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THE DIFFERENTIAL PRODUCTION MODEL WITH QUASIFIXED INPUTS: A PANEL DATA APPROACH TO U.S. BANKING By GRIGORIOS T. LIVANIS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 Copyright 2004 by Grigorios T. Livanis This dissertation is dedicated to my parents, Theodosios and Konstantina; my brothers, Harilaos and Ioannis; and the love of my life, Maria Chatzidaki, who made this happen. ACKNOWLEDGMENTS First and foremost, I would like to express my deep gratitude and sincere appreciation to my advisor, Dr. Charles B. Moss, for his outstanding guidance, encouragement, and advice during my graduate studies and the development of this dissertation. He has always been a source of motivation and inspiration. I would like especially to acknowledge Dr. Elias Dinopoulos for the endless discussions, advice, and encouragement during the research process that contributed to the quicker completion of this dissertation. Sincere appreciation is also extended to the other members of my committee Dr. James Seale, Dr. Timothy Taylor and Dr. Mark Flannery for their guidance, and constructive criticisms that led to improvements in this dissertation. I would like to express my immeasurable gratitude to my parents, Theodosios and Konstantina Livanis; and my brothers, Harilaos and loannis Livanis, for their continuous love and moral support, despite the distance. I especially thank my parents, who taught me that I could achieve anything that I committed myself to fully. In the last years of my studies I was privileged to have my brother, loannis, studying at the same University. His humor and support made those years more enjoyable. Finally, I would like to express my deepest love and gratitude to my partner in life, Maria Chatzidaki, for all of her love, support and sacrifice. Without her by my side, I would not have reached my goals successfully. Words cannot express how thankful I am to be sharing my life with someone so loving, patient, and thoughtful. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................................................................................. iv LIST O F TA BLES ............................................................................................................ vii ABSTRACT............................................................ viii CHAPTER I INTRODUCTION AND OBJECTIVES ................................................................. 1 1. 1 Introduction....................................... .......................................................... 1 1.2 O objectives .............................................................................................. ........ 4 1.3 O overview ............... ........................................................................................ 5 2 M ETH O D O LO G Y .................................................................................................. 7 2.1 Introduction................. .................................................................................. 7 2.2 The Case of Multiple QuasiFixed Inputs .......................................................8 2.3 Cost M inim ization ....................................................................................... 11 2.3.1 Returns to Scale and Elasticities of Variable Cost............................12 2.3.2 Factor and Product Shares .............................................................17 2.3.3 Marginal Shares of Variable Inputs ..................................................20 2.3.4 Input Demand Equations................................................................21 2.3.5 Comparative Statics in Demand.....................................................29 2.4 Conditions for Profit Maximization ................................... ...................33 2.4.1 O utput Supply ..................................................................... .......... 35 2.4.2 Comparative Statics in Supply......................................................41 2.5 Rational Random Behavior in the Differential Model ...................................43 2.6 Comparison to the Original LT Model ........................................................46 3 PARAMETERIZATION AND ALTERNATIVE SPECIFICATION .......................48 3.1 Input Demand Parameterization ..................................................................48 3.1.1 The Case of Multiple QuasiFixed Inputs...........................................50 3.1.2 The Case of One QuasiFixed Input ...........................................55 3.2 Output Supply Parameterization..................................................................56 3.3 Alternative Specification for the CostBased System ....................................57 3.4 Capacity Utilization and QuasiFixity ......................................................... 61 4 ESTIMATION METHODS.....................................................................................64 4.1 Choice of Estimation Method......................................................................64 4.2 Fixed Effects and Pooled Model ..................................................................73 4.3 Random Effects ...........................................................................................84 5 APPLICATION TO U.S. BANKING INDUSTRY ..............................................89 5.1 Introduction......................... ........................................................................ 89 5.2 The US Banking Industry in the 90s............................................................91 5.3 Brief Literature Review ...............................................................................92 5.4 D ata D description .......................................................................................... 96 5.5 Em pirical M odel ........................................................................................ 102 5.6 Em pirical R esults.........................................................................................109 6 SUMMARY AND CONCLUSIONS ..................................................................122 APPLNDIX ANALYTICAL GRADIENT VECTOR ..............................................128 A.1 Gradient Vector for Section 4.1 ................................................................128 A.2 Gradient Vector for Section 4.2................................................................ 129 LIST OF REFERENCES ............................................................................................. 130 BIOGRAPHICAL SKETCH ......................................................................................... 137 LIST OF TABLES Table page 51 Financial indicators for the U.S. banking industry, 19902000............................92 52 Definition of variables and descriptive statistics (mean and standard deviation) .................................................................................... 100 53 Parameter estimates and standard errors for the differential model, 19902000 .................................................. ................................................1...... 54 Parameter estimates and standard errors for the translog, 19902000.................114 55 C oncavity test..................................................................................................... 116 56 AllenUzawa elasticities of substitution........................................................... 