a nx


a2T Olnx, 0
8 In x8 In x, a In w,


2 T
_ 0
, 1ny,


A2 02T
VC =nx,1nz,0
VC a Inx, ln zk


SIn x,
OIn y,

SI anx,
a In w,


SOlnx,
,=1 aIn zk


Now the following matrices can be defined


F=diag(f,,...,f,,), H= a2T H,= 0 2T1
ainx,a Inxj, lnxa,1n y, n
nnxn r nxm


and H = -2T T
a Inx,a In zk ,

Therefore Equations 2-49 to 2-54 can be written for all combinations of inputs, outputs

and quasi-fixed inputs, in matrix form, as


(F- In x
lnx
(F Y H) ny
8 In y


SaIn A
-F a in y H,
In y'


a In x a In 2
(F-y7H) -l F i,, =n -F
a In w a In w


SIn x,
a 1nyr


a 1n 2
Sln y,


,f Ilnx,
0 In w


Olnx,
8 Inz,


(2-49)



(2-50)


An a2T a nx,
VC k 1nx, Inx, Inz,


(2-51)



(2-52)



(2-53)



(2-54)


(2-55)


(2-56)








a In x 0 In 2
(F-y,H)l F -i,, an= 7 H3
SIn z' In z


i,-F = 7, g'
8 In y1

a In x
i, F -0
ainF =0


alnx
a,- ,- = i^ In


(2-57)


(2-58)


(2-59)


(2-60)


Now, premultiply Equations 2-55 to 2-57 by F-' and combine with Equations 2-58 to

2-60, to form Barten's fundamental matrix equation


S1nnx
7, H) F-' i,, n yln
, 0 a in A
a In y'
i:, aaln /


F alnx
F--
a n w'
a n A
a n w'


F alnx1
a In x
a In z'


[YF-iHi
71g:


,F- 'H3
7/,f I


and solving for the matrix of the decision variables we obtain


F a1nx
a In y'
aIn 2
a n y'


F a1nx F aInx
a n w' a n z'


aIn 2
a ln w'


ain 2
aln z'


F-'(F-y,H)F-' iN 7,FH,
i' 0 y, g'


-I yF-'H3
0 7,j/'


From Magnus and Neudecker (1988), if A is a non-singular partitioned matrix


defined as A =[L A] [ (F
A,] A


-. H)F- i and the matrix D= A A2 'A
0 22 21


is also non-singular, then the inverse of matrix A is given by

A I' + A- 'A2D-'A21AI-' -A,' A- D-'
A-' A 1 -1 1
-D-'A H,,-' D-'


SF- (F


(2-61)








It follows then, that

* D = -i,F (F-y,H)- F i, which is a scalar. Using the property of the inverse of a

scalar, we get that D-' =
-iF(F-yH)-' F.i,

S- (F(F-IH) F.i,,iF(F- H) F
A AA'A,2D-'A,A=I, =F(F-z)H)-F F
i F(F ,H)- Fi)

F(F-y,H), Fi,
-A^ -Ai,(D- =

i:F(F-y,H)-' F.i

iF(F-y,H)- F.i,

As in LT, define / = i,',F(F yH)i Fi,, which is a positive scalar and implies that


Dl = --1. Then define the nx n matrix 0 = 0, 1 as


=1 F(F-Y,H)-'F (2-62)


This matrix is symmetric positive definite due to H being symmetric and positive

definite (sufficient condition in order to obtain a cost minimum). The above definitions

imply that 0 is normalized so that its elements add up to one:

n n
-)*i,, = 1 1,, (2-63)


Then we can define the n -element vector 0 as the row sums of 0:

n
0 = -i,,, 41, = 0 -1, (2-64)
I=1









F(F-y, H)-' F-i
Equation 2-64 can be written equivalently as 6 = i, = (F-yH)-' F which
iF(F-;,H) F-i,

implies that = -D-'A,,AI-,, (1 x n). Also, simple algebra shows that the following

relationships hold:


(2-65)


At this point there is one important distinction between this model and the LT

analysis. In the LT model 0 = 0, where 0 is the vector of the marginal shares 0, defined

in Equation 2-38. However, in the present model this relationship does not hold since the

proof, provided by LT. is conditional on the production function having the output

homogenous of degree one property.

Using these relationships (Equations 2-62 to 2-65) the inverse of matrix A can be

written as


A-'= 1


and so Equation 2-61 transforms to

F 1nx F a1nx F alnx 1 0
a In x a In x W In x
F F F
a In y' 1nw' 81nz' y F-'H,
InA 9ln2 8InA L -- Y1g'
L In y' a In wv' a In z'

Solving for the individual terms we obtain the following relationships

0 In x
F ,=x 7,1V ( ')F-'H, +7yg'
Sln y'

F In x
0 In w


-I y,F-'H,3
0 Yp'





(2-66)


(2-67)


i,,'o=1, 'i, ,=1, I,, -. = and i,,( ') =0, ( ') i =0









F =lnx (0 00')F-'H3 + 0y,u' (2-68)
a In z

0 In A ,i 1
= ',F-'H, -- g' (2-69)
a In y'
&ln y
n -0' (2-70)
0 In w

8 1n F 1
a = 0' HF 71 (2-71)
S1nz yz

Since the optimum variable input-demand equations are given by x* = x'(w, y,z)

then the differential demand for variable inputs can be found by taking the total

differential of this expression (logarithmic):

8 In x O In x O In x
dlnx= dn d y-- d n w+ -- dlnz
a In y' I1n w' 9 n z'

Premultiplying now this expression by F and using the solutions above, Equations 2-66

to 2-68, we obtain the system of differential input-demand equations:

Fdlnx= y,7V(0- 0')F-'H,+ +7y,g']dln -y/(Q(D- ')dlnw+

+[y7,/(qO- ')F-'3 + ypu']dlnz (2-72)

The coefficient of the output needs further transformation in order to have some

economic interpretation. For this reason, let g' = i',G, where G is an mx m diagonal

matrix with (g,,...,g,,) on the diagonal. Then, it is easy to show that

[y,y/(- ')F-'H, + 0,g']=y,[y(0--0')F-'H,G-' +i.]G (2-73)

From Equation 2-37 we have that

_= (wx,) /y, w,x, alnx, 1 VCf 1 lnx,
o VC / r VCI/y, alny, y, aVC/yry, yr 8n y,







which from Equation 2-31 can be rewritten as

SVC f 8lnx, f a Inx,
Ag,r aln yr 7gr lny,

The last member of this equation is the (i,r)' element of y, F G y' where
a n y')

A-
VC

Thus, from Equation 2-73, [,'] becomes


[or] ,F a lnx G y [q (- ')F-'HIG-' + i: GG-'
a In y

or equivalently,

[o:]= [V (- 0')F-' H,G-'+ i(]

The last expression can be rearranged to

[0,"]-. 1, =i1'((- O')F-'H,G-' (2-74)

Therefore from Equations 2-73 and 2-74 we can write the coefficient of d In y as

r, [[" ] + ] G = y [r :] G (2-75)

Following similar analysis for the coefficient of the quasi-fixed input let p' = i'M,

where M is an l x I diagonal matrix with (p,,..., p) on the diagonal. Then, as before, it

holds that

[y7,V/(O- ')F-'H, + y,p p'] =7, [V/( 0')F-'H3M-' +i,;]M (2-76)

In Equation 2-39 it was shown that the marginal share of the quasi-fixed input is given by

8 (w,x,) / k wx, a lnx, 1 VC. fa In x, ,ic i i
a k /zk z which further implies that
SVC/z, VCIz, 9\nz, z, zk Q kn z,








S8alnx h emInxet
k = lnx This is the (i,k)" element of y,' F nzM Combining then this
ry,u a Inzk

relationship and Equation 2-76, we obtain a simplified expression for [,k as


[ '] 1-1 F 31n M-1=Y1, 1 (a)- 0')F-' HVM-'+ Oi]MM-'.

This expression can be further simplified to

[ ] = [ _((D ') F- VHM-i + i']

Rearranging terms in this expression, we obtain

[ k ]-0,= (O-( ')F-'H3M-' (2-77)

Therefore the coefficient of d In z, using Equations 2-76 and 2-77, becomes

y [y(s- ')F 'HM-'+Iil M= 7, [ --+-]M =y [,k]M (2-78)

Finally, using Equations 2-75 and 2-78, the system of variable input-demand

equations can be written as Fd In x = y7 [' Gd In y + y, [ ] MdIn z -v(D 0')d n w,

with the i"' equation given by

fdlnx, = Ogd In y, + 7, ,k/pkdInz ,-VZ((, )dln w (2-79)
r=\ kt=l /=

2.3.5 Comparative Statics in Demand

The variable factor demand equation (Eq. 2-79) describes the change in the firm's

demand for variable inputs due to changes in input prices, output quantities and quasi-

fixed input levels. If all input price changes are proportional so that d In w, in Equation

2-79 can be put before the summation sign then the price term vanishes. This is obvious

bn n ?I
by noting the following relationship 1 (, j )= (,j)- 0, 1 ) = 0, =0, since
J=1 j=I J=1









(i)=1 from Equation 2-65. Therefore if output and quasi-fixed inputs remain


unchanged and all variable input prices change proportionately then the demand for

variable inputs remains unchanged. This property just verifies that the variable input

demands must be homogeneous of degree zero in input prices. Further, if (-, ) is

less than zero then the firm will increase the use of the i'h factor, when absolute price of

the j'' factor increases, ceteris paribus.

Turning now to volume changes, the total variable input decision of the firm can be

obtained by summing the factor demand, Equation 2-79, over i


.fdlnx, ,,g,=: dlny +yy ,'kpkdlnzk (j -,0 )ln w1
t=1 t=1 r=l i=1 k=l 1=1 J=1

n n n
Noting that 8r = 1, i^ = 1, and that V' (,, ,)= 0 from Equation 2-65, (last
t=l r=1 I=1

relationship), then the above equation can be written as


fd lnx, = y, g,d In y, + 7,Z/kdln Zk (2-80)
i=1 r=l k=l

This is the total variable input decision of the multiproduct firm and is equivalent to the

total differential of the production technology of the firm. At the optimum, it has been

aT T
shown that gr -- and pk Using these relationships, Equation 2-80
9 In y,. In z,

becomes





Here LT analysis uses the relative prices equation instead of the absolute price version
of the model, Equation 2-79, (see Laitinen and Theil (1978), pg. 41-45). However, this
does not affect our results.









f M T a.- 9T
d Inx, = In y, I dln zk
,= 1 rY l1 In yr k=1 a zk

f wx, VC 9T
Using now the definition of y, then =w VC where the last term follows
7, VC A dlnx,

from the first-order condition. Therefore, the equivalent form of the above equation is

S T m' T 9T
S d-dlnx +I --+flnyr+ dlnz, =0
a In x, r=, In yr k=1 lnzk

This is simply the logarithmic total differential of the production technology.

However, the factor demand and the total variable input decision can be written

into an equivalent form, which are more useful for the parameterization and estimation. If

we proceed by multiplying the first and second term of the right hand side of Equation

s, Y Pk
2-80 by = 1.- = 1 respectively, then the total variable input decision is



transformed to


fd Inx,=y g In y,+y Pk k d In z
iS r r1igS k k=1 -I P.z
S e

The Divisia volume index of variable inputs, outputs and quasi-fixed inputs have been


defined as d In X = fd n x, dln Y = g-d Iny, and dInZ = d In z
r=I I= gr k=1 kpe
v e

respectively. Further. by the definition of y, (Equation 2-15), gr (Equation 2-32) and
r

p/4 (Equation 2-34), we have the following expressions
k








n VC

72 =711g,= 1n ( y, 8T a n (2-81)

aT aln8 a ln y,n
r la In yr

I InVC VC '
y3-711 c9 I n yr k I)lnVC
(3 In y, k= a In yz


Therefore, we can write the total variable input decision of the firm (Equation 2-80) as

d InX =2d InY+yd In Z (2-83)

where 72, 7y are the elasticities of variable cost with respect to proportionate output

changes and quasi-fixed input changes, respectively.

Using the same technique as above, for the factor demand equation we obtain an

equivalent form of Equation 2-79:

fdlnx,=y2 tr 3d ;- Iny,+7Z" k d In zk (, )dln w (2-84)
,=I Kg" k=l P- e J=1

This expression is going to be useful for the parameterization of the factor demand.

The variable input allocation decision of the firm (when output changes are not

proportionate) can be found by multiplying Equation 2-83 by 0,, which gives

O,d In X y20,d In Y y730d In Z = 0, and putting this expression back into Equation 2-79:


fdlnx, =0,dlnX+y, 0gddIny,-1 g, 0, ny, +
r==1 = z g,


,Z d -r I q dln k )d I- n w,
k=1 k k=1 Z e Y=1
e


This expression is simply









fdlnx =,.dlnX + y,(O,'-O,)g,d In y, + y, ( dlnz,
r=l k=l


(0, 0)dIn wJ (2-85)
/=1

This is the input allocation decision of the firm. This decision describes the change in the

demand for the i'h input in terms of the Divisia volume index d In X change in output,

changes in the input prices and changes in the quasi-fixed inputs.

2.4 Conditions for Profit Maximization

Assume now that the firm's objective is to maximize profits (plus quasi-fixed

costs) for given input and output prices. That, is the firm wants to


max I(pwz) p,,- w,x such that T(x,y,z)=0 (2-86)
X,'y ( r I

Given the assumptions on the production technology (in the beginning of this

chapter) the profit function is non-negative and well defined for all positive prices and

any level of the quasi-fixed factors. Further, it is continuous, linear homogeneous and

convex in all prices, it is continuous, non-decreasing and concave in the quasi-fixed

factors and finally it is non-decreasing (non-increasing) in output prices (input prices) for

every fixed factor (McKay et al. 1983).

Assuming that we have a first-stage of cost minimization, which gives us the input

demands, then in the second stage we can maximize profits as a function only of y.

Therefore, the problem that the multiproduct, multifactor firm faces is transformed to


max (p,w,z) p,ry, -VC(w,y,z)
r=\


The first-order conditions of this maximization problem are









an aVc' VC
= P, = 0= = P, (2-87)
y, y, y,


Using Equation 2-31 for g,, where gr = Y then Equation 2-87 becomes


Ag, = p,.y,. Summing this expression over r and using the second term of Equation

2-31 we obtain the following

P, y. R R
A=- r-=- (2-88)

In y,. r In y,

where R = prYr denotes total revenue of the firm. Also, we obtain that the share of the
r


r'' product in total revenue, multiplied by -T is
8 Ilnyr

g r, OT (2-89)
R a In y,


Since gr = a notice that -g- Pr denotes the revenue share of the r'
In y,r g, R


product of the multiproduct firm. Further, using Equation 2-87, Equation 2-37 can be

a(wx, )
rewritten as 6,' = ) which is the additional expense on the i'1 input, incurred for
i( p, y,. )

the production of an additional dollar's worth of the r"' output.

2HI
For the second-order conditions to be valid, it must hold that is negative


i iVC
definite, for which it is sufficient that -- is symmetric positive definite, because
Oyay'









a2 R
= 0 follows from the assumption that the price vector is given. Therefore, we will
ayayi

8'VC
make the assumption that is symmetric positive definite.


This maximization problem will give us the unconditional output-supply equations

of the form y = y (p, w, z). Taking the logarithmic total differential of the output supply

we have


dlny i ln yy dlnp i n p ln dln w+ n y dlnz (2-90)
S9lnp 8 Ilnw 8l nz

2.4.1 Output Supply

The output supply of the multiproduct-multifactor firm has the form provided by

Equation 2-90. However, we need analytic expressions for the coefficients of

d In p, d In w and d In z in order to provide an estimable, with economic meaning form.

Proceeding the usual way, as in the derivation of the input-demand equation, we write the

first-order condition as an identity and then we totally differentiate with respect to its

arguments:

8VC(y(p. 1.z).w,z)
P, -0 (2-91)


Then taking the total differential of Equation 2-91 with respect to p,, w, and z, we

obtain the following relationships

in a2vc any,, a2vc a___
"' 8OyC Op2 "' cVC 0ln y,, 3r1pI (2-92)
P,: =.. y = 5,,P, (2-92)
v=1 ry 'V, aP, v=i yy, c In p,

02VC' "' 'VC ay, y ( 2VC 2VC
w, : 0 > -- (2-93)
y ,, y,.ayo w, Ow' ,y y') yOw'









2VC "' d VC ay, dy ('VC 'VC
zk: a-- + -0 :> (2-94)
k yZk yry aZk az' yWy) Oyaz'

However, Equation 2-92 needs further modification before it gets a familiar form.

Thus, solving for y, from Equation 2-89 we get

Rg I
y, = aT (2-95)
d In y,

Substituting this expression back into Equation 2-92 we obtain


a D2VC Rg,. 1 dln y,
SV',ay., pT a In p,
SIn y, ,

which for all (r.s) pairs in matrix form becomes


1 02VC a1ny
R _P-'G =P
Tar J yTy' In p'
0 In y,.

In this expression. P denotes an mx m diagonal matrix with the output prices on the

diagonal, G = diag(g ) and p is the vector of output prices. However, from Equation

aT
2-32 we have that Tr(G) = g,. =- ,where Tr denotes the trace operator.
d 1n yr

Therefore, the above equation can be written as

R VC P-'G, = P
Tr(G) Dyay' In p'

which is simplified to the following expression








G O1ny = V-1
Tr(G) a In p'P avyy'

Finally, simplifying the right-hand side of this expression, we get

G 1n y 1 p82VC .
G- Olnp= P ) P = C@ (2-96)
Tr(G) a In p' R ay8y' )

a2VC -' (8 VC
where we let V = -p p > 0 and ** = P\ 1 P .
R y') R [ayy')

At this point we need to bring Equations 2-93 and 2-94 into the same form as in

Equation 2-96. Beginning with Equation 2-93, pre-multiplying by P and post-
R

multiplying by W(= diag(w )), we get

I ay W j2VC 82VC
iP P' W (2-97)
R 8w' R yy') Syw'


Solving Equation 2-95 for R (for all (r,s) pairs) and using Tr(G)= _g = T
d lnyr

we obtain the following relationship for the total revenues of the multiproduct firm:

PY
R = Tr(G) (2-98)
G

Substituting Equation 2-98 back into Equation 2-97 we obtain

G y p VC 2VC
P YTr (G) aw' R oy y') 9yw'

After canceling terms in the left-hand side of the equation, this can be simplified to

G 1 y 1 (2VC )'2VC
Tr(G) Y Ow' W R yy' 9-y8w'

However, the left-hand side of this expression can be further simplified to get









G 8lny _1 p VC VC
Tr(G) anw' R [y y' 9yow'

Using then the definition of 'y*, we obtain

G Olny -_ 'K', where K=W P (2-99)
Tr(G) a In w Owy')


Using similar analysis for Equation 2-94, that is pre-multiplying by P and post-
R
2 -1
1 8 y I 8V 'C 8VC
multiplying by Z. we obtain P Z -- 1 Z, which from Equation
R dz' R 8y -8 yz'

2-98 becomes

G Oln y (2-C
-- Oy' ', where = Z P- (2-100)
Tr(G) alnz' 9z8y'

G
Therefore, pre-multiplying the differential output supply by and using the
Tr(G)

solutions from Equations 2-96, 2-99 and 2-100 we obtain

d In y = 1''d In p y O'K'd In w q'/ O'd In z (2-101)
Tr(G)

where 0 = [0' is an m x m symmetric positive definite matrix, which is normalized so

m m
that its elements add up to one, 0*, = 1. However, there is no clear interpretation for
r=l s=1

the coefficients of d In w and d In z. Starting with the input price coefficient, define K

as the nx m matrix that has the marginal shares 0, (Equation 2-37) as its (i,r)' element.

9VC
Then, from Shephard's lemma in vector form we have that =- x. If we differentiate
8w









a2VC ax
this relationship with respect to y' we obtain = However, from Equation 2-73
wawy' hy'

we have that


c 1n x
F nx
SIn y'


Substituting for [09,], this expression simplifies to

F ln xy ]G
SIn y' [O

Using the definitions of the terms in both sides of the equation, this expression can be

also written as


WX ax Y
v -Y=yKG =
VC 9y'X X


where G =


-PY
R )


2-88, respectively, and T;,


R 1 K
VC yT K


PY
R-
R *


R 1
and y, =- are derived from Equations 2-95 and
VC- T


BT
= a -T After some algebra the above equation can be
r 9 In Yr


8x ax
transformed to W =x K P. This expression can be solved for or K, in order to get
ay' ay'

ax avVC 2 VC
-- W-'K P and K = W P-', respectively. Therefore, the matrix of
ay' away' away'

marginal shares 6,r can be written as


or] = K = W P-
9way'


(2-102)


where P is an m x m matrix with the output prices or marginal costs on the diagonal,


depending on which are defined.


=7,[y/()- )F-'H,G-' +i,]G









Given Equation 2-102, we can write the s'h element of K'd n w as

n
d In W = -d nw, 2 (2-103)
1=1

This is the Frisch variable input price index (this is denoted by the superscript F). For

the coefficient of the quasi-fixed inputs notice that the (s, k)'h element of 'Q'd In z can

be written as

nil I ZVCIV1 aV
02VC d1lnz, =yy1'02, dlnzk (2-104)
Sk=1 a(Py,)Oanzk k=1 s=1 a(p,y)a lnzk

Then we can define

"' a2VC
4rk = *, (2-105)
"= 9(psy, ) Inz,

This can be interpreted as the sum of the changes in the marginal costs of the various

products due to the changes in the availability of quasi-fixed inputs, where the weights

are the coefficients 0,, which define the substitution or complementarity relationship in

production (see next section).

C g,
Noting that the r'h component of G is equal to and using Equations
Tr (G) g,


2-103 and 2-105 we can write the r' equation of the output supply, Equation 2-101, as

m m I
g, dlnYr =V/' ,dlnp.,- 0'VdlnW -y/rrkdlnzk (2-106)
S, .S=1 s=\ k=l


or in an equivalent form



2 F denotes that this is a Frisch price index, given that it has a marginal share as a weight
instead of a budget share in a Divisia index.










dny, 0 fd In yl n-i'd InwZk (2-107)



The variable in the left hand side of Equation 2-107 is d In y which is the
Sg,


contribution of the r'" product to the Divisia volume index of outputs. Note also, that


g P, Yr, which is the revenue share of the r' product.
-g, R


2.4.2 Comparative Statics in Supply

The supply Equation 2-107 describes the change in the firm's supply of the r'

product as a linear combination of all output price changes, each deflated by its own

Frisch input price index and all quasi-fixed input changes. For the output-supply system,

the following hold:

* If all input prices are unchanged then d n W" = 0. Then Equation 2-107 becomes
ni i
g' dlny, = I O,.',dln p, qs qrkd nzk.
g,
9, r=l k=1


* If the prices of all variable inputs and all outputs increase proportionately then

d In W' = d n p, and thus Eq. 2-107 becomes d In y, = -*V r7kd In zk.



To find the total output decision of the firm, define 0j = 01 and note that
s=1

m m
0* = 1 is implied by the normalization yZ ,, = 1. Therefore, the weighted means of
rr=l s=.


the logarithmic price changes that occur in Equation 2-106 are d In P' = 0 d In p, ,
r=l








d In W" = 0,d in Wr Correspondingly, let for the coefficient of the quasi-fixed input


l7, = k Next, we sum Equation 2-106 over r and use the symmetry of 0' to obtain
r

dinY=j Y 'd InF1` \-,Vd1nzk (2-108)
= ) k=\

This is the total output decision of the firm, which shows that '* is the price elasticity of

total output (j' > 0). Next, multiplying Equation 2-108 by 0, and putting the result back

into Equation 2-107. we obtain the output allocation decision

+ p PF
dlny, =OdlnY+q'* Od In -q,,,d Inl
I 1
Vqr*,..dlnz, +y *O' rd In zk
k=1 k=1

or equivalently,

-g dlny, =OdlnY+V/ Od (lnp +,, 7 o rk-)dnzk (2-109)


P1:
The deflator in the price term is d In = d In P" d In W '', which is the same for
W "

each input-deflated, output price change in Equation 2-109. If these corrected output

price changes are proportionate then the second term in the right hand side of Eq. 2-109

is equal to zero. This shows that in Equation 2-109 only relative input-deflated output

price changes have a substitution effect. Therefore, if 0, < 0,r # s then r'n and sh

products are specific substitutes, while if O, > 0,r # s then r' and s' products are

specific complements. Further, 0, < 0,r # s implies that an increase in the s' relative








input-deflated output price leads to a decrease in the production of the r'h product.

Finally, the Divisia elasticity of the r"' output is obtained from Equation 2-109 as

d =ln yr 0
SdinY g, /r g


If this Divisia elasticity is negative (De < 0) then the specific output is inferior, since

when firm increases total output the particular output decreases.

2.5 Rational Random Behavior in the Differential Model

According to the theory of rational random behavior (Theil 1975), economic

decision-makers actively acquire information about uncontrolled variables, such as prices

of inputs in the case of cost minimization and prices of outputs in the case of profit

maximization, or both prices. However, this information is costly, implying that the

decision-makers have incomplete information. To account for this non-optimality, Theil

(1975) suggested adding a random term to the decisions of the firm. He further, showed

that if the marginal cost of information is small then the decision variables of the firm

(input and output levels in our case) follow a multinormal distribution with a mean equal

to the full information optimum and a covariance matrix proportional to the inverse of the

Hessian matrix of the criterion function.

Chavas and Segerson (1987) criticized Theil's approach to rationalize the stochastic

nature of choice models because it relies on a quadratic loss function for the decision-

maker. That is the error term is not an integral part of the optimization problem of the

decision-maker. They instead provided a method to include it in the cost function of the

firm. In this study we will follow the rational random behavior theory since otherwise it

would unnecessarily complicate the analysis. Notice though that the covariances of the









error terms in both systems are independent of the inclusion of quasi-fixed inputs. That is,

under this theory the short-run model has the same covariances as the variable LT model.

The proof is almost the same as provided by Laitinen (1980, page 209) and it will not be

reproduced here.

Therefore, relying on the theory of rational random behavior, an error term is added

to the variable input-demand equation (Eq. 2-84) to get


fd Inx, = y O, 'gd n y,. +y yk ,d Inz, + r,,d n w, + (2-110)
r=l k=l i=1


where g g = u and = -/ .



Then ,..... ,, have an n -variate normal distribution with zero means and variances-

covariances of the form

Cov( ) = (7 ), ij =,...,n (2-111)

These covariances form a singular nx n matrix, that is the sum of ,E,...,n, has zero


variance since (0,, -- ,)= 0, from Equation 2-65. This further, implies that the


total input decision of the firm continues to take its non-stochastic form (Equation 2-80),

when the theory of rational random behavior is applied to the firm.

In the case of profit maximization, the rational random behavior theory implies that

a disturbance e, must be added to the system of output-supply equations of the firm

M M ?I /
g'd n y, = y/',ild In p, j 9~j' "dln w, 'rldlnz, + (2-112)
,=1 v=1l =1 k=I









where the above expression was derived by taking into account Equations 2-106 and

2-103. Further, E' ..., ',,, have an m -variate normal distribution with zero means and

variances-covariances of the form


Cov'(,.,)= O- 0 with rs = ,...,m (2-113)
7Y2

Notice that the 02 is the same coefficient as in Equation 2-111. The vectors

e = (?,... ,,)' and = (',...,,,)' are independently distributed. This implies that the

system consisting of the input-demand equations and that of the output-supply equations,

constitute a two-stage block-recursive system (Laitinen 1980). The first stage consists of

Equation 2-112, which yields the m output changes and the second consists of Equation

2-110, which yields the n input changes for given changes in output. The independence

of the input and output disturbances can be interpreted as meaning that the firm gathers

information about the two sets of prices independently.

In the case of output supply, however, summation of i ,...,,, over r is not equal

to zero. This implies that the total output decision of the firm (Equation 2-108) takes a

stochastic version, when the theory of rational random behavior is applied to the firm.

This is also obvious, below


dln Ydln 1- -l/, Ikdlnzk + (2-114)


2 *
where E' = J ,, and from y 0* = 1, it follows that Var(E) -
r r 7)2








2.6 Comparison to the Original LT Model

In this section a brief comparison of the original LT model with the extended

model (ELT) developed in the previous sections is provided. Laitinen and Theil (1978)

assumed that the production function is negatively linear homogenous in the output

vector, which implies that

aT
y -1 (2-115)
Ilnyr

This relationship is not crucial for the derivation of the input-demand and output-supply

equations, but for the definition of the coefficients in those equations. Taking into

account the expression (Equation 2-9) for the returns to scale it is obvious that Equation

2-115 imposes a restriction to this measure, namely that the denominator is equal to

negative unity, while in the ELT model no such assumption is imposed. As mentioned

before, the main difference between the LT and ELT models relies on the coefficients g,,

n, and p,. Specifically, in the LT model g, is the share of the r'h product in total

variable cost, but in the ELT model this is true for g' = g, / g' 3. In the case where


quasi-fixed inputs are introduced to the model then similar results hold for the definition

of p,/. Concerning the price coefficients r,,, in the LT model these coefficients were

decomposed to n,, = -,/(9,, 0,, ), where 0, is the marginal share defined in Equation

2-38. This relationship is entailed from assumption 2-115 and that the second derivatives

of Equation 2-115 with respect to output and variable inputs are equal to zero (Laitinen

1980, page 180). In contrast, this relationship does not hold for the ELT model were no


3 This is obvious from Equation 2-31. See also discussion below this equation.









such assumption is made and ,,, = -q(u,, ,j). However, as was shown in Equations

2-63 to 2-65 and the discussion below these equations, the same properties hold for both

decompositions, as far as it concerns summation of these coefficients across input-

demand equations or over all inputs in the same input-demand equation. The systems of

equations for both models are represented below

LT Model


ID: fd In x, = y 8 g,d In y, + + -,,d In w, + c,
r=l /=1


OS: grdln y = y',*dln p, yO-,0,dlnw, + c


ELT Model

/ t l t
ID: j;d In x, = d In y, + In z, + 7rd In + ,
ID: fdlnx, = 7Y2O g, diny.+y3 dinzk + dinw +g
r=l k=I =


OS: g, dlny,= O*,dln p,- yj O,;,dlnw, y'qrk dnz, + E
=I s=1I =l k=1

Notice that in the ELT model, there are more terms in both input-demand and

output-supply systems of equations, corresponding to the quasi-fixed inputs (zk). This is

one of the generalizations pursued in this study. Further, as it was shown above, there is

no need to make the assumption 2-115 in order to obtain the two systems. For instance,

7, in the LT model is equivalent to 72 in the ELT model where both coefficients are

defined as the revenue-variable cost ratio or as the elasticity of variable cost with respect

to outputs of the firm. This assumption serves into easier derivation of the equations but it

imposes a restriction in the returns to scale.














CHAPTER 3
PARAMETERIZATION AND ALTERNATIVE SPECIFICATION


3.1 Input Demand Parameterization

In order to estimate the variable input-demand and output-supply systems of the

multiproduct firm, there is a need to parameterize them since both depend on the

infinitesimal changes in the natural logarithms of prices and quantities. Laitinen (1980)

provided a parameterization for the LT model, which is extended in the section to

account for quasi-fixed inputs and the non-output-homogeneous production technology.

Thus, a finite change version of the differential d In q is defined as Dq, = In q, -In q-_,,

where q refers to all prices and quantities relevant to the firm and q, is the value at time

t. Further, an error term is appended to each variable input-demand equation as depicted

in Equation 2-84, relying on the theory of rational random behavior (Theil 1975):

r I I nI
fdlnxY = 2 gSrd In y, + y3 kpdIn z, + x,,d n w, + E, (3-1)
r=\ k=1 J=I

where the following relationships were defined or proved in the previous chapter:


S Revenue share of the firm, g' P from Equation 2-89;
r sg R


* Cost share of the firm, f = W'x-
VC

* Share of the k'" quasi-fixed input shadow value in total shadow value of the quasi-
fu A VC/Q1nz,
fixed inputs. k VC/alnzk from Equation 2-34;
z _,, aVC/olnz,
e e









* Negative semidefinite price terms of rank n-1, known as Slutsky coefficients in
the Rotterdam model = -y (oi, 0,0) ;

* Revenue-Variable Cost ratio or elasticity of variable cost with respect to outputs,
aInVC R
y, = y7, g, = = from Equations 2-81 and 2-87;
I2 Ia In y, VC

* Elasticity of variable cost with respect to the quasi-fixed inputs, defined as
a In VC
7Y = Pk = ln- =F. from Equation 2-82;
k k Olnz,

* Share of i"' variable input in the shadow price of quasi-fixed input zk, defined as


aVC/ az,

* Share of i'"' variable input in the marginal cost of the r'h product, defined as
a (wx,) / y,



VC/ r k
* 71= ; = 1; XA = 1: Cg. =1; X = 1; Z ,,=O;.


* Covariance of the error terms, Cov (,, ) = c2 (0, ).

There are two existing problems with the estimation of the demand system. First,

y3,,k Pk,, I tk are not observable since they involve derivatives of the variable cost
k

function with respect to the quasi-fixed inputs. They would be observable if quasi-fixed

SVC
inputs were at their full equilibrium levels, since at that point --- = v with vk being
aZk

the ex-ante market rental price of the quasi-fixed input. This in turn, would transform the

model to a long-run with no quasi-fixed factors. A solution to this problem is to leave

Y7, pk, pk as unknowns and estimate one coefficient bk = y3,ukk'. However, as is
k

usual with demand systems, the estimation method requires dropping one equation from








the system due to singularity of the disturbances, as was shown in Section 2.5.

