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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ix, 137 leaves : ill. ; 29 cm.
Livanis, Grigorios T
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Thesis (Ph. D.)--University of Florida, 2004.
Includes bibliographical references.
Statement of Responsibility:
by Grigorios T. Livanis.
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Copyright 2004


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.


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.



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

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

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


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


3.1 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


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



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.


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


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


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

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

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.


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


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


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


* 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

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

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

-- (Y. = v. which states that a necessary condition for a firm to be in long-run

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)

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,

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,


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

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

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

term by y Equation 2-12 becomes

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

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

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,

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

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

alny In xy

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

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

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

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

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


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

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:

k lnzk
k alnz,

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



k kan z





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,

S9lnz, yClnVC
S9T In yr
a In y,


which can also be written as


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

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


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)

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

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)


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

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

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):

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)

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

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,

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

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)

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)

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

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,

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


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

w,x, (w, y, z)- 2(w, y, z) -0 (2-41)

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,


= 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


* 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
A = wx, .
1 Inx,

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

These transformations give the following relationships

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,

,=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
(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

8 Inz,



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







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

a,- ,- = i^ In





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

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

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

F alnx1
a In x
a In z'


,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


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:

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

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:


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
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'



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')


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,


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

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

n VC

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

aT aln8 a ln y,n
r la In yr

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

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)

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)

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'' 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.

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

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

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


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

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-

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:

R = Tr(G) (2-98)

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-
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'

Therefore, pre-multiplying the differential output supply by and using the

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)

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.

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

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 =

R )

2-88, respectively, and T;,

R 1 K

R *

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

= 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-


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

d In W = -d nw, 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)'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

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.
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

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, ,

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

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)

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)

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

y -1 (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 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.


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-

* 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,
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

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

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

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

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

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

g, = Y, while the average factor share of the i't input in t and t -1, and the average

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

total input decision (Equation 3-2) holds without disturbance. Since 73 is not observable,

Equation 3-2 cannot be solved for 2,. and thus employing 2= from Equation 2-87

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

= vk, then at any point different than this optimum, it must hold that

= vk + 6, where 5k denotes the deviation between the ex-ante market rental price

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


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


, .=

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-

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

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


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,

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)

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

+ r,d ln w, +s,

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)

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

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

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.


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


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


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


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


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

9L y y, Mq, q,
= ,Nv,, =0 (4-14)
x,, x,-Nv, ON

where = -K'

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


The iterative procedure that Laitinen (1980) suggests, consists of the following


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


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 (,


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

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


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

where i, is a vector of ones, and we impose a, = 0 to avoid the dummy variable trap,

since /f contains an overall constant in each equation. Further, we make the following


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
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

information matrix for = L, are given by

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

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


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


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

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


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

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.

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


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



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