118 57 Economies of scale for the mean size U.S. bank, 19902000 ......................... 121 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE DIFFERENTIAL PRODUCTION MODEL WITH QUASIFIXED INPUTS: A PANEL DATA APPROACH TO U.S. BANKING By Grigorios T. Livanis August 2004 Chair: Charles B. Moss Major Department: Food and Resource Economics This study assesses the empirical and policy implications of using the differential approach in opposition to dual specifications for the decisions of the multiproduct firm. In applied production analysis, the dual specifications of the firm's technology usually fail to satisfy the theoretical properties of the cost or profit function. If the validity of those properties is not examined, then empirical results should be interpreted with caution. On the other hand, the differential production model of the multiproduct firm has rarely been tested empirically, since it was first developed by Laitinen and Theil in 1978. The novelty of this study is that it generalizes the differential production model for the multiproduct firm to account for quasifixed 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 19902000. To assess the empirical results of the differential model (and to provide a direct comparison with a dual specification), a translog cost function was applied to the same dataset. Results indicated that the differential model is consistent with economic theory, while the translog specification failed to satisfy the concavity property of the cost function for each year in the sample. Concerning the Allen elasticity of substitution both models found similar results. One disadvantage of the differential model was the assumption of perfect competition, which resulted in total revenue over total cost being the measure of scale economies. CHAPTER 1 INTRODUCTION AND OBJECTIVES 1.1 Introduction This study extends the multiproduct differential production model, developed by Laitinen and Theil (1978), to incorporate quasifixed 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 firstorder conditions in a cost or profit optimization problem, to attain the inputdemand and outputsupply equations, respectively. In contrast, the dual approach involves specifying a flexible functional form for the cost or profit function, to describe the firm's technology, which yields a system of equations to be estimated (e.g., a translog cost function with respective input shares). Thus, it can be considered as an approximation in the space of the variables (quantities and prices), while the differential approach is an approximation in parameter space. The disadvantage of the dual approach is that usually different functional forms lead to different results for the same dataset, as Howard and Shumway (1989) indicated, often failing to satisfy parameter restrictions. Especially, concavity restrictions tend to be nonlinear and more difficult to impose (Diewert and Wales 1987); and as a result, few 2 empirical studies examine the concavity of their results in detail (exceptions are Featherstone and Moss 1994, Salvanes and Tjotta 1998, see also Shumway, 1995 for a recent survey of studies testing various parameter restrictions). Numerous models have been developed for analyzing consumer demand based on the differential approach (Rotterdam, AIDS, CBS, NBR). Further, as demonstrated by Barten (1993), Lee et al. (1994) and Brown et al. (1994) a number of competing systems can be generated from alternative parameterizations of the differential system of demand that was originally introduced by Theil (1965, 1976, 1980). Thus the form of consumer demand can be selected through simple parameter restrictions. In applied production analysis, a similar differential inputdemand system was developed by Theil (1977) and Laitinen and Theil (LT, 1978). The Theil (1977) model concerns oneoutput 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 quasifixed inputs. Davis (1997) provided an application of the Theil (1977) model; while Fousekis and Pantzios (1999) generalized the Theil (1977) parameterization by including Rotterdamtype, CBStype, and NBRtype effects. Recently, Washington and Kilmer (2000, 2002) applied the LT model in international agricultural trade. However, they assumed inputoutput separability and independence, which transformed the model into a single output model. Our study extends the LT model to account for quasifixed 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 inputoutput 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 quasifixity, based on a simple Hausman specification test (Schankerman and Nadiri 1986) or on direct test of the coefficients of the estimated model. The proposed model was applied to the U.S. banking industry, giving specific attention to the concavity property of the cost function. The banking industry was selected because, most probably, the assumption of perfect competition in both input and output markets of a specific bank will hold; and because of the availability of data. The proposed model was compared with a standard transcendental logarithmic (translog) specification with quasifixed 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 quasifixed 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 costbased system (inputdemand system of equations). This will be useful for deriving a new test for asset quasifixity. 3. To provide alternative econometric procedures using Maximum Likelihood estimators for balanced and unbalanced panel data, for estimating the extended LT model. 4. To apply the extended LT multiproduct model to the U.S. banking industry and to econometrically estimate the system of deriveddemand and outputsupply equations using the econometric methods developed in this study. 5. To compare the results of the extended LT model with those of a flexible functional form specification, such as the translog. Specific attention was given to the concavity property of the cost function in input prices. The two models were also compared in terms of Allen elasticities of substitution and degree of economies of scale. 1.3 Overview Chapter 2 provides the mathematical derivation of the basic model used in this analysis. It borrows heavily from the derivation techniques as presented by Laitinen (1980), but differs in terms of the added generalizations of a nonhomogeneous, in the output vector, production technology; and of the existence of quasifixed inputs. Also, the extended model was compared with the original LT model, showing that the assumption of outputhomogeneous production technology affects only the inputdemand 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 quasifixed inputs in the inputdemand system, and the development of a "supermodel" for the costbased system of equations. Specifically, the coefficients of the quasifixed inputs are a function of the respective shadow price of the quasifixed input. To parameterize those coefficients, the procedure of MorrisonPaul and MacDonald (2000) was used, whereby shadow prices are decomposed to their exante market rental prices plus a deviation term. The "supermodel" for the costbased system, developed in this chapter, accommodates for a new test for asset quasifixity 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 