Proceeding this way, though, the coefficient of the quasi-fixed input in the dropped

equation cannot be recovered, since h,b is still an unknown constant. Therefore, a


complete demand system estimation method must be employed. An alternative is to

transform the coefficients of the quasi-fixed inputs in order to add up in a known

constant. Both methods will be discussed in the next chapter, at the choice of the

econometric procedure, Section 4.1.

So far there is no distinction between the cases of one and multiple quasi-fixed

inputs. As it is going to be shown in the next section, the one quasi-fixed input is a

special case of the multiple quasi-fixed inputs case and the estimation method does not

differ. Berndt and Fuss (1989), in their measures of capacity utilization showed that in the

case of multiple inputs and multiple outputs the long-run economic capacity outputs

cannot be uniquely determined unless additional demand information is incorporated in

the model, such as the equality of marginal revenue with the long-run marginal cost of

the firm. An alternative method though, is to consider perfect competition and specify a

variable profit function as in the case examined by the present study.

3.1.1 The Case of Multiple Quasi-Fixed Inputs

Summing Equation 3-1 over i and using the definitions of the Divisia indexes as

presented in Chapter 2, we obtain the total input decision of the firm:

d n X = y7d n Y + yd In Z (3-2)


In Equation 3-1 the factor and product shares f = wx' and g = PrY, are observable
VC R

and can be calculated for any period from price and quantity data. As in the Rotterdam








model or Laitinen (1980), arithmetic means are employed for these shares, since they are

used to weight logarithmic changes between two periods. Therefore, by using a subscript


t to denote time, the factor and product shares at period t are given by f, = w, and
VC,


g, = Y, while the average factor share of the i't input in t and t -1, and the average
R,

revenue share of the r"' product in t -1 and t are given respectively by


f, = (f,, + /,, ,); g., = (g +g,-) (3-3)

Further, define Dx, = In x, In x,_,, Dy, = In y, In y,_-, Dz, = In z, In z,_, as the finite-

changes version of the variables in the model, which imply that the finite-change version


of the Divisia indexes can be written as DX, = /,Dx,,, DY, = _,Dy, and
=1l r=l


DZ, = 7Z, Dzk, respectively. According to the theory of rational random behavior the
k=1

total input decision (Equation 3-2) holds without disturbance. Since 73 is not observable,

R
Equation 3-2 cannot be solved for 2,. and thus employing 2= from Equation 2-87
VC

we define its geometric mean as


= R, -R,_ (3-4)
vc, .VC,1

Then, the total input decision in its finite change version can be written as

DX, = ,,DY, + ,,DZ, (3-5)

To solve the problem of identification of ,, one could proceed in two ways. First,

Equation 3-5 could be solved for ,, = (DX, -2,DY,)/ DZ,. However, the possibility of








DZ, being zero and that it requires specification of the unobservable term ft, this

solution becomes unattractive. Instead, an approximation for 73, seems to be more

plausible. Remembering that at the full equilibrium level of the quasi-fixed input

aVC
= vk, then at any point different than this optimum, it must hold that
azk

QVC
= vk + 6, where 5k denotes the deviation between the ex-ante market rental price
zk

vk and the shadow price of the quasi-fixed input (Morrison-Paul and MacDonald 2000).

It follows from this definition that if 5k = 0, then the quasi-fixed input is at its full

equilibrium level, while if 5k > or < 0 then we have undercapacity or overcapacity

utilization of the specific quasi-fixed input, respectively. Therefore, we could use the

following approximation
OVC zk ,
3, = VCz v Zk k Z VktZk' + ey (3-6)
Zk VC k V, VC, k VC,

Then taking the geometric mean of y3, and accounting for the error of the approximation

Ey, we have that


SVC VCv -

Further, to solve the problem of identification of puk,, we follow the same technique as in

y3, and define its approximation as


k VC/, -VktZk + k,e=l,...,1 (3-8)
S'(VC/aze,,)z, v,,z,,
e C


(3-7)


, .=








while we use its arithmetic mean in our parameterization using the same argument as in

the case of f, and gr,:


= (4, +4,-,)+ (3-9)

A problem with the finite change version DX, = y2,DY, + 3,DZ, is that it will

usually be violated by the definitions of DX,, DY,, DZ, 2, and y3, in the previous

page. One possible explanation, as noted by Laitinen and Theil (1978), is technical

change since Equation 3-2 is the total differential of the production function and Equation

3-5 entails changes from period t -1 to t. This could be a generalization of Hicks neutral

technical change. However, in this model there is one more explanation, which is the

approximation of p/, and y3, by the use of market rental price for the quasi-fixed input

since shadow price is unknown. To account for these possibilities and the errors induced

by the approximation of p', and y3, ( *, E* respectively), we need to add a residual in

the finite-change version of the total variable input decision:

DX, = y,,DY, + y3,DZ, +E, (3-10)



where 73, r= 2jl ,j t( P + -1) and E, contains s,,s\.
VC, VC, 2

From this equation the residual E, can be calculated as

E, = DX, ,DY, -y,DZ, (3-11)

The input changes are then corrected by computing

,, = (Dx,, E,) (3-12)








This correction amounts to enforcing the finite-change version of the total input decision,

since summation of the correct input over i yields

x,, = DX, E, = 7,DY +y,,DZ, (3-13)


Taking into account Equation 3-12 for the residual correction and the

parameterizations of the quantities and prices, the finite-change version of the variable

input-demand (Equation 3-1) can be written as


x,, = Or, +, + k + Z ;,,w,, + ,, (3-14)
r=l k=l I=1

In this formulation we have defined the terms ,, = yg',Dy, zk, = 73k, Dzk, and

T,,i = -/ (i, 0,, ) as before. Further, it is assumed that 0,", (k t, and o2 are constant

over time so that Cov (,, e )= o-2_ ( i,) implies that the contemporaneous

covariance matrix of demand disturbances covariancee that concerns disturbances of

different equations but of the same year) is the same in each period. The effect of the

correction in variable input levels, as appears in Equation 3-12, is to make Equation 3-13,

which can also be written as I 2, = I r, + I k hold. This, in turn, gives that
i r k

summation of Equation 3-14 over all inputs i will yield ,' = 1, = 1, =0


and c,, = 0. Therefore, the variable input demand (Equation 3-14) satisfies the


following properties:

* Adding up: 0,' = 1, ,' = 1 and r, = 0, where i,j = 1,...,n and k = 1,...,1.


* Homogeneity: =r,, = 0.
]









* Symmetry: 7,, = ni, .

* Negative semi-definite matrix of the price parameter (r, ) of rank n-1, implying
that the underlying cost function is concave in input prices.

3.1.2 The Case of One Quasi-Fixed Input

As it is going to be shown below this is a special case of the multiple quasi-fixed

inputs case. Notice, that when the firm employs only one quasi-fixed input then by

definition p/ = 1, and so the variable input-demand equation (Eq. 3-1) becomes


f;dlnx,=y ,y 'g'dllny +y ,dlnzk + -7,dlnw +e, (3-15)


In this case, the Divisia index of the quasi-fixed input degenerates to

d In Z = ik- d In zk = d In zk and the elasticity of the variable cost with respect to quasi-
k

fixed input becomes 7y = 9 In VC / a In zk, since k = 1. Disregarding for a moment the

error term and summing Equation 3-15 over all i, we obtain the total input decision

dlnX = 2dlnY+ yd lnzk, k=1 (3-16)

Proceeding then, as in the case of multiple quasi-fixed inputs, the following variable

input-demand equation is obtained:


x,, = 8', + k, + ZI, Dwi, +, (3-17)
r=l 1=1

This differential variable input demand satisfies the same properties as Equation

3-14. For instance, the assumption Zk still holds. The only difference with the case of


multiple quasi-fixed inputs is that the residual term (E,) used to correct the variable input

does not contain anymore error due to approximation of /,, since /4, = 1. Further, if the









changes in the level of the quasi-fixed input are not zero then there is no need to use the

approximation for y;, since it could be obtained from ,3 = (DX, yDY,)/ DZ,.

3.2 Output Supply Parameterization

As in the input demand case, we rely on the theory of rational random behavior to

append an error term in the supply equation of the firm (Eq. 2-106) in order to obtain

M n I
gd ln y,= y',6 dln p,- s dlnw, /rkdlnzk ,+ (3-18)
<=1 i=l k=l

where the following definitions were provided in the previous chapter:

* Price elasticity of total output, q'l with /* > 0.

* Substitution or complementarity relationship in production denoted by 0O.

* The sum of the changes in the marginal costs of the various products due to the
changes in the availability of quasi-fixed inputs, weighted by the coefficients 09,
"I. a2VC
as r,. = 0,
=i (pSy, )anzk


* Normalization condition, Y O, = 1
r=l N=l


* Covariance of the error terms, Cov(c;, 2) = Ors
72

Similarly, a finite-change version of the output-supply system (multiplied by 72, in order

to make it homoscedastic) is


72,gr,'Dyr, = YZ y ;, Dp,, rkDzk r (3-19)
s=l <=1) k=1

If it had been assumed that the coefficients 'y/* were constants then an autoregressive

scheme (AR) would be present in the supply system, since this assumption would imply

that the variance-covariance matrix of the disturbances depends on 72,, which varies over








time. Multiplying though, each equation in the system by 72, the disturbances become

homoscedastic and now it is assumed that the coefficients a,. = y,,W*O and


pfrk = Y2,/*rrk are constant. The covariance of the disturbances is then given by

Cov(c,,E ,) = ecr-y2, 0 = a,, which is constant. The supply system can be written

then in a more compact form, as

In m I
Y,, = ar,Dp, a,, ,, rDzk +E (3-20)
.=1 =1 S=1 k=l

The properties of the output-supply system are:

Output supply is homogeneous of degree zero in both input and output prices.

The coefficient matrix of the output prices, [a,], must be negative definite of rank
m, implying that the profit function is convex in output prices.

Symmetry condition: [a,,]= [a, ].

Nonlinear symmetry condition: If linear symmetry conditions are imposed in both
systems then the nonlinear coefficients of the input prices are not free parameters.

3.3 Alternative Specification for the Cost-Based System

The variable input-demand system as represented by Equation 3-1 assumes

constant price effects, output and quasi-fixed effects. However, there is no reason to ex-

ante impose such restrictions on the system. Fousekis and Pantzios (1999) provided a

generalization of Theil's (1977) parameterization for the one product firm, based on

different parameterizations for the Rotterdam model. In this section their results are

extended to the multiproduct, multifactor firm.

To allow for variable output effects, 0,', let us define


f =a, +m,' In X


(3-21)








where / is the cost share and In X is the variable inputs Divisia index. Note that, since

f = 1, it must hold that a, = 1 and that m," = 0. Multiplying then Equation 3-21


by variable cost (VC ) and differentiating with respect to y,, we get

8(w,x,) aVC VC VC a In X
-' = a,- + m In X --+m---
a, y, y, y, Iln y,

VC 9 In X QVC
Noting from Equation 2-10 that = then the above equation is
Y, Oln Y, oy,

transformed to

a(w,x,) avc
/ =a, + m, In X + m
ayr y,

Making use now of Equation 3-21 and the definition of 9, (see below Equation 3-1) we

have that

9 (wx,) aVC
0,"- I =/ f, + m, (3-22)
r @aY,.


Therefore, the i"' input demand with variable output effects becomes


fd In x, = Y2 (f +m, )gd In y, + y ,ud In z, + njd In w, + (3-23)
r=l k=1 J=1

To allow for variable effects in all coefficients, let us define now


f = a, + m, In X + s s, In w (3-24)
J=1

Since f/ = 1 it must hold that a, = 1, m, = 0 and also that s,, = 0, Zs,, = 0,


s,, = s,, where i, = 1...,n .

Totally differentiating Equation 3-24 we have









df = m'd In X + sd In w, (3-25)


From Equation 2-28 it holds that the total differential of the variable cost ratio is equal to

df = fd In w, + fd In x, fd In VC. Also summing this expression over all inputs i, it

holds that d In VC = fd In w, + d In X. Combining these two expressions we obtain


df =fd ln w, + fd n x, f fd ln w, fd lnX (3-26)


Equating now Expressions 3-25 and 3-26; and after some algebra we get


fd n x, =(m," +f)d In X + (s, f(,j f )dn w,) (3-27)


where ,, is the Kronecker delta. To verify that the input price terms in Equation 3-27

satisfy the adding-up property we sum Equation 3-27 over all inputs i, to obtain that


(s f (3, -f )d In w)= 0, which verifies that the adding-up property holds for
I=1 J=1

the input price terms.

Equation 3-27 is a system of input demands that must be equal with the input-

demand system presented in Equation 3-1. Forcing this equality we have

n m I
(m, +f)dlnX+ (s,, -f (,, -f, )dln w )= 7,2or gd ny +y3 k,4dlnz,
S=1 r=l k=1

n
+ r,d ln w, +s,
j=1

Summing this expression on both sides over i and using the previous results, we verify

the total input decision of the multiproduct firm:

d In X = y2d InY + y3d n Z (3-28)








Substituting now Equation 3-28 back into Equation 3-27 we obtain

oI / n
fdlnx,=y, 2(mI,'+)gdlny+ .(mr+ f)pdlnzk+ ( -f(s -f )dlnw)
r=l k=l J=1

By rearranging terms, we get an allocation-type differential system of input demands:


fd lnx, = y2(m; + f)g'd lnYy +Y3 m dlnZ + fd lnzk +
r=1 k=1


+(s, (, )dln w,) (3-29)
J=I

Letting now m,' = ," f, we get


fdlnx, =y7O,'g'ddln y, + y,,d ln Z + s, -f (5 -f )dln w) (3-30)
r=1 I=l

Then we could combine Equations 3-23 and 3-29 into one general equation, since

the left-hand side variables are the same but the right-hand side variables differ. This

implies that the models are not nested. Therefore,




/ A
f,d In x, = 2 '(+ e,f,)gd In y, +y ( + e,f)p';d In z, +


+ tr(, -ef ( ,, f )dln w,)


where e,,e, are two additional parameters to be estimated and the additional restriction

S(m + uk) = 1 e is imposed in the estimation.

Using a likelihood ratio test one could test which of the following restrictions are

valid and so, which differential input-demand system fits the data better:

1. If e, = e, = 0 then we get our original differential system.

2. If e = e2 = 1 then we have all coefficients variable, Equation 3-29.








3. If e = 1, e, =0 then we have only variable output effects, Equation 3-23.

4. If e = 0, e, = 1 then we have only input price effects being variable, Equation 3-30.

Note that the presence of d In Z in Equation 3-29 may create problems of

multicollinearity, so an instrumental variable approach is suggested for the estimation of

the system. Also Equation 3-31 seems more plausible than Equation 3-29 since it

alleviates the problem of multicollinearity.

3.4 Capacity Utilization and Quasi-Fixity

The most appealing alternative parameterization of the differential model is given

by Equation 3-30, since it allows us to test for quasi-fixity and capacity utilization.

Decomposing the Divisia index of the quasi-fixed factor in Equation 3-30, we get the

following equation


fdlnx, = ,2O,'gdlnyr +73,y .;dlnz,+ (s, f (, f)dln w) (3-32)
r=l k=l I=1

Using the definitions of 73, O,' and /u it is easy to show that

VVC
73k'k 0, 1 dn VC I In zan d k
730 ,' dlnzk :,' kl((- ~-

k aInzk


=,'d Inzk = C1 d ln zk
k= In zk k=-

Substituting now this term back into Equation 3-32, it transforms the input-demand

system into

'g n S
fdnx,: d ny,.+ ,, dlz, +( -f(8,,-f)dlnw,) (3-33)
r=I k=l ,=i








/
Then the total input decision of the firm, d In X = y2d In Y + C cr,,, d In zk, is
i k=1

obtained by summing Equation 3-33 over i and using the previous result that -,' = 1.


The most important result is that the summation of Equation 3-33 over i for a

specific quasi-fixed input gives us an estimate of the elasticity of variable cost with

respect to the level of that quasi-fixed factor. This estimate,

ge,.( = (VC/ zk)(zk /VC), provides a tool to test for quasi-fixity of input k.


Specifically, a testable hypothesis for quasi-fixity is Ho : (aVC / z) + vk = 0, where vk is

the ex-ante market rental price of the quasi-fixed input. If H0 holds then it implies that

the quasi-fixed input is at its full equilibrium level and should not be included in the

right-hand side of the demand equation.

Given that we have an elasticity estimate we need to transform the null hypothesis

H VC z VC
into Ho: Ho: VC Zk + vk = 0, where the first term in the parenthesis is the
zk VC zk

estimate from the input demand estimation and is being multiplied by (VC / z ) at each

data point at the sample. If the null holds at some data point then the quasi-fixed input k

is at its full equilibrium level and the model is misspecified, while deviations from H0

show that the input is quasi-fixed. A one way t-test could be developed to find the sign of

capacity utilization. Note that we can test at each observation on the sample, like the

Kulatilaka, (1985) t-test, providing the whole path of changes between full static

equilibrium and short-run equilibrium for the input zk. If one uses the average of the








observations in the sample to construct (VC/zk) then H0 provides a joint test for quasi-

fixity for all observations.

Schankerman and Nadiri, (1986) provided a test for quasi-fixity through a

Hausman test for specification error in a system of simultaneous equations, where their

system consisted of a restricted cost function, short-run demand for variable inputs and

long-run demand for fixed factors. Given that in this study we do not have a functional

form for the cost function we cannot apply their test for the differential model. However,

a specification test between the conditional demands for variable inputs, Equation 3-1,

and the long-run demands for the quasi-fixed inputs can be obtained. This would be a

simultaneous-equations error specification test.

However, the estimation of Equation 3-33 requires complete system of estimation

methods, since the disturbances add up to zero, implying that their variance matrix is

singular. If, we would proceed by deleting one equation from the system then the

coefficients of the quasi-fixed inputs in the deleted equation could not be recovered since

they do not add up to a known constant. Since the focus of the present study is on the

comparison of the differential model with a translog specification we will not test for

quasi-fixity. However, we provide directions for the estimation methods for such systems

in the following chapter.













CHAPTER 4
ESTIMATION METHODS


4.1 Choice of Estimation Method

In Chapter 3, a model for the decisions of a multiproduct firm over a period of time

was presented. While this formulation seems to be restrictive for real applications, it

should be noted that it can be transformed to reflect different situations. For instance, the

one firm could represent one sector of the whole economy, such as agriculture. Further, if

one was considering the input-demand system, then it could be transformed to reflect

situations in International Trade or Marketing. Specifically, in International Trade

variables in the left hand side of the equation could denote the international trade of flows

of imports of a specific country from different import sources, which necessarily add up

to total imports. In marketing analysis they could represent the market shares of all

brands of a specific product, which add up to unity.

The purpose of this chapter is to present and develop different methods of

estimation for the differential model. Specifically, in this section we present the

econometric procedure for the joint estimation of the input-demand and output-supply

system of a multiproduct firm, as provided by Laitinen (1980). It will form the basis for

the econometric procedures in the next sections, which concern multiple multiproduct

firms; that is, panel data structures. In those sections maximum likelihood estimation

methods for time-specific, fixed-effects and firm-specific, random-effects panel data are

developed. The novelty in those sections is the consideration of systems of equations,








which are nonlinear in the parameters and have nonlinear cross-equations restrictions. For

convenience, we reproduce the systems of equations


x,, = ,Y,, + ,Zk, + ~ ,Dw,, + r, (4-1)
r=l k=1 k=l


an = aDp,, a,, ,s Dw,, ,rk Dzk, + (4-2)
s=1 1=1 s=1 k=1

where i 1,..., n and s, r = 1,..., m denote number of equations and we have assumed

constant coefficients, 0,', a,, rk and a,,O," over time. Also, as was shown in

Chapter 3, both systems of equations have homoscedastic covariance matrices, which are

denoted as Cov(e,, ) = -2' = o a,, and Cov( ,, ,) o=2 (" ,). Further,

the following changes in notation have been made: x,, = f,(Dx,, E,), = y,,~'Dy,,


,k = 3 z iDZk,, -, i2 (s -0 ) a,, = 7,y2 s, f ,, = Y q ,, c,, = a,,o .

Note that, since we have assumed that the matrix [ar ] is constant over time, then

y/, which is the price elasticity of the firm's total supply, is proportional to the cost-

revenue ratio. This can be seen from equations 2 = R / VC and I a,, = 72,y' An
r S

alternative parameterization can be formulated with constant q,* (Theil 1980). This could

happen if we divide both sides of Equations 4-1 and 4-2 by 72, and treat tr, /72, and


a,, /Y2, as constants. The disturbances e,, /72, and / y2, are still homoscedastic.

Another problem with the parameterization of Equations 4-1 and 4-2 is that we

have assumed constant technology for the firm, but this can be resolved by adding a

constant term in both systems. Laitinen (1980, page 118) suggested that these terms








would represent systematic changes in the firm's technology (Hicks neutral technical

change).

Before we proceed into the estimation method for the joint system of Equations 4-1

and 4-2, we need to impose the adding-up restrictions, symmetry, and homogeneity

properties of the two systems. We choose to impose those restrictions in order to reduce

the number of coefficients to be estimated. To satisfy the adding-up property in the input-

demand system, which creates the problem of a singular variance matrix of disturbances

in the system of Equations 4-1. we drop one equation from this system. Following this

method to deal with singular disturbances necessitates the use of a maximum likelihood

(ML) estimator, which gives estimates invariant to the dropped equation (Barten, 1969).

Recently, there have been developed methods for estimating a complete system of

equations with singular covariance matrix of disturbances (Equation 4-1) that do not

require dropping one equation; and so do not rely on the invariance property of the ML

estimator. Dhrymes (1994) considered the case of autoregressive errors in singular

systems of equations. His estimation method relies on the use of a generalized inverse

(Moore-Penrose) for the variance of the disturbances and on a formulation of an Aitken

Minimand. Shrivastava and Rosen (2002) provided a ML estimator for a complete system

of equations with unknown singular covariance matrix of disturbances. Complete system

of equations estimation with singular covariance of disturbances in seemingly unrelated

regression methods (SUR) and three stage least squares (3SLS) framework was provided

by Kontoghiorges (2000) and Kontoghiorges and Dinesis (1997), respectively.

The initial approach was to estimate the joint system of input-demand and output-

supply equations (Equations 4-1 and 4-2, respectively) by employing one of the








previously mentioned methods for the input-demand system and then to provide a joint

method of estimation for both systems. However, the nonlinear cross-equations

restrictions on the parameters and most importantly the need for a panel data method led

to the use of the more standard method, of simply dropping one equation. The

transformation for the quasi-fixed inputs -aVC / azk = vk + ,k used in the

parameterization of the input-demand system (Equation 4-1) serves that purpose, since

the summation of zk over i adds-up to a constant and thus the last equation can be

dropped.

The homogeneity property of the input-demand system (Equation 4-1) in input

prices and output-supply system (Equation 4-2) in both input and output prices is

imposed by subtracting the input price that corresponds to the dropped equation from all

prices in both systems. Symmetry is an important property that needs to be imposed or

tested. Given the adding-up conditions, symmetry can not be tested without homogeneity

already imposed. In the joint system of equations we have symmetry conditions for the

price terms in the input-demand system and for the price terms in the supply system.

Symmetry in the price terms of the two systems of equations (homogeneity restricted)

can be imposed by including on the coefficient vector only the unique elements and

rearrange the exogenous variables matrix to correspond to those elements. For instance,

consider the case of one firm utilizing three variable inputs, two quasi-fixed inputs and

four outputs. Then, the homogeneity and symmetry imposed input-demand and output-

supply system will have the following form

Eo ] y (i w) (w2 3) 0 0 0 o
x1, 0 0 0 (W14 -W3) y1 y2 (2-3) 21 22
"2 0 0 (wi*)Y l"~ -e wz) y,, y2, 2, 3 ~ 72]









Y~ P ( P (wi-) (w 2-w3) 0 0 0 ,
( ] p0 0 0 p2 (w,1-0)(w -0w3)][l1 a12 C11 12 22 21 22J
2 P, 0 0 P 2 (WI (W2 W3) 1c1]

In this formulation, we have omitted two outputs and the quasi-fixed inputs to save

space, and we have dropped one input-demand equation due to the singularity of the

disturbances. Notice that the input price parameters in the supply system are allowed to

vary freely when the model is unrestricted or homogeneity restricted. However, under the

homogeneity and symmetry restricted model, as above, these parameters are fixed (not

free), since it is required c,,, = a,0,'. Therefore, imposing symmetry in the joint system

transforms the input price terms in the output supply to nonlinear, creating an additional

complexity in the estimation procedure. Then, for our example that turns out to be the

estimated model in the next chapter, the vector of coefficients has fifty free parameters in

the homogeneity restricted model, including an intercept for each equation, while in the

homogeneity and symmetry restricted model consists of thirty five free parameters.

Having showed how to impose adding-up, linear symmetry and homogeneity

restrictions, the input-demand and output-supply systems can be written in a stacked-

equation form. To account for the singular covariance matrix of disturbances in the input-

demand system, the last equation was deleted. Therefore, Equations 4-1 and 4-2 are

written in matrix form as

x, = Oy, + Kz, + Dw, +, = Nv, + i = ,...,n-1 (4-3)

y, = Ap, + Cw, + Fz + = Mq, + s"*, r = 1,.... m (4-4)

which is subject to the following restrictions

Homogeneity

Ai,, + Ci, = 0, in output supply (4-5)








Di, = 0, in input demand (4-6)

Linear Symmetry Conditions

A = A', in output supply (4-7)

D = D', in input demand (4-8)

Nonlinear Symmetry

C = -A K', in input demand and output supply (4-9)

The adding-up property of the input-demand system has been imposed by deleting

the last equation. Homogeneity in both systems has been imposed by subtracting the

input price that corresponds to the dropped input-demand equation, i.e. for i = 3, from all

prices in both systems. The linear symmetry conditions have been imposed as shown

before, but the nonlinear symmetry condition is left for the estimation procedure.

Accordingly, the following conventions in the notation have been made

A = [a, ].,,,, C= [aO;' ],r,,, F = [f 4r ] k = O=[ r]n- ,, D = [ ]_7,,,


K -=[k Ln-l

z, =(Dz,,....,Dzk, ) =( ,..., _)', and ", = ( ",...,m .


The price vectors, w, = (Dw, ...,DI V,_,) = (Dw,,..., D _,,), p, = (Dpt,,..., Dpm,)',

denote the modified prices, where Dw,, has been subtracted from every price. Finally,

N = [0 K D] and M = [A C F] are partitioned matrices, and v' = (y,, z,, w'),

q, = (p', w', z').

The joint system, as presented in Equations 4-3 and 4-4, without the nonlinear

symmetry restrictions, is a triangular system. Further, relying on the theory of rational








random behavior the disturbances in the demand system (Equation 4-3) are stochastically

independent of those of the supply system (Equation 4-4), making the joint system block

recursive. This has two implications. First, it implies that the decisions of the firm take

place in two separated phases. First the output-supply decision is taken and then given

this decision the input demands are determined. Accordingly, we can view ;, as a

predetermined variable. However, we can observe that the marginal shares of the inputs,

0,r occur not only in the demand system (Equation 4-3) but also in the supply system

(Equation 4-4). Therefore, in spite of independence of the disturbances of the two

systems, a joint method of estimation of Equations 4-3 and 4-4 is more appropriate in

order to impose these restrictions on the marginal shares. Further, in the supply system

4-4 the parameters are nonlinear if we impose symmetry and homogeneity.

Let us denote the variance across equations in the input-demand system and output-

supply system, as

E(, ,*") = ',,,,, and E(e,c,) = c n,_) (4-10)

By relying on the rational random behavior theory, the above systems form a block

recursive system and under normality we have that the joint system error covariance

structure is

.. 0 ,x _-]
E(, *, ) = -v ,, ,(,,,+,,I =) ( 4-11)


However, we choose not to force the off-diagonal elements of the covariance

matrix to be zero. Bronsard and Salvas-Bronsard, (1984) suggested to test for the

exogeneity of y, in the input-demand system, by estimating the joint system one time

with Equation 4-11 imposed and one without, and then form a likelihood ratio test for the








covariance restricted versus the unrestricted model. Assuming that the disturbances are

independent in different periods, Laitinen (1980, page 120), writes the log likelihood

function of the joint system as


T(m + n 1)2- T 1 -Mq y, (4-12)
2 2 2 ,x,-Nv, x, -Nv,

From Magnus and Neudecker, (1988) we have that

1n =- and a'a-'a
-= and = aa
ay-, as-'

Then for given M, N, the first-order condition with respect to Y-' is given by


L T'2 1 y, Mq, y, Mq,
ay-' 2 21 x, Nv, x, Nv,

which gives the following expression for the covariance matrix,

S1 i y, Mq, y, Mq,
T _x, Nv, x, Nv,

If one wanted to assume that Y follows the assumption of Equation 4-11 then the

off-diagonal elements in Equation 4-13 would be zero. In this case one could use a two-

step estimator, where in the first step Q*i and Q are estimated from each system

separately (impose homogeneity in each system at this step) and then use those as an

initial estimator of x, where now impose the linear and nonlinear cross equations

restrictions.

To apply the nonlinear symmetry constraints, it is convenient to regard the

elements of M and N as functions of a vector u that contains only the free parameters

in the joint system. Further, we choose to substitute the nonlinear symmetry restrictions,








C = -A- K' at the objective (likelihood function). Another, equivalent way would be to

include it as a constraint and maximize the constrained log-likelihood function. Magnus

(1982) proposes the latter method but he also suggests substituting a large value, like

1000, to the lagrangean multiplier.

Then for given Y the first-order conditions with respect to the i'h element of vector

p/, are given4 by

aM
IFM-q,
9L y y, Mq, q,
= ,Nv,, =0 (4-14)
x,, x,-Nv, ON



where = -K'
N K D

In Appendix A. 1, the analytical derivatives of OM /ap, and ON/ap, are provided for a

system that consists of three variable inputs, two quasi-fixed inputs and three outputs.

Also, notice that we have made the substitution C = -A K' in the supply system.

Finally, Laitinen (1980, page 124) shows that the information matrix for the

parameters has the following form

[M ~ M
,The inverse of the information matrix will yield an asymptotic estimate of the


covariance matrix for the parameters (4-15)

The inverse of the information matrix will yield an asymptotic estimate of the

covariance matrix for the parameters that maximize L. Then for a given vector ,u, we


4 Laitinen (1980) has already derived these conditions and we reproduced them here.








define ao as the vector with i"' element L / 8pu,, given in Equation 4-14, and E the


square and symmetric matrix, whose (ij)element is -E given by Equation


4-15.

The iterative procedure that Laitinen (1980) suggests, consists of the following

steps:

Compute i using Equation 4-13 with M, N evaluated at the given vector u .

Use Equations 4-14 and 4-15 to evaluate c and E.

Let A/ = E-'ow. Then if Ap < 0.000001 use the given vector / as the vector that
maximizes L and E-' as its asymptotic covariance matrix. If the previous
condition does not hold then update the vector pu by using u,,ew = olId + E-co.

Get /t, .

Then from Equation 4-12, the concentrated log-likelihood function at the optimum

becomes

Lm T(m n -l)ln2re n I (4-16)
mx2 2

While Laitinen (1980) does not suggest an initial estimator for the vector pu, a consistent

initial estimator for the joint system under the assumption of independent disturbances of

the two systems and without linear or nonlinear symmetry restrictions imposed, could be

obtained by separate iterative SUR in each system. If disturbances are not independent

then a consistent estimator would be an iterative SUR in the joint system.

4.2 Fixed Effects and Pooled Model

The previous econometric procedure considers only one firm over multiple years,

but our dataset consists of a large number of firms observed over a small period of time.








Therefore, we need to consider panel data techniques for the estimation of the differential

model. In this section we analyze fixed-effects models and pooling across years or cross-

sectional units. Recently, Baltagi et al. (2000) showed that for a dynamic specification of

the demand for cigarettes, pooling was superior to heterogeneous estimators.

Before we proceed to the estimation methods, it should be noted that if one does

not want to impose the disturbances in the input-demand system to be independent of

those in the output-supply system, then there is no need to follow the structure of

equations as initially presented in Equations 4-3 and 4-4, and was followed until Equation

4-16. Instead, we could consider that the two systems of (n-1) plus m equations as one

system with G equations and an equation index g, g = 1,...,G Further, we make the

following conventions regarding notation in this and subsequent sections. We suppose

that i = 1,..., N refers to the number of firms in our sample; t = 1,..., T is the number of

years that each firm is observed (balanced case); g, is the index for the equations of the

input demands with g, = 1,.., G and g, is the equation index for the output-supply

system with g = 1,....G, and G = G + G2. Then the firm i at time t we have



g=1 =I g==I


j12 O9,2,h+x i ag,,2g0,'W' g2kDZ2 +6 1+2 (4-18)
,9= =1 = g2=1 k=1

where h, k = #.k ag1 2 g, = 1,...G,1 and g, = g2 = 1...,G.