timespecific, fixedeffects, and individualspecific, randomeffects 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 timespecific, fixed effects econometric method, presented in Chapter 4 is adapted for estimating the extended LT model and the translog specification for the banking industry. Then the results of both models are compared in terms of rejection (or not) of concavity, and elasticity measures. Finally, Chapter 6 provides a summary, conclusions of the present study, and presents unresolved issues for future research. CHAPTER 2 METHODOLOGY 2.1 Introduction The LaitinenTheil (LT, 1978) model extends previous studies by Hicks (1946) and Sakai (1974), to explicitly account for inputoutput separability, input independence, homotheticity and nonjointness of production. It concerns longrun behavior of risk neutral multiproduct firms under competitive circumstances. Moreover, it is generally applicable, since it does not require specific assumptions, such as inputoutput 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 inputoutput 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 inputoutput separability, the model became a single output model (Theil 1977). The advantage of the LT model is that it avoids the use of a functional form for the dual specification (either cost or profit functions). That is, it does not specify a functional form for the true technology of the firm. However, the parameterization of the model provided by Laitinen (1980) implies constant price effects, and implies that the change in the cost share of the i'h input due to the change in r'h product is also constant. Therefore, there is a need for parameterization allowing for variable output and price effects. Fousekis and Pantzios (1999) provided such a general model but for the single output firm. This chapter provides the general methodology and derivation of the shortrun system of inputdemand and outputsupply 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, inputoutput 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 quasifixed inputs. Apart from Clements (1978) and Rossi (1984), who used a transformation technology separable in the fixed inputs, there is no other attempt to specify or extend and test a more general model. 2.2 The Case of Multiple QuasiFixed Inputs Let the production technology of a multiproduct, multifactor (MPMF) individual firm be represented by a transformation function: T(x,y,z) = 0 (21) 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 quasifixed 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 singleequation multiproduct, multifactor in an implicit form production function, is not as general as it was thought to be. The production function shown by Equation 21 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 KuhnTucker conditions do not hold. Therefore, our study did not examine separability of that form; and instead left it for future research. Assume that a MPMF firm minimizes variable costs of producing the vector of outputs y, conditional on the vector of quasifixed inputs z and fixed prices w for the variable inputs. This shortrun or restricted cost function can be denoted as VC = VC (y, w; z), and it is assumed that it satisfies the following properties (Chambers 1988): * VC (y, w; z) is monotonically nondecreasing, homogeneous of degree one and concave in w. * VC (y, w; z) is non decreasing and convex in y. * VC(y,w; z) is non increasing and convex in z. * VC(y, w; z) is twice continuously differentiable on (w, y; z). Applying Shephard's lemma on the restricted cost function, the conditional factor aVC demands are then obtained as x, = = VC (y, w; z). If v denotes the vector of ex 0w, ante market rental prices of the quasifixed inputs, then the shortrun total cost of producing the vector y is given by SC = VC(y, w; z)+ v z'. The longrun cost function C(w,y) of the multiproduct firm is then obtained by minimizing shortrun total cost with respect to quasifixed inputs, while holding the variable inputs and the level of output at the observed costminimizing levels. That is, C(w,y) minSC min (VC(y, w;z)+ v z') The firstorder condition of this minimization problem implies that aSC OVC(y,w;z) + = +v=0 az az* where z* denotes the static equilibrium levels of z. This condition can be written as QVC(y,w;z')  (Y. = v. which states that a necessary condition for a firm to be in longrun 9z" equilibrium is that the shadow prices of the quasifixed inputs be equal to the observed exante 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 fullequilibrium demand levels for the quasifixed inputs are equal. The same result holds for the shortrun and longrun marginal cost and demands for variable inputs of the multiproduct firm. 2.3 Cost Minimization For the multiproductmultifactor 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"' quasifixed factor of production (k = 1,...,1) with an exante market rental price denoted by vk. Assume a production function in an implicit form that is not separable into the quasifixed 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 (22) Then in the shortrun, 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 quasifixed input levels. Thus, the problem that the firm faces is min VC(w,x) w,x,:T(x,y,z)=0 (23) S=1 n The Lagrangean of the above problem can be written as L = w,x, AT(x,y,z) and the firstorder conditions needed to attain a minimum are given by the following equations: 9L 9T(.) aL w x, T() =0 (24) SIn x, In x, 6L T(x,y, z)=O 0 (25) 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 24 and 25 are assumed to yield unique positive values for x, and A; and Equation 24 is a vector of n x 1. The secondorder conditions are given by the following equations: a2L 82T = 8, w,x, A (26) 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 23 gives the conditional or compensated shortrun demands of the inputs as a function of all input prices, output quantities, and quasifixed 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 shortrun, it is sufficient that the matrix of the second order derivatives that has a size n x n (Equation 26), is symmetric and positive definite. The minimum shortrun cost is then given by VC(w,y,z) = w,x,(w,y, Z) (27) 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, dlnzk = (28) ,= 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 quasifixed. Letting this be the case, and defining x" = [x, z], then d In x, = d In x, = d In zk = d In z, and d In y, = d In y, can each be put before its summation sign. Then we have dln T +dlnz T z dln y, T cInx, = lnzk = alnyr This can also be written as d nT 'x T "' TFy Inx + +dlny, =0 ,1 ln x, k1 a nzk r= In y, Therefore. R dlInyr y a In, x, k=1 anzk RTS nx' (29) 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 27) with respect to output Y : OVC Ox, VC In x, = I = _Y f, (210) Yr, y,. Yr, In y, where f = is the variable cost share of input i and the last expression has been VC derived from the second by multiplying the second term by (yVCI/y,VC) and noting that 0 In x, = I x,. Also, notice that the above equation can be written as x, SnVClnx, (211) 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 quasifixed inputs constant, we get aT 81n x BT T ax, +0= 0 (212) ,= In x, In y, In y, aT wxI However, by using the firstorder condition W= w' and by multiplying the first SIn x, A Y VC term by y Equation 212 becomes yVC y VC wx, ln x, aT w,x, + = 0, where f Ayr y, VC ln y, a In y, VC Using now Equation 210 the above expression can be written as y, 9VC aT Y + = 0 (213) SaY, a In y,. If we sum Equation 213 over r then we get 8VC I 3VC '" aT r In y S0 or A =T l (214) SZl Y ,=, Q1n y,. ST A a In yaT n anyr Letting = then from Equation 214 we have that VC 8 In VC In yr 7 alnY (215) VC n Y a In y, The elasticity of variable cost with respect to proportionate output changes, holding quasifixed inputs constant, is obtained by substituting in Equation 215 the expression for the lagrangean multiplier (A) from the firstorder condition (Equation 24). That is, fT we substitute A = VC/ in Equation 215 to obtain SIn x, ST 8 ln VC 1y lnVC_ Olnyr (216) alny In xy llnx, To find the elasticity of variable cost with respect to proportionate quasifixed 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 quasifixed input we obtain QVC x VC 8 Inx ac W Ox = VC ln (217) zk zk k IOlnZk QVC Notice that = w, denotes the shadow price of the quasifixed input. Also, from the azk analysis in Section 2.1, in order for the firm to be in longrun equilibrium, it has to be the 8VC case that z = vk where v, is the exante market rental price of the quasifixed input. azk Further, Equation 217 can also be transformed into the following expression 8 In VC alnx, =  f (218) 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 quasifixed inputs, and outputs constant, we get a T Inx, aT Sn, nz+ , lnx, ln z, 8d In z, Again, using the firstorder condition, i 9 In x, = w,, multiplying the first term of the 2 VC above equation by z and using Equation 217 we obtain the following relationship z VC zk VC OT + z nz 2 az In zA Summing this equation over k. we obtain the second interpretation for 2: Ia(" 3VC SaT k lnzk k alnz, However. Equation 221 must be equal to Equation 214 implying the following relationship SVC IaT k kan z alnyr (220) (221) (222) I' 8lny Solving for the elasticity of cost with respect to proportionate quasifixed input change from the above equation, we obtain al n FC ln nz, aT S9lnz, yClnVC S9T In yr a In y, (223) which can also be written as (219) OT z 1n z e = V, k (224) ,r In y, or equivalently, Equation 223 (through the use of Equation 216) can be written as aT 8 In VC zk C VC k (225) S n zk In x, Finally, taking into consideration Equations 216 and 225, the degree of returns to scale (RTS) in terms of derivatives of the variable cost function (Equation 29) can be written as O T + OT 'nlnVC ,= anx, k= anzk a In z RTS k (226) 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 (227) VC Taking the total differential of Equation 227 we have df = fd In w, + fd In x, fd In VC (228) Summing Equation 228 over i and noting that = 1 and so d f = 0, we have I I d n VC = fd In w, + fd In x, (229) 1 / or in a more compact form d In VC = d in W + d In X (230) 18 where d n W = fd In w,, d n X = fd In x, are the Divisia indexes of variable input I I prices and variable input quantities, respectively (Divisia input price index and Divisia input volume index). Then considering Equation 214 for A, define as in Laitinen and Theil (1978) gR Y, a aVC/aInyr I yT (231) 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 linearhomogeneous production function in the aT output vector, which implies that I 1, as in LT, then g, would be just the in y, share of the r'h product in total variable marginal cost. It is the case though that at the point of the firm's optimum (from Equation 213): aT gr = n (232) 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 221 for define Similarly, considering Equation 221 for A, define zk OVC aVC/alnzk T (2)T Pk ( I3) t A ~ )zk J6VC/Olnzk nk (233) k as the share of the k"' quasifixed 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 222 we obtain the ratio of the k'h quasifixed input shadow value in AT the variable marginal cost of m outputs, multiplied by O a In y, OVC//1nz T Pk = v_ k9 C In 2 IC OT (234) a VC / n y, a in y, I' Using now Equation 231, the above equation transforms to 8VC / In z, Pk VCalnz r=l,..., m (235) SVC / ln y,. Also, at the point of the firm's optimum (from Equations 233 and 220), it holds 9T k aln (236) a In z, As in the case of outputs, note that / =  Therefore, we can define the k k alnzk share of the k'1 quasifixed input shadow value in total shadow value of the quasifixed 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 quasifixed 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 (237) VCl/gy, Then multiply Equation 237 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 / (238) r Equation 238 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 nonnegative. In a similar fashion define the share of i'h variable input in the shadow price of quasifixed input zk as k =(w,x,)/ (239) a VC / Ozk Then multiply Equation 239 by and sum over k to get the share of the i'h Lue variable input in variable marginal cost of m outputs: SPk k =_ a VC/lnzk a(w,x,)/zk k P"Z. k aVC/alnz, a VC / z, e G which can be simplified to Sa(w,x,) / a In k k (240) ", _OVC / lnz, k As in the case of the outputs summation of 'k, over i is always unity but need not be nonnegative. 2.3.4 Input Demand Equations The first step is to write the firstorder 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 quasifixed input quantity zk, in order to determine how the optimum changes in response to changes in these given variables. Therefore, the first order conditions as identities are aT(x(;w,y,z),y,z) w,x, (w, y, z) 2(w, y, z) 0 (241) xlnx, T(x(w. y, z). y, z) 0 (242) Totally differentiating Equation 241 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 (243) 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 (244) \alnw c In x, a In w, = ,1 In x, In x, 8 In wJ alnx, a1nA2 T n 82T alnx, ( )2 wUx A AA 0 (245) 0,lnzk alnzk, lnx, =, lnx,Olnx Olnzk In x,a In zk Notice that Equation 243 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 244 and 245 represent nx n and nxl (k = 1,...,I) distinct equations, respectively. Then totally differentiating Equation 242 with respect to In y,, In w,, and In zk we have, respectively = lnx, ln y, a nyr 0T 0 (247) 8, In x, 8 In w aT 81nx 8T T+ 0 (248) ,=a 1nx, 8l nzk, lnzk Since we differentiate with respect to each output, input price and quasifixed input level, Equations 246 to 248 are vectors of dimension m x 1, n x 1 and l x 1, respectively. The next steps for the derivation of the inputdemand system consist of the following * Divide Equations 243 to 245 by variable cost (VC), use the definition of the cost shares ', = f,, and use from the firstorder conditions the relationship VC 9T A = wx, . 