The notation is further simplified by assuming


Ygi = XgrPJ", + U (,


(4-19)








where i = 1,..., N. = 1,..., T, g = 1,...,G and G = G, + G2 ; g, is the dependent variable

in the g"' equation. For instance, the vector of exogenous variables is denoted as

[Y,,' Y ;,, ]' = [x,,, -..., x,,,,,,,,-, y,,, ]'; x,, is the matrix of exogenous variables for

the g'h equation and /, is the coefficient vector of the equation (if symmetry has been

imposed then 8, has no duplicate terms).

Then stacking all G equations for each observation (i, t), we obtain

y1' xt,, O .1 O A u~w
y2,, 0 O0*
Y = + I (4-20)

Y(,;", 0 ... ... X;, ; u ;it

This can be written in a compact form as

Y, = X,, +U,, (4-21)

where Y,, is a G xl vector, X,, is a G x K matrix of exogenous variables and f is a


K x vector with K = I K, and Kg is the number of regressors in the g' equation
g=I

including a constant; U,, is a G x 1 vector of the error terms. Since we want to impose

symmetry and some coefficients appear in at least two equations, then we redefine f as

the complete coefficient vector, which is nonlinear and does not contain any duplicates,

apart from the nonlinear terms (see example in Appendix A.2). Further, we redefine

X,, = [xI',,X...x, x,,,]', where the k'h element of x ,, is redefined to contain the observations

on the variable in the g'' equation which corresponds to the kth coefficient in 8 If the

latter does not occur in the gth equation, then the k'' element of x,, is set to zero.









A pooled model would then consist of regressing Equation 4-21 for all i and t.

However, it is implicitly assumed that all firms have the same intercepts and slopes over

the entire period, which is a very restrictive assumption. One way to account for

heterogeneity across individuals or through time is to use variable intercept models. So

following Baltagi (2001, page 31) let us decompose the disturbance term in Equation

4-21 as a two-way error component model:

U,, = a, + v,, i=l ...,N, t= ,...,T (4-22)

where a* denotes all the unobserved, omitted variables from Equation 4-21, which are

specific to each firm and are time invariant; a, denotes all unobserved, omitted variables

from Equation 4-21 that are period individual-invariant variables. That is, variables that

are the same for all cross-sectional units at a given point in time but that vary through

time. Finally, v,, is white noise.

It is this ability to control for all time-invariant variables or firm-invariant variables

whose omission could bias the estimates in a typical cross-section or time-series study

that reveals the advantages of a panel. The way we treat a, and a,, it then differentiates

between fixed-effects and random-effects models. Specifically, if we treat a, as fixed

parameters to be estimated as coefficients of firm-specific dummies in the sample, then

we follow a fixed-effects approach. Instead, if we assume that a, are random variables

that are drawn from a distribution we have a random-effects model. The same arguments

are true for a,. Random-effects models are considered in Section 4.3.








So suppose that we formulate a fixed-effects model for N multiproduct firms,

where the effects of omitted, unobserved, firm-specific variables are treated as fixed

constants over time. Then, Equation 4-21 becomes

,, =a, +X,, + U, i = 1,...,, t=, ..., T (4-23)

In this formulation a7 represents the firm-specific effects. For instance, in banking it

could account for all differences such as location, management skills or persistent X-

inefficiency, that permanently affect the demand for inputs and supply of outputs of a

particular bank relative to some other bank that face similar conditions. However, we

could reject the use of a firm-specific, fixed-effects model in this study for two reasons.

Our sample consists of T -> fixed and N -+ large and a fixed-effects approach would

result in a huge loss of degrees of freedom (df = NT N K +1). Secondly, our model is

already first difference, which sweeps out the individual effects. For instance, our Y, is

equal to (In Y, In Y,,_) and from Equation 4-22 it is obvious that a, are difference out.

However, one could argue that fixed effects exist between Y,, = In Y, In Y,_, and

Y,1 = In ,_, In ,,-2 For that purpose we consider a random-effects model in the next

section.

A more appropriate fixed-effects model would be to consider time-specific effects

a, as fixed parameters and estimate them as coefficients of time dummies (Dum, ) for

each year in the sample. That is, to consider the model


Y' = i,; a,Dum, +X,,p+U,, =X,,fl+U,, i== ,...,N, t=I,...,T (4-24)
\ t=l








T
where i, is a vector of ones, and we impose a, = 0 to avoid the dummy variable trap,
t=1

since /f contains an overall constant in each equation. Further, we make the following

assumptions

Assumption 4.1: The error terms of Equation 4-24 are independent and identically

distributed as

U,, IIN,; (0(,,, ,,) (4-25)

Assumption 4.2:

X, and U,, are uncorrelated (4-26)

Notice, that we made the assumption of normality since we are going to use a

maximum likelihood method. If a generalized least squares method was to follow, then

one should replace Assumption 1 with the following,

E,, ifi=j,t=s
E(U,,) = O, and E(U,,U,)= 0 if i j,t = s (4-27)
0 ifi j,t s


0-1", ... cr,
0I 1 '" 1G
Also, ,, is defined as l,, = w '. which is the correlation across equations
.-l" 0-"

for an individual at time t and is positive definite. We assume no correlation between

individuals for the same year and no contemporaneous correlation across years, since we

imposed in the parameterization of the model (Chapter 3) the disturbances to be

homoscedastic. Notice, that the formulation in Equation 4-24 implies that only intercepts

vary over time. It further implies that there are common shocks in the demand for inputs








and supply of outputs for all firms in a specific year. This could be clearer by stacking the

observations by year first, so

Y=X, X + U, (4-28)

where Y, X, and U, are now the stacked (GTxl) vector, (GTx K) matrix, with K

including the time dummies and (GTx 1) vector of Y's, X's and U's respectively,

corresponding to the T observations of individual i. That is,



Y,= : X,= ; U (4-29)
Y17 ,,I_ U,,,

And let V =[U,,,..., U,,](;7, with U, = vecV,, i= ,...,N (4-30)

Then making use of Equation 4-25, the (GT xl) vectors U, are distributed as IIN(0, ),

with variance matrix

E(U,, U) = 1,. 1,, = Q (4-31)

Y, O 0
0
Q= =17,
0 I. Y E

O C- -

The model as presented in Equation 4-28 is simply a seemingly unrelated

regression (SUR), first considered by Zellner (1962) but nonlinear in the parameters. We

may then formulate the following proposition.

Proposition 4.1: The log-likelihood associated with the linear model (Equation 4-28),

but nonlinear in the parameters, under the Assumptions 4.1 and 4.2, is given by


C= L,,with L, =-GTln2r- -Inl ,, U,;.U,,
1=1 2 2 2 ,==








The proof of this proposition is simple and is based on Magnus (1982). The

probability density of YI takes the form (see Appendix of Magnus, 1982)


f(Y, I 1, )= 27r-2 0-12 e-1u ') (4-32)

The log-likelihood function for firm i is then,

1 11
L,= -GTln 2r- ln n (U/,'-'U,) (4-33)
2 2 2

Notice however that 0 = I1, 0 Y,, and so its inverse is equal to 2-' = I, O YU' and the

determinant is equal to Q = T U, (Theil 1971). Then the log-likelihood function can be

written as

1 T
L, = GT n27- In I -- U,,'U,, (4-34)
2 2 2 ,=1

Now we can formulate the next proposition for the gradient vector and information

matrix of the model considered in Proposition 4.1.

Proposition 4.2: Consider the linear model in Equation 4-28, but nonlinear in the

parameters, under the Assumptions 4.1 and 4.2. Then the gradient vector, and the

N
information matrix for = L, are given by
1=1


Ac NT 1I T? 8N 7
= 1 1+ U"UU' ,= _I I X,, ,__ (Y,-X,,_f) (4-35)
aX-1 2 2 ,=1 ,=1Ph Ph



S- X -
1= h (4-36)

0 NT 2
2








To prove this proposition notice that for a given vector /, we can differentiate

Equation 4-34 with respect to the covariance matrix to get

aL T 1 '
S Z,, +- U,,U,, = 0
a1,1 2 2 ,=1

Summing this equation over all individuals it gives Equation 4-35. Further, we can solve

for 1,, in the above first-order condition, to get


1 =- .UU', =- iv;' (4-37)
S=1 1 NT =

Before we differentiate with respect to the nonlinear vector fl, note that U,, = Y X,/3

and p contains no duplicates. Then, for given ,, we differentiate Equation 4-34 with

respect to the h'" element of /, Vh, h = 1,...,K to get


yL, _y 'X,,Y + 4 X,,'Z-.'X,, YX_''XX,, =0
)/A 2 ,,, 8 / -h) a/3


which simplifies to


= X, I Y,-X,1) =0 (4-38)


Summing this gradient vector over all individuals it gives Equation 4-35. In Section A.2

of the Appendix we provide the analytical gradient vector for the example of this chapter.

Notice that if / was linear in the parameters then Equation 4-38 would give us the GLS

estimator,


GS = XI-:x'X, X^ZY, (4-39)








In order now to find the Hessian we need to take the second-order derivatives. We begin

with the covariance matrix

02L, T 2 02 NT 2
'- 2 and so = (4-40)
aC I gy 2 9 ,9l 1 2

Taking the second-order derivative of the coefficient vector, after some algebra we obtain

the following expression


Yfi,, N)',, (_,l, (4-41)
af a)


Since E(Y, X,,f) = 0, the expectation of the above expression gives the information

matrix of the parameters


11 ,, (4-42)


Given that the information matrix, which is defined as I = -E(D2 (/, u)), is block

diagonal (Heymans and Magnus, 1979), and combining Equation 4-42 with Equation

4-40 we get the expression in Equation 4-36.

Further, the inverse of the information matrix gives the asymptotic covariance of

the estimates and the disturbances. Notice, that the asymptotic covariance matrix for fl

can be obtained independently from that of 0, since the information matrix is block

diagonal. The iterative procedure to find the estimates that maximize the likelihood

function is similar to the one in the previous section and is based on the multivariate

Gauss-Newton method (Harvey, 1993). Thus, define co as the vector with h"' element

C/ 3l and I, as the square matrix in the upper left corer of the information matrix.








Then, the multivariate Gauss-Newton iterative procedure consists of the following

steps:

Get initial consistent estimates of the vector f, using the GLS estimator presented
in Equation 4-39 by disregarding the linear and nonlinear symmetry conditions.
Impose though homogeneity and obtain the relevant estimates.

Compute i' using Equation 4-37 at the given vector /.

Use Equations 4-35 and 4-42 to evaluate co and I1 .

Let A/8 = Ip-1co. Then if A/8 < 0.000001 use the given vector 0 as the vector that
maximizes L and Ip-' as its asymptotic covariance matrix. If the previous
condition does not hold then update the vector /f by using f,,n = fo + I-l1 and
go back to step 2. Continue until convergence.

Get /, 1,,.

To avoid potential confusion between the Scoring method and the multivariate

Gauss-Newton method as presented above, notice that in the case examined above those

methods coincide. Specifically, if y/ denotes the vector that includes the parameters and

the variance to be estimated in the model, ) is the initial estimate of this vector and / *

is the revised estimate, then for step 4 in the procedure above the method of scoring

consists of calculating -/* = y + I- (y )D In (q). The Gauss-Newton method starts by

minimizing the sum of the squared error terms and in the multivariate case for systems of


equations, it turns out that the updating procedure is /* = y + Z,-'Z' ZI'I ,


where Z, is equal to -as,' / Q/ (Harvey 1993, page 139). Given that the log-likelihood

function is concentrated with respect to i,, at step 2 in the procedure, then it obvious that








1
the inverse term in the last equation is Z,I 'Z,' = I,, while the last term of this


equation is equal to -Z,Y-X = C .


From Proposition 4.1 we have that the log-likelihood function is

1 NT 1 N
= GNT In 27 In I U','U,,
2 2 ,=U =1

Substituting in this expression the estimates / and ,,, the concentrated log-likelihood

N1 7 1 I
function is obtained. Since = Tr( (NTU') = -GNT then the
2 ,=1 1=1 2 2

concentrated likelihood function at the optimum becomes

1 NT
max = GNT(In27r -1)- In E, (4-43)
2 2

The previous method of estimation accommodates unbalanced panel-data designs,

since it is simply a pooling of the observations across years, through the use of time-

specific, dummy variables. If the data were balanced then for time-specific or firm-

specific, random-effects panel data, the Magnus (1982) method could be used for the

estimation of the model. Further, in the case of unbalanced panel data and random

effects, but with linear symmetry conditions the maximum likelihood estimator is a

straightforward extension of the one provided by Magnus (1982). Wilde et al. 1999

provide an application of this procedure.

4.3 Random Effects

In this section we consider that the individual-specific effects are random variables

that follow the normal distribution. Our proposed estimation method under symmetry and

nonlinear restrictions on the parameters is a special case of the Magnus (1982) maximum









likelihood estimation for a balanced panel. Biom (2004) has provided a stepwise

maximum likelihood method for systems of equations with unbalanced panel data.

For convenience, we rewrite Equation 4-21 for the joint input-demand and output-

supply system as

Y, = X,, + ,, (4-44)

e,, = a, + u,, (4-45)

In this formulation we assume that a, are firm-specific, random-effects, and E,, are

random errors. Further the coefficient vector / has no duplicates and includes an overall

intercept. The matrix of exogenous variables is assumed to have the form


X, = [x',,,..., xI;,, ] Concerning the distributional form of the random variables we

assume






Y ifi=j,t=s
u,, IIN(; (0G,, ), that is E(u,,)= 0Gx and E(u,u ;,)= 0 ifi j,t=s (4-47)
0 ifi j,t s

X,,, a, and e,, are uncorrelated (4-48)


1 Fi _a _a
L *** Wl IG
Then we have = '. and a = '.



From Equations 4-46-4-48 it is easily shown that



5 The error terms in this section are not related to any disturbances in the previous
sections.









Z1,u+ ifi=j,t=s
E(E,,) = 0(,, and E(,,E',) = if i = j, t s (4-49)
0 ififj,t s

As before, we stack the observations by time to get

Y, = X,p + i u, a, + u,, = X,, + s, (4-50)

where Y, is a (GTxl) vector, X, is a (GTxK)matrix and e, is a (GTxl) vector,

corresponding to the T observations of firm i. Also i, is a vector of ones. It follows that

,, IIN,; (O(0 2Q) with

Q= I,. 0 ,, + J, Y,, (4-51)



since E(E,e') Q= = Q and I, is a T dimensional identity

Y,, ... ... Y,, + E,

matrix and J, = iri,' is a T x T matrix with all elements equal to one. Then according to

Biorn (2004), we could rewrite Q as

S= B, ,, +A, (Y,, + T,) (4-52)

1 1
where A, = J, and B, = I,- J, are symmetric and idempotent matrices.
T T

Following Magnus (1982) the log-likelihood for the i'h individual is given by

GT 1 1 )
L,=- In27r 1ln (Y-,)'Q (Y-X ) (4-53)
2 2 2

Defining Y, = ,, + TS, we have that

Q = JA, 0 + B, 9 Y,, (4-54)








Using the property that A, and B, are symmetric and idempotent matrices, Magnus

(1982) shows in his lemma 2.1 that

= 1, I -1 (4-55)

Also note that


,(I, 0 ,2)e, =) e,',j,, and e,(A. (91' -- 2'))' =(1/T)D (,1 -')E,
=1 rs=l

Using then the above expressions and Equation 4-55 we can rewrite the log likelihood as

GT 1 1 1 T 1 (4-56)
L,- =-2 In2 2n'-(7-1)n2 e, e' (-I'- )e,, (4-56)
2 2 2 2 ( ts=\

For given covariance matrices, Y,, and E,, we take the first and second order conditions

of Equation 4-56 with respect to the h'h element of the nonlinear vector of parameters.

Using the same techniques as in the previous section, the gradient vector and information

matrix of the coefficient vector are given, respectively, by

0 _C N 7( ,,fl )' _' YE" +l 7"' ( -1 (4-57)



I= XT X ,11 ,l a -=l ,Ph


Asymptotic covariance matrix of / is obtained by taking the inverse of the information

matrix above. For given coefficient vector ,, we do not need to derive the first- and

second-order conditions with respect to the covariance of the error terms, since the

nonlinearity is in the parameters of the model. Therefore, we adapt the results from

Magnus (1982), since in that paper there was a time-specific error component and not a

firm-specific, to get






88

1 N1 N
L V,(IT A,)V,', L = ) V,(TAr-lr)V,' (4-59)
(T-(-)NA, T(T-1)N,

An iterative procedure could be employed as in the previous section for the estimates that

maximize the log-likelihood function. Further, to prevent the solution to converge

towards a local maximum, Magnus (1982) suggests ensuring that E, and 1a are positive

semidefinite.













CHAPTER 5
APPLICATION TO U.S. BANKING INDUSTRY


5.1 Introduction

One of the objectives of this study is to utilize the differential production model as

means of estimation of the input demand, output supply and efficiency measures of US

banks. To examine the robustness of the differential model results and to highlight the

differences in the description of the technology that are induced by fitting the differential

model, a comparison is provided against a commonly used in the literature parametric

specification (translog). The discussion of the results focuses on three aspects of

technology: concavity, returns to scale and input substitution as measured by the Allen-

Uzawa elasticities of substitution.

The differential model is based on the total differentiation of the first-order

derivatives of any arbitrary cost or profit function given a technological constraint. As it

was shown in Chapter 2. this provides an input-demand and output-supply system of

equations for the multiproduct-multifactor firm. The restrictive assumption of the

differential assumption as presented in Chapter 2 is the one of perfect competition that

may not hold in the "empirical world" and thus limiting its applications. A dual approach,

instead, involves specifying a flexible functional form that achieves a second-order

approximation of any arbitrary twice differentiable cost function at a given point

(Diewert 1971). The translog, which was developed by Cristensen, Jorgenson and Lau

(1973), can be interpreted as a Taylor series expansion and is the most popular of the








Diewert flexible forms. However, White (1980) has shown that while second-order

approximations allow us to attain any arbitrary function at a given point, there is no

implication that the true function is consistent at this point. Moreover, different functional

forms lead to different results for the same dataset, as Howard and Shumway (1989)

indicated; and often fail to satisfy parameter restrictions.

In the empirical banking literature some of the major concerns are related to the

functional form specification and to the validity of the efficiency measures obtained from

such specifications. For instance, Berger and Humphrey (1997) have shown that a local

approximation, such as the translog, usually provide poor approximations for banking

data that are not near the mean scale and product mix. The geographic restrictions on

branching that have contributed to the proliferation of banks in the United States and the

large amount of mergers happened when a state allowed for branching, stimulated the

interest on correct efficiency measures such as economies of scale. However, early

findings on economies of scale were contradictory and naturally led to the use of non-

parametric measures of efficiency.

In the next section a brief review of the performance and structure of the U.S.

banking industry is provided, while in Section 5.3 previous findings on the "puzzle" of

economies of scale, functional form specification and the controversy on what constitutes

a bank's inputs and outputs, are presented. The data used for the analysis are described in

Section 5.4, while the empirical model is presented in Section 5.5. Empirical results and

comparison of the differential model and translog specification in terms of satisfying

concavity, Allen elasticities of substitution and economies of scale are provided in

Section 5.6.









5.2 The US Banking Industry in the 90s

The banking industry constitutes a major part of the U.S. economy and it can be

described as a competitive industry. In recent years, the number of commercial banks in

the U.S. has begun to fall dramatically. It has decreased from 14,095 in 1984 to around

8,337 in 2000 (Table 5-1) and most of the banks exiting have been small (less than $100

million in assets). Moreover, the large banks' share of assets has increased to almost one

third, while the small banks' share has decreased to less than 5% (Dick 2002).

Bank failures played an important, but not predominant, role in the decline in the

number of commercial banks during 1985-1992, and bank failures have played an almost

negligible role in the continuing decline seen since 1992 (Berger and Mester 1997). The

primary reason for the decline in the number of commercial banks since 1985 has been

bank consolidation. Until the passage of the Riegle-Neal Interstate Banking and

Branching Efficiency Act (1994), U.S. commercial banks were prohibited from

branching across states. This Act permitted nationwide branching as of June 1997, while

some states had already allowed for intrastate and interstate branching (as early as 1978).

Recently, the Gramm-Leach-Bliley Act in 1999 allowed U.S. commercial banks to

participate in securities activities, such as investment banking (underwriting of corporate

securities) and brokerage activities involving corporate securities.

Table 5-1 illustrates the number of banks for the period 1990-2000 along

profitability measures, such as return on equity and return on assets for the "average"

bank in each year. Profitability in the banking sector, as measured by the mean return on

equity rose by 1.2% from 5.44% in 1990 to 11% in 2000. An alternative measure of

profitability, mean return on gross total assets, rose from 0.61% in 1990 to 1% in 2000. It




Full Text

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THE DIFFERENTIAL PRODUCTION MODEL WITH QUASI-FIXED INPUTS: A PANEL DATA APPROACH TO U.S. BANKING By GRIGORIOS T. LIV ANIS A DISSERTATIO PRES E NTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQ U IR E MENTS FOR THE D EGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004

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Copyright 2004 by Grigorios T. Livanis

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This dissertation is dedicated to my parents Theodosios and Konstantina; my brothers Harilaos and Ioannis ; and the love of my life Maria Chatzidaki who made this happen.

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ACKNOWLEDGMENTS First and foremost I would like to express my deep gratitude and sincere appreciation to my advisor Dr. Charles B. Moss for his outstanding guidance encouragement and advice dw-ing my graduate studies and the development of this dissertation. He has always been a source of motivation and inspiration. I would like especially to acknowledge Dr. Elias Dinopoulos for the endless discussions advice and encow-agement during the research process that contributed to the quicker completion of this dissertation. Sincere appreciation is also extended to the other members of my committee Dr. James Seale Dr. Timothy Taylor and Dr. Mark Flannery for their guidance and constructive criticisms that led to improvements in this dissertation. I would like to express my immeasurable gratitude to my parents Theodosios and Konstantina Livanis; and my brothers Harilaos and Ioannis Livanis for their continuous love and moral support despite the distance. I especially thank my parents who taught me that I could achieve anything that I committed myself to fully. In the last years of my studies I was privileged to have my brother, Ioannis studying at the same University. His humor and support made those years more enjoyable. Finally I would like to express my deepest love and gratitude to my partner in life, Maria Chatzidaki for all of her love, support and sacrifice. Without her by my side, I would not have reached my goals successfully. Words cannot express how thankful I am to be sharing my life with someone so loving patient and thoughtful. IV

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TABLE OF CONTENTS ACKNOWLEDGMENTS ................. ... .... ............ ........... ..................... ...... ......... ... .......... iv LIST OF TABLES ........... . ... ......... ....... .... .............. . . .......... ................. ....... .... ............. vii ABSTRACT ... ... ... .......... ..... .......................... ... ... . ............. .......................... ... .............. viii CHAPTER 1 INTRODUCTIO AND OBJECTIVES ..... . ..... ................ ...... ........... ....................... 1 1.1 Introduction ......................... . ... ................. ... ... .. ................. .... ...................... .... 1 1.2 Objecti ves ................... ....... ...................................................... . ... .......... ......... 4 1.3 O v erview .......... ..... ... .... ................................................................................... 5 2 M ETH ODOLOGY .......... .............. ........ ............ ....... ...... ... ............ .. ........................ . 7 2.1 Introduction ........... . ............ .......... ....... ... ... .... ... .... .... . ........ ....... ............ ... ...... 7 2.2 The Case of Multiple Quasi-Fixed Inputs ..... ...... ... ....... ..................... ............... 8 2.3 Cost Minimization .................................................... ...................... . ... .... ... .... 11 2.3. I Return s to Scale and E la sticities of Variable Cost.. ......... ......... ........ .12 2 3.2 Factor and Product Shares ......... .......... ......................... ......... ... .... ... .17 2 .3.3 Margina l Shares of Variable Inputs ..... .............................. ............ ... 20 2.3.4 Input Demand Equations ... ...... ... .... ... ........ .................... .... ...... ......... ... 21 2.3.5 Co mparati ve Statics in Demand ... .... ................ .................................. 29 2.4 Conditions for Profit Maximization ...... ................ ... .............. ....... ....... ... .... ... 33 2.4.1 Output Supply .................................................................................... 35 2.4.2 Comparative Statics in Supply ....... ... .. .......... ... .... ...... ..... ................. .41 2.5 Rational Random Behavior in the Differential Model ....... ...... ... ................... .43 2 .6 Co mp ariso n to the Ori g inal LT Model .... ...... ........ ... .... .......................... .... ..... .46 3 PARAM ETERIZATION AND ALTERNATIVE SPECIFICATION ... .... . .... ....... .48 3.1 Input D e mand Parameterization ..... ........ ....... ............ . ... .............. ...... .... .... .48 3.1.1 The Case of Multiple Quasi-Fixed Input s .......................... ............... 50 3.1.2 T he Case of On e Quasi-Fixed Input ................ ......... ... ...... ............... 55 3 2 Output Supply Param e t e rization .......... ... .... ........ .... . . .... ............ .... .............. .... 56 3 3 Alternative Specification for the Cost-Based System ... ........... ............ ....... . 57 3.4 Cap acity Ut ili za tion and Quas i-Fixit y ... ..... .... ... ... ....... .............. ........ .......... 61 V

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4 ESTIMATION METHODS ..... .... ... ........... ... ... ... ...... ... ...... .... ........... . . .... ........ ... 64 4.1 Choice of Estimation Method . ... . .... ...... .... . . ... ... ... . ..... .... ............ ... .... ..... 64 4.2 Fixed Effects and Poo l ed Mo d e l.. . .... ... . ... ... . ......... ... .... ...... .... ............. ...... ... 73 4.3 Random Effects ...... . .... ....... ....... ... ........ ............ ... ................. . .... ... . . . .... . ... 84 5 APPLICATION TO U.S BANKING IN DUSTR Y ... . ..... ... . ..... .... . . ... ...... ........ 89 5.1 lntroduction .................. . .... ... ....... ..... . .... ..... ........... .... ... ... ....... . . .... . . ...... . 89 5.2 The US Banking Industry in the 90s . .................. ... . ........ ..... .... .... ... ... ........ 91 5.3 Brief Literature Review . . ... ...... .... .... . ... .... . ... ... ..... ...... .... . . .... . ..... . . ........ 92 5 .4 Data Description . ................ .... ... .... . .... ...... ..... . . .... ........... ..................... ...... 96 5.5 Empirical Model ... ... ... ........ ..... ....... . ..... .... ..... . . ...... ...... ... ....... . .... .... ...... 102 5 .6 Empirical Results ......... .... . .... ..... ...... ... ... ...... .... . .... ... . ....... ... ....... .. ... ... ........ 109 6 SUMMARY AND CONCLUSIONS .... ... ... ...... ...... ........ ....... ... ..... ............... .... 122 APPE DIX ANALYTICAL GRADIENT VECTOR .... ... . .... ... ... ............ . ........ .... 128 A. l Gradient Vector for Section 4.1 . ... ........ ......... ... ...... . . .......... .... ........ .. .... 128 A.2 Gradient Vector for Section 4 2 ........... ..... . ......... .... ........ .... ...... ... . ..... ..... 129 LIST OF REF E RENCES .... ................ . .... ...... ... .... ... ... ... .... ... . .... ... .... .... . ......... ...... . 130 BIOGRAPHICAL SKETCH .................................. .... . .... ... . ... . ....... .... ... . ... ........ ..... 137 VI

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LIST OF TABLES 5 1 Financial indicators for the U.S. b anking in d ust r y, 1990 2000 ......... .... ............. .. 92 5-2 Definition of variables and descriptive statistics (mean and standard deviation) ...... ....................................................... . ............ .... .... ..... 100 5-3 Parameter estimates and standard errors for the differential model 1990-2000 .................................... .......... ... ...... ... ......... .. ..... ............. .... ........ . .... 111 5-4 Parameter estimates and standard errors for the translog, 1990-2000 ............... ... 114 5-5 Concavitytest ...... ........ .................. .... ...... ..... ........ ................. ...... ........... .... ........ 116 5-6 Allen-Uzawa elasticities of substitut i on .... ... ...... ... ... .... .... ........ ................... .... . 118 5-7 Economies of scale for the mean size U.S bank 1990-2000 ......... .... ...... .... ... . 121 Vll

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE DIFFERENTIAL PRODUCTION MODEL WITH QUASI-FIXED INPUTS: A PA EL DAT A APPROACH TO U.S. BANKING By Grigorios T. Livanis August 2004 Chair: Charles B. Moss Major Department: Food and Resource Economics This study assesses the empirical and policy implications of using the differential approach in opposition to dual specifications for the decisions of the multi product firm. In applied production analysis, the dual specifications of the firm's technology usually fail to satisfy the theoretical properties of the cost or profit function. If the validity of those properties is not examined then empirical results should be interpreted with caution. On the other hand the differential production model of the multi product firm has rarely been tested empirically since it was first developed by Laitinen and Theil in 1978. The novelty of this st udy is that it generalizes the differential production model for the multiproduct firm to account for quasi-fixed inputs in production; and to account for production technologies that are not output homogeneous as assumed in the original model. Another objective of this study was to provide alternative parameterizations of the differential model to account for variable coefficients over time. For this reason a supermodel was developed that contains different specifications that can be tested by V111

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simple parameter restrictions. Further, maximum likelihood estimators were provided for the case of panel data in the differential model. The contribution of these estimators to the econometrics literature was the consideration of nonlinear symmetry constraints for the differential model under balanced and unbalanced panel data designs. The extended differential production model was applied to the U.S. banking industry for the period 1990-2000. To assess the empirical results of the differential model (and to provide a direct comparison with a dual specification), a translog cost function was applied to the same dataset. Results indicated that the differential model is consistent with economic theory while the translog specification failed to satisfy the concavity property of the cost function for each year in the sample. Concerning the Allen elasticity of substitution both models found similar results. One disadvantage of the differential model was the assumption of perfect competition which resulted in total revenue over total cost being the measure of scale economies. IX

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CHAPTER 1 INTRODUCTION AND OBJECTIVES 1.1 Introduction This study extends the multiproduct differential production model, developed by Laitinen and Theil (1978) to incorporate quasi-fixed inputs and applies this formulation to the U.S. banking industry. The differential approach differs from the dual specifications of cost and profit functions that have become the cornerstone of the literature in applied production analysis. Specifically in the differential approach there is no particular spec ification of the firm's true technology and thus it can describe different technologies without being exact for any particular form. The differential approach entails differentiation of the first-order conditions in a cost or profit optimization problem to attain the input-demand and output-supply equations respectively. In contrast the dual approach involves specifying a flexible functional form for the cost or profit function to describe the firm s technology which yields a system of equations to be estimated (e.g. a translog cost function with respective input shares) Thus it can be considered as an approximation in the space of the var iabl es (quantities and prices) while the differential approach is an approximation in parameter space. The disadvantage of the dual approach is that usually different functional forms l ead to different results for the same dataset as Howard and Shumway (1989) indicated often failing to satisfy parameter restrictions. Especially concavity restrictions tend to be nonlinear and more difficult to impose (Diewert and Wales 1987) ; and as a result few

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empirica l studies examine the concavity of their results i n detail ( exceptions are Featherstone and Moss 1994 Salvanes and Tjotta 1998 see also Shumway, 1995 for a recent survey of studies testing various parameter restrictions). Numerous models have been developed for ana l yzing consumer demand based on the differential approach (Rotterdam AIDS CBS NBR). Further as demonstrated by Bart en ( 1 993 ) Lee et al. (1994) and Brown et al. ( 1994) a number of competing systems can be generated from alternative parameterizations of the differential system of demand that was originally introduced by Theil (1965, 1976 1980). Thus the form of consumer demand can be selected through simple parameter restrictions 2 In applied production analysis a similar differential input-demand system was developed by Theil (1977) and Laitinen and Theil (LT 1978). The Theil (1977) model concerns one-output transformation technologies while the LT model extends to the multi product case However neither mode l ( especially not the LT mode l ) has been used much in empirical anal y sis because of their complexity. Exceptions include Rossi (1984) who extended the LT model to account for fixed inputs. However he assumed that the production function was separable into variable and quasi-fixed inputs. Davis (1997) provided an application of the Theil (1977) model ; while Fousekis and Pantzios (1999) generali z ed the Theil (1977) parameterization by including Rotterdam-type CBS type and NBR-type effects. Recently Washington and Kilmer (2000 2002) applied the LT model in international agricultural trade However they assumed input-output separability and independence which transformed the model into a single output model. Our stud y extends the LT model to account for quasi-fixed inputs that are not s eparable from th e variable inputs in the firm's technolo gy. The model nests the Rossi

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3 (1984) model and a testable hypothesis is this separability Further in order to generalize the LT model the output homogeneous assumption for the transformation technology is relaxed and a comparison to the LT model is provided. Testable hypotheses were input independence output independence, and input-output separability, as in the LT model. In the empirical section the usual parameter restrictions of homogeneity and symmetry of the cost or profit function are imposed; and the concavity of the cost function in input prices and the convexity of the profit function in output prices were tested. Going one step farther alternative specifications of the extended model were provided forming the base for a test for quasi-fixity based on a simple Hausman specification test (Schankerman and N adiri 1986) or on direct test of the coefficients of the estimated model. The proposed model was applied to the U.S. banking industry giving specific attention to the concavity property of the cost function. The banking industry was selected because most probably the assumption of perfect competition in both input and output markets of a specific bank will hold ; and because of the availability of data. The proposed model was compared with a standard transcendental logarithmic (translog) specification with quasi-fixed inputs which is the most common specification applied to banking data. Comparison of the two models centers on whether the concavity property of the cost function is rejected. Contributions in the field of production analysis often check whether concavity is fulfilled by the estimated parameters of the cost function. Since the seminal paper s of Lau (1978) and Diewert and Wales (1987) concavity is often directly imposed either locally or globally on the parameters. More recently Ryan and Wales ( 1998 2000) and Mochini ( 1999) discussed further techniques to impose

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4 concavity. Symmetry and homogeneity properties of the cost function can be regarded as technical properties since they are a result of the continuity property and the definition of the cost function respectively. On the other hand concavity is the first property with true economic context since it is a result of the optimization process For instance, Koebel (2002) showed that a priori imposition of concavity may lead to estimation biases, when aggregation across goods is considered. Further, a radical failure in concavity may in fact be attributable to an inappropriate specification of the functional form. Finally traditional measures of efficiency (such as economies of scale) were provided for the differential model and other measures of substitutability or complementarity in th e input and output sectors of the banks such as Allen elasticities of substitution. These measures were compared with those of the translog cost specification. 1.2 Objectives Specific objectives of our study can be summarized in the following: 1. To mathematicall y derive the LT differential production model of multi product firms under th e assumption of quasi-fixed inputs and use a more general production technolo gy that is not output homogeneous as in the LT model. 2. To provide alternative parameterizations of the extended LT model especially for the cost-based system (input-demand system of equations). This will be useful for derivin g a new test for asset quasi-fixity. 3. To provide alternative econometric procedures using Maximum Likelihood estimators for balanced and unbalanced panel data for estimating the extended LT model. 4. To apply the extended LT multiproduct model to the U.S. banking industry and to econometrically estimate the system of derived-demand and output-supply equations using the econometric methods developed in this study. 5. To compare the results of the extended LT model with those of a flexible functional form spec ification s uch as the translog. Specific attention was given to the concavity property of the cost function in input prices. The two models were also compared in terms of Allen elasticities of substitution and degree of economies of scale.