1 Inx, * Multiply Equations 246 to 248 by and use the following relationships VC aT 9T A g, = /k = and = y,. a In y, 8 In/p, VC These transformations give the following relationships aIn2z ln zk, 2A l VC 8 1n c In A A ~'anw, VC a2T a nx x 2 x, lnx, l n yr VC a nx a2T Olnx, 0 8 In x8 In x, a In w, 2 T _ 0 , 1ny, A2 02T VC =nx,1nz,0 VC a Inx, ln zk SIn x, OIn y, SI anx, a In w, SOlnx, ,=1 aIn zk Now the following matrices can be defined F=diag(f,,...,f,,), H= a2T H,= 0 2T1 ainx,a Inxj, lnxa,1n y, n nnxn r nxm and H = 2T T a Inx,a In zk , Therefore Equations 249 to 254 can be written for all combinations of inputs, outputs and quasifixed inputs, in matrix form, as (F In x lnx (F Y H) ny 8 In y SaIn A F a in y H, In y' a In x a In 2 (Fy7H) l F i,, =n F a In w a In w SIn x, a 1nyr a 1n 2 Sln y, ,f Ilnx, 0 In w Olnx, 8 Inz, (249) (250) An a2T a nx, VC k 1nx, Inx, Inz, (251) (252) (253) (254) (255) (256) a In x 0 In 2 (Fy,H)l F i,, an= 7 H3 SIn z' In z i,F = 7, g' 8 In y1 a In x i, F 0 ainF =0 alnx a, , = i^ In (257) (258) (259) (260) Now, premultiply Equations 255 to 257 by F' and combine with Equations 258 to 260, to form Barten's fundamental matrix equation S1nnx 7, H) F' i,, n yln , 0 a in A a In y' i:, aaln / F alnx F a n w' a n A a n w' F alnx1 a In x a In z' [YFiHi 71g: ,F 'H3 7/,f I and solving for the matrix of the decision variables we obtain F a1nx a In y' aIn 2 a n y' F a1nx F aInx a n w' a n z' aIn 2 a ln w' ain 2 aln z' F'(Fy,H)F' iN 7,FH, i' 0 y, g' I yF'H3 0 7,j/' From Magnus and Neudecker (1988), if A is a nonsingular 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 nonsingular, 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 (261) It follows then, that * D = i,F (Fy,H) F i, which is a scalar. Using the property of the inverse of a scalar, we get that D' = iF(FyH)' F.i, S (F(FIH) F.i,,iF(F H) F A AA'A,2D'A,A=I, =F(Fz)H)F F i F(F ,H) Fi) F(Fy,H), Fi, A^ Ai,(D = i:F(Fy,H)' F.i iF(Fy,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(FY,H)'F (262) 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,, (263) Then we can define the n element vector 0 as the row sums of 0: n 0 = i,,, 41, = 0 1, (264) I=1 F(Fy, H)' Fi Equation 264 can be written equivalently as 6 = i, = (FyH)' F which iF(F;,H) Fi, implies that = D'A,,AI,, (1 x n). Also, simple algebra shows that the following relationships hold: (265) 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 238. 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 262 to 265) the inverse of matrix A can be written as A'= 1 and so Equation 261 transforms to F 1nx F a1nx F alnx 1 0 a In x a In x W In x F F F a In y' 1nw' 81nz' y F'H, InA 9ln2 8InA L  Y1g' L In y' a In wv' a In z' Solving for the individual terms we obtain the following relationships 0 In x F ,=x 7,1V ( ')F'H, +7yg' Sln y' F In x 0 In w I y,F'H,3 0 Yp' (266) (267) i,,'o=1, 'i, ,=1, I,, . = and i,,( ') =0, ( ') i =0 F =lnx (0 00')F'H3 + 0y,u' (268) a In z 0 In A ,i 1 = ',F'H,  g' (269) a In y' &ln y n 0' (270) 0 In w 8 1n F 1 a = 0' HF 71 (271) S1nz yz Since the optimum variable inputdemand 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 266 to 268, we obtain the system of differential inputdemand equations: Fdlnx= y,7V(0 0')F'H,+ +7y,g']dln y/(Q(D ')dlnw+ +[y7,/(qO ')F'3 + ypu']dlnz (272) 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(00')F'H,G' +i.]G (273) From Equation 237 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 231 can be rewritten as SVC f 8lnx, f a Inx, Ag,r aln yr 7gr lny, The last member of this equation is the (i,r)' element of y, F G y' where a n y') A VC Thus, from Equation 273, [,'] 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' (274) Therefore from Equations 273 and 274 we can write the coefficient of d In y as r, [[" ] + ] G = y [r :] G (275) Following similar analysis for the coefficient of the quasifixed 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 (276) In Equation 239 it was shown that the marginal share of the quasifixed 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 276, we obtain a simplified expression for [,k as [ '] 11 F 31n M1=Y1, 1 (a) 0')F' HVM'+ Oi]MM'. This expression can be further simplified to [ ] = [ _((D ') F VHMi + i'] Rearranging terms in this expression, we obtain [ k ]0,= (O( ')F'H3M' (277) Therefore the coefficient of d In z, using Equations 276 and 277, becomes y [y(s ')F 'HM'+Iil M= 7, [ +]M =y [,k]M (278) Finally, using Equations 275 and 278, the system of variable inputdemand 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 (279) r=\ kt=l /= 2.3.5 Comparative Statics in Demand The variable factor demand equation (Eq. 279) 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 279 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 265. Therefore if output and quasifixed 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 279, 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 265, (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 (280) 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 280 9 In y,. In z, becomes Here LT analysis uses the relative prices equation instead of the absolute price version of the model, Equation 279, (see Laitinen and Theil (1978), pg. 4145). 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 firstorder condition. Therefore, the equivalent form of the above equation is S T m' T 9T S ddlnx +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 280 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 quasifixed inputs have been defined as d In X = fd n x, dln Y = gd Iny, and dInZ = d In z r=I I= gr k=1 kpe v e respectively. Further. by the definition of y, (Equation 215), gr (Equation 232) and r p/4 (Equation 234), we have the following expressions k n VC 72 =711g,= 1n ( y, 8T a n (281) aT aln8 a ln y,n r la In yr I InVC VC ' y3711 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 280) as d InX =2d InY+yd In Z (283) where 72, 7y are the elasticities of variable cost with respect to proportionate output changes and quasifixed input changes, respectively. Using the same technique as above, for the factor demand equation we obtain an equivalent form of Equation 279: fdlnx,=y2 tr 3d ; Iny,+7Z" k d In zk (, )dln w (284) ,=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 283 by 0,, which gives O,d In X y20,d In Y y730d In Z = 0, and putting this expression back into Equation 279: fdlnx, =0,dlnX+y, 0gddIny,1 g, 0, ny, + r==1 = z g, ,Z d r I q dln k )d I n w, k=1 k k=1 Z e Y=1 e This expression is simply fdlnx =,.dlnX + y,(O,'O,)g,d In y, + y, ( dlnz, r=l k=l (0, 0)dIn wJ (285) /=1 This is the input allocation decision of the firm. This decision describes the change in the demand for the i'h input in terms of the Divisia volume index d In X change in output, changes in the input prices and changes in the quasifixed inputs. 2.4 Conditions for Profit Maximization Assume now that the firm's objective is to maximize profits (plus quasifixed 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 (286) X,'y ( r I Given the assumptions on the production technology (in the beginning of this chapter) the profit function is nonnegative and well defined for all positive prices and any level of the quasifixed factors. Further, it is continuous, linear homogeneous and convex in all prices, it is continuous, nondecreasing and concave in the quasifixed factors and finally it is nondecreasing (nonincreasing) in output prices (input prices) for every fixed factor (McKay et al. 1983). Assuming that we have a firststage of cost minimization, which gives us the input demands, then in the second stage we can maximize profits as a function only of y. Therefore, the problem that the multiproduct, multifactor firm faces is transformed to max (p,w,z) p,ry, VC(w,y,z) r=\ The firstorder conditions of this maximization problem are an aVc' VC = P, = 0= = P, (287) y, y, y, Using Equation 231 for g,, where gr = Y then Equation 287 becomes Ag, = p,.y,. Summing this expression over r and using the second term of Equation 231 we obtain the following P, y. R R A= r= (288) In y,. r In y, where R = prYr denotes total revenue of the firm. Also, we obtain that the share of the r r'' product in total revenue, multiplied by T is 8 Ilnyr g r, OT (289) 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 287, Equation 237 can be a(wx, ) rewritten as 6,' = ) which is the additional expense on the i'1 input, incurred for i( p, y,. ) the production of an additional dollar's worth of the r"' output. 2HI For the secondorder conditions to be valid, it must hold that is negative i iVC definite, for which it is sufficient that  is symmetric positive definite, because Oyay' a2 R = 0 follows from the assumption that the price vector is given. Therefore, we will ayayi 8'VC make the assumption that is symmetric positive definite. This maximization problem will give us the unconditional outputsupply 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 (290) S9lnp 8 Ilnw 8l nz 2.4.1 Output Supply The output supply of the multiproductmultifactor firm has the form provided by Equation 290. 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 inputdemand equation, we write the firstorder condition as an identity and then we totally differentiate with respect to its arguments: 8VC(y(p. 1.z).w,z) P, 0 (291) Then taking the total differential of Equation 291 with respect to p,, w, and z, we obtain the following relationships in a2vc any,, a2vc a___ "' 8OyC Op2 "' cVC 0ln y,, 3r1pI (292) P,: =.. y = 5,,P, (292) v=1 ry 'V, aP, v=i yy, c In p, 02VC' "' 'VC ay, y ( 2VC 2VC w, : 0 >  (293) y ,, y,.ayo w, Ow' ,y y') yOw' 2VC "' d VC ay, dy ('VC 'VC zk: a + 0 :> (294) k yZk yry aZk az' yWy) Oyaz' However, Equation 292 needs further modification before it gets a familiar form. Thus, solving for y, from Equation 289 we get Rg I y, = aT (295) d In y, Substituting this expression back into Equation 292 we obtain a D2VC Rg,. 1 dln y, SV',ay., pT a In p, SIn y, , which for all (r.s) pairs in matrix form becomes 1 02VC a1ny R _P'G =P Tar J yTy' In p' 0 In y,. In this expression. P denotes an mx m diagonal matrix with the output prices on the diagonal, G = diag(g ) and p is the vector of output prices. However, from Equation aT 232 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 = V1 Tr(G) a In p'P avyy' Finally, simplifying the righthand side of this expression, we get G 1n y 1 p82VC . G Olnp= P ) P = C@ (296) 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 293 and 294 into the same form as in Equation 296. Beginning with Equation 293, premultiplying by P and post R multiplying by W(= diag(w )), we get I ay W j2VC 82VC iP P' W (297) R 8w' R yy') Syw' Solving Equation 295 for R (for all (r,s) pairs) and using Tr(G)= _g = T d lnyr we obtain the following relationship for the total revenues of the multiproduct firm: PY R = Tr(G) (298) G Substituting Equation 298 back into Equation 297 we obtain G y p VC 2VC P YTr (G) aw' R oy y') 9yw' After canceling terms in the lefthand side of the equation, this can be simplified to G 1 y 1 (2VC )'2VC Tr(G) Y Ow' W R yy' 9y8w' However, the lefthand 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 (299) Tr(G) a In w Owy') Using similar analysis for Equation 294, that is premultiplying by P and post R 2 1 1 8 y I 8V 'C 8VC multiplying by Z. we obtain P Z  1 Z, which from Equation R dz' R 8y 8 yz' 298 becomes G Oln y (2C  Oy' ', where = Z P (2100) Tr(G) alnz' 9z8y' G Therefore, premultiplying the differential output supply by and using the Tr(G) solutions from Equations 296, 299 and 2100 we obtain d In y = 1''d In p y O'K'd In w q'/ O'd In z (2101) Tr(G) where 0 = [0' is an m x m symmetric positive definite matrix, which is normalized so m m that its elements add up to one, 0*, = 1. However, there is no clear interpretation for r=l s=1 the coefficients of d In w and d In z. Starting with the input price coefficient, define K as the nx m matrix that has the marginal shares 0, (Equation 237) as its (i,r)' element. 9VC Then, from Shephard's lemma in vector form we have that = x. If we differentiate 8w a2VC ax this relationship with respect to y' we obtain = However, from Equation 273 wawy' hy' we have that c 1n x F nx SIn y' Substituting for [09,], this expression simplifies to F ln xy ]G SIn y' [O Using the definitions of the terms in both sides of the equation, this expression can be also written as WX ax Y v Y=yKG = VC 9y'X X where G = PY R ) 288, respectively, and T;, R 1 K VC yT K PY R R * R 1 and y, = are derived from Equations 295 and VC T BT = a T After some algebra the above equation can be r 9 In Yr 8x ax transformed to W =x K P. This expression can be solved for or K, in order to get ay' ay' ax avVC 2 VC  W'K P and K = W P', respectively. Therefore, the matrix of ay' away' away' marginal shares 6,r can be written as or] = K = W P 9way' (2102) 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 2102, we can write the s'h element of K'd n w as n d In W = d nw, 2 (2103) 1=1 This is the Frisch variable input price index (this is denoted by the superscript F). For the coefficient of the quasifixed 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 (2104) Sk=1 a(Py,)Oanzk k=1 s=1 a(p,y)a lnzk Then we can define "' a2VC 4rk = *, (2105) "= 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 quasifixed 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, 2103 and 2105 we can write the r' equation of the output supply, Equation 2101, as m m I g, dlnYr =V/' ,dlnp., 0'VdlnW y/rrkdlnzk (2106) 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 ni'd InwZk (2107) The variable in the left hand side of Equation 2107 is d In y which is the Sg, contribution of the r'" product to the Divisia volume index of outputs. Note also, that g P, Yr, which is the revenue share of the r' product. g, R 2.4.2 Comparative Statics in Supply The supply Equation 2107 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 quasifixed input changes. For the outputsupply system, the following hold: * If all input prices are unchanged then d n W" = 0. Then Equation 2107 becomes ni i g' dlny, = I O,.',dln p, qs qrkd nzk. g, 9, r=l k=1 * If the prices of all variable inputs and all outputs increase proportionately then d In W' = d n p, and thus Eq. 2107 becomes d In y, = *V r7kd In zk. To find the total output decision of the firm, define 0j = 01 and note that s=1 m m 0* = 1 is implied by the normalization yZ ,, = 1. Therefore, the weighted means of rr=l s=. the logarithmic price changes that occur in Equation 2106 are d In P' = 0 d In p, , r=l d In W" = 0,d in Wr Correspondingly, let for the coefficient of the quasifixed input l7, = k Next, we sum Equation 2106 over r and use the symmetry of 0' to obtain r dinY=j Y 'd InF1` \,Vd1nzk (2108) = ) 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 2108 by 0, and putting the result back into Equation 2107. 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 (2109) P1: The deflator in the price term is d In = d In P" d In W '', which is the same for W " each inputdeflated, output price change in Equation 2109. If these corrected output price changes are proportionate then the second term in the right hand side of Eq. 2109 is equal to zero. This shows that in Equation 2109 only relative inputdeflated 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 inputdeflated 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 2109 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 decisionmakers 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 decisionmakers have incomplete information. To account for this nonoptimality, 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 decisionmaker. 