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5 1.3 Overview Chapter 2 provides the mathematica l derivation of the basic model used in this analysis. It borrows heavily from the derivation techniques as presented by Laitinen (1980), but diffe rs in terms of the added generalizations of a non homogeneous in the output vector, production technology; and of the existence of quasi-fixed inputs. Also, the extended model was compared with the original LT model showing that the assumption of output-homogeneous production technology affects only the input demand system and do es not need to be imposed The basic parameterization of the extended model closel y following Laitinen (1980) is provided in Chapter 3 The novelty in this is the parameterization of the coefficients of the quasi fixed inputs in the input-demand system, and the development of a "supermodel for the cost-based system of equations. Specifically the coefficients of the quasi-fixed inputs are a function of the respective shadow price of the quasi-fixed input. To parameterize those coefficients the procedure of Morrison Paul and MacDonald (2000) was used whereby shadow prices are decomposed to their ex-ante market rental prices plus a deviation term The "supermodel for the cost-based system developed in this chapter accommodates for a new test for asset quasi-fixity and different assumptions on the estimated coefficients through simple parameter tests. Chapter 4 concerns the econometrics of the differential approach. Section 4 1 presents the econometric issu es related to the differential model, and the two step Maximum Likelihood procedure provided by Laitinen (1980). Since this procedure does not conform to the data used in the empirical analysis Maximum Likelihood estimators were developed for time-specific fixed-effects and individual-specific random-effects panel data bas e d on previou s studies by Magnus (1982) and Biorn (2004). These

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procedures are useful for systems of equations with balanced or unbalanced panal. data designs with nonlinear restrictions on the parameters Chapter 5 covers the empirica l part of the present study The time-specific, fixed effects econometric method presented in Chapter 4 is adapted for estimating the extended LT model and the translog specification for the banking industry. Then the results of both models are compared in terms ofrejection (or not) of concavity and elasticity measures Finally Chapter 6 provides a summary conclusions of the present study and presents unresol v ed issues for future research. 6

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CHAPTER2 METHODOLOGY 2.1 Introduction The Laitinen-Theil (LT 1978) model extends previous studies by Hicks (1946) and Sakai (1974) to explicitly account for input-output separability, input independence, homotheticity and non-jointness of production. It concerns long run behavior of risk neutral multiproduct firms under competitive circumstances. Moreover, it is generally applicable since it does not require specific assumptions such as input-output separability or constant elasticities of scale or substitution. Before the LT model Pfouts (1961 1964 and 1973) had extended the Hicks model to account for fixed inputs but it was a special case since he assumed input-output separability and output independence In the empirical literatme the LT model has hardly been applied. To my knowledge only Rossi (1984) extended the LT model to account for fixed inputs (but he assumed separability between variable and fixed inputs) and applied the model in Italian farms. Washington and Kilmer (2000 2002) were two other studies that used the LT model in international trade of agricultural products. However by assuming input and output independence and input-output separability the model becan1e a single output model (Theil 1977). The advantage of the LT model is that it avoids the use of a functional form for the dual specification (either cost or profit functions). That is it does not specify a functional form for the true technology of the firm. However the parameterization of the model 7

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8 provided by Laitinen (1980) implies constant price effects and implies that the change in the cost share of the i 'h input due to the change in r'" product is also constant. Therefore, there is a need for parameterization allowing for variable output and price effects. Fousekis and Pantzios (1999) provided such a general model but for the single output firm. This chapter provides the general methodology and derivation of the short-run system of input-demand and output-supply equations for a multiproduct firm, under perfect competition in both markets of the firm The model used was developed by LT, but it was transformed to account for a more general transformation technology that does not impose any restrictions on the returns to scale of the firm ; nor imposes any restriction on homogeneity homotheticity, input-output separability or any other separability assumptions. These assumptions could be tested through parameter restrictions of the model. Further the LT model was extended to account for quasi-fixed inputs. Apart from Clements ( 1978) and Rossi ( 1984 ) who used a transformation technology separable in the fixed inputs there is no other attempt to specify or extend and test a more general model. 2.2 The Case of Multiple Quasi-Fixed Inputs Let the production technology of a multiproduct multi factor (MP-MF) individual firm be represented by a transformation function: T(x,y,z) = O (2-1) where y E iR; denotes a vector of variable outputs x E IR: a set of variable inputs and z E IR: a set of quasi-fixed inputs (inputs that are difficult to adjust). Strictly positive

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9 prices of outputs and inputs are denoted by p E JR: + and w E JR: + respectively. This transformation technology satisfies certain regularity conditions (Lau 1972): The domain of T ( x y, z) is a convex set containing the origin T ( x, y, z ) is convex and closed in {y, x, z} in the nonnegative orthant JR++ T ( x y, z ) is continuous and twice differentiable in y x and z T ( x y z) is strictly increasing in y and strictly decreasing in x Mittelhammer et al. (1981) showed that a single-equation multiproduct, multifactor in an implicit form production function is not as general as it was thought to be. The production function shown by Equation 2 1 restricts each output to depend on all inputs, and other outputs that appear as arguments in the implicit form Further they showed that it cannot represent separability in the form of two independent functional constraints such as T(.) = g1 (.) + g2(.), on the arguments of T(x, y ,z). In such cases the gradient vector of T(x, y z) is zero which further imp l ies that the Kuhn-Tucker conditions do not hold. Therefore our study did not examine separability of that form; and instead left it for future research. Assume that a MP-MF firm minimizes variable costs of producing the vector of outputs y conditional on the vector of quasi-fixed inputs z and fixed prices w for the variable inputs. Thi s s hort-run or restricted cost function can be denoted as VC = VC (y, w; z ) and it is assumed that it satisfies the following properties (Chambers 1988): V C ( y w; z ) i s monotonically non-decreasing homogeneous of degree one and concave m w VC ( y w; z ) i s non decreasing and convex in y.

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10 VC (y, w; z ) is non increa sing and convex in z. VC (y, w; z ) is twice continuously differentiable on ( w, y; z). Applying Shephard's lemma on the restricted cost function, the conditional factor demands are then obtained as x, = ave = VCw (y, w; z). If v denotes the vector of ex-0H1, ante market rental prices of the quasi-fixed inputs, then the short-run total cost of producing the vector y is given by SC= VC ( y w; z) + v z'. The lon g-run cost funct ion C ( w, y ) of the multi product firm is then obtained by minimizing short-run total cost with respect to quasi-fixed inputs while holding the variab le inputs and the level of output at the observed cost-minimizing levels That is C( w ,y ) = m}n SC= m}n(VC(y, w;z)+v z') The first-order condition of this minimization problem implies that asc avc(y ,w;z) --=--'---+v= O az az where z denote s the static equilibrium le vels of z This condition can be written as a vc(y, w;z) = v which states that a necessary condition for a firm to be in long-run az equ ilibrium i s that the shadow prices of the quasi-fixed inputs be equal to the observed ex-ante market r e nt a l pric es v (Samuelson 1953). Therefore the shadow price of a quasi fixed input i s defined as the potential reduction in expenditures on other variab le inputs that can be achieved b y using an additiona l unit of the input under consideration while maintaining the level o f outputs. Further Berndt and Fuss (1989) showed that when this condition holds, temporar y and full-equilibrium demand levels for the quas i-fixed inputs

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are equal. The same result holds for the short-run and long-run marginal cost and demands for variable inputs of the multi product firm. 2.3 Cost Minimization For the multiproduct-multifactor fim1, let Y be the rt h product ( r = 1 ... m) to which corresponds a price p Let x be the / factor of production ( i = 1, . n) whose price is denoted b y w and z k be the k1" quasi-fixed factor of production (k = 1 ... ,l) with an ex-ante market rental price denoted by v k Assume a production function in an 11 implicit form that is not separable into the quasi-fixed inputs as in Rossi (1984), nor is it negatively linearly homogeneous in the output vector as in LT (1978) ; and assume that it satisfies the properties mentioned in Section 2.2. Thus it can be written as T(x, y,z) =0 (2-2) Then in the short -run the firm s objective is to minimize variab le cost ( VC) subject to its transformation technology, by varying the input quantities for given output and input prices and for given quasi-fixed input levels. Thus the problem that the firm faces is mJn{VC(w x) = t w x :T(x y,z)-o} (2-3) 11 The Lagrangean of the above problem can be written as L = L w x AT ( x y z) and the 1 = ) first-order conditions needed to attain a minimum are given by the following equations: (2-4) 8 L al= -T(x,y, z ) = 0 (2-5)

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12 In this formulation /4 > 0 is implied by the positivity of x and the assumption that the marginal physical product of each input is positive ( BT( .) I a 1n x > 0 ). Further, Equations 2-4 and 2-5 are assumed to yield unique positive values for x, and /4; and Equation 2-4 is a vector of n x 1 The second -order conditions are given by the following equations: a2L a2T ----= 8lj w,x, -,i----a ln x ,a ln x j 8lnx,8lnx1 (2 -6 ) { l i = j where 8 i s a Kronecker delta That is 8IJ = . 0 1 '* J The solution of the minimization problem described in Equation 2-3 gives the conditional or compensated short-run demands of the inputs as a function of all input prices output quantities and quasi-fixed inputs. That is x5 R = x5 R (w,y, z) and ,1,8 R = ,1, SR ( w, y, z ) where x8 R denotes the vector of inputs and /4 SR is the Lagrangean multiplier. To obtain a minimum cost in the short-run it is sufficient that the matrix of the second order derivatives that has a size n x n (Equation 2 -6 ) is symmetric and positive definite. The minimum short-run cost is then given by VC(w,y,z) = L w,x,(w,y,z) (2-7) 2.3.1 Returns to Scale and Elasticities of Variable Cost Consider first the total differential of T(x, y z) = 0 in natural logarithmic form: n 8T m 8 T I 8 T I--d lnx+I--dln~+I--dln~=O l = I a In x I r = I a In Y, k = I a 1n z k (2-8)

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13 The degree ofreturns to scale (RTS) is defined as the proportional increase in all outputs resulting from a proportional increase in all inputs variable and quasi-fixed. Letting this be the case and defining x0 = [ x, z ] then d In x = d In x1 = d 1n zk = d In z1 and d In Y = d In Y., can each be put before its summation sign. Then we have 11 8 T I 8 T Ill 8 T dlnx I--+dlnzkI--+dlny, I--=0 l=I 8 In x k=I 8 In zk r=I 8 In Y, This can also be written as d l O ( f, 8 T 8 T J d I 8 T O nx1 L.J--+ L.J + ny,L.J = ,=181nx k=,8Inz k =181ny, Therefore ( 8 T 8 T J -I-+IRTs= dlny, = 1 = 1 8lnx, k = I olnzk dlnx ; f 8 T r=I fJlny, (2-9) otice that thi s rel a tion s hip for the returns to scale is the same as the relationship derived by Cav es et al. ( 1981 ) The marginal co s t of the rth output can be found by taking the derivative of the optimum variabl e cost function (Equation 2-7) with respect to output Y, : ave= I w ox,= VCIJ, 81nx, OJI, I I oy, Y, I I a In Y, (2-10) where J, = w ,x, i s th e variable cost share of input i and the last expression has been vc derived from the s e cond by multiplying the second term by (y,V C I y,VC ) and noting that 81n x, =_!_ox, Als o notice that the above equation can be written as x,

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a 1n ve = I 1, a 1n x alny r I I alny r 14 (2 11) Ne x t differentiatin g the optimum transformation technology T(x, y, z ) = 0 with respect to ln Y r holdin g input prices oth e r outputs and quasi-fixed inputs constant we get I ___E_ a In x + aT = O l = l alnx, alny r a!nyr (2 12) Howe ve r b y u s in g th e fir s t-order condition ___E_ = w,x, and b y multipl y ing the first alnx A term b y Y r ve E quati o n 2-12 becomes y,Ve Y r -Ve f, w x atn x a T O h w x J, ~---+--= were= A Y,. l = l ve a In Y r a In Y,. ve I Us in g now E qu a tion 2 -10 th e abo v e expression can be written as Y r ave+ a T = O A 01r alny r If we sum E quati o n 2 -13 o ve r r then we get ave 1 111 a v e III a r I a1ny I y + I =OorA.=-' r A r = l 01,. r = l a In Y,. L aT ,. a !n Y,. L e ttin g r = __3__, th e n fro m E quation 2 -14 w e hav e that v e I a1nve A ,. alny,. r, = ve = a r alny,. (2-13) ( 2-14 ) (2 -15 )

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15 The elasticit y of v ariable cost with respect to proportionate output changes holding quasi-fixed inputs con s tant is obtained by substituting in Equation 2-15 the expression for the lagrangean multiplier ( J) from the first-order condition (Equation 2-4) That is we substitute J = ve IL_!!__ in Equation 2-15 to obtain I alnx, I a r = I a In ve = -r a In Y, ,. a1ny,. I_E_ I a1nx, (2-16) To find the ela s ticity of variable cost with respect to proportionate quasi fixed input changes we follow similar anal y sis as above holding output constant. Therefore taking the derivative of the optimum variable cost function with respect to a quasi-fixed input we obtain (2-17) otice that -ave= w k denotes the shadow price of the quasi fixed input. Also from the azk anal ys i s in S e cti o n 2.1, in order for the firm to be in long-run equilibrium it has to be the ave case that -= vk, w here v k is the ex-ante market rental price of the quasi-fixed input. azk F urther E quation 2-17 can also be transformed into the following expression a1nve = If, a ln x a In Z k I I a 1n Z k (2-18) Now, takin g the deri v ati v e of the optimum production technology T(x,y,z) = 0 with respect to In z k, holdin g input prices other quasi-fixed inputs and outputs constant we ge t

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f __!!_ alnx, + a r =0 =1 a In x a In z k a In z k 16 (2-19) Again using the first-order condition __!!_ = w x , multiplying the first term of the alnx, A. above equation by z k ve and using Equation 2-17 we obtain the following relationship z kVe 2 ave+ a r =O A. a z k alnz k umming this equation over k we obtain the second interpretation for A.: However Equation 2-21 must be equal to Equation 2-14 implying the following relationship I ave = k a1nz k I a r I ave a1ny, ,. alny,. (2-20 ) (2-21) (2-22 ) Solving for the elasticity of cost with respect to proportionate quasi-fixed input change from the abo ve equation we obtain (2-23) which can also be writt e n as

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17 (2 -24 ) or equivalently E quation 2-23 (through the use of Equation 2-16) can be written as I ~ 5 ="""' 8 lnVC =-k 81nz k v c = a In ar k zk I-, 8 ln x (2-25) Finally, taking int o consideration E quations 2-16 and 2 25, the degree of returns to scale (RTS) in term s of derivatives of the variab l e cost function (E quation 2 -9 ) can be written as ( I ~+ ar J 1 I a1nvc RTS = / : I a In x k : I a In z k = k a In z k I aT I 8 lnVC r : I 8 ln Y r r a In Yr (2-26) 2.3.2 Factor and Product Shares We ha ve a l ready defined the varia ble cost share of input i as f, = ;; (2-27) Taking the total differential of E quation 2 27 we have df, = f,d In w + f, d In x f,d In VC (2-28) Summing Eq u ation 2-28 over i and noting that I f, = 1 and so d IJ, = 0 we have dlnVC = I J,dln w + I J,dlnx (2-29) or in a more compact form d In VC = d ln W + d In X (2 30)

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18 where dlnW = L J,dln w,, dlnX = Lf,dlnx, are the Divisia indexes of variable input prices and var iabl e input quantities respectively (Divisia input price inde x and Divisia input vol ume index). Then considering Equation 2-14 for A, define as in Laitinen and Theil (1978) (2 -31 ) as the share of the /" product in total va riable marginal cost multiplied b y L BT B ln y, Notice that if we had assumed ne ga tively linear-homogeneous production function in the output vector, which impli es that L B T = l as in LT then g, would be just the Blny,. share of the r'" product in total var iable marginal cost. It is the case though that at the point of the firm's optimum (fro m E quation 2-13): BT g =---r Bln Y,. (2-32) Noting that L g = L BT we can defin e the s hare of the rth product in total , a ln Y, var iable mar g inal cost as g, BVC/ Blny "\"' w ith r, s = l ... m Li g,. Li ave I a ln Y., s T h ese s har es are nec ess arily po s itive and have unit sum over r. F urther we can d efine the Divi s ia volume index of o utpu ts as d ln Y = L .f: d In Y, r L.i g s s Similarl y considerin g E quat ion 2-2 1 for A, defin e

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19 A = 2 ave = ave I a ln zk L a T A azk Iavel alnzk k alnzk (2 -33 ) k as the share of the k1" quasi-fixed input shadow value in total shadow value of the quasi fixed input s, multiplied b y I a T Further substituting for I a T its equivalent k alnz k k ainzk form from Equation 2-22 we obtain the ratio of the k1" quasi-fixed input shadow value in the variable mar g in a l cost of m outputs, multiplied b y I a T : r a lny, ave I a1nzk L a T k = I ave I a1ny,. ,. a1ny Us ing now E quation 2 3 l the above eq u ation transforms to ave/alnzk k = ave I ainy,. g ,.' r =1 ... m Also at the point of the firm s optimum (from Equations 2 -33 and 2 2 0) it holds (2-34) (2-35) (2 -36 ) As in the case of outp ut s not e that I A = I a T Therefore we can define the k k alnzk share of the kth qu as i-fi xe d input shadow value in total shadow value of the quasi-fixed inputs as ave/alnzk = Iave1 a1nz e with k,e=1, ... 1 e e which are po s iti ve an d ha ve unit s um over k. Further as in the case of outputs the Di v isia volume index of quasi-fixed inputs i s defined as d In Z = ( I ~>. ) d In z

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20 2.3.3 M ar g inal S h a r es o f Varia bl e Input s Like in LT model define the share of ;th variable input in the marginal cost of the r111 product as 0 = a(w, x ) I oy, ave I oy,. Then multiply Equation 2-37 by J,. and sum over r to get L.g, 0 = I _&_0 = I ave 1a1ny, acw, x ) loy, Lgs I ave 1a1nys ave 1a1ny, s The above equation can be written as I acw, x ) 1 a1ny,. 0 = _,_. ---/ I ave1a1ny, (2 37) (2-38) Equation 2-38 defines the share of the /1 input in variable marginal cost of outputs. Finally as Laitinen and Theil mentioned s ummation of 0 ;', or 0 over i gives a l ways unity but need not be non-negative. In a similar fashion define the share of /1 variable input in the shadow price of quasi fixed input zk as c;k = a (w,x,) lazk I a v e I azk Then multiply Equation 2-39 by ~ k and sum over k to get the share of the ;th L.A e variable input in variable marginal cost of m outputs: (2 39)

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21 e e which can be simplified to IB(w, x ) / B in z k <; = -'k==-------1 IBvc1 B1nz k (2-40) k As in the case of the outputs summation of f, {; over i is always unity but need not be non-negative. 2.3.4 Input Demand Equations The first step is to write the first-order conditions as identities and then to differentiate them with respect to their arguments That is with respect to each output Y , input prices w , and qu as i-fixed input quantity z k in order to determine how the optimum changes in response to changes in these given variables. Therefore, the first order conditions as identities are ( ) 1 ( )BT(x(w,y,z),y,z) _0 wx w y,z -,fl, w,y, z -------= ' Blnx, (2-41) T(x(w,y,z),y, z ) = 0 (2 -42 ) Totally diff ere nti a tin g E qu a tion 2-41 with re s pect to 1n Y,, In w J and In zk, it gives the followin g relationship s, respectively Blnx, Blnl B T B2T Blnx1 B2T wx ---/4-----A,I ------/4---=o (2-43) 'Blny, B lny, B ln x J=I Blnx,B1nx1 Blny B ln x ; Blny s: w B ln x 1 __.!!._ Blnl u,j W,X; + 1X1 B 1n W J /l, B ln x B ln W J (2-44)

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22 wx 8 ln x1 -J-a_In_J_ 8T -JI--a_2T ___ a_ln_x.c..._1 -A 82T =0 ( 2 45 ) 1 alnz k alnzk 8 ill X1 j = I alnx18 lnX J 81nz k 8lnx18lnzk Notice that Equation 2-43 represents n distinct equations equal to the number of inputs However if we consider all the outputs we are going to have n x m distinct equations. Similar arguments can be used to show that Equations 2-44 and 2-45 represent n x n and n x l ( k = 1 ... l ) distinct equations respectively. Then totall y differentiating E quation 2-42 with respect to 1n Yr, In w J and 1n zk we have respectively t a1nx1 + ar = 0 1 = 1 alnxl 8lnyr 8lnyr I ~ 8 ln x1 =0 i = I 8 ln x1 81n w1 t ~ a1nx1 + a r =0 1=1 8lnx1 8 ln z k 81nzk (2 -46 ) (2-47) (2-48) Since we differentiate w ith respect to each output input price and quasi-fixed input level Equations 2 46 to 2 -48 are vectors of dimension m x 1 n x 1 and l x 1 respectively. The n ext steps for the derivation of the input-demand sys tem consist of the following Di vide E quation s 2-43 to 2 -45 b y variable cost (VC) us e the definition of the co s t s h a r es w1x1 = J,, a nd use from the first-order conditions the relationship vc I ar l--=wx. al I I n x1 Multiply Eq u at i ons 2 -46 to 2 -48 by __i_ and use the following relationships vc ar a r gr = -a 1n k = -a 1n Yr A A, and =Y1 vc

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23 These transformations g ive the following relationships (2-49) (2 50) 8lnx 8ln,1, ,1, [~ 82T 8lnx1 ] ,1, 82T J, I -J,---~----------:0 1 a in z k 1 8lnzk vc j=I ainx,alnxj a inzk vc 8 ln x,8lnzk (2-51) (2-52) (2-53) (2-54) ow the followin g matrices can be defined [ a2r ] [ a2r ] F = diag(fi . ,!,, ) H = H1 = 8 ln X;8 ln x j 8 ln x,a In Y, n x n n x m [ 82T ] and H3 = ----. a 1n x ,a In zk n x i Therefore Equations 2-49 to 2-54 can be written for all combinations of inputs outputs and quasi -fix ed input s, in matrix form as (2-55) ( F -H) 8Inx -Fi 8ln,1, = F Yi 81n w' n 81n w' (2-56)

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( F H) 8 ln x F i 8 ln ,1 = H Y1 8 ln z' n 8 ln z Y1 3 olnx I in F. 8 ln y = Y1 g { F 8 ln x =O n 81n w' 8lnx z n F 8 ln z' = Y1 (1 x !) 24 (2 -57 ) (2 -58 ) (2 -59 ) (2 60) ow, premultipl y Equations 2-55 to 2-57 by F -1 and combine with Equations 2-58 to 2 60, to form Barten's fundamental matrix equation F olnx F 8 ln x F 8lnx [r'(F-;,H)r' ~] o ln y I 8 lnw' 81nz' =[r,r'~, -I r.r'~ ] l,, 8 ln,1 8 ln-1 8 ln-1 Y1g 0 r1 -----olny' 8 ln w I 8lnz' and solving for the matrix of the decision variables we obtain F olnx F 81nx F 8 ln x olny' 81n w I 8 ln z = 8 ln-1 8 ln A 8 ln-1 ------8 ln y I 8 ln w I 8 ln z [r' (F 7,H)r' i,, nr.r'~, I r.r'~ ] 0 (2-61) ll 0 Y1g r1 From Magnus and eudecker (1988), if A is a non-sin g ular partitioned matrix i s also nonsi n g ul ar then th e inverse of matrix A is give n by [A-1 A -1A n 1A A -1 A -1A n -1 ] A I = 11 + 11 1 2 21 11 11 1 2 D -1A A -1 n 1 2 1 11

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25 It follows then that D = -i:,F ( F y1Hf' F i n which is a scalar. Using the property of the inverse of a 1 scalar we get that n -1 = _1 -(F(F-y1H) F-i11 F(FH)-1 F-i i'F(FH)-1 F A I +A-IA n l A A I = F (F H)I F Yi n n Yi 11 11 1 2 2 1 II Yi ., F(FH)1 F 1n Y1 1n As in LT define 1/1 = ( F ( F y1Hf' Fi11, which is a positive scalar and implies that n -1 = J_. Then define the n x n matrix
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26 ( ) -1 F F -yH F i Eq uation 2-64 can be written equivalently as = ~') 1 ][rr'~, 8 ln y 8lnw 81n z' -I rr'~, ] 8 ln J 8lnJ 8lnJ yg 0 r1 -----VI I 8 lny' 8lnw' 8 ln z Solving for the individual terms we obtain the following relationships (2-66) F ::~, =-v,r(
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27 (2-68) (2 -69) alnl =-,1,' alnw' 'f' (2 -70) (2 71) Since the optimum variable input-demand equations are given by x = x (w ,y, z) then the differential demand for variable inputs can be found by taking the total differential of this expression (logarithm ic): alnx alnx 8lnx d In x = --d ln y + --d 1n w +--d ln z alny' a1n w' alnz' Premultiplying now this expression by F and using the solutions above, Equations 2 66 to 2-68 we obtain the system of differential input-demand equations: (2-72) The coefficient of the output needs further transformation in order to have some economic interpretation. For this reason let g' = t:,,G, where G is an m x m diagonal matrix with (g, ... g,,,) on the diagonal. Then it is easy to show that (2-73) From E quation 2 37 we have that 0,. = a (w,x,) I By, = w,x, 8lnx; 1 = ve-J, 1 alnx, I ave I By, a v e I By, a ln Y,. Y ave I By, Y, a ln Y r

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28 which from Equation 2-31 can be rewritten as 0 = ve f, a In x = _L_ a ln X ; I Ag, a In Y r,g, a 1n Y The last mem berof this equation is the ( i, r )"' element of y,, ( F : : ;, ) c where Thus, from Equation 2-73 [ 0 ,' ] becomes [ 0 ] = F a 1n x a -1 = [ (
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29 <;/ =_L_ alnx, This is the (i,k) element of y,, (F alnx, ) M i. Combining then this y k alnzk alnz relationship and Equation 276, we obtain a simplified expression for [ <;,k] as [,t:k]= -i(Fa lnx ) M -i= I [ (
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n _I(1 )=l from Equation 2-65. Therefore if output and quasi-fixed inputs remain J=I unchanged and all variable input prices change proportionately then the demand for variable inputs remains unchanged. This property just verifies that the variable input demands must be homogeneous of degree zero in input prices. Further if ( u -; 1 ) is 30 less than zero then the firm will increase the use of the ith factor, when absolute price of the /' factor increases, ceteris paribus. Turning now to volume changes, the total variable input decision of the finn can be obtained by summing the factor demand, Equation 2-79, over i n nm n l nn IJ;dlnx = r,IIB,' g dln y, +r,IIt/ *din z k -lf LL(1j -;jiln w1 1 1 = 1 1 = 1 r=I i = I k = I i=I J=I n 11 n oting that _Ie,' = 1 _Ii;/= 1 and that If _I(u 1 ) = 0 from Equation 2-65, (last t = I 1= 1 t=I relationship) then the above equation can be written as n m I IJ;dlnx, =r, I g, dln y,+r,IAdlnzk (2-80) 1=1 r=I k=I This is the total variable input decision of the multi product firm and is equivalent to the total differential of the production technology of the firm. At the optimum, it has been shown that g = oT and k = o T U sing these relationships Equation 2-80 oln Y r o In zk becomes 1 Here LT anal ys i s uses th e r e lative prices equation instead of the absolute price version of the model E quation 279 ( s ee Laitinen and Theil (1978) p g 41-45 ) However, th i s does not affect our r e sults.