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 quasifixed inputs. That is, under this theory the shortrun 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 inputdemand equation (Eq. 284) to get fd Inx, = y O, 'gd n y,. +y yk ,d Inz, + r,,d n w, + (2110) 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 (2111) These covariances form a singular nx n matrix, that is the sum of ,E,...,n, has zero variance since (0,,  ,)= 0, from Equation 265. This further, implies that the total input decision of the firm continues to take its nonstochastic form (Equation 280), 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 outputsupply equations of the firm M M ?I / g'd n y, = y/',ild In p, j 9~j' "dln w, 'rldlnz, + (2112) ,=1 v=1l =1 k=I where the above expression was derived by taking into account Equations 2106 and 2103. Further, E' ..., ',,, have an m variate normal distribution with zero means and variancescovariances of the form Cov'(,.,)= O 0 with rs = ,...,m (2113) 7Y2 Notice that the 02 is the same coefficient as in Equation 2111. The vectors e = (?,... ,,)' and = (',...,,,)' are independently distributed. This implies that the system consisting of the inputdemand equations and that of the outputsupply equations, constitute a twostage blockrecursive system (Laitinen 1980). The first stage consists of Equation 2112, which yields the m output changes and the second consists of Equation 2110, 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 2108) 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 + (2114) 2 * where E' = J ,, and from y 0* = 1, it follows that Var(E)  r r 7)2 2.6 Comparison to the Original LT Model In this section a brief comparison of the original LT model with the extended model (ELT) developed in the previous sections is provided. Laitinen and Theil (1978) assumed that the production function is negatively linear homogenous in the output vector, which implies that aT y 1 (2115) Ilnyr This relationship is not crucial for the derivation of the inputdemand and outputsupply equations, but for the definition of the coefficients in those equations. Taking into account the expression (Equation 29) for the returns to scale it is obvious that Equation 2115 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 quasifixed 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 238. This relationship is entailed from assumption 2115 and that the second derivatives of Equation 2115 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 231. See also discussion below this equation. such assumption is made and ,,, = q(u,, ,j). However, as was shown in Equations 263 to 265 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 inputdemand 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 inputdemand and outputsupply systems of equations, corresponding to the quasifixed 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 2115 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 revenuevariable cost ratio or as the elasticity of variable cost with respect to outputs of the firm. This assumption serves into easier derivation of the equations but it imposes a restriction in the returns to scale. CHAPTER 3 PARAMETERIZATION AND ALTERNATIVE SPECIFICATION 3.1 Input Demand Parameterization In order to estimate the variable inputdemand and outputsupply 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 quasifixed inputs and the nonoutputhomogeneous 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 inputdemand equation as depicted in Equation 284, 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, (31) 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 289; r sg R * Cost share of the firm, f = W'x VC * Share of the k'" quasifixed input shadow value in total shadow value of the quasi fu A VC/Q1nz, fixed inputs. k VC/alnzk from Equation 234; z _,, aVC/olnz, e e * Negative semidefinite price terms of rank n1, known as Slutsky coefficients in the Rotterdam model = y (oi, 0,0) ; * RevenueVariable Cost ratio or elasticity of variable cost with respect to outputs, aInVC R y, = y7, g, = = from Equations 281 and 287; I2 Ia In y, VC * Elasticity of variable cost with respect to the quasifixed inputs, defined as a In VC 7Y = Pk = ln =F. from Equation 282; k k Olnz, * Share of i"' variable input in the shadow price of quasifixed input zk, defined as aVC/ az, * Share of i'"' variable input in the marginal cost of the r'h product, defined as a (wx,) / y, VC/ r k * 71= ; = 1; XA = 1: Cg. =1; X = 1; Z ,,=O;. * Covariance of the error terms, Cov (,, ) = c2 (0, ). There are two existing problems with the estimation of the demand system. First, y3,,k Pk,, I tk are not observable since they involve derivatives of the variable cost k function with respect to the quasifixed inputs. They would be observable if quasifixed SVC inputs were at their full equilibrium levels, since at that point  = v with vk being aZk the exante market rental price of the quasifixed input. This in turn, would transform the model to a longrun with no quasifixed factors. A solution to this problem is to leave Y7, pk, pk as unknowns and estimate one coefficient bk = y3,ukk'. However, as is k usual with demand systems, the estimation method requires dropping one equation from the system due to singularity of the disturbances, as was shown in Section 2.5. Proceeding this way, though, the coefficient of the quasifixed 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 quasifixed 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 quasifixed inputs. As it is going to be shown in the next section, the one quasifixed input is a special case of the multiple quasifixed 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 longrun 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 longrun 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 QuasiFixed Inputs Summing Equation 31 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 (32) In Equation 31 the factor and product shares f = wx' and g = PrY, are observable VC R and can be calculated for any period from price and quantity data. As in the Rotterdam model or Laitinen (1980), arithmetic means are employed for these shares, since they are used to weight logarithmic changes between two periods. Therefore, by using a subscript t to denote time, the factor and product shares at period t are given by f, = w, and VC, g, = Y, while the average factor share of the i't input in t and t 1, and the average R, revenue share of the r"' product in t 1 and t are given respectively by f, = (f,, + /,, ,); g., = (g +g,) (33) 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 finitechange version of the Divisia indexes can be written as DX, = /,Dx,,, DY, = _,Dy, and =1l r=l DZ, = 7Z, Dzk, respectively. According to the theory of rational random behavior the k=1 total input decision (Equation 32) holds without disturbance. Since 73 is not observable, R Equation 32 cannot be solved for 2,. and thus employing 2= from Equation 287 VC we define its geometric mean as = R, R,_ (34) vc, .VC,1 Then, the total input decision in its finite change version can be written as DX, = ,,DY, + ,,DZ, (35) To solve the problem of identification of ,, one could proceed in two ways. First, Equation 35 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 quasifixed input aVC = vk, then at any point different than this optimum, it must hold that azk QVC = vk + 6, where 5k denotes the deviation between the exante market rental price zk vk and the shadow price of the quasifixed input (MorrisonPaul and MacDonald 2000). It follows from this definition that if 5k = 0, then the quasifixed input is at its full equilibrium level, while if 5k > or < 0 then we have undercapacity or overcapacity utilization of the specific quasifixed input, respectively. Therefore, we could use the following approximation OVC zk , 3, = VCz v Zk k Z VktZk' + ey (36) Zk VC k V, VC, k VC, Then taking the geometric mean of y3, and accounting for the error of the approximation Ey, we have that SVC VCv  Further, to solve the problem of identification of puk,, we follow the same technique as in y3, and define its approximation as k VC/, VktZk + k,e=l,...,1 (38) S'(VC/aze,,)z, v,,z,, e C (37) , .