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31 II f m ar / a r L _!._d In x1 = I--.d In Yr I-d In zk 1=) Y1 r=I a In Yr k=I a In zk Using now the definition of y1 then f, = w;x1 VC = ___!!___, where the last term follows y1 VC A 8lnx1 from the first-order condition. Therefore, the equivalent form of the above equation is 11 a r m a r I ar I -dlnx + I--dlny + I--dlnz k =0 1=1 8 In x i I r=I a In Y r r k=I a In zk This is simply the logarithmic total differential of the production technology. However the factor demand and the total variable input decision can be written into an equivalent form which are more useful for the parameterization and estimation. If we proceed by multipl yi ng the first and second term of the right hand side of Equation L g r 2-80 b y _r_ = 1 Lg, transfonned to L A = 1 respectively then the total variable input decision is L.A T he Divisia vo lume inde x of variable inputs outputs and quasi-fixed inputs have been II Ill g I d efi ned as dlnX=~fdlnx1 dlnY=~L~sdlnyr and dlnZ=~L~e dlnzk, respectively. F urther, b y the d efin ition of y1 (Equation 2 -15 ) L gr (Equation 2-32) and Lk (Equat ion 2-34) we have th e following expressions k

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32 ainve a 1n y ( ar J m a ln ve r2=r1I g =-ar I ain = Iain =&v c.y r I-r Y r r : I Y, a in Y (2 81) I ave k a 1 n z k I ave ( I ar J = I a in ve = 6 (2 82 ) r aJny, k : I ainzk VC,. a in Y, Therefore we can write the tota l variable input decision of the firm (Equation 2-80) as (2-83) where y2 y3 are the elasticities of variable cost with respect to proportionate output changes and quasi-fixed input changes respectively. Using the same technique as above, for the factor demand equation we obtain an equivalent fom1 of E quation 2 -79: This expression is going to be useful for t h e parameterization of the factor demand The variable input allocation decision of the firm (when output changes are not proportionate) can be found by multiplying E quation 2-83 by 0,, which gives 0,d In X y20,d In Y y30 d In Z = 0 and putting this expression back into Equation 279: This expression is simp ly

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33 m I J,d In x = 0 d In X + r,I( 0,' 0 )g,d In Yr+ r,I(/ -0, )Adln z k ~, k~ 11 -w L(u -, 1 yiln w1 (2-85) J=I This is the input allocation decision of the firm. This decision describes the change in the demand for the i'" input in terms of the Divisia volume index d 1n X, change in output changes in the input prices and changes in the quasi fixed inputs. 2.4 Conditio n s for Pro fit Max i mizat i o n Assume now that the firm s objective is to maximize profits (plus quasi fixed costs) for given input and output prices. That is the firm wants to rr:!ax(TI(p,w,z ) =LP,Y,-Lw, x ; J suchthat T(x,y,z)=O J r I (2-86) Given the assumptions on the production technology (in the beginning of this chapter) the profit function is non-negative and well defined for all positive prices and any level of the quasi-fixed factors. Further it is continuous linear homogeneous and convex in all prices it i s continuous non-decreasing and concave in the quasi-fixed factors and finall y it i s non-decreasing (non-increasing) in output prices (input prices) for ever y fixed factor (McKay et al. 1983) A s sumin g that w e have a first-stage of cost minimization which gives us the input demands then in the second stage we can maximize profits as a function only of y. Therefore th e probl e m that the multiproduct multifactor firm faces is transformed to m;x( TI(p, w,z ) t.P,Y, -VC( w,y,z) J Th e fir s t ord e r condition s of thi s maximi z ation problem ar e

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34 (2 87) Using Equation 2-31 for g,, where (g, =2:'.!:_ ave], thenEquation287 becomes 1 oy, lg,= P ,Y, ummin g this expression over r and using the second term of Equation 2-31 we obtain the following LP,Y, R R 1=-r = = I ~ T L 8 T I g , 8 ln Y,. 8 ln Y,. r (2 88) where R = LP,Y, denotes total revenue of the firm. A l so we obtain that the share of the rt11 product in total revenue, mult i p l ied by I BT ,. 81ny,. IS (2 89) Since Lg, = L B T notice that J' = P,Y, denotes the revenue share of the r'11 , 8lny, ~gs R s product of the multi product firm Further using Equation 2-87 Equation 2-37 can be 0 B(w, x ) h. h h dd. l h .,1, d ..-rewritten as = ( ) w 1c 1st ea 1t1ona expense on t e z mput mcurre 1or 8 P ,Y, the production of an additional dollar s worth of the r'" output. For the second-orde r conditions to be valid it must hold that 8 2 I1 is negative oyoy' definite for which it is s ufficient that 82VC is symmetric positive definite because oyoy'

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35 82 R = 0 follows from the assumption that the price vector is given. Therefore, we will oyoy' ak h h a2vc . . d fi m et e assumption t at --1s symmetnc pos1t1ve e llllte. oyoy' This maximization problem will give us the unconditional output-supply equations of the form y' = y' (p, w, z) Taking the logarithmic total differential of the output supply we ha ve dlny = (alny' ) d lnp+(alny' )d ln w+(alny ]d ln z 8lnp 8lnw 8 ln z (2-90) 2.4.1 Output Supply The output supply of the multiproduct-multifactor firm has the form provided by Equation 2-90. However we need analytic expressions for the coefficients of d In p d In w and d In z in order to provide an estimable with economic meaning form. Proceedin g the u s ual way as in the derivation of the input-demand equation, we write the fir s t-order condition as a n identity and then we totally differentiate with respect to its arguments: avc(y(p, w,z), w ,z ) P -OJl, = 0 (2-91) Then takin g the total differential of E quation 2-91 with respect to P s, w and zk we obtain th e following r e l at ion s hip s (2-92) a 2vc III a 2vc oy w --+ "---' = 0 => I O)l,OW, i = I O)l,OJls OW, oy (a2vc)-1 a2vc ------ow' oyoy' oyow' (2-93)

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36 ay (a2vc)_ a2vc ------az' oyoy' oy8z' (2-94) However Equation 2-92 needs further modification before it gets a familiar form. Thus solving for y, from Equation 2-89 we get Rg I Y v =-aT P, I -, 8lny v (2-95) Substituting this expression back into Equation 2-92 we obtain 111 a2vc R g I -V v= I GY,GY,, P v 1 I a T V a InyV ainyv -8 a l -rsP np, which for all ( r s) pairs in matrix form becomes I aT V a1n Y v In this expression P denotes an m x m diagonal matrix with the output prices on the dia gona l G = diag (g, ) a nd p is the vector of output prices. However, from Equation aT 2-32 we have that Tr( G) = Ig,. =-I--, where Tr denotes the trace operator. , 8 ln Y, Therefore the above equat ion can be written as 1 R 82VC p -'G 81ny = p Tr ( G) oyoy' 8 ln p' which is s implifi ed to the following expression

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37 G 8lny = P(R 82VC p -']_, Tr ( G) 8 ln p' 8y8y' Finally, simplifying the right-hand side of this expression, we get G alny =_!__P(a2vc] P= 0 Tr ( G) 8 ln p' R 8y8y' If (2 96) ( J 1 ( 2 J-1 1 a -vc . 1 a vc where we let If/ = -p --p > 0 and If 0 = P --P R 8y8y I R 8y8y' At this point we need to bring Equa tions 2-93 and 2-94 into the same form as in Equat i on 2-96 Beginning with Equatio n 2-93, pre-multiplying by _!__ P and post R multiplying by W ( = diag ( w ,)), we get (2-97) SolvingEquation2-95for R (fora!! (r,) pairs)andusing Tr(G)=I g ,=-I aT , 8lny, we obtain the following relationship for the total revenues of the multi product firm: PY R=-Tr(G) G Substituting Equation 2-98 back into Equation 2 -97 we obtain c P ay w = _!__ P a vc a vc w ( 2 ] I 2 p. y Tr ( G) 8w' R 8y8y 8y8w' (2-98) After canceling terms in the left-hand side of th e eq u ation this can be s implified to G 1 ay w = _!__ P a v c a vc w ( 2 ] I 2 Tr ( G) Y 8w' R 8y8y 8y8w' However the left hand side of this expression can be further simp lifi ed to get

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38 c a1ny =-J_P a2vc a2vc w ( J1 Tr ( G) 8 ln w' R 8yc3y' ayaw' Using then the definition of 1;/0 *, we obtain G alny =-' e 'K' where K=W(a2VC]p-1 Tr ( G) 8 ln w' tfl away' (2-99) Using similar analysis fo r Eq uation 2-94 that is pre multiplying b y J_ P and postR multiplying b y Z we obtain J_p ay Z = _J_p a-vc a VC Z w hich from Equ at ion ( ? J-1 2 R 8z' R 8yc3y' 8y8z' 2-98 becom es G 8 ln y = 0 0 Tr ( G) 8 ln z' tfl ( a2vcJ where Q = Z -p-i azay' (2 100) T h erefore, pre-multiplying the di ffe rential output supply b y ) and u s ing the Tr G so lution s from Equat ion s 2-96, 2-99 and 2-100 we obtain ) d In y = tfl e d In p -tfl e K'd In w-tf1 0 01d ln z Tr G (2 -101 ) whe re e = [ 0 ;, ] is an m x m sy mmetric pos itive definite matrix which is normalized so m m that it s elements add up to one LL 0;, = 1 However there is no clear interpretation for r=I s=I the coeffic i ents of d In w and d In z. Starting with the input price coefficient, define K as the nx m matrix that has the m arg inal s hare s 0,' (Equat ion 2-37) as its (i r )"' element. Then from Shephard s lemma in vector form we have that ave= x. If we differe nti ate aw

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39 h l h. h b a2VC ax H c. E 2 3 t 1s re ations 1p wit respect to y we o tam --= -. owever 1rom quat10n -7 away' ay' we have that Substituting for [ 0,'] this expression simplifies to Using the definitions of the terms in both sides of the equation, this expression can be also written as WX ax I_= KG =-_!!__ I _K(PY 'VT J vc ay' X Yi vc I TY, R Y, where G = ( -PY 'VT J and y1 = _}!__LI are derived from Equations 2-95 and R )', VC T r Y, 2-88, respectively and IT = L aT After some algebra the above equation can be r Y, r a In Yr transformed to W ax = KP. This expression can be solved for ax or K in order to get ay' ay' ax a2vc -1 a2vc 1 -=--= W KP and K = W--P respectively. Therefore the matnx of ay' away' away' marginal shares 0,' can be written as [ e ] = K = w a2vc p 1 I away/ (2-102) where P is an m x m matrix with the output prices or marginal costs on the diagonal, depending on which are defined.

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40 Given Equation 2-102, we can write the s 'h element of K 'd 1n w as n dlnW./ = Ie/dln w, 2 (2 103) /=) This is the Frisch variable input price index (this is denoted by the superscript F ). For the coefficient of the quasi-fixed inputs notice that the (s,k)"' element of 0n1dlnz can be written as (2-104) Then we can define m a2vc 77,k = I e, s ac )a 1n = I P s Y s zk (2-105) This can be interpreted as the sum of the changes in the marginal costs of the various products due to the changes in the availability of quasi-fixed inputs where the weights are the coefficients 0:.,, which define the substitution or complementarity relationship in production (see next section). oting that the r'" component of Tr~G} is equal to I~, and using Equations 2-103 and 2-105 we can write the r'h equation of the output supply Equatio n 2-101, as (2-106) or in an equivalent form 2 F denotes that this is a Frisch price index, given that it has a marginal share as a weight instead of a budget share in a Divisia index.

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41 (2 -107) s The variab l e in the left hand side of E quation 2-107 is J d 1n Y,, w hich is the L.g s contribution of the r'h pro duct to the Di v i s i a vo lume index of outputs ote also that J = P,.Y, which i s the re ve nue share of the r1h product. L.g, R 2.4.2 Comparative Statics in Supply T h e supply E qu a tion 2 -107 d esc ribes the change in the firm s su ppl y of the r1h product as a linear combination of a ll output pric e changes each deflated by its own Frisch input pric e index and all quasi-fixed input chan ges. For the output-supply system the follo wing hold: If all input prices a r e unchan ge d then d ln W/" = 0. Then E quation 2 -107 become s g m I L~, dlny, = v / ~ 0:.,dlnp, -lf,lB77, kdlnzk s If th e pric es of all va riable inputs and all outputs incr ease proportionately then I d ln W F = d lnp,. andthusEq. 2-107becomes J dlny,=-lf,l L7J,kdlnzk L_.g_ k= I m To find th e tota l output decision of the firm define e ; = I e:s, and note that .1'= 1 m m I e : = 1 i s impli ed b y the normali z at i on IIe:., = 1. Therefore the weig h ted means of r = I s = I m th e lo ga rithmic price c h anges th a t occur in E quation 2-106 are d In p F = Le; d In p , r=I

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42 m d ln WFF = L 0 ; d 1n W,1' Corresponding l y, let for the coefficient of the quasi fixed input r = I L 77,k = 7Jk ext we sum Equat ion 2-106 over r and use the symmetry of e to obtain (2 108) This is the total output decision of the firm, which shows that 1./f is the price elasticity of total output ( 1./f > 0 ). ext multiplying Equation 2-108 by 0; and putting the result back into Equation 2-107 we obtain the output allocation decision m ( J ( p F J g, P s d ln Y, 0 d In Y + 1./f L 0,sd ln---,::-1./f 0 d In~ L,.g, s=I W W I I -1./f L 77,kd 1n zk + 1./f 0; I 7J.-d l n z k k=I k=I or equivalently s }' The deflator in the price term is d ln :;F = d 1n p F -d 1n WFF, which is the same for each input-deflated output price change in Equation 2-109. If these corrected output price changes are proportionate then the second term in the right hand side of Eq. 2-109 is equal to zero. This shows that in Equation 2-109 only relative input-deflated output price change have a substitution effect. Therefore if 0:s < 0 r -:Is then r'h and s11' products are specific substitute while if 0:., > 0 r -:1s then r'h and s'h products are specific complements. Fmther, 0;, < 0 r -:ts implies that an increase in the s h relative

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input-deflated output price leads to a decrease in the production of the rth product. Finally, the Divisia elasticity of the r11r output is obtained from Equation 2 109 as D = d In Y, = e: e dlnY /"\:' g ~gf If this Divisia elasticity is negative ( D e < 0) then the specific output is inferior s i nce when firm increases total output the particular output decreases. 2. 5 R atio nal Random Beha v ior in th e Diffe rential M od e l According to the theory of rationa l ran dom behavior (Theil 1975) econom i c 43 decision-makers actively acquire information abo u t u ncontrolled variab l es s u ch as prices of inputs in the case of cost minimization and prices of outputs in the case of profit maximization or both prices However t h is information is costly, imp l y i ng that the decision-makers have incomplete information. To account for this non optimality Theil (1975) suggested adding a random term to the decisions of the firm. He further showed that if the marginal cost of information is small then the decision variables of the firm (input and output levels in our case) follow a multinormal distribution with a mean equal to the full information optimum and a covariance matrix proportional to the inverse of the He sian matrix of the criterion function. Chavas and Segerson (1987) criticized Theil's approach to rationalize the stochastic nature of choice models because it relies on a quadratic loss function for the decision maker. That is th e error term is not an integra l part of the optimization prob lem of the decision-mak er. They instead provided a method to include it in the cost function of the firm. In this s tudy we will follow the rat i onal random behavior theory since otherwise it would unnecessarily complicate the analysis. Notice though that the covariances of the

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44 error terms in both systems are independent of the inclusion of quasi-fixed inputs. That is, under this theory the short-run model has the same covariances as the variable LT model. The proof is almost the same as provided by Laitinen (1980, page 209) and it will not be reproduc ed here. Therefore relying on the theory of rational random behavior, an error term is added to the variable input-demand equation (Eq. 2-84) to get m I n J,d In x, = Y2L0,r g;d In Yr+ r]I,;/ ;d In z k + L nljd In w j +, (2 -110) r = I k = I J = I e Then 1 11 have an n -variate normal distribution with zero means and variancescovariances of the form (2 -111) These covariances form a singular n x n matrix that is the sum of ... 11 has zero n variance since I Ip' ( u -, 1 ) = 0 from Equat ion 2-65. This further implies that the 1 = 1 total input decision of the firm continues to take its non-stochastic form (Equation 2-80) when the theory of rational random behavior is applied to the firm. In the case of profit maximization the rational random behavior theory implies that a disturbance ,~ must be added to the system of output-supply equations of the firm m m n I g;d ln Yr = L If/ 0,~,d In P -LL Ip' 0;, 0 d 1n w L Ip' 17rkd In zk + & ; (2-112) < = I s = I 1 = 1 k=I

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45 where the above expression was derived by tiling into account Equations 2-106 and 2-103. Further s ;, ... s,:, have an m -variate normal distribution with zero means and variances covariances of the form 2 C51.f . Cov(sr,s J =--0rs with r s = 1, ... m (2-113) Y 2 Notice that the CY2 is the same coefficient as in Equation 2-111 The vectors s = (s1 s n )' ands = (s; ... ,s,:,)' are independently distributed. This implies that the system consisting of the input-demand equations and that of the output-supply equations constitute a two-stage block-recursive system (Laitinen 1980). The first stage consists of Equation 2-112, which yields the m output changes and the second consists of Equation 2-110, which yields the n input changes for given changes in output. The independence of the input and output disturbances can be interpreted as meaning that the firm gathers information about the two sets of prices independently. In the case of output supply however, summation of s; ... ,s;, over r is not equal to zero This implies that the total output decision of the firm (Equation 2-108) takes a stoc hastic version, when the theory of rational random behavior is applied to the firm. This is also obvious below (2-114) 2 where E = 2:S: and from IIe: = 1 it follows that Var(E*) = CY l.f/ r Y2

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2.6 Comparison to the Original LT Model In this section a brief comparison of the original LT model with the extended model (EL T) developed in the previous sections is provided Laitinen and Theil (1978) assumed that the production function is negatively linear homogenous in the output vector, which impli es that 46 I a r =-1 r 81n Y r (2-115) This relationship is not crucial for the derivation of the input-demand and output-supply equations, but for the definition of the coefficients in those equations. Taking into account the expression (Equation 2 -9 ) for the returns to scale it is obvious that Equation 2-115 imposes a restriction to this measure namely that the denominator is equal to negative unit y whi l e in the ELT model no such assumption is imposed. As mentioned before, the main difference between the LT and EL T models relies on the coefficients g , ;rlJ and A. Specifically in the LT model g r is the share of the rth product in total variable cost but in the ELT model this is true for g; = g r / L g s 3 In the case where s quasi-fixed inputs are introduced to the model then similar results hold for the definition of k Concerning the price coefficients ;rlJ, in the LT model these coefficients were decomposed to ;rlj = -1.j/(0lj 0 ,0), where 0 is the marginal share defined in Equation 2-38. This relationship is entailed from assumption 2-115 and that the second derivatives of Equation 2 -115 with respect to output and variable inputs are equal to zero (Laitinen 1980 page 180). In contrast this relationship does not hold for the EL T model were no 3 This is obviou s from E quation 2 31. See also discussion below this equation.

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47 such assumption is made and 1r1J = -tf (IJ 1). However, as was shown in Equations 2-63 to 2-65 and the discussion below these equations the same properties hold for both decompositions, as far a it concerns summation of these coefficients across input demand equations or o er all inputs in the same input-demand equation. The systems of equations for both models are represented below LT Mode l m n ID: J,d In x = Y1L 0,r grd In Yr+ L7Z',i In w +&; r = I 1=1 n, m n "'"" "'"" "'"" s OS: grdlnyr = L,;lf 0r,dlnp_ L,;L,;tf 0rs 0 dlnw, +&r 1 = S= I / = I ELT Model m I n ID: J,d In x = Y2L0,r g;d In Yr+ r 3 I t/ ;d In zk + Inljd In w +&, r = I k=I J=I Ill Ill II I OS: g;dln yr = Llf e:,dlnp, LLl// 0 : ,0,sdln w LlfTi1rkdlnzk +&; = I s = I 1 = 1 k = I Notice that in the EL T model there are more terms in both input-demand and output-supply systems of equations corresponding to the quasi-fixed inputs (z k). This is one of the generalizat i ons pursued in this study. Further as it was shown above, there is no need to make the assumption 2-115 in order to obtain the two systems. For instance y1 in the LT model is equivalent to y2 in the EL T model where both coefficients are defined as the r eve nu e va riabl e cost ratio or as the elasticity of variable cost with respect to outputs of the firm. This assumption serves into easier derivation of the equations but it impos es a restriction in the returns to s cale.

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CHAPTER3 PARAMETERIZATION AND ALTERNATIVE SPECIFICATION 3.1 Input Demand Parameterization In order to estimate the variab l e input-demand and output-supply systems of the multiproduct firm there is a need to parameterize them since both depend on the infinitesimal changes in the natural lo garithms of prices and quantities. Laitinen (1980) provided a parameterization for the LT model, which is extended in the section to account for qua si-fixed inputs and the non-output-homogeneous production technology. Thus a finite change version of the differential d In q is defined as Dq1 = In q1 -In q1 _1 where q refers to all prices and quantities relevant to the firm and q1 is the value at time t. Further an error term is appended to each variable input-demand eq uation as depicted in E quation 2 -8 4 rel y in g on the theory of rational random behavior (Theil 1975): m I 11 J;d In x, = Y2L 0,' g;d In Yr+ r 3 Ii;/ Jt;d In zk + I nijd In W J + C ; (3-1) r=I k=I J=I w here the following relationships were defined or proved in the previous chapter : R evenue share of the firm g' = _L = P,Y from Equation 2-89 "' R g,. w.x Cost s hare of the firm J; = -1 1 ; vc Share of the e h qua s i-fi xe d input shadow value in total shadow value of the quasi-fi A avc1 a1nzk fr 1 xe d input s, A = ="' om E qu ation 2-3 4 ; ~ A ~aVC!alnz e e e 48

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Negative semidefinite price terms of rank n -1, known as Slutsky coefficients in the Rotterdam model, 1r u = -w ( u 1 ) ; Revenue-Variable Cost ratio or elasticity of variable cost with respect to outputs, a1nve R r2 = r1I g = I---=from Equations 2 -81 and 2-87; , alny ve Elasticity of variable cost with respect to the quasi-fixed inputs, defined as '' '' a ln Ve Y3 =r1L,; k = L,;---=&1-c,: from Equat1on2-82; k k alnz k Share of i 'h variable input in the shadow price of quasi-fixed input z k defined as k a ( w,x, ) azk c;, = ave I azk ; Share of i'" variable input in the marginal cost of the r'" product, defined as 0 = a (w,x,) loy,. ave I ay, Covariance of the e1Tor terms eov ( t:,, t:1 ) = o-2(// ( iJ 1 ) There are two existing problems with the estimation of the demand system. First, y 3 c;,k, A, Lk are not observable since they involve derivatives of the variable cost k 49 function with respect to the quasi-fixed inputs. They would be observable if quas i-fixed h full b 1 1 h ave h b mputs wer e at t e1r equi I nurn eves, smce at t at pomt ---= v k wit v k emg azk the ex-ante market rental price of the quasi-fixed input. This in turn, would transform the model to a long-run w ith no quasi-fixed factors. A solution to this problem is to leave y3 c;,k, A Lk as unknown and estimate one coefficient b,k = y3c;,k / However, as is k u s u a l with demand systems, the estimation method requires dropping one equation from

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the system due to singularity of the disturbances as was shown in Section 2.5. Proceeding this way though the coefficient of the quasi-fixed input in the dropped equation cannot be recovered, since I b;k is still an unknown constant. Therefore, a complete demand system estimation method must be employed. An alternative is to transform the coefficients of the quasi-fixed inputs in order to add up in a known constant. Both methods w ill be discuss e d in the next chapter, at the choice of the econometric procedure Section 4. 1 50 So far there is no distinction between the cases of one and multiple quasi-fixed inputs. As it is going to be shown in the next section the one quasi-fixed input is a special case of the multiple quasi-fixed inputs case and the estimation method does not differ. Berndt and Fuss ( 1989 ), in their measures of capacity utilization showed that in the case of multiple input s and multiple outputs the long-run economic capacity outputs cannot be uniquely determined unless additional demand information is incorporated in the model, such as the equality of marginal revenue with the long-run marginal cost of the firm. An alternative method though is to consider perfect competition and specify a variable profit function as in the case examined by the present study. 3.1.1 The Case of Multiple Quasi-Fixed Inputs Sununing Equation 3-1 over i and using the definitions of the Divisia inde xes as presented in Chapter 2 we obtain the total input decision of the firm: (3-2) In Equation 3-1 the factor and product shares J, = ;;; and g; = P;, are observable and can be calculated for any period from price and quantity d ata. As in the Rotterdam

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51 model or Laitinen (1980 ), arithmetic means are employed for these shares since they are used to weight logarithmic changes between two periods Therefore by using a subscript t to denote time the factor and product shares at period t are given by J,1 = w u x ; i and vc1 g;1 = PnJn, while the average factor share of the it h input int and t-1, and the average I revenue share of the r11' product in t -1 and t are given respectively by z = (J,1 + J,,,_ 1 ) ; / 1 ( I I ) g,t = 2 g,, + g ,,,1 (3-3) Further define Dx, = In x, -ln x _1 Dy = ln y -ln y,_, Dz, = In z, -ln z,_1 as the finitechanges version of the variab les in the model, which imply that the finite-change version n n of the Divisia indexes can be written as DX,= 'Il,Dx;,, DY,= Lg;1Dy r1 and I DZ = LJi~1D zk,, respectively. According to the theory of rational random behavior the k= I total input decision ( Equation 3-2) holds without disturbance. Since y3 is not observable E quation 3-2 cannot be solved for y2 and thus emp lo ying y2 = ___!!__ from Eq uation 2 -87 vc we define its ge ometric mean as R -R VC, -VC,_1 Then the total input d e cision in its finite change version can be written as DX, = fi,DY, + 13,DZ, (3-4) (3-5) To solve the problem of identification of 131 one could proceed in two ways. First E quation 3-5 could b e s olved for 131 = (DX, -y2, DY,) I DZ, However the possibilit y of

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DZ1 being zero and that it requires specification of the unobservable term ;1 this solution becomes unattractive. Instead an approximation for 731 seems to be more plausible. Remembering that at the full equilibrium level of the quasi-fixed input ave = v k, then at any point different than this optimum, it must hold that azk 52 ave ---= v k + 8 k, where 8k denotes the deviation between the ex-ante market rental price azk v k and the shadow price of the quasi-fixed input (Morrison-Paul and MacDonald 2000). It follows from this definition that if 8 k = 0 then the quasi-fixed input is at its full equilibrium level while if 8 k > or < 0 then we have undercapacity or overcapacity utilization of the specific quasi-fixed input respectively. Therefore, we could use the following approximation (3-6) Then taking the geometric mean of y3 1 and accounting for the error of the approximation t: r we have that (3-7) Further to solve the problem of identification of ;1 we follow the same technique as in y3 1 and define its approximat ion as k,e=l ... l (3 -8)

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53 while we use its arithmetic mean in our parameterization using the same argument as in the case of f,1 and g~1 : -/ 1 ( , ) kt= 2 kt+ k t 1 + (3-9) A problem with the finite change version DX1 = y2 1D~ + y31DZ1 is that it will usually be violated by the definitions of DX1 DY,, DZ, y2 and y3 1 in the previous page. One possible explanation as noted by Laitinen and Theil (1978) is technical change since Equation 3-2 is the total differential of the production function and Equation 3-5 entails changes from period t 1 to t. This could be a generalization of Hicks neutral technical change. However in this model there is one more explanation, which is the approximation of ;1 and y31 by the use of market rental price for the quasi-fixed input since shadow price is unknown. To account for these possibilities and the errors induced b y the approximation of ;1 and y31 (&:, ; respectively) we need to add a residual in the finite-change version of the total variable input decision: (3-10) ( L V k1zk1 )(I vk,1-1 2k,1-1) 1 h -k k (, ) dE w ere y31 = -'-------'--'------'-; k = k + k 1 _1 an contams VC VC 2 r I t-1 From this equation the residual E1 can be calculated as (3-11) The input changes are then corrected by computing (3-12)

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54 This correction amounts to enforcing the finite-change version of the total input decision since summation of the correct input over i yields (3-13) Taking into account Equation 3-12 for the residual correction and the parameterizations of the quantities and prices the finite-change version of the variable input-demand (Equat ion 3-1) can be written as m I n xii= L 0,'ji,, + I?/zk, + InljDwj, +,, (3-14) r = I k = I J=I In this formulation we have defined the terms Yn = fi,g;1Dy ,1 zk, = YJ,Ji;1D z k1 and u = -lj/ ( 1J -A) as before. Further it is assumed that 0/, i;/ 1r1J and a-2 are constant over time so that Cov ( , 1 ) = a-21// ( u i 1 ) implies that the contemporaneous covariance matrix of demand disturbances ( covariance that concerns disturbances of different equations but of the same year) is the same in each period The effect of the correction in va riabl e input l eve ls as appears in Equation 3-12 is to make E quation 3-13 which can also be written as 2)\ =LY,,+ Izk1 hold. This in tum, gives that r k summation of E quation 3-14 over all inputs i will yield IB,' = 1 It/"= 1 L"ii = 0 and Ll'11 = 0. Ther efo re the variable input demand (Eq uation 3-14) satisfies the following prop erties: Ad din g up : :z:e,r = 1 It,K = 1 and L"u = 0 where i,j = 1 ... n and k = 1 ... ,1. Homo ge neit y : I n u = 0 J

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55 Symmetry: T[u = 1[1 ; Negative semi-definite matrix of the price parameter (1[ ii) ofrank n -1, implying that the underlyin g cost function is concave in input prices. 3.1.2 The Case of One Quasi-Fixed Input As it is going to be shown below this is a special case of the multiple quasi-fixed inputs case. Notice, that when the firm employs only one quasi-fixed input then by definition ; = 1 and so the variable input-demand equation (Eq. 3-1) becomes nl II J,d In x, = y2 I 0 ,' g ; d In Y, + y3c;/ d In zk + I 1[;1d In w1 + & (3-15) r=I J=I In this case the Divisia index of the quasi-fixed input degenerates to d In Z = I;d ln z k = d In z k and the elasticity of the variable cost with respect to quasik fixed input becomes y3 = o 1n VC I o In zk, since k = 1. Disregarding for a moment the error term and summing Equation 3-15 over all i we obtain the total input decision (3-16) Proceeding then as in the case of multiple quasi-fixed inputs the following variable input-demand equation i s obtained: ,,, n x,t = I B,'ji,t +c;/zkt + InuDwJI +&i t (3-17) r=I ;=I This differential variable input demand satisfies the same properties as Equation 3-14 For instance the assumpt ion I<;/ still holds. The only difference with the case of multiple quasi-fixed inputs is that the residual term ( E ) used to correct the variable input does not contain an y mor e error due to approximation of ;1 since ;1 = 1 Further if t he

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56 changes in the level of the quasi-fixed input are not zero then there is no need to use the approximation for y3 since it could be obtained from f3 = (DX1 -f2D~)/ DZ1 3.2 Output Supply Parameterization As in the input demand case, we rely on the theory of rational random behavior to append an error term in the supply equation of the firm (Eq 2-106) in order to obtain (3 -18 ) where the following definitions were provided in the previous chapter: Price elasticity of total output, If with If > 0 Substitution or complementarity relationship in production denoted by e ; s The sum of the changes in the marginal costs of the various products due to the changes in the availability of quasi-fixed inputs, weighted by the coefficients e;s 111 a2vc as T/rk = I e,, ----,=, 8(p,yJ8 ln z k Ill ,,, Nom1alization condition, LL e;, = 1 r = I s=I 2 CYlf/ Covanance of the error terms, Cov(c:r, c:J =--0rs Y 2 Similarly a finite-chan g e version of the output-supply system (multiplied by f2 1 in order to make it homos cedastic) is (3-19) If it had been assumed that the coefficients If e;, were constants then an autoregressive scheme (AR) wou ld be present in the supply system since this assumption would imply that the variance-covariance matrix of the disturbances depends on y21 which varies over

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57 time. Multiplying though each eq u ation in the system by f2 the disturbances become homoscedastic and now it is assume d that the coefficients ars = f2,lfl e;s and /3,k = f2,l/f r; ,k are constant. The covariance of the disturbance s is then given by Cov(t:;,, t:_ :,) = CY2f2,l/f e,:, = CY2a,s, which is constant. The supply system can be written then in a more compact form, as m n m I Yn = L a , .DPsr -LLa,,0,5Dw;, -L/3,kDzkt +t:;; (320) s=I 1= 1 s=I k=I The properties of the output supp l y system are: Output supp l y i s homogeneous of de gree zero in both input and output prices. The coefficient matrix of t he output price s, [ a,J must be ne gat i ve definite of rank m implying that the profit func tion i s convex in output p ric es Symmetry condition: [ ars] = [a.,,] onlinear symmetry condition: If linear symmetry conditions are imposed in both systems then the non lin ear coeffic ient s of th e input prices are not free param eters 3.3 Alternative Specification for the Cost-Based System The variable input-demand system as repr ese n ted by Equation 3 1 assumes constant price effects output and quasi-fixed effects. However, there is no reason to ex ante impose such restrictions on the system Fousekis and Pantzios (1999) provided a genera li za tion of Theil s (1977) parameterization for the one product firm based on different parameterizations for the R otter dam model. In this sect i on the i r results are extended to the multiproduct, multifactor firm. To allow for varia ble output effects 0 ,', let us define J, = a + m; In X (3-21)

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58 where J, is th e cost s hare and 1n X is the variable inputs Divisia index. Note that since If= 1 it mu s t hold th a t Ia, = 1 and that Im; = 0. Multiplying then E quation 3-21 b y v ariable co s t ( ve) and differentiating with respect to y, we g et a(w,x, ) = a ave + m r ln X ave +mr Ve atnX O)lr O)lr O)I, Yr a 1n Y fr E 2 l O h Ve a 1n X ave l h b . otmg o m quati o n t at --= --, t 1en t e a o v e equation 1s Yr a ln Y,. oy r tr a n s formed t o a (w,x,) I ave ,. l X r --= a + m n + m O)I, oyr I I I Making u s e now of E qu at ion 3 2 1 and the definition o f 0,' ( see below Equation 3-1) we h ave that 0 = a(w, x ) I ave = J, r I ---,+m oy r O)I,. Th e r efo re th e i 'h in p ut d e mand with v ariable output e ffects become s m I n J,d In x = r22)J, + m ; )g; d In Y r + r3I~: ; d In z k + I n u d In W J + r=I k = I ;=I T o allow for varia bl e effe ct s in all co e ffici e nt s let us define no w II J, = a + m ; ln X + I s u ln w1 J=I (32 2) ( 3-23 ) ( 32 4 ) Since I f = 1 it mu s t hold th a t Ia,= 1 Im;= 0 and a lso that Is!i = 0 I s!i = 0 I I I j su= Jl,w herei, j = l ... ,n. To t a ll y di ffe r e nti a tin g Eq u a ti o n 3 2 4 we have

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59 df, = m ; dinX + I s u d In w1 (325) J From Equation 2 -28 it holds that the tota l differential of the variable cost ratio is equal to df, = J,d 1n w + J,d In x J,d In VC. Also summing this expression o v er all inputs i, it holds that d In VC = L J,d 1n w + d 1n X Combining these two expressions we obtain df, = J,d In w + J,d In x J,Lfd I n w, J;d In X ( 3-26) Eq uatin g now Express i ons 3-25 and 3-26; and after some algebra we get n f,d ln x = (m ; + J,)d In X + I( si/ -J, ( t5u -.,0 )d 1n w1 ) J=I (3-27) where t5u i s the Kronecker delta To verify that the input price terms in Equation 3 27 satisfy the adding-up property we sum E quation 3-27 over all inputs i, to obtain that n n I I ( s u f, ( t5u f;) d In w1 ) = 0 which verifies that the adding-up propert y holds for t=I J=I the input price terms. Equat ion 3 27 is a syste m of input demands that must be equal with the input demand system presented in Equa tion 3-1. F orcin g this e qu al it y we have n m I (m ; + f,)dlnX + I(s u -J,(t5u -l;)dlnw j ) = r 2 I B/g;dtny +r3 I ~ / ;d1nzk J=I r = I k= I 11 + In, lln w1 +, J=I Summing thi s expression on both sides over i and usin g the previous results, we v eri fy the total input decision of th e multi product firm: (3-28)

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60 Substituting now Equation 3-28 back into Equation 3-27 we obtain m I n f,d In x = y2 L)m; + f,)g;d In Y + Y3L (m; + f,);d In z k + L( s1 J; ( 8iJ )d ln w1 ) r=I k=I J=I By rearranging terms, we get an allocation-type differential system of input demands: + I ( S u -J, ( 8 u ~) d In W1 ) J=I (3-29) Letting now m ; = 0,' -J, we get n, n f,dln x, = y2L0,' g;_dlny, + y30;' dlnZ + L( s u J; ( 8 u )din w1 ) r=I J=I (3-30) Then we could combine Equations 3-23 and 3-29 into one general equation, since the left-hand side variables are the same but the right-hand side variables differ. This implies that the models are not nested. Therefore, m I J,dlnx, = Y2L(m; +e1J,)g;d1ny +r3L(m; +e1J;) ; d1nz k + r = I k = I n + L ( n u e2f, ( 8 u ~) d ln w1 ) ;=I where e1 e2 are two additional parameters to be estimated and the additional restriction L ( m ; + u,k) = 1 -e1 is imposed in the estimation. Using a lik elihood ratio test one could test which of the following restrictions are valid and so which differential input-demand sys tem fits the data better: 1. If e1 = e2 = 0 then we get our original differential system. 2 If e1 = e2 = 1 then we ha ve all coefficients variable, Equation 3-29.

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61 3. If e1 = 1 e2 = 0 then we have only variable output effects, Equation 3-23. 4. If e, = 0, e2 = 1 then we have only input price effects being variable, Equation 3-30. Note that the presence of d In Z in Equation 3-29 may create problems of multicollinearity so an instrumental variable approach is suggested for the estimation of the system. Also Equation 3-31 seems more plausible than Equation 3-29 since it alleviates the problem of multicollinearity. 3.4 Capacity Utilization and Quasi-Fixity The most appealing alternative parameterization of the differential model is given by Equation 3-30, since it allows us to test for quasi-fixity and capacity utilization. Decomposing the Divisia inde x of the quasi-fixed factor in Equation 3-30 we get the following equation m I n f,d In x = Y2_L0, g;d ln y, + r3B,,.I;d1n z k + I( SI) -J, ( 01) I; )d In w j ) (3-32) r; I k;I J;I Using the definitions of y3 0,' and ; it is easy to show that ave 0,.~ 'di = 0,. ( a In Ve] ~ aln zk dl r 3 ,L,;k J1Zk I L,; L,;( J nzk-k;I k alnzk k ; J I ave k aln zk 0,. a ln Ve d In 0 d 1n = I L,; zk = I L,; 6vc,zk 2k k=I a In zk k ; I Substituting now this term back into Equat ion 3-32 it transforms the input-demand system into m I n f,dlnx, =r2_LB ,'g;dlny, +0,'_L&vc,:;dlnzk + I (siJ-J,(olJ-f;)dlnw1 ) (3-33) r=I k=I J=I

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62 I Then the total input decision of the firm, d In X = y2d 1n Y +LL &vc.zk d In z k is I k=l obtained by summing Equation 3-33 over i and using the previous result that L 0;' = 1. The most important result is that the summation of Equation 3-33 over i for a specific quasi-fixed input gives us an estimate of the elasticity of variable cost with respect to the level of that quasi-fixed factor. This estimate, I &vc . = ( ave I azk )( z k I ve)' provides a tool to test for quasi-fixity of input k Specifically a testable hypothesis for quasi-fixity is H0 : ( ave I azk ) + v k = 0, where v k is the ex-ante market rental price of the quasi-fixed input. If H0 holds then it implies that the quasi-fixed input is at its full equilibrium level and should not be included in the right-hand side of the demand equation. Given that we ha ve an elasticity estimate we need to transform the null hypothesis into H0 : H0 : [ ave _!_i__) ve + v k = 0 where the first term in the parenthesis is the azk ve zk estimate from the input demand estimation and is being multiplied by (Ve I z k ) at each data point at the sample. If the null holds at some data point then the quasi-fixed input k is at its full equilibrium level and the model is misspecified while deviations from H0 show that the input is quasi-fixed. A one way t-test could be developed to find the sign of capacity utili za tion Note that we can test at each observation on the sample like the Kulatilaka (1985) t-test providing the whole path of changes between full static equilibrium and short-run equilibrium for the input zk. If one uses the average of the

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63 observations in the sample to construct (VC I z k ) then H0 provides a joint test for quasi fixity for all observations. Schankerman and adiri, (1986) provided a test for quasi-fixity through a Hausman te st for specification error in a system of simultaneous equations where their system consisted of a restricted cost function short-run demand for variable inputs and long-run demand for fixed factors. Given that in this study we do not have a functional form for the cost function we cannot apply their test for the differential model. However a specification test between the conditional demands for variable inputs Equation 3-1, and the long-run demands for the quasi-fixed inputs can be obtained. This would be a simultaneous-equations error specification test. However the estimation of Equation 3-33 requires complete system of estimation methods since the disturbances add up to zero, implying that their variance matrix is singular. If, we would proceed by deleting one equation from the system then the coefficients of the quasi-fixed inputs in the deleted equation could not be reco ve red since they do not add up to a known constant. Since the focus of the present study is on the comparison of the differential model with a translog specification we will not test for quasi-fixity. However we provide directions for the estimation methods for such systems in the following chapter.

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CHAPTER4 EST IM AT ION METHO DS 4.1 Choice of Estimation Method In Chapter 3, a model for the decisions of a multiproduct firm over a period of time was presented. Whi l e this form u lat ion see m s to be restrictive for real ap plications it should be noted that it can be transforme d to reflect different situations For instance the one firm cou ld represent o n e sector of the whole economy such as agriculture F urth er, if one was considering the input -d emand sys tem then it could be t rans formed to reflect situations in International Tra d e or Marketing Specifically in International Trade varia bles in the l eft hand side of the equation cou ld denote the international trade of flows of import s of a specific country from different import sources w hich necessarily add up to total imports. In marketing ana lysis they could represent the market shares of all brands of a specific product whic h add up to un ity The purpose of this chapter i s to present and develop different met hod s of estimation for the differential model. Specifically in this section we present the econometric procedure for the joint estimation of the input-demand and output-suppl y system of a multi product firm as provided by La itinen (1980). It will form the basis for the econo m etric procedures in the next section s, which concern multiple multiproduct firms ; that i s panel data structures. In those sections maximum likelihood estimat ion methods for time-specific fixed-effects an d firm-specific random-effects panel data are developed. The novelty in those secti ons i s the cons id erat i on of systems of equations 64

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65 which are nonlinear in the parameters and have nonlinear cross equations restrictions For convenience we reproduce the systems of equations m I n x,, = IB,'y,1 + It/zk1 + In-l)Dwjl +&,, (4-1) r ; J k;J ;;J m n m I Y n = Ia, .DP.,1 -LLa,s0/Dw,, LfJ,kDzk, +&: (4 2 ) S;J /;) S;J k;J where i = 1 ... n and s, r = 1 ... m denote number of equations and we have assumed constant coefficients 0,", r;,k 1r1J, a,s, /J,.k and a,s0,s over time. Also as was shown in Chapter 3, both systems of equations have homoscedastic covariance matrices which are the following changes in notation have been made: i,1 = J,JDx,, -E,), Yn = y21g~Dyr1, Note that since we have assumed that the matrix [a,s] is constant over time, then If/, which i s the price e lasticity of the firm's total supply, is proportional to the costrevenue ratio. This can be seen from equations y2 =RI VC and I I a,s = y211j1 An r s alternative paramet e ri zat ion can be formulated with constant If/ (Theil 1980). This could happen if we divide both sides of Equations 4-1 and 4-2 by y2 and treat ,riJ I y2 and a,s I y21 as constants The disturbances &u I y2 and ;; / y2 are still homoscedastic. Another problem with the parameterization of Equations 4-1 and 4-2 is that we have assumed constant technology for the firm but this can be resolved by adding a constant tern1 in both syste m s Laitinen (1980 page 118) suggested that these terms

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would represent systematic changes in the firm's technology (Hicks neutral technical change). 66 Before we proceed into the estimation method for the joint system of Equations 4-1 and 4-2, we need to impose the adding-up restrictions, symmetry, and homogeneity properties of the two systems. We choose to im pose those restrictions in order to reduce the number of coefficients to be estimated To satisfy the adding-up property in the input demand system which creates the problem of a singular variance matrix of disturbances in the system of Equations 4-1 we drop one equation from this system. Foil owing this method to deal with singular disturbances necessitates the use of a maximum likelihood (ML) estimator which gives estimates invariant to the dropped equation (Barten, 1969). Recently there have been developed methods for estimating a complete system of equations with singular covariance matrix of disturbances (Equation 4-1) that do not require dropping one equation; and so do not rely on the invariance property of the ML estimator. Dhrymes (1994) considered the case of autoregressive errors in singular systems of equations. His estimation method relies on the use of a generalized inverse (Moore-Penrose) for the variance of the disturbances and on a formulation of an Aitken Minimand Shri v astava and Rosen (2002) provided a ML estimator for a complete system of equations with unknown singular covariance matrix of disturbances. Complete system of equations estimation with singular covariance of disturbances in seemingly unrelated regression methods (SUR) and three stage l east squares (3SLS) framework was provided by Kontoghiorge s (2000) and Kontoghiorges and Dinesis (1997) respectively. The initial approach was to estimate the joint system of input-demand and output supply equations ( E quations 4-1 and 4-2 respectively) by employing one of the

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67 previously mentioned methods for the input-demand system and then to provide a joint method of estimation for both systems. However, the nonlinear cross-equations r estrictions on the parameters and most importantly the need for a panel data method led to the use of the more standard method, of simply dropping one equation. The transformation for the quasi-fixed inputs -8VC I 8zk = vk + 5 k used in the parameterization of the input-demand system (Equation 4-1) serves that purpose, since the summation of z k over i adds-up to a constant and thus the last equation can be dropped. The homogeneity property of the input-demand system (Equation 4-1) in input prices and output-supply system (Equation 4-2) in both input and output prices is imposed by subtracting the input price that corresponds to the dropped equation from all prices in both systems. Symmetry is an important property that needs to be imposed or tested Given the adding-up conditions symmetry can not be tested without homogeneity already imposed. In the joint system of equations we have symmetry conditions for the price terms in the input-demand system and for the price terms in the supply system. Symmetry in the price terms of the two systems of equations (homogeneity restricted) can be imposed by including on the coefficient vector only the unique elements and rearrange the exogenous variables matrix to correspond to those elements. For instance, consider the case of one firm utilizing three variable inputs two quasi-fixed inputs and four outputs. Then the homogeneity and symmetry imposed input-demand and output supply system will have the following form Y 2 ( W I W 3 ) ( W 2 W 3 ) 0 0 0 0 ( WI W 3 ) Y1

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68 (w1 w3 ) ( w2 -w3 ) 0 0 0 Pi In this formulation we have omitted two outputs and the quasi-fixed inputs to save space and we have dropped one input-demand equation due to the singularity of the disturbances. otice that the input price parameters in the supply system are allowed to vary freely when the model is unrestricted or homogeneity restricted. However under the homo g eneity and s y mmetr y restricted model as above these parameters are fixed (not free), since it i s r e quir e d c,,s = arsB/. Therefore imposing s y mmetry in the joint system transforms the input price term s in the output supply to nonlinear creating an additional comple x ity in the estimation procedure Then for our example that turns out to be the estimated model in the next chapter the vector of coefficients has fifty free parameters in the homo g enei ty r es t r i c ted model including an intercept for each equation while in the I homo g en e it y and sy mm e tr y re s tricted model consists of thirt y five free parameters. Having showed how to impose adding-up linear symmetry and homogeneity restrictions the input-d e mand and output-supply systems can be written in a stacked e quation form. To ac c o unt for th e s in g ular cov a riance matrix o f disturbances in th e input d e mand sys t e m th e la s t e quation was deleted. Therefor e, Equa tions 4-1 and 42 are written in matri x form a s x = 0y, + Kz, + Dw, + e, = N v + e , Y, = Ap, + C~, + F ~ + e, .. = Mq + e .. which i s s ubj ec t t o th e fo llowin g r es triction s Homogeneity Aim + Ci11 = 0 in o utput s upply i = l ... n l (4-3 ) r = l ... m (44) (4-5 )

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Di n = 0 in input demand Linear Symmetry Conditions A= A' in output supply D = D' in input demand Nonlinear Symmetry C = -A K', in input demand and output supply (4-6) (4 -7 ) (4-8) (4-9) 69 The adding-up property of the input-demand system has been imposed by deleting the last equation. Homogeneity in both systems has been imposed by subtracting the input price that con-esponds to the dropped input-demand equation i.e. for i = 3, from all prices in both systems. The linear symmetry conditions have been imposed as shown before but the nonlinear symmetry condition is left for the estimation procedure. Accordingly the following conventions in the notation have been made A = [ a , ],,,x,,,, C = [ a,.,e,s ],,,xn, F = [,B,k ],,,x k 0 = [ 0,' L t xm, D = [ 7riJ ] n l x n, The price vectors w, = ( D w1,, Dwn_11 )', :!', = ( Dw1 , Dwn -i, )', P = (Dp1,, Dpm, )', denote the modified prices where Dwm ha s been subtracted from every price. Finally N = [0 K D] and M = [ A C F] are partitioned matrices, and v; = (i, z;, w;), ( , ') q, = P,,:!'.1:',,~ The joint system as presented in Eq uations 4-3 and 4-4 wit hout the nonlinear symmetry res trictions i s a triangular system. Further relying on the theory of rational

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70 random behavior the disturbances in the demand system (Equation 4-3 ) are stochastically independent of those of the supply system (Equation 4-4) making the joint system block recursive This has two implications. First, it implies that the decisions of the firm take place in two separated phases. First the output-supply decision is taken and then given this decision the input demands a re determined Accordingly we can view Yrr as a predetermined variable. However, we can observe that the marginal shares of the inputs 0,'" occur not only in the demand system (Equation 4-3) but also in the supply system (Equation 4-4 ). Therefore, in spite of independence of the disturbances of the two systems a joint method of estimation of Equations 4-3 and 4-4 is more appropriate in order to impose these restrictions on the marginal shares. Further in the supply system 4-4 the parameters are nonlinear if we impose symmetry and homogeneity. Let us denote th e va riance across equations in the input-demand system and outputsupply system as (4-10) By rel y ing on the rational random behavior theory the above systems form a block recursi ve system and under normality we have that the joint system error covariance st ructure is Q [ E(c1 &1 ) = I c111+n-1 J x C111+ n 1 J = O n l x m O,,~' ] ( 4-11) However we choose not to force the off-diagonal elements of the covariance matrix to b e zero Bronsard and Salvas-Bronsard, (1984) suggested to tes t for the exogeneity of y1 in the input-demand system b y estimating the joint system one time with Equa tion 4-11 imposed and one without, and then form a likelihood ratio test for the

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71 covariance restricted versus the unrestricted model. Assuming that the disturbances are independent in different periods Laitinen ( 1980 page 120), writes the log likelihood function of the joint syste m as ( 4-12) From Magnus and eudecker (1988) we have that 8 ln 1z:1 1 aa'z:1 a = Z:' and -= aa' az:1 az:1 Then for given M,N the first-order condition with respect to z:-1 is given b y which gives the following expression for the covariance matri x, (4-13) If one wanted to assume that Z: follows the assumption of Equation 4-11 then the off-diagonal e l ements in Eq uation 4 -13 would be zero. In this case one could use a twostep estimator where in the first step 6.* and Q are estimated from each system separately (impose hom ogeneity in each system at this step) and then use those as an initial estimator of Z: where now impose the linear and nonlinear cross equations restrictions. To apply the nonlinear symme try constraints, it i s convenient to regard the elements of M and N as functions of a vector th at contains only the free parameters in the joint system. Further we choose to substitute the nonlinear symmetry restrictions

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72 C =-A K' at the objective (likelihood function). Another equivalent way would be to include it as a constraint and maximize the constrained log-likelihood function. Magnus (1982) proposes the latter method but he also suggests substituting a large value, like I 000 to the la gra n gea n multiplier. Then for g iven Z: the first-order conditions with respect to the i'h element of vector 4 b are given y BM ~=t[y,-Mq,]z:1 8 q = 0 a, l = I x, -Nv, aN -v a I [Ml [A -AK' DF]. where = N 0 K (4-14) In Appendix A. I th e a nalytical derivatives of BM I 8 and 8N I BA are provided for a system that con s ist s of three variab l e inputs two quasi-fixed inputs and three outputs Also, notice that we ha ve made the substitution C = -A K' in the supply system. Finally Laitinen (1980, page 124) shows that the information matrix for the param ete r s ha s th e follo w in g form BM q [ 82 L ) r a E -I J a a 1 / = I aN v 8 j-t1 I (4 15) The inve r se of the information matrix will yield an as y mptotic estimate o f the covariance m atr ix for t h e paramet e r s that maximize L Then for a g i ve n vector we 4 Laitinen (1980) ha s already derived these conditions and we reproduced them here.

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73 define OJ as the vector with t h element 8L I 8 , given in Equation 4-14, and E the d . h (. )1h 1 E( 8 2 L J b E . square an symmetnc matnx, w ose z, J e ement 1s ----, given y quat10n 8 8 ) 4-15. The iterati ve procedure that Laitinen (1980) suggests consists of the following steps: Compute i: usin g Equation 4-13 with M,N evaluated at the given vector. Use Equations 4-14 and 4-15 to evaluate OJ and E. Let /j_ = E -10J. Then if /j_ < 0.000001 use the given ve ctor as the vector that maximizes L and E -1 as its asymptotic covariance matrix. If the previous condition does not hold then update the vector by using Aiew = A id + E -1 OJ Get, i:. Then from Equation 4-12 the concentrated log-likelihood function at the optimum becomes L =-T(m + n l)ln21r e T lnjfj max 2 2 (4-16) While Laitinen (1980) does not suggest an initial estimator for the vector a consistent initial est im ator for the joint syste m under the ass umption of independent disturbanc es of the two sys tem s and without linear or nonlinear symmetry restrictions imposed could be obtained by separate iterative S U R in each system. If disturbances are not independent then a consistent estimator would be an iterative SUR in the joint system 4.2 Fixed Effects and Pooled Model The pr ev iou s econome tric procedure considers only one firm over multiple years but our dataset consists of a large number of firms observe d over a small period of time.

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74 Therefore we need to consider panel data techniques for the estimation of the differential model. In this section we analyze fixed-effects models and pooling across years or cross sectional units. Recentl y, Baltagi et al. (2000) showed that for a dynamic specification of the demand for cigarettes pooling was superior to heterogeneous estimators. Before we proceed to the estimation methods, it should be noted that if one does not want to impose the disturbances in the input-demand system to be independent of those in the output-suppl y system then there is no need to follow the structure of equations as initiall y pre sented in Equations 4-3 and 4-4 and was followed until Equation 4-16. Instead we could consider that the two systems of (n-1) plus m equations as one system with G equations and an equation index g g = l ... ,G. Further we make the following conventions regarding notation in this and subsequent sections. We suppose that i = 1 ... refers to the number of firms in our sample ; t = 1 ... T is the number of years that each fim1 is observed (balanced case) ; g is the index for the equations of the input demands with g1 = I ... G, and g2 is the equation index for the output-supply system with g 2 = 1 ... G2 and G = G, + G2 Then the firm i at time t we hav e G_ / G1 x = g g y + ;:k i + n -Dw + & g, 1 g , g,1 ':, g, 1kt g,g, ,g 1 g, 1 ( 4-17 ) K = I k=I g1 = I (4-18 ) The notation is further s implified by assuming ( 4-19)

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75 where i = 1 ... N, t = 1 ... T g = 1, ... G and G = G1 + G2 ; y g11 is the dependent variable in the g'h equation. For instance the vector of exogenous variab l es is denoted as [y1,,, y011 ]' = [ x111, x0111, j\,, ... ji0211 ]'; x g,, is the matrix of exogenous varia b les for the g'h equation and /Jg is the coefficient vector of the eq u ation (if symmetry has been imposed then /Jg has no duplicate terms). Then stacking all G equations for each observation (i,t), we obtain Y111 X1,1 0 0 /31 ulil Y211 0 = + (4-20) Y c11 0 Xe;,, /Jc U Git This can be written in a compact form as Y,, = x,,/3 + u11 (4-21) where Y,, is a G x 1 vector, X11 is a G x K matrix of exogenous variables and /J is a G K x 1 vector with K = L Kg and K g is t h e num b er ofregressors in the g'h equation g=I including a constant ; U11 is a G x 1 vector of the error terms Since we want to impose symmetry and so m e coefficients appear in at least two equations then we redefine f3 as the complete coefficient ve ctor which is nonlinear and does not contain any duplicates apart from the nonlinear terms (see examp l e in Appendix A .2). Further, we redefine X11 = [x ;;,, ... x~,,J', where the k'h e l eme n t of x g11 i s redefined to contain the observations on the variable in th e g'h equation which corresponds to the k'h coefficient in /J. If the latt e r do es not occ ur in the g'h e quation then the kth element of x g11 is set to ze ro

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76 A pooled model wo uld then consist of regressing Equation 4-21 for all i and t. However, it is implicitl y assumed that all firms have the same intercepts and slopes over the entire period which is a very restrictive assumption. One way to account for heterogeneity across indi v iduals or through time is to use variable intercept models. So following Baltagi (200 I page 31) let us decompose the disturbance term in Equation 4-21 as a two-way e1Tor component model: U,, = a,* + a + v,, i=l, ... ,N, t=l, ... T ( 4-22) where a,* denote s all the unobserved omitted variables from Eq uation 4-21, which are specific to each firm and are time invariant ; a, denotes all unobserved omitted variables from Equat ion 4-21 that are period individual-invariant variables. That is, variables th at are the same for a ll cross-sectional units at a given point in time but that vary through time. Finally v,, i s w hite noise It is this ability to control for all time-invariant variables or firm-invariant variables whose omission could bias the es timates in a typical cross-section or time-series stud y that rev ea l s th e advantages of a panel. The way we treat a,* and a , it then differentiates between fixed-effects and random-effects models. Specificall y, if we treat a ; as fixed parameters to be es timated as coefficients of firm-specific dummies in the sample then we follow a fixed-effects approach. Instead if we assume that a ; are random v ariables that ar e drawn from a di s tribution we have a random -effe cts model. The same arguments are true for a,. Randomeffec ts models are considered in Section 4.3.

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So suppose that we formulate a fixed-effects model for N multiproduct firms, where the effects of omitted, unobserved, firm-specific variables are treated as fixed constants over time. Then, E quation 4-21 becomes 77 Y,, =a: + X,,/J + U,, i = l ... ,N, t = 1 ... T (4-23) In this formulation a ; represents the firm-specific effects. For instance, in banking it could account for all differences such as location management skills or persistent Xinefficiency that permanently affect the demand for inputs and supply of outputs of a particular bank relative to some other bank that face similar conditions. However we could reject the use of a firm-specific fixed-effects model in this study for two reasons. Our sample consists of T fixed and N lar ge and a fixed-effects approach would result in a huge loss of degrees of freedom ( df = NT N K +I). Secondly our model is already first differenced which sweeps out the individual effects. For instance, our Y,1 is equal to (ln Y,, -ln Y,,_1 ) and from E quation 4-22 it is obvious that a ; are differenced out. However one could argue that fixed effects exist between Y,, = 1n Y,, -ln Y,,_1 and Y,,_1 = 1n Y,,_1 -ln Y,,_2 For that purpose we consider a random-effects model in the next sec tion. A more appropriate fixed-effects model would be to consider time-specific effects a, as fixed parameters and estimate them as coefficients of time dummies ( Dum, ) for each year in the samp l e That i s, to consider the model Y,, = i c 0(f a ,Dum,J+x;,/J +U;, =-K_;,f!.. +U;, t = I i = l ... N t = 1 ... T (4-24)

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78 T where i G is a vector of ones, and we impose I a, = 0 to avoid the dummy variable trap, t=l since /J contains an overall constant in each equation. Further, we make the following assumptions Assumption 4.1: The error terms of Equation 4-24 are independent and identically distributed as (4-25) Assumption 4.2: K,, and U11 are uncorrelated (4-26) Notice that we made the assumpt ion of normality since we are goi ng to use a maximum likelihood method. If a genera lized least squares method was to follow then one should replace Assumption 1 with the following 11 {z: zif i = 1 t = s E(Uu)=O G x i and E(Uuu;s)= 0 ifi-:t:J,t=s (4 -2 7) 0 ifi-:t:j t-:t:s l 11 0"11 Also, Z:11 is defined as Z:,, = : II (J" G I 11 ] (J"IG : which is the correlation across equations a-" GG G x G for an individual at time t and is positive definite. We assume no correlation between individuals for the same year and no contemporaneous correlation across years since we imposed in th e parameterization of the model (Chapter 3) the disturbances to be homoscedastic. otice that the formulation in Equation 4-24 implies that only intercepts vary over time. It further implies that there are common shocks in the demand for inputs

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79 and supply of outputs for all firms in a specific year This could be clearer b y stacking the observations by year first so Y, =K,/}_+U, (4-28) where Y,, K and U are now the stacked ( GT x l) vector, ( GT x K) matrix, with K including the time dummies and ( GT x l) vector of Y's, X's and U 's respectively corresponding to the T observations of individual i. That is Y = : ; X = : ; U = : [Y,11 [K,Jl [uill Y,,. K,.,. uiT (4 -2 9) And let V, =[U ,l' ... U,,.Jcx T with U, =vecV, i=l, ... ,N (4-30) Then making u se of Equatio n 4-25, the ( GT x 1) vectors U ; are distributed as IIN ( 0 Q), with variance matrix E( U,,U,') = I.,. L11 = Q ( 4-31) L,, 0 0 0 0= = l.,.L11 0 L GTxGT The model as presented in Equation 4-28 is simply a seemingly unrelated regression (SUR) first considered by Zell ner (1962) but nonlinear in the parameters. We may then formulate the following proposition. Proposition 4.1: The lo g -likelihood associated with the linear model (Equation 4-28) but nonlinear in the parameters under the Assumptions 4.1 and 4.2 is given by N l T l T =IL, with L =-GTln2n--inlL,,I -I u:,L~1UII 1 = 1 2 2 2 t = I

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The proof of this proposition is simp l e and is based on Magnus (1982) The probability density of Y, takes the form (see Appendix of Magnus, 1982) The log likelihood function for firm i is then 1 1 I I 1 I L =-GTln2.1r--ln Q --(UQ U) I 2 2 2 I I 80 (4 32) (4 33) Notice however that Q =Ir L11 and so its inverse is equal to Q-' =Ir I ~ and the determinant is equal to I Q I = TII11I (Theil 1971). Then the log-likelihood function can be written as (4-34) ow we can formulate the next proposition for the gradient vector and information matrix of the model considered in Proposition 4 .1. Propos i tion 4 .2: Consider the linear model in Equation 4-28 but nonlinear in the parameters under the Assumptions 4.1 and 4.2. Then the gradient vector and the N information matrix for =LL, are given by 1 = 1 I a =-NTL +_!_ U ~ =-~~[x al}_) I -'(r X /J) a I ~ 2 II 2 6-6' II ii al}_" -fr~ -11 al}_h II II i i ( 4-35) I= [[ N T [ 8/J )' 8/J ]l K,, ai_.. L ;'K,, ai_, L 0 (4-36) 0

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81 To prove this proposition notice that for a g i ven vector /J, we can differentiate Equation 4-34 with respect to the covariance matrix to get aL; =-TI +_!_ fiuu' =O aI-1 2 II 2 II 1 1 II 1-1 Summing this equation over all individua l s it gives Equation 4 35. Further we can solve for I in the above first-order cond ition t o get 11 f = -1-~fiu U = -1 -~VV' 11 NT II II NT I I (4 37) Before we differentiate with respect to the nonlinear vector f.!.., note that U ;1 = _f;1 Kit/.!.. and /J contains no duplicates Then, for given L11 we differentiate Equation 4 34 with respect to the h1" element of /J, V h, h = l . K to get which simplifies to [ l r a/3 a L1 = X-=-I -1 Y X =0 af] I ( _, 1 af] ) II ( _;,f.!..) _h l = I h (4-38) Summin g this g radient v e ctor over all indiv i duals it gives Equation 4-35. In Section A 2 of the Appendix we provide the ana l yt ical grad i ent vector for the examp l e of this chapter. Notice that if /J was linear in the parameters then Equation 4-38 would give us the GLS estimator (4-39)

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82 In order now to find the Hessian we need to take the second order deri vat ives. We begin with the covariance matrix (4-40) Taking the second -order derivative of the coefficient vector, after some algebra we obtain the following expression 8J..., = """'"""' X --=---L I (Y -X /3) -"""'"""' X -=--L 'x -=-r [ N r ( 82/J J ] [ N r ( 8/J J' 8/J ] 81}_1,81}_:,, .s, 11 81}_1,8!}_, ; 11 it -II_ .s, -II 8l!_m 11 i i 8/}_h ( 4-41) Since E(Y;, Ji,,/}_)= 0 the expectation of the above expression gives the information matrix of the paran1eters N r 8/J 8/J I X -=-L -1X -=P --" a13_., J " a13_, [ I ] (4 -42) Given that the information matrix, which is defined as I = -E ( D2 L (I}_, L11)), is block diagonal (Heymans and Magnus 1979) and combining Equation 4-42 with Equation 4-40 we get the expression in Equation 4-36 Further the inverse of the information matrix g ives the asymptotic covariance of the estimates and the disturbances. Notice, that the asymptotic covariance matrix for /J can be obtained independently from that of Q since the information matrix is block diagonal. The iterative procedure to find the estimates that maximize the likelihood function is s imilar to the one in the previous section and is based on the multivariate Gaussewton method ( Harvey 199 3). Thus define OJ as the vector with H element 8 / 8 /J and I P as the square matrix in the upp er left comer of the information matrix

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83 Then the multivariate Gauss-Newton iterative procedure consists of the following steps: Get initial consistent estimates of the vecto r /J, using the GLS estimator presented in Equation 4-39 by disregarding the l inear and nonlinear symmetry conditions Impose though homogeneity and obtain the relevant estimates. Compute i11 usin g Equation 4-37 at the given vector I}__. Use Eq uation s 4-35 and 4-42 to evaluate w and I P Let !)./}__ = l p-1w. Then if!)./}__< 0 000001 use the given vector I}__ as the vector tha t maximizes L and r p -1 as its asymptotic covariance matrix. If the previous condition does not hold then update the vector I}__ by using /}__new = /}__0,d + r;/ w and go back to step 2. Continue until convergence. To avoid potential confusion between the Scoring method and the multivariate Gauss-Newton method as presented above notice that in the case examined above those methods coincide. Specifically if 1/f denotes the vector that includes the parameters and the variance to be estimated in the model ij) is the initial estimate of this vector and 1/f is the revised esti mate then for step 4 in the procedure above the method of scoring consists of calculating 1/f* = ij) + r 1 ( ij)) D In ( ij)) The Gauss-Newton method starts b y minimizing the sum of the squared error terms and in the multi var iate case for systems of equations it turns out that the updating procedure is \If* = \V + [ Z, L -1 z; r Z, L-1 , where Z1 i s equal to -a; I al/f ( Harve y 1993 page 139). Given that the log-likelihood function i s concentrated with respect to ~" at step 2 in the procedure then it obvious that

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84 the i averse term in the last equation is [ z, L I z; r = Ip while the last term of this equation is equal to Iz,z:-1&1 =OJ. I From Proposition 4.1 we have that the log-likelihood function is = I_GNT 1n 21rNT 1n1z: I _!_~ f, uz:-1u 2 2 u 2 II u II 1=1 1 = 1 Substituting in this expression the estimates /J and I.11, the concentrated log-likelihood concentrated likelihood function at the optimum becomes 1 NT = -GNT(ln21r-l)--lnlZ: I mu 2 2 u (4-43) The previous method of estimation accommodates unbalanced panel-data designs since it is simply a pooling of the observations across years, through the use of time specific dummy variables. If the data were balanced then for time-specific or firm specific random-effects panel data the Magnus (1982) method could be used for the estimation of the model. Further, in the case of unbalanced panel data and random effects but with linear symmetry conditions the maximum likelihood estimator is a straightforward extension of the one provided by Magnus ( 198 2). Wilde et al. 1999 provide an application of this procedure. 4.3 Random Effects In thi s section we consider that the individual-specific effects are random variables that follow the normal distribution. Our proposed estimation method under symmetry and nonlinear re s triction s on the parameters is a spec i a l case of the Magnus (1982) maximum

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likelihood estimation for a balanced panel. Biorn (2004) has provided a stepwise maximum likelihood method for systems of equations with unbalanced panel data 85 For convenience, we rewrite Equation 4 -21 for the joint input-demand and output supply system as (4-44) &,r = a +u/{ (4-45) In this formulation we assume that a are firm-specific, random effects and & ;1 are random errors. Further the coefficient vector /3 has no duplicates and includes an overall intercept. The matrix of exogeno u s varia bles is assumed to have the form X,1 = [ x;,1 x;,1 ]'. Concerning the distributional form of the random variables we assume ( ) { I if i = j a -JJN G (0cxl L a ) that is E(a, ) = 0 Cxl' E a ; a / = o a if i j (4-46) ifi=j,t=s if i =t:-j, t = s ( 4-4 7) ifi=t:-j t=t:-s X,1 Q1 and 11 are Uncorrelated [ 11 0"11 Then we have I,, = : 11 O"c;1 a ] 0"1c O"a CG CxC From Equations 4-46-4-48 it is easily shown that 5 The error terms in this section are not related to any disturbances in the previous sections. (4-48)

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86 {L.11 + L.0 if i = J, t = S E(E:,,)=0c x l and E(E:,,E:~ s)= L,a ifi=j,f-:t;S 0 ifi-:1;J ,t-:1;s (4-49) As before we stack the observations by time to get Y, = X fJ + i r a + u,, = X,, fJ + E:, (4-50) where Y, is a (GTxl) vector X is a (GTxK)matrix and E:, is a (GTxl) vector corresponding to the T observations of firm i Also i T is a vector of ones. It follows that (4-51) since E(E:1 ; ) = = Q and I T is a T dimensional identity matrix and J r = i T i / is a T x T matrix with all elements equal to one. Then according to Biorn (2004) we could rewrite Q as (4-52) where A T = _!_Jr and B T = I r _!_JT are symmetric and idempotent matrices. T T Followin g Mag nu s (198 2 ) the log-likelihood for the i 'h individual is given by ( 4-53) Defining I.1 = I.11 + TI.0 we have that (4-54)

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87 Using the property that A T and B T are symmetric and idempotent matrices, Magnus (1982) shows in his lemma 2.1 that (4 55) Also note that T T :Ur r :'), = Ic;/r: ';, and c ;(Ar(I~' -I:')) =(1 / T) I c; /(I~' -r:'),s /;I t,s;I Using then the above expressions and Eq uation 4 -55 we can rewrite the log likelihood as T T GT 1 I I 1 I I 1 L 1 L _ L =--ln2;,r--ln I -(T-l)ln I I -(I -I ) (4 56) 2 2 2 ,, 2 / ( 11 2T 11 ,, IS t-1 t,s-1 For given covariance matrices, I.11 and I.0 we ta ke the first and second order conditions of Equation 4-56 with respect to the h1h element of the nonlinear vector of parameters. Using the same techniques as in the previous sect ion the gradient vector and information matrix of the coefficient vector are given respectively by ( 4-57) ( 4-58) A s ymptotic cov ar iance matrix of /J is obtained by taking the inverse of the information matrix above For given coefficient vector /J, we do not need to derive the firstand second-order conditions with respect to the covariance of the error terms since the nonlinearity is in th e parameter s of the model. Therefore we adapt the results from Magnus ( 1982) since in that paper there was a time-specific error component and not a firm-specific to g et

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88 z: = l ~V(J. 11 )V' Z: = l ~V(T11 -I ) V u (T -l)N f:( T ~ i a T(T -l)N ''r T i (4-59) An iterative procedure could be employed as in the previous section for the estimates that maximize the lo g -likelihood function. Further to prevent the solution to converge towards a local maximum Magnus (1982) suggests ensuring that Z:u and Z:0 are positi v e semidefinite.