= while we use its arithmetic mean in our parameterization using the same argument as in the case of f, and gr,: = (4, +4,,)+ (39) 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 32 is the total differential of the production function and Equation 35 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 quasifixed 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 finitechange version of the total variable input decision: DX, = y,,DY, + y3,DZ, +E, (310) 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, (311) The input changes are then corrected by computing ,, = (Dx,, E,) (312) This correction amounts to enforcing the finitechange version of the total input decision, since summation of the correct input over i yields x,, = DX, E, = 7,DY +y,,DZ, (313) Taking into account Equation 312 for the residual correction and the parameterizations of the quantities and prices, the finitechange version of the variable inputdemand (Equation 31) can be written as x,, = Or, +, + k + Z ;,,w,, + ,, (314) 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 )= o2_ ( 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 312, is to make Equation 313, which can also be written as I 2, = I r, + I k hold. This, in turn, gives that i r k summation of Equation 314 over all inputs i will yield ,' = 1, = 1, =0 and c,, = 0. Therefore, the variable input demand (Equation 314) 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 semidefinite matrix of the price parameter (r, ) of rank n1, implying that the underlying cost function is concave in input prices. 3.1.2 The Case of One QuasiFixed Input As it is going to be shown below this is a special case of the multiple quasifixed inputs case. Notice, that when the firm employs only one quasifixed input then by definition p/ = 1, and so the variable inputdemand equation (Eq. 31) becomes f;dlnx,=y ,y 'g'dllny +y ,dlnzk + 7,dlnw +e, (315) In this case, the Divisia index of the quasifixed input degenerates to d In Z = ik d In zk = d In zk and the elasticity of the variable cost with respect to quasi k fixed input becomes 7y = 9 In VC / a In zk, since k = 1. Disregarding for a moment the error term and summing Equation 315 over all i, we obtain the total input decision dlnX = 2dlnY+ yd lnzk, k=1 (316) Proceeding then, as in the case of multiple quasifixed inputs, the following variable inputdemand equation is obtained: x,, = 8', + k, + ZI, Dwi, +, (317) r=l 1=1 This differential variable input demand satisfies the same properties as Equation 314. For instance, the assumption Zk still holds. The only difference with the case of multiple quasifixed 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 quasifixed 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. 2106) in order to obtain M n I gd ln y,= y',6 dln p, s dlnw, /rkdlnzk ,+ (318) <=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 quasifixed inputs, weighted by the coefficients 09, "I. a2VC as r,. = 0, =i (pSy, )anzk * Normalization condition, Y O, = 1 r=l N=l * Covariance of the error terms, Cov(c;, 2) = Ors 72 Similarly, a finitechange version of the outputsupply system (multiplied by 72, in order to make it homoscedastic) is 72,gr,'Dyr, = YZ y ;, Dp,, rkDzk r (319) 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 variancecovariance 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 ,) = ecry2, 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 (320) .=1 =1 S=1 k=l The properties of the outputsupply 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 CostBased System The variable inputdemand system as represented by Equation 31 assumes constant price effects, output and quasifixed 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 (321) 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 321 by variable cost (VC ) and differentiating with respect to y,, we get 8(w,x,) aVC VC VC a In X ' = a, + m In X +m a, y, y, y, Iln y, VC 9 In X QVC Noting from Equation 210 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 321 and the definition of 9, (see below Equation 31) we have that 9 (wx,) aVC 0," I =/ f, + m, (322) 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, + (323) 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 (324) J=1 Since f/ = 1 it must hold that a, = 1, m, = 0 and also that s,, = 0, Zs,, = 0, s,, = s,, where i, = 1...,n . Totally differentiating Equation 324 we have df = m'd In X + sd In w, (325) From Equation 228 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 (326) Equating now Expressions 325 and 326; and after some algebra we get fd n x, =(m," +f)d In X + (s, f(,j f )dn w,) (327) where ,, is the Kronecker delta. To verify that the input price terms in Equation 327 satisfy the addingup property we sum Equation 327 over all inputs i, to obtain that (s f (3, f )d In w)= 0, which verifies that the addingup property holds for I=1 J=1 the input price terms. Equation 327 is a system of input demands that must be equal with the input demand system presented in Equation 31. Forcing this equality we have n m I (m, +f)dlnX+ (s,, f (,, f, )dln w )= 7,2or gd ny +y3 k,4dlnz, S=1 r=l k=1 n + r,d ln w, +s, j=1 Summing this expression on both sides over i and using the previous results, we verify the total input decision of the multiproduct firm: d In X = y2d InY + y3d n Z (328) Substituting now Equation 328 back into Equation 327 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 allocationtype 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,) (329) J=I Letting now m,' = ," f, we get fdlnx, =y7O,'g'ddln y, + y,,d ln Z + s, f (5 f )dln w) (330) r=1 I=l Then we could combine Equations 323 and 329 into one general equation, since the lefthand side variables are the same but the righthand 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 inputdemand 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 329. 3. If e = 1, e, =0 then we have only variable output effects, Equation 323. 4. If e = 0, e, = 1 then we have only input price effects being variable, Equation 330. Note that the presence of d In Z in Equation 329 may create problems of multicollinearity, so an instrumental variable approach is suggested for the estimation of the system. Also Equation 331 seems more plausible than Equation 329 since it alleviates the problem of multicollinearity. 3.4 Capacity Utilization and QuasiFixity The most appealing alternative parameterization of the differential model