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CHAPTERS APPLICATION TO U.S. BANKING INDUSTRY 5.1 Introduction One of the objectives of this study is to utilize the differential production model as means of estimation of the input demand, output supply and efficiency measures of US banks. To examine the robustness of the differential model results and to highlight the differences in the description of the technology that are induced by fitting the differential model a comparison is provided against a commonly used in the literature parametric specification (translog). The discussion of the results focuses on three aspects of technology: concavity returns to scale and input substitution as measured by the Allen Uzawa elasticities of subs titution. The differential model is based on the total differentiation of the first-order derivati ves of any arbitrary cost or profit function given a technological constraint. As it was s hown in Chapter 2 this provides an input-demand and output-supply system of equations for the multiproduct-multifactor firm. The res t rictive assumption of the differential assumption as presented in Chapter 2 is the one of perfect competition that may not hold in the "empirical world" and thus limiting its applications. A dual approach, inst ead, involves specify in g a flexible functional form that achieves a second-order approximation of any arb itrar y twice differentiable cost function at a g iven point (Diewert 1971 ). T he tran s log which was developed by Cristensen, Jorgenson and Lau (1973) can be interpreted as a Tay lor series expansion and is the most popular of the 89

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90 Diewert flexible forms. However, White (1980) has shown that while second order approximations allow us to attain any arb i trary function at a given point there is no implication that the true function is consistent at this point. Moreover different functional forms lead to different results for the same dataset, as Howard and Shumway (1989) indicated ; and often fail to satisfy parameter restrictions. In the empirical banking literature some of the major concerns are related to the functional form specification and to the validity of the efficiency measures obtained from such specifications. For instance Berger and Humphrey (1997) have shown tha t a local approximation, such as the translog usually provide poor approximations for banking data that are not near the mean scale and product mix. The geographic restrictions on branching that ha ve contributed to the pro l iferation of banks in the U nited States and the large amount of mergers happened when a state allowed for branching stimulated the interest on correct efficiency measures such as economies of scale. However early findings on economies of scale were contradictory and naturall y led to the use of non parametric measures of efficiency. In the next section a brief review of the performance and structure of the U.S. bankin g industr y is provided while in Section 5.3 previous findings on the "puzzle" of economies of scale functional form specification and the controvers y on what constitutes a bank's input s and o utput s, a r e presented. The data u se d for the analysis are describ ed in Section 5.4 whi l e the em pirical mod e l is presented in Section 5.5. Empirical results and comparison of th e differential model and translog specification in terms of satisfying concavity Allen elasticities of s ubstitution and economies of scale are provided in Section 5.6.

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91 5 2 T h e US Ba nkin g I ndu stry i n t h e 90s The banking industry constitutes a major part of the U.S. economy and it can be described as a competitive industry In recent years, the number of commercial banks in the U.S. has begun to fall dran1atically. It has decreased from 14,095 in 1984 to around 8 ,33 7 in 2000 (Table 5-1) and most of the banks exiting have been small (less than $100 million in assets) Moreover, the l arge banks' share of assets has increased to almost one third while the small banks share has decreased to less than 5% (Dick 2002). Bank failures played an important, but not predomi nant role in the decline in the number of commercial banks during 1985-1992, and bank failures have played an almost negligible role in the continuing decline seen since 1992 (Berger and Mester 1997). The primary reason for the decline in the number of commercial banks since 1985 has been bank consolidation. Until the passage of the Riegle-Neal Interstate Banking and Branching Efficiency Act (1994), U.S. commercial banks were prohibited from branching across states This Act permitted nationwide branching as of June 1997 while some states had already allowed for intrastate and interstate branching (as early as 1978). Recentl y th e Gramm-Leach-Bliley Act in 1999 allowed U.S. commercial banks to participate in securities activities, such as investment banking (underwriting of corporate securities) and brokerage activities involving corporate securities. Table 5-1 illu strates the number of banks for the period 1990 -2000 along profitability measures such as retwn on equity and return on assets for the "average" bank in each year. Profitability in the banking sector as measured by the mean return on equity rose b y 1.2% from 5.44% in 1990 to 11 % in 2000. An alternative measure of profitability mean return on gross total assets rose from 0 .61 % in 1990 to 1 % in 2000. It

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92 is obvious though that both of these meas ur es have been stable from 1993 to 2000, even if gross total assets are increasing steadily Table 5 -1. Financial indicators for the U.S. banking industry 1990-2000 Year umber of Banks ROE ROA GTA 1990 12,306 5.44 0.61 350,175 1991 11, 917 6.82 0.68 349,659 1992 11, 456 10.90 0 98 357 869 1993 11, 064 11.97 1.11 380 463 1994 10, 660 11.55 1.07 420,658 1995 10, 004 11.78 1.12 465,465 1996 9 ,593 11.86 1.12 487 833 1997 9 167 11.83 1.13 544,932 1998 8,804 11.19 1.09 618,883 1999 8 606 11.21 1.02 652,281 2000 8 337 11.09 1.01 727,936 ROE refers to return on equity ROA to return on assets and are percentages while GTA is the gross total assets o f the bank. All these measures were calculated from the Call Reports and are means for all banks in a g i ven year. All financia l data are in real 2000 values. 5.3 Bri e f L iteratur e R eview Existing efficienc y studies in the banking literature can be considered as a mixture of measurin g cost e f fici e nc y or profit efficiency employment of parametric or nonparametric methods and utilization of frontier analysis or more traditional techniques (economies of scale and scope). Regarding comparisons between frontier methods and traditional methods Ber g er Hunter, and Timme (1993) suggested that although frontier estimation method s are theoretically correct studies that have compared results of frontier methods to those of more traditional estimation methods have found only small differences for scale measures. However they showed that noticeable differences exist in scope measur e s betw ee n frontier and more traditional estimation methods. They also conclud e d th a t th e tran slog s p e cification for examinin g scope e conomies causes problems due to the multiplicati ve nature of the outputs

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93 Berger and Humphrey (1997) reviewed 130 studies, which examined efficiency by using frontier methods and the y concluded that nonparametric methods yield slightly lower mean efficient estimates and seem to have greater dispersion than the results of the parametric methods. Clark (1988) reviewed thirteen studies that measured economies of scope and scale for commercial banks credit unions and savings and loans associations. He concluded that overall economies of scale exist at low levels of input, but no consistent evidence of economies of scope was found. Further, some evidence of cost complementarities also exists. However, Hunter, Timme and Yang (1990) tested for economies of scale and scope in large banks by using a minflex Laurent functional form but the y found no cost complementarities. For robustness of their results they included deposits as inputs and as outputs. Noulas et al. (1990) examined returns to scale and input substitution for large U.S banks. They rejected the hypothesis of short-run or long-run returns to sca le and suggeste d that economies or diseconomies of scale are not large enough to support the creation of only a few large banks based on cost economies. Berger Hancock and Humphrey (1993) estimated scope measures using a profit function approach. Th e y argued that a profit function method should be used to examine "optimal scop~ economies (see also Berger 1995) and they emphasized the importance of the imposition of cur v ature properties. McAllister and McManus (1993) found that banks face increasin g returns to sca l e to about $500 million of assets and Berg er and Mester (1997) who compare banks within size ranges concluded that in all ranges the mean bank operates at a less than efficient scale. Featherstone and Moss (1994) measured economies of scale and scope in agricultural banking and they found that economies of scale and scope do not e xist for small s i ze banks which include the agricu ltural banks.

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94 More recently Hunter and Timme (1995) suggested that cost inefficiencies dominate the impact of scale and scope diseconomies. Turning now to the problem of selection of flexible functional form, Ellinger and Neff (1993) concluded that the most commonly used functional forms to measure banks costs are the translog genera lized translog and minflex Laurent. Lawrence (1989) tested the robustness of competing flexible functional forms by using a generalized functional form i.e. a Box-Cox transformation of all the variables in a cost function. He rejected the Box-Cox transformation of the ge neralized translog but indicated that translog specifications provide an adequate fit to bank cost data. Nonetheless a disadvantage of the log-quadratic output structure of the translog is its inabilit y to model cost behavior when any output is zero. As a result the estimated translog cost function cannot be used to estimate economies of scope or product specific economies of scale (P ulle y and Braustein 199 2). This led Pulley and Braustein (1992) to propose a composite cost function for multi product finns in order to estimate economies of scale in banking. McAllister and McManus (1993) compared the results of fitting a Fourier functional form and a translog to bank data. They found that banks exhaust scale economies at a much l arger output level under a Fourier specification than suggested b y globa l estimation of a translog cost function. They concluded that the translog cost function specification g ives a poor approx imation when applied to all bank sizes suggesting that nonparametric estimation procedures should be examined. Mitchell and Onvural (1996) a1Tived at the same conclusions about the translog specification when it is fitted across bank sizes.

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95 Berger and Mester (1997) attribute differences in estimates of scale economies to a fundamental shift in bank costs over time that is probably associated with regulatory or technological changes. They found that for the period 1990-1995 banks operated at smaller than efficient scale irrespective of whether a global translog function or a Fourier functional form was used to estimate the bank's cost function. It should be noted though, that the Fourier flexible functional form is not free of troubles. For instance, it is left to the researcher whether to augment the underlying translog function with trigonometric terms or orthogonal polynomials, and the number of such terms to include for estimation. The results of the aforementioned studies indicate that the average cost curve for banks may be U-shaped and that economies of scale exist only for small banks. The findings of scope economies are inconclusive. It appears that this criticism of the standard methodology is partially based on the inability of previous U.S. studies to find strong evidence of economies of scale for the largest banks. Recent evidence though suggests that sizable economies of scale exist that increase with bank size (Hughes and Mester 1998 Berger and Mester 1997). Another problem that concerns the banking literature is that estimates of bank cost characteristics depend on changes in the definition of inputs and outputs. Specifically, concerns exist on whether deposits are inputs or outputs and which factors of production are quasi-fixed. Flannery (1982) argued that factors such as transaction and information costs which could result into a significant proportion of the inputs into the bank s production function are quasi-fixed in the short-run. In particular, retail bank deposits should be considered as quasi-fixed factors of production because both banks and their costumers incur setup costs or "transaction specific investments costs. However there

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96 is no consensus in the definition of inputs and outputs. For example, Hunter and Timme (1995) used core deposits and physical capital as quasi-fixed inputs to examine bank scale economies but compared this case with a case where some of the core deposits were treated as outputs and with a case where all inputs were variable. They found that for large size banks the ray scale economy estimates were not considerably di ffe rent under the three specifications, but the mean efficiency indices were significantly different. Further Berger and Mester (2003) used ph ys ical capital and financial equity capital as fixed inputs but treated core deposits as a var iable input in order to examine bank performance. Hughes et al. (2001) provided an answer to whether deposits are inputs or outputs by simply testing the sign of the partial deri va tive of the operating cost function with respect to the quantity of deposits. Their results strongly suggest that deposits are inputs. 5.4 Data Description There are numerous studies that specify different inputs and outputs of a multiproduct bank and then compare which specification better fits the data. Particularl y, there are three alternative approaches in defining inputs and outputs. These are (1) the intermediation or asset approach, (2) the user cost and (3) the va lue-added approach (Berger and Humphrey 1992). According to the intermediation approach banks act as financial intermediaries between borrowers and l en ders (Sealey and Lindley 1977). In this framework purchased funds and core deposits are considered as inputs while bank outputs are loans and other assets. Physical inputs as labor and premises are specified as inputs that generate costs. Total costs include operating and interest expenses of the bank. The u ser cost approach classifies the output or input of a bank according to its net contribution to bank revenue, while the value-added approach determines outputs based

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97 on those having the largest value added. Thus, under the value-added approach deposits are specified as outputs since they have a large impact on value added. In tlus study we view the banking firm as an intermediary (Diamond 1984), operating in competitive markets and using a multiple input-output technology. Trus assumption naturally leads to the selection of the intermediation approach wruch is also more compatible with profit maxinuzation (Berger and Mester 2003). Trus is because deposits can be considered as reducing profits because the bank borrows these funds and has an interest expense ( except for checking accounts) wrule loans and other assets generate positive cash flows and profits. The data are taken from Reports of Condition and Income (Call Reports) for the period 1990 2000 which contain balance sheet and income statement data for all U.S. commercial banks. Since the data contain quarterly observations for each bank, it is desirable to aggregate them into annually observations. Specifically all the income statement data are in year to date format and so we used the last quarter of the year while the balance sheet represents snapshots of the bank at the end of each quarter and so when selecting a variable for the balance sheet we used the average of the four quarters in each year for this variable. Observations for banks which involved in a merger were deleted for that quarter in which the merger occurred. The specification of inputs and outputs follows closely the one of Berger and Mester (2003) with some differences though since we have not included off-balance sheet items but have calculated a market rental price of equity capital. Table 5-2 gives the definitions of the variables that are going to be used in the empirical specification their sample mean s and s t a ndard deviations for the years 1990 1995 and 2 000. Specifically

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98 we assume that the bank transforms five inputs three variable and two quasi-fixed, into four outputs. The variable inputs are purchased funds core deposits and labor, while quasi-fixed inputs are the physical capital of the bank and the equity capital. The outputs of the bank are commercial loans business loans, real estate loans and securities. The prices of the variable inputs were found by dividing the interest expense on that input over the stock of the input. For outputs, the prices are defined in a similar way and are the interest income from the specific loan category or security over the stock of that output. Berger and Mester (2003) suggest that this way of calculation is problematic since these prices may be endogenous. They suggest, instead for each price of a specific bank to substitute the average price that all other banks received in the same market. This method however ma y solve the problem of endogeneity but it creates another problem that of measurement error since local market area prices are only an approximation to the effective costs and income that are reflected in the banks balance sheet and income statement. For the differential model it was necessary to specify market rental prices for the two quasi-fixed inputs. Fo r the market rental price of the physical capital was used a state average of the ratio of occ upan cy expense over the stock of physical capital. Before taking the state average outliers were deleted both in the upper and lower bound of the distribution of physical capital. This measure reflects also local conditions for the market rental price of physical capital but it may introduce also a measurement error for bank holdin g companies (BHC) or banks that operate in different states. The market rental price of equity was obtained b y an average return on equity in the U.S state that each bank belongs. The observations with negativ e or very high return on equity were d e l ete d

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99 before taking those averages. Then for each bank the state average was substituted. These approximations, however, are taken into account into the differential model since all the inputs of the bank are corrected for this approximation as was shown in the parameterization of the model. Instead, McAllister and McManus (1993) arbitrarily picked a required return on equity, which they assumed was identical across all banks. Clark (1996) used the Capital Asset Pricing Model to determine a market-based required return on equity. Hughes et al. (2001) argue that a plausible range of market return for banks equity is between 0.14 and 0.18 and they evaluate cost minimization (optimality) at this range of prices. Observations with less that 1 % of capital to equity ratio were deleted since they look suspicious as Berger and Mester (2003) noted. Also observations with zero values for physical capital were deleted. The only environmental variable that was available from the Call Reports and was used in this study was a one-digit code indicating the chartering authority of the bank. Specifically banking-type entities, thrifts, credit unions and Edge corporations have federal charters while agreement corporations have state charters. We used banks with state charters as the base case. Regulatory variables such as unit branching state were not made available and so they were not included in the study. Finally the use of the differential model necessitates first differencing of the individual series and so we deleted the observations of the banks that did not exist in the previous or after y ear of analysis. This resulted to a loss of 15, 000 observations, leaving us with 96 584 observations for the period 1990 2000. Instead in the translog where there is no differencing the total number of observations was 111, 908.

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100 Table 5-2. Definition of variables and descriptive statistics (mean and standard deviation) Symbol Definition 1990 1995 2000 Cost and Revenue vc Variable cost 27,683 25 ,185 40,620 (324 861) (258,501) (545,901) R Revenues 33 757 37,822 58,878 (384 240) ( 396,330) (806 564) Variable input quantities Number of employees 125 146 192 x,, (955) (1,145) (2,360) Quantity of purchased funds (time deposits over $100 000 foreign deposits federal funds purchased demand notes issued to US 118 737 167 ,161 312 929 X pf' Treasury, trading liabilities other (1,949 831) (3,025,916) (5, 781 246 ) borrowed money mortgage indebtedness and obligations under capitalized leases and subordinated notes and debentures) Quantity of core deposits (time X cD deposits under $100 000 domestic 198 770 250 320 340 506 transaction accounts and savings (1,181, 062) (1, 525,891) (3,859 352 ) deposits) Variable input prices Bank s price of labor (Salaries and 34 7558 37.9986 43.3118 w,, employee benefits/Number of (8 7998) (9.7581) (12.3015 ) emplo y ees) Bank s price of purchased funds 0.0 755 0.0539 0.056 7 Wp,.(expense of purchased funds / x PF) (0.0165) ( 0.0113) ( 0.008 ) Bank s price of core deposits 0.0627 0.041 0.04 22 WCD ( expense of core deposits / Xco) (0.0085) (0.0068) (0.0079) Variable output quantities Loans to individuals in domestic 36 391 47 520 57 315 Y a offices (284 169) (385 290) (653 672 ) All loans oth e r than consumer 92 036 107 ,721 186 389 Ys1, loans real estate loan s (1, 350 062) (I 770 320) (3, 585 618 ) Re a l estat e loans in bank s 88 ,701 1 2 0 489 201, 930 Y11E domestic offices (656 245) ( 817 765) (2 363 300 )

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101 Table 5-2. (Continued) Symbol Definition 1990 1995 2000 Securities: All other non financial 129,364 183,781 273 778 Y s assets (1,203 502) (2, 130,830) (4 237,691 ) Variable output prices Bank's price of consumer loans (interest income on consumer loans 0.1032 0.0923 0.0914 P n less provisions for loan and lease (0 0339) (0 0288 ) (0.0304 ) losses and allocated transfer risk allocated to consumer loans / YCL) Bank s price of business loans (interest income on business loans 0 1291 0.1123 0.106 PB/, less provisions for loan and lease (0 0351) ( 0.0291) (0.0291 ) losses and allocated transfer risk allocated to business loans / y BL) Bank's price ofreal estate loans (interest income on real estate loans 0.0958 0.0825 0.0803 PRE less provisions for loan and lease (0.023) ( 0.0192) (0.0195) losses and allocated transfer risk allocated to real estate loans) Bank s price of securities = interest 0.0786 0.0612 0.0628 P s income on securities I y (0.0141) (0.0133) (0.0147) Quasi-fixed input quantities Z 1 Premises and fixed assets 5 086 6 577 8 879 (49 190) (65, 107 ) (109 805 ) Quantity of financial equity capital 22 274 36 767 60 178 Z2 (156 851) (302 486) (767 529) Quasi fixed input market rental prices Prox y for the price of z1 (State 0.32 2 1 0.3114 0.295 w : avera ge o f occupancy expense / z1 ) (0.0452) (0.0402) (0.0478) Prox y for the price of z2 (Modified 0.1196 0 1303 0.1298 WZ2 State average Return on Equity) (0.0168) (0.0158) (0.0165) All financial variabl e s ar e measures in 1 000s of constant 2000 dollars by using the implicit GDP price d e flator. All prices are interest rates

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102 5.5 Empirical Model Following the parameterization of Chapter 3 the empirical differential model for three variable inputs two quasi-fixed and four outputs consists of the following system of input-demand and output-suppl y equations 4 2 2 xii =a, + IB,'jirl + Ic;/ zk, + Inl)Dwj l + ,,, i j = 1,2 (5-1) r ; I k ; I ; ; 4 2 4 2 j\1 = a,*+ I a,.sDPs -LLa,sB,'Dw;1 -LfJ, k D z k1 +;; r,s = 1 ... 4 ( 5-2 ) .<; 1 / ; l S;l k;l where we have superseded the firm subscript for notational convenience and i, j = l .. 2 denote input s of a multi product firm and r s = l ... 4 denote its outputs. The output variables are defined as ji,, = y21g;1Dy,,, while the quasi-fixed inputs as z k1 = YJ,Ji;1D z k1 For any v ariable we define the finite, first difference as Dq1 = In q1 In q,_1 Prices of v ariable inputs and outputs are g iven by Dw; and Dps1 respectively. Then l,, = ( J,, + J,, l ) and g;, = ( g;1 + g;, _1 ) are the arithmetic means of the variable cost ratio and the revenue-cost ratio respectively. In this formulation we hav e also corrected the variable input s x11 for the approximation in the quasi-fixed inputs and for the difference approximation, since we use xii = J;i( Dx,1 E ) where the correction E, = DX, -r2,Dl ~ -J'31DZI comes from the total input decision of the firm. Further, n III I D X1 = I ZDx11, D~ = I g ;1Dy,1 and D Z,= LJi~ Dzk1 denote the Divisia indexes of / ; l r;l k;l v ariabl e input s o utput s an d qua sifix e d inputs respecti ve l y. For th e quasi-fixed inputs we ha ve defin e d ;1 = ( ;1 + ;,1_1 ) as the arithmetic mean o f the market rental price

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103 ratio ( v" z" I v , z , ) where v" is the market rental price of quasifixed input k at time t. Similarly we have defined 1]1 as the geometric mean of the market rental value of quasifixed in puts over var iable cost ratio ( v., z" ) IVC, The geometric mean of the revenue variable cost ratio is given by y2 1 = R, R1-1 Furth h ,----. er we assume t at the vc1 -vc1 _1 coefficients 0 ,', t, ,cl), a,s and /3,k are constant while we have shown that the disturbances in both systems are homoscedastic. The coefficients of the differential model do have economic interpretation unlike other param etric models. Specifically n u are negative semidefinite price terms of rank n 1 known as Slutsky coefficients in the Rotterdam model ; c;,k is the share o f ith v ariable input in the shadow price of quasi-fixed input ; 0;' is the share of th variable input in the mar gi nal cost of the rth product ; a,, are the output price terms which mus t be positi ve definite ; a and a ; are technolog y shifters in the input-dem a nd and outputsupply system re s pecti ve ly ; and /3, k are the summation of the changes in the marginal costs of the various products due to the changes in the availability of quasi fixed inputs weighted b y t h e coeffic i e nt s 0 ; a nd total output price elas ticity 1;/, where /3,k = y2 11;/ TJ,k Lait in en (1980) s how ed how one could recover the 0;s and If/ terms. The adding up r estr ictions L 0,' = 1 L t;;K = 1 L 1r iJ = 0 and La; = 0, where i j = 1 ... n and k = 1 ... ,l, h omoge n e it y in th e input-demand sys t e m L7ru = 0 and j

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104 homogeneity in the output -suppl y system have been imposed b y subtracting the price of labor from all input prices Dw ,,, and output prices Dp,,. Symmetry in the input-demand system 1r1J = n1,, in the output supply [a,J = [as,] and the nonlinear symmetry conditions for the coefficients of the input prices in the output-supply system were imposed in the estimation procedure as was shown in Chapter 4. Considering the above specification of the input-demand and output-supply system we could derive the elasticity of /' input with respect to the r 'h output (&x y), own and cross price elasticities ( & xw) fixed factor elasticities ( & x J, cost elasticity with respect to fixed factor (&1,c,= ; ) cost elasticity with respect to r 'h output (&vc,y,), and technology shifts in the input-demand and output-supply curves, respectively as = dlnx, = y2g, 0 C ~ -I d ln Y J, (5-3) d In x nlj & =---= x w dlnw J, j I (5 4 ) = d In x = y3 k ,Ek & r ':,, -dlnz k J, (5-5) (5-6 ) (5-7 ) ~ a a & = and & = a J, a gr (5-8 ) Similarl y, one can find the elasticities from the output-supp l y s y stem. Since a comparison with the translog specification is the main focus of this study estimates for all these

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105 elasticities are not going to be provided. Estimates are going to be provided for t he returns to scale (RTS) and for Allen elasticity of substitution derived by Laitinen (1980, page 42) for the differential model, as A 7r,j =~, i-:tj xw J,~ (5-9 ) Similarly Allen elasticities of transformation for the output-supply system can be written, as *A a = _r. ,_ r s yp -, g,.g, (5-10) The system of Equations 5-1 and 5-2 is estimated by using the maximum likelihood procedure, outlined in Chapter 4 Section 4.2 under the assumption of normal distributed disturbances for a panel of banks for the period 1990-2000. Consequently, we append time dummy variables in each equation to capture time specific effects that are not captured by the overall constant term representing common shocks in the technology of all foms in a given year. As mentioned in Chapter 4, the differential model is a first differences model implying that any firm-specific, time-in variant fixed-effects have been eliminated. However we have appended one firm-specific variab le which is the charter type of the bank. While we have assumed that first differences make each se ries stationary there is no a priori reason for a one period la g in each time series One could test for an ARIMA (0 d 0) model and then difference each time series accordingly. Given that the sample used is highly unbalanced and tests should b e performed for each bank in the sample for the whole period we assume that the differential follows an ARIMA (0 1 0) and so the first-differences model i s appropr i ate Therefore the differential model has already

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106 accounted for heterogeneity in size, age, management, employee's education, technology and location under the assumption that these effects can be represented by a bank-specific fixed intercept term and this term is not correlated with the intercept terms. Estimation of the model as presented in Equations 5-1 and 5 2 for all banks and for the period 1990 2000 makes the implicit assumption that all banks have the same slope coefficients and intercept for each year in the sample and the san1e slope coefficients across years This seems to be a quite odd assumption, given earlier resu l ts in the literature suggesting that economies of scale differ across bank sizes and product mix. Berger and Mester (2003) estimated a Fourier flexible functional form for each year in the sample in order to allow for all the coefficients to vary over time and captured heterogeneity across firms through the use of firm specific dummies (bank belongs to unit banking states limited branching states primary regulator of the bank and other ) However these dwnmy variables account only for changes in the intercept of the mode l for a given cross section and so implicitly they made the assumption of same slope coefficients for all firms in a given year. Differences on asset size and product mix may be also manifested in the slope coefficients. The slope coefficients in the differential model are actuall y marg inal shares e.g. 0/ is the share of /' v ariable input in the marginal cost of the r11' product of a specific firm. Making the assumption of common slopes across firms it implies that these fim1s should have a homogeneous technology in average To accow1t for slope heterogeneity a random coefficients model could be implemented (Hsiao 1986 page 1 2 8) but this task is left for future research. To capture s i z e effects in the intercept of the differential model two more dummy variables relating to asset si z e have been included. Specifically we defined small banks

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107 as those with asset size below 300 million US dollars, medium scale banks as those with asset size from 300 million to 2 billion and we excluded the large banks wjth asset size above 2 billion as the base case. In order to capture scale effects these dummies were no t defined in real 2000 asset values but in their annual values. The discussion of the results focus on three different years in the sample: (1) 1991, which is in the midpoint of the "credit crunch period ( 1989 1992) (2) 1998, which is right after the enactment of the Riegleeal Interstate Banking and Branching Efficiency Act (1994) that permitted nationwide branchin g in 1997 and (3) in 2000 which is the y ear after the Grarnm-Leac h Bliley Act of 1999 that permitted commercial banks in participating in securities activities such as investment banking. Another reason for centering the discussion on these years was th e obs e r v ation from Table 5-1 of stabilit y in the mean return o f equit y and assets from 1993 and after. Finally note that these years in the differential model are actually changes from the previous years (i.e 1991 is the change from 1990 to 1991 ) The translog cost specification with the input shares equations where homogenei ty ha s be e n imp ose d b y di v idin g the input prices and the variable cost b y the price of labor has the followin g form 144 2 122 2 4 + LLg,s ln y lny_~ + L h ln z k + LLh ln z k 1n z + LLlh;, In( w /wi)ln y 2 r=I s= I k=I 2 k=I i = I 1=1 r=I 2 2 [ W ) 4 2 II 2 I I 4 1 1 + ~Bks;k In w;_ ln z k + ~BP,k lny ln z k + ~d, D + ~~d11 In w ; D + ~~c,, lny D 2 II 3 2 3 4 3 2 3 + LLekt In zkD + L a s A s + LLa,s In W ; A s + LLh,s lny A s + LL mks lnz k A s ( 5-11) k=I t = I s = I 1 = 1 s = I r=I s=I k=I s=I

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108 and by Shephard's lemma the input shares equations are given by (512) The usual symmetry conditions have been imposed on the estimation procedure while we have dropped on share equation due to singularity of the disturbances. In this specification we have included time dummies T, for each year in the sample, which capture technological changes between the periods. The 1998 year dummy was dropped as a base period. In 1997 the Riegle-Neal Act b ecame effective and therefore a comparison with the other years could reveal any possible effects of the Act in the technology or economies of scale of banks. To test for technologica l c han ges between the base year and any other year in the sample, it is only required a Wald test res tric ting all the coefficients for the dummy that needs to be tested to zero. Further A is a set of three dummies that capture asset size of banks and we have dropped the lar ge banks as a base. In this specification we allow not only for variable intercept tem1s but a l so for variable s lop e coefficients in the input prices the output level s and in the quas i -fixed in puts Economies of scale (SCE), also called Ray sca le economies measure the elasticity of cost with respect to proportional changes in the scale of outputs holding the product mix unchanged (Baum.al, Panzar and Willig 1988). That is 4 1 1 4 3 + LLC,-1D1 + LLh,s ln y,.A s (5-13 ) r = I t = I r = I s = I

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The Allen Uzawa elasticities of substit u tion fo r t h e translog can be calc u lated (Binswanger 1974) as follows b b + S (S 1 ) t:A = 1J + 1 for i -:1: j and t: .. = u 1 1 u S, + S1 11 S ~ 5.6 Empirical Results 109 (514 ) The differential system of equations, Equations 5 1 and 5 2, and the translog cost function (Equation 5-11) along with the input share equations (E quation 5-12) were estimated b y the nonlinear Full Information Maximum Likelihood (FIML) method developed in Chapter 4 Section 4.2. T h e mult i variate Gauss Newton procedure for time specific fixed-effects pane l data was imp l emented i n SAS by us ing the proc model and fit s tatements. The symmetry and homogeneity restrictions were imposed on the estimation procedure as it was shown in Chapter 4 while the concavity property of the cost function was tested for both models. In Table 5-3 the parameter estimates for the differential model are presented. All the coefficients are statistically s i g nificant at the 1 % level of confidence and the own input price coefficients 1r11 are negative satisfying the monotonicity assumption of the w1derlying cost function Insignificant terms were mostly the coefficients of the time dummy variables. In particular insignificant were the following terms : t he 1991-92 and 1996 97 period in the core deposits (C D) input-demand equation; the 1990-91 and 1992 93 in the consumer loan s (CL) output-supply equation ; the 199 3 94 1996-97 and 1998 99 in the business loans (BL) equation and 1991 92 in the securities (SR) equation. These results indicate that only systematic changes in the technology exist between these periods and 199 7 98 period a nd those are captured b y the common intercept term. However in the CL e qu at ion the int e rcept is ins i g nificant indicating no systematic

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110 changes in the technology for this output in a Hicks neutral sense. The CL and BL had the lowest fit in the data as measured by the equation adjusted R-square, while the inpu t demand equations had the largest fit with the CD equation explaining 88% of the cases in the sample. One of the advantages of the differential system is that the coefficients of the model have economic interpretation. For instance in the output supply system, the coefficients of the output prices are the product of the price elasticity of total output !// and the terms 0:,. which reveal the substitut ion or complementarity re lation in production. In Table 5-3 this product appears as a,.. and since !// is always positive (proved in Chapter 2), the sign of those coefficients shows whether the outputs are specific substitutes or complements For instance, consumer loans and business loans are specific substitutes (since a1 2 < 0 ) which implies that an increase in the relative input-deflated price of business loans will result to a decrease in the production of consumer loans. T he parameter estimates and standard errors for the translog cost function and the input share equations are presented in Table 5-4. It appears that only nine parameters are insignificant at levels of confidence grea ter than 10% while all other estimates are s i gnificant at the 1 % level of confidence. This result was expected given the lar ge sample which consists of 111, 908 observations for the period 1990 2000 The insignificant parameters are the cross terms of outputs and of quasi-fixed inputs with time dummies in the trans log cost function. This implies that th e slope coefficient of the specific output or quasi-fixed input and for the specific year does not change with respect to the base y ear. A lso the estimate of medium size banks ( a2 ) is insignificant, implying that the intercept of the cost function does not vary with respect to medium size banks.

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111 Table 5-3. Parameter estimates and standard errors for the differential model, 1990-2000 Equation Parameter Standard Equation Parameter Standard estimate error estimate error CD 0' I 0.4242 0.00245 LB 0' 3 0.2345 0.00175 CD 0 2 I 0 5779 0.00198 LB 0 2 3 0.2547 0.00141 CD (),3 I 0.6415 0.00168 LB 0 3 3 0.2124 0.00119 CD (),4 I 0.6907 0.00181 LB 04 3 0.1434 0.00129 CD ?i 0.4348 0 00568 LB c;~ 0.5770 0.00399 CD ? 1 2 0.1952 0.00531 LB c;32 0.5854 0.00375 CD n,, -0 0773 0.00093 LB JZ'l 3 0 0823 0.00068 CD n,2 0.0049 0.00061 LB 7Z'23 0.0188 0.00066 CD c, 0.0128 0.00069 LB 7Z'33 0.1011 0.00068 CD a o -0.0196 0 .00 113 LB
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112 Table 5-3. (Continued) Equation Parameter Standard Equation Parameter Standard estimate error estimate error CL C3 -0.0013* 0 00086 BL C4 0.0212 0.00108 CL a11 -0.0038 0.00045 BL a12 -0 0014 0 00039 CL C11 a 0 0003* 0.00038 BL a22 -0.0123 0.00066 CL C12 a -0 .0008 0.00015 BL C21 a -0 .0053 0.00049 CL /311 0.0147 0.00071 BL C22 a -0 0019 0.00016 CL /312 0.0947 0.00142 BL /321 0.0372 0 00089 CL a2 0.0002* 0.00143 BL /322 0.1442 0.00177 CL d 3 -0.0036 0.00127 BL a3 0.0164 0.00178 CL d 9 -0.0058 0.00141 BL -0 0001 0.00159 CL p 3 -0 0056 0 00089 BL d10 -0.0086 0.00175 CL e 3 -0.0013* 0 00088 BL p4 -0.0326 0 00111 CL [3 0.0100 0.00088 BL e4 0.0251 0.00110 CL g3 0.0074 0.00091 BL [4 -0.0005* 0.00110 CL h 3 0.0075 0 00090 BL g4 -0.0021 0.00113 CL k 3 0.0039 0.00091 BL -0.0033 0.00112 CL Il3 0.0054 0.00093 BL -0.0004* 0.00114 CL r 3 0.0000* 0.00043 BL Il4 -0.0006 0.00116 CL S 3 0.0067 0.00094 BL r 4 -0.0024 0 00054 BL S4 0.0057 0.00117 RE C s -0.0110 0.00126 RE a13 -0.0008 0.00046 RE P s -0.0179 0.0013 0 RE a23 -0.0007* 0.00056 RE es -0 .02 05 0.00128 RE a33 -0.01 29 0.00093 RE fs -0.0034 0.00128 RE a -0.0029 0.00060 RE -0.0075 0.00132 C31 gs RE C32 a -0.0008 0.00018 RE h s -0.0023 0.00131 RE /33 1 0.0583 0.00103 RE k s 0 0068 0.00133 RE /332 0 .2 583 0 00206 RE n s 0.0189 0.00136 RE 0.0155 0.00208 RE r s -0.0028 0.00063 RE d s 0.0155 0.00186 RE S 5 0.0288 0.00137 RE d11 0.0171 0.00205

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113 Table 5 3. (Continued) Equation Parameter Standard Equation Parameter Standard estimate error estimate error SR a,4 0.0048 0.00051 SR d,2 -0.0083 0.00190 SR a24 0.0042 0 00063 SR P6 0.0009* 0.00121 SR aJ4 0.0089 0 .0 0078 SR e6 0 0200 0.00120 SR C 6 0 0047 0.00117 SR f6 -0 .0297 0.00119 SR a44 -0.0212 0.00120 SR g6 0.0306 0.00123 SR C41 a -0.0044 0.00063 SR h6 0 0077 0.00122 SR C42 a 0.0001 0.00019 SR k 6 0.0201 0.00123 SR /J4 1 0.0252 0.00095 SR n 6 0.0012 0.00126 SR /J42 0.2672 0.00190 SR r6 0.0013 0.00058 SR as 0.0256 0.00193 SR S6 0.0215 0.00127 SR d 6 -0 .0197 0.00172 Insignificant even at the 10% le ve l of confidence. All other estimates are significant at the 1 % level of confidence a Denotes C;s = L a,s0/, i = 1, 2 The period 1998 s 1997 was dropped as the base PF=Purchased Funds, CD=Core Deposits, LB=Labor Equation CL = Consumer Loans with subscript 1 BL= Business Loans with subscript 2, RE=Real Estate Loans with subscript 3, SR=Se curities with subscript 4 For CD PF, CL, BL, RE, SR at that order and respectivel y across equat i ons the coefficients are as follows : Intercept terms = ao-a5 ; For instance <14 is the intercept term for the RE equation. Dummies: r 1 r 6 = charter-type; d 1 -d 6 = Asset size below 300 million ; d7-d 1 2 = Asset size above 300 million but below 2 billion; c 1 -c 6 = 1990-1991 ; p 1 -p 6 = 1991 1992 ; e,-e6= 1992-1993; f1f6= 1993 1994 ; g,-g6= 1994-1995 ; h1-h 6 = 1995 1996 ; k 1 -k 6 = 1996-1997 ; n,-116= 1998 1999 ; s,-s6= 1999 2000.

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114 Table 5-4. Parameter estimates and standard errors for the translog 1990-2000 Symbol Parameter Standard Symbol Parameter Standard estimate error estimate error cl{) 3.7163 0.07361 p31 0.0053 0 00017 b, 1.3322 0.00426 p32 0 0338 0.00031 b2 0.0436 0.00440 p41 -0.0068 0.00044 b11 0.2129 0.00060 p42 -0 .0 798 0.00099 b22 0.0574 0.00059 a, -0.7527 0.04660 b12 0.0628 0.00047 a 2 0.0463* 0.03620 g, 0.2002 0.00324 d, -0 0348 0.01110 g2 0.1365 0.00087 d 2 -0.1612 0 01110 g3 0.1520 0.00348 d 3 -0 2322 0.01130 g4 0.0180 0 00874 d 4 -0.1971 0.01110 g11 0.0486 0.00010 d s 0 0584 0.01130 g22 0.0052 0.00000 d 6 -0.0754 0 01130 g33 0.1113 0.00015 d 7 -0.0685 0.01080 g44 0.2584 0.00104 d, o 0.0265 0.01050 g,2 -0.0026 0.00002 d o -0 0419 0.01090 g13 -0.0 288 0 .0 0010 a11 0.1092 0.00150 g14 -0. 0640 0.00023 a12 0.0839 0.00121 g23 -0.0075 0.00002 a21 -0.0957 0.00141 g24 -0.0004 0.00007 a22 -0.0754 0.00104 g34 -0.09 27 0.00027 h11 0.0029 0.00111 f, -0.1098 0.00537 h12 0.0104 0.00076 f2 0.4282 0 01042 h 2 1 0.0048 0.00039 f11 0.0061 0 00027 h22 -0.0039 0 00033 f22 0.0051 0 00128 h31 0 0829 0.00120 f12 -0.0121 0.00049 h32 0 0236 0.00077 th11 -0.0048 0.00014 ~, 0.1468 0.00332 th12 -0.0017 0 00004 ~2 0.0542 0.00244 th13 0.0267 0.00018 m 1 I 0 0243 0.00212 th14 0.0355 0.00036 m12 0.0048 0.00155 th 2 1 0.0017 0 .0 0013 m21 0.2070 0 00434 th22 0.0007 0.00004 m22 -0.1034 0.00322 th23 -0.0097 0 00016 d11 0 0062 0.00098 th24 -0.0051 0.00033 d, 2 0.0186 0.00098 ks11 -0.0 300 0.00021 d13 0.0283 0.00103 ks,2 -0.0 220 0.00042 d14 0.0218 0.00105 ks21 0.0007 0.00019 d, s 0.0168 0.00103 ks22 0.0326 0.00040 d16 0.0154 0.00104 P11 0.0052 0 .0 0014 d17 0.0100 0.00105 P12 0.0409 0.00029 dig 0 0025 0.00103 P 2 1 0.0005 0.00004 d110 -0.0064 0.00104 p22 0.0014 0.00009 d11 I -0 0192 0.00104

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115 Table 5 -4. (Continued) Symbol Parameter Standard Symbol Parameter Standard estimate error estimate error d 2 1 0.0040 0.00094 C37 -0.0069 0.00097 d22 -0.0129 0.00097 C3g 0.0070 0 00084 d23 -0.0378 0.00105 C 3 1 0 0.0074 0 00088 d24 -0.0487 0.00109 C31 I 0.0123 0.00108 d2s -0 0420 0.00104 C41 0 0137 0.00221 d26 -0.0240 0.00100 C42 0.0061 0.00230 d27 -0 .0165 0 00102 C43 0 0149 0.00242 d2s -0.0066 0.00099 C44 0.0054 0.00246 d 210 0.0083 0.00098 C45 -0.0024* 0.00227 d211 0.0322 0.00094 C46 -0 0034 0.00249 C12 0.0055 0 00076 C47 0.0061 0.00242 C13 0.0019 0.00087 C4g 0.0132 0.00237 C14 0.0016 0.00079 C410 0.0113 0.00243 C15 0.0047 0 00071 C411 0.0166 0.00240 C16 -0.0019 0 00079 e11 -0.0016* 0 00144 C17 0.0025 0.00079 e,2 -0 0038 0 00143 C1g 0.0008* 0.00069 e13 0 0076 0 00150 C110 0.0061 0.00071 e14 -0 0034 0.00149 C111 0.0050 0 00077 e,s 0.0062 0.00139 C21 0 0026 0 0002 1 e16 -0 0024 0.00151 C22 0.0007 0 00020 e17 -0.0013* 0.00151 C23 0 0003* 0.00023 e,s 0.0065 0.00137 C24 -0.0009 0.00022 e110 -0.0029 0.00136 C25 -0.0011 0 00022 e1 II -0.0039 0.00144 C26 -0.0008 0 00023 e21 -0.0022* 0.00223 C27 -0.0009 0.00024 e22 0.0049 0.00230 C2g -0.0005 0 00023 e23 0.0065 0.00238 C 2 1 0 -0.0002* 0.00023 e24 0.0032 0.00242 C 2 1 I 0.0013 0.00044 e2s 0.0147 0.00226 C 3 1 -0.0050 0.00089 e26 0.0183 0.00260 C32 -0 0033 0.00085 e21 0.0050 0 00250 C33 0.0058 0 00084 e2s 0.0127 0.00238 C34 0.0007 0.00082 e210 -0.0208 0 .00 249 C35 -0 .0042 0.00084 e211 -0.0306 0 00271 C36 -0.0068 0 00088 d 9 -0 0294 0.01070 Insignificant even at th e 10% leve l of confidence. All other estimates are significant at the 1 % leve l of confidence. The year 1998 was dropped as the base.

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116 The striking result is that concavity is rejected for the mean size bank in each year of our sample under the translog specification. Table 5-5 presents a comparison between the translog specification and the differential in terms of satisfying or rejecting concavity. Table 5-5. Concavity test Year Translog Di fferential 1990 Reject 1991 Reject N.s.d 1992 Reject N.s.d. 1993 Reject N.s.d. 1994 Reject N.s.d. 1995 Reject N s.d. 1996 Reject N.s.d. 1997 Reject .s .d. 19983 1999 Reject Reject 2000 Reject N.s.d. a Base year excluded from the estimation. In the different ial, year denotes the change between cw-rent and previous year. The concavity test was an examination of the eigenvalues of the Hessian of the cost function with respect to input prices evaluated for the mean size bank and for the average product mix in the industry for the translog specification. In the differential model the test for concavity requires the price matrix in the input demands [ni.l] to be negative semidefinite. Since these coefficients are constant through time, accepting concavity for the pooled model may not imply that concavity is accepted for each year in our sample. Further rejection of concavity implies possible misspecification of the model. For that purpose separate regressions for each year in our sample were performed ( cross-section) and provided further evidence that concavity is rejected for each year in the translog specification, whi l e in the differential, concavity is rejected only for the period 1998-1999. Further we p e rformed the same test for the price matrix [a,s] w hich should be

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117 positive definite in the output-supply system. Our results indicate that convexity in the output prices is rejected for every year in the sample, not a promising result for the differential since it already satisfies concavity. However this may be attributed to a number of reasons as we have not included in the analysis, off-balance sheet items that appear to be significant outputs of the U.S. commercial banks in the past decade. Quasi fixity of some outputs including the off-balance sheet items of the banks is another reason that may have driven this rejection. It should be noted though that the fit of the consumer loans and business loans output-supply equations was poor. For instance consumer loans explained only 7% of the cases in our sample, while the business loans explained only 12%, as represented by the adjusted R-square of each equation. These results impl y pos s ible misspecification in the output-supply system where we come back to what constitutes inputs and what outputs in banking. In terms of Allen-Uzawa elasticities of substitution significant differences are found between the two models Evaluating Equation 5-14 for the translog at the sample mean for each year in our sample and E quation 5-9 for the differential again at the sample mean of each year we obtain the results of Table 5-6 It appears that the translog specification overestimates the elasticities between purchased funds and labor and between core deposits and purchased funds (or the differential underestimates). In both models labor is an Allen substitute for purchased funds but in the trans l og it appears to have the greater substitutability among the inputs while this substitutability is reduced through the period analyzed. Instead in the differential greater substitutability among the inputs appear to be between both pairs of labor-core deposits and labor-purchased funds.

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118 Table 5-6. Allen-Uzawa elasticities of s ub stitution Translog Differential Year CD-PF CD LB P F LB CD-PF CD L B PF-LB 1990 0.2322 -0.0614 1.1750 1991 0.1017 0.0181 1.1901 -0.0639 0.5650 0.4653 1992 -0.2116 0.1685 1.1922 -0.0802 0.4950 0.4849 1993 -0.4483 0.2363 1.1838 -0.1018 0.4370 0.4814 1994 -0.2955 0.2250 1.1562 -0.1038 0.4236 0.4317 1995 0.0147 0.1464 1.1398 -0.0853 0.4470 0 3742 1996 0.0579 0.1353 1.1330 -0.0733 0.4735 0 .347 7 1997 0.1128 0.1201 1.1228 -0.0697 0.4803 0.3271 1998a 1999 0.1730 0 .1061 1.1033 -0.0629 0.4937 0.2782 2000 0.2903 0.0345 1.0927 -0.0574 0 5130 0.2489 a Base year excluded from the analysis. In the differential model the e lasticities are the percentage change between the current and previous year. In the case of core deposits and labor there is an increasing degree of substitutability over the years The striking result is in the case of purchased funds and core deposits where both models find them to be Allen complements for the period 1992 to 1994 which i s the period after the credit crunch and before the Riegle-Neal Act in 1994. However the differential is consistent through the whole period and reports complementarity between those two inputs. Recalling t hat the Allen measure of substitution in the differential is the ratio of the cross price elasticity over the product of the average cost ratio of each input, and that this term is statistically significant, it implies that the differential provides a purer measure However, the assumption of constant price effects through time that is these own and cross price elasticities are constant through time might have driven this implausib l e r es ult. Therefore one could use the variable price effects developed in Chapter 3 in order to verify the robustness of these results in the differential model.

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119 The final point of comparison between the two models is in terms of economies of scale. Before we start with the discussion of the results we need to provide an equation for the economies of scale in the differential model given that there is no explicit functional form of the cost function as in the translog. Economies of scale (SCE) measures the elasticity of cost with respect to a proportionate increase in all outputs and is given by SCE = I olnTC r=I O lny, (5-15) Increasing returns to scale (costs increase proportionately less than output increases) decreasing or constant returns to scale exist if Equation 5-15 is less than, greater than or equal to unity respectively (Baumol Panzar and Willig 1988 page 50). Notice that in the case of profit maximization the first-order conditions imply equality between the price of each output and their corresponding marginal costs This result was used by Laitinen and Theil (1978) for the long-run model in order to parameterize the coefficient which is equivalent to our y2 coefficient in the input-demand and output-supply system and in their model was also equal to the returns to scale. By assuming profit maximization in t he s hort-run this parameterization transfers through in the short-run model. Substituting then the first-order condition in the scale economies measure SCE, it simply becomes the revenue total cost ratio of the multi product firm. To make this result obvious note that Equation 5-15 can be written as SCE = I a In TC = I oTC L r = I a In Y r = I O)l, TC Given the assumption of quasi-fixed inputs the total marginal cost can be written as

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arc= ave= vc I1, a 1nx ; 8y, 8y y, 1 alny, 120 By substituting this relationship iri the SCE equation and using the results from equations (2-66) and (2-75) we get that SCE = f a ln TC = VC f I1, alnx; = VC f I r201, g r = I a ln y, TC r = I a ln y, TC r=I From the adding-up properties of the input-demand system the term inside the summation is equa l to y2 which from the profit maximization case is equal to the revenue variable cost ratio. Making s ub stitution of these results into the scale economies measure it yields that in the differential model under the assumption of profit maximization is equal to the revenue total cost ratio. This further implies that constant returns to scale exist when there is a zero profit condition. Therefore, in the differential model t here is no explicit measure for the economies of sca le if profit maximi zation is assumed. T he system of demand equations could be estimated separately and regard the y2 term as a parameter to be estimated. However this implies that a complete system of demand equations should be implemented and in the case of panel data it is cumbersome. Table 5-7 presents the results for the economies of scale derived from the translo g model (Equation 5-13) for the mean size bank in the sample and also those calculated from the revenue-total cost ratio (for the differential model). While most of the previous st udie s that use the translog specificat ion ( except Stiroh 2000, Hughes and Mester 1998 Berger and Mester 1997) could not find economies of scale for the banking industr y, in this study usin g a tran slog cost function specification with v ariable slopes for the outputs quasi-fixed inputs and input prices we found significant economies of scale evaluated at the empirical mean of our sample.

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121 Table 5-7. Economies of scale for the mean size U.S bank, 1990-2000 Year Translog Total revenue-Total cost Economies of scale ratio 1990 0.898922 1.035225 1991 0.892359 1.045724 1992 0.899825 1.066944 1993 0.877201 1.072805 1994 0.858988 1.090447 1995 0.848583 1.073222 1996 0.861026 1.059456 1997 0.877344 1.052327 1998a 1.049784 1999 0.876549 1.044226 2000 0.881262 1.037623 a No estimates for the translog, since 1998 was dropped as t he base year. This result can explain the wave of mergers that happened in the industry. Comparing the overall economies of sca le results of our study with those of Stiroh (2000), who examined economies of scale for Bank Holding Companies and by asset size, we fow1d similarities w ithin a range of 0.01-0.14, with our results showing stronger economies of scale. However concavity has b een rejected in the translog specification wh ich may be caused from heterogeneity in the data. Specifically Berger and Mester (1997) argued that there is a scale bias by including banks of different sizes in a single regression and one needs to correct for possible heteroscedasticity Moreover the economies of scale as measured in the translog are actually the returns to the quasi-fixed factors which may explain the finding of unusually high scale economies. Finally, we observe that in our sample the total revenue-total cost ratio is almost equal to one over the whole period implying constant returns to scale or slight diseconomies of scale for the U.S. commercial banks if the measure of economies of scale from the differential model is adapted.

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CHAPTER6 SUMMARY AND CONCLUSIONS The Lajtinen-Theil ( 1978) or differential model concerns long-run behavior of a multi product firm under the assumptions of perfect competition and output homogeneous production technology. The purpose of this study was to provide an extension of the Laitinen-Theil ( 1978) model for the multiproduct-multifactor firm by examining firm behavior in the short-run. Specifically the extended model accounted for quasi-fixed inputs in the transformation technology of the multi product firm, wrule the assun1ption of output homogeneous technology was relaxed. It turns out that the assumption of output homo geneou technology does not play a crucial role in the derivation of the input demands apart from the returns to scale measure and in the price terms of the input demand system. The mathematical derivation of the system of input-demand and output-supply equations is presented in Chapter 2 where for the convenience of the reader a detailed representation of the derivations followed. To capture stochastic behavior of the multiproduct firm we appended disturbances in the system of input-demand and output supply equations relying on the theory of rational random behavior, developed by Theil (1975). It appears that quasi-fixity does not play a role in obtaining the covariance matrices of the two systems of equations since both in our study and in Laitinen-Theil (1978) are the san1e Further as in Laitinen and Theil (1978) the covariance matrix of the disturbances in the input-demand system was singular. 122

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123 Chapter 3 deals with the parameterization of the joint system of input-demand and output-supply equations, when quasi-fixity is present. In the original model of Laitinen and Theil the parameterization of the input-demand system was adding up to the total input decision of the firm with all coefficients being observed. However in the parameterization of the short-run model developed in Chapter 2, the coefficients of the quasi-fixed inputs in the total input decision of the firm were unobserved, since t hey entailed their shadow prices. To overcome this "pro blem" a technique devised by Morrison-Paul and MacDonald (2000) was used. In sort, it can be assumed that the shadow price of the quasi-fixed input is equal to its market rental price plus a deviation term. The input demand system then can be corrected for this introduction of the error term the same way that Laitinen and Theil corrected for the finite differences approximation in their model. However this method is not required for the estimation of the model. This method was only used in order for the input-demand system to add-up into known constants, and thus deletion of one equation (s ince disturbances are singular) would impose no problem in the estimation. Recentl y vario u s econometric techniques have been developed for the estimation of the full system of input-demand equations that one could follow. However if a joint est imation of the input-demand and output-supply system of equations is desired then it is more parsimonious to follow the above described method, since when symmetry is impo sed the nonlinear restrictions on the parameters of the model should also be considered.

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124 As in the Rotterdam model there is a similar misspecification in the differential model of production since it is assumed that all coefficients of the input-demand and output-supply systems are constant through time. To alleviate this restriction, alternative parameterizations of the differential model were developed, accounting for variable price and output effects in the input-demand system. It turns out that all alternative parameterizations could be tested through parameter restrictions in a more general model developed for that purpose. Fu1iher one of the alternative parameterizations was used for the development of a test for quasi-fixed inputs in the multi product firm. This test could be implemented at each sample point to provide us with the behavior of the quasi-fixed input over time. In Chapter 4 the econometrics of the differential model were presented. Several econometric methods were provided for the estimation of the differential model. In Section 4 1 the Laitinen (1980) maximwn likelihood procedure was presented, which concerned the estimation of the differential model for one multi product firm over time. However the need to account for panel data structures led to the development of two new procedures. In Section 4.2 the cases of time-specific and firm-specific effects panel data were considered. It was argued that firm-specific fixed effects may be already accounted for in the differential model if first differences of each time series are observed (i.e. ARIMA (0 1 0)). This would imply that these firm-specific fixed effects enter multiplicatively in the unknown cost function of the firm which futiher implies variable slopes in the cost function of the firm over time, in a Hicks neutral sense. Therefore, a maximum likelihood method was provided only for the case of time-specific, fixed effects panel data ; g i ve n al s o that the available dataset had large cross-sectional units and

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125 few time periods. This method is a sub-case of the maximum likelihood method for systems of equations and panel data, developed by Magnus (1982). However, it was adapted to account for time-specific, fixed effects panel data, since it originally concerned time-specific random-effects panel data; and also to account for the nonlinear cross-equations symmetry restrictions. In the last part of Chapter 4, a firm specific, random effects maximum likelihood method was presented which also accounts for the nonlinear symmetry restrictions. It is left for future research a random coefficients approach and the case of random effects with unbalanced panel data for the differential model. In Chapter 5 the developed model was applied to the U.S. banking industry for the period 1990-2000 where the discussion of the results focused in the technology that is implied by the differential model. To provide a direct comparison with dual functional forms, a translog cost specification was also implemented and tested for the same dataset. The two models were compared in terms of, rejecting or satisfying the concavit y property of the cost function Allen elasticities of substitution and an efficienc y measure such as economies of scale. The most str ikin g result of Chapter 5 is that even if the translog fails to satisfy concavity for each year in our sample, the differential model satisfies conca v ity for the whole period. Considering that in the differential model the test for concavity is a simple test on the price coefficients in the input-demand system of equations that are assumed to be constant over time the differential model was regressed for each year in the sample. Results indicated that the differential model satisfied the concavity propert y for all years, apart from 1999 However the differential system failed to satisfy convexity in the output

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126 supply system which could simply imply that for the specific dataset used there is a misspecification on the supply side or that significant variables were excluded from the analysis, as indicated in Chapter 5. For instance, possible quasi-fixed outputs that ha ve not been included or included as variab le could have driven this result. In terms of Allen elasticities of s ubstitution the differential model pro v ide s consistent estimates through time while there was an inconsistency between the models in one pair of inputs. That is, the translog finds that purchased funds and core de posits are Allen substitutes apart from three years in the sample, but the differential model always finds that they are Allen complements However both results are inconsistent with previous findings in the literature where these inputs are found to be Allen substitutes. For the differential model i t may impl y that the price effects are not constant thr o ugh time as assumed and one should use the variable price effects parameterization of the differential developed in Chapter 3. One of the disadvantages of the parameterization that was used to estimate the differential model is that the measure for economies of scale becomes the total revenue total cost ratio, restricting this way the analysis. A descriptive analysis of the samp le showed that this ratio was almost equal to one or slightly higher, implying constant or decreasing returns to scale under the assumptions of the differential model. It also implies that in the U.S. banking industry a zero profit condition holds given the data used in the analysis To alleviate this restriction on the returns to sca le one of the alterna ti ve parameterizations and a comp l ete system estimation method could be followed It should be noted finally, that when using th e translog model significant economies of scale were found for the mean-size U .S. commercial bank over the period 1990 2000 providing an

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127 explanation for the mergers activity in the industry. Considering that the measme of economies of scale in the short-run is simply a measure ofreturns to the quasi fixed factors the economies of scale that were found from the translog system were surprisingly high and further analysis s h o ul d b e do n e in this area Moreover, t he concavity property of the cost funct i on was rejected under the translog specification for all of the years in the analysis imp l ying t h at the results for the econom i es of scale measure should not be valid. In the bank i ng literature is a common pract i ce ( with few exceptions) not to test for the validity of t h e concavity property but one shou l d empirically test for all the theoretical propert i es of the cost or profit function. Failure to do so may lead to incorrect policymak i ng decisions.

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APPENDIX ANALYTICAL GRADIENT VECTOR A.1 Gradient Vector for Section 4.1 Consider one firm that uses three variabl e inputs two quasi-fixed inputs in the production process and produces four outputs. After droppin g one input-demand equation and imposing homogeneity and symmetr y t h e model presented by equations 4 3 and 4.4 has the following coefficient matrices a11 a12 a13 a14 /J11 /J12 A= a12 a22 a23 a24 F= /J 2 1 /J22 [ ff" ff,, l D= /J 3 1 /J32 a13 a23 a 3 a34 lr12 722 a14 a24 a34 a44 /J41 /342 0-[ 0,, 012 013 0 ] 021 022 023 024 (a11011 +a12012 +a13013 + a14014) -(a110 2 1 + a12022 + a13023 +a14024) -(a12 0 1 1 + a22012 +a23013 + a24014 ) C = -(a13 011 + a23012 + a330 1 3 + a34014 ) -(a12 B 2 1 + a22022 + a23023 +a2 4 024) -(a13 0 2 1 + a23022 + a33023 + a34024) ( a14011 + a24012 +a340 u +a44014 ) -(a14 0 2 1 + a24022 + a340 n + a44024) The other matrices are s imilar The W1ique e lement vector consi sts of -{011,0 1 2 0 1 3 014, ,lr11>lr 12,0 2 022,023, 024,,,lr 22,} a11, a 12,au, a14 /J11, /J 12, a n a23, a 2 4 /J 2 , /J22 a33, a 3 4 /J 3 1 /J32, a44, /J 4 1 /J42 aM aA a c a F Then recall that --= -Pi + w + z and differentiate with r espect to the a a a a e l ements of th e uniqu e ve ctor. For instance 128

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129 -a11 0 8M -a12 0 --v = WI since 0 1 1 appears in matrix C in the output supply. 00)) I -a13 0 -a14 0 1 0 0 0 -011 -02 1 8M 0 0 0 0 0 0 --v = P i + w, QQII I 0 0 0 0 0 0 0 0 0 0 0 0 The partial derivatives of M with re s pect to elements that belong uniquely to the input demand are all zero. All the other can be found similarly. A.2 Gradient Vector for Section 4.2 The uniqu e coefficient vecto r now consists of { 01) '012 013' 014 ,~11, ~12' Jr) ) 7l"12' 0 2 1 022, 023,024 '~21 ~22' 7l"22' } /3' = a11 a12,a 13,a14,C11,c12,/J11 /J12, a22,a 23, a24,c21 C22 /J2i,/J22, a33, a 3 4 C31 C32, /33 1 /332 a 44 C 41 C 42 /J4 1 /342 where now we ha ve more e l ements con-espon di ng to the nonlinear terms. That is C11 C12 -(a11011 +a12012 +a13013 +a14014) -(a1102 1 +a12022 +a13023 +a14024) C21 C22 -(a12011 +a22012 +a23013 +a24014) -(a1202 1 + a22022 + a23023 + a24024) = C31 C32 -( a13011 + a23012 + a33013 + a34 014) -(a13021 +a23022 +a3 3 023 +a34024) C41 C42 -(a14011 +a24012 +a34013 +a44014) -(a14 021 + a24022 + a34023 + a44024) The gradient vecto r then has to take into consideration the above decomposition of the clJ t erms. T hu s ::,1 = {~'~.-~ : ~ ~ ~;::~~;~:,~: :::::::::::::::::::~:} 0, ..... ............... 0

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BIOGRAPHICAL SKETCH Grigorios Livanis was born on September 4 1974, in Athens, Hellas. After graduating from high school he worked for 1 year in a pri vate company in Athens. His quest for knowledge made him take the national examinations for entrance into a Hellenic university. In 1993 he was admitted to the Department of Energy Technology at the Technological Institute of Athens. Howe ve r he was not satisfied wit h their program of study and he decided to retake the national examinations. Higher scores admitted him in fall 1994 to the Department of Agricultural Economics at the Agricultural University of Athens. His devotion to academic excellence and merit was demonstrated b y ranking first in his department for the entire duration of his studies. For this reason he received annually recognition awards and scholarships from the State Scholarships Foundation of the Hellenic R epublic After receiving his B.Sc ./ M.Sc. combined degree with high honors in the s ummer of 1999 he was offered assistantships to pursue graduate studies in agricultural economics at the Un iversity of Wisconsin Michigan State U niversity U ni vers i ty of Missouri and the Un i versity of Florida (UF). His choice to join the gra duate school at UF was highly influenced by the fact that it was the only one where both he and his fiancee Maria Chat z idaki were offered a scholarship among their combined university choices. In the fall of 1 999, he enro lled in the Department of Food and Resourc e Economics at the Un iversity of Florida. He rec eive d his M.Sc. degree with a minor in mathematics in D ecember 2000 He co ntinu ed on toward hi s doctoral degree in spring 2001, in the same 1 37

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138 department with fields of specialization in financial and production economics and applied econometrics. In spring 2004 he received a Presidential Recognition Award for his outstanding achievements and contributions to the University of Florida. He fulfilled all the requirements and coursework with a 3.95 overall G.P.A. and was awarded a Doctor of Philosoph y degree in August 2004.

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. fMis~-air-Professor of Food and Resource Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is full y adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy J LSeale Professor of Food and Resource Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. othy G. Tay lor Professor of Food and Resource Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality a s a dissertation for the degree of Doctor of Philosophy. Elias Dinopoulos Professor of Economics I certify that I have read this study and that in m y opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Flannery Barnett Banks Eminent Sc Insurance and Real E state of Finance

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This dissertation was submitted to the Graduate Faculty of the College of Agricultural and Life Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy August 2004 ~-..---:--Dean Graduate School

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UNIVERSITY OF FLORIDA 111111111111111111111111111111111111 1111111111111111111111111111 3 1262 08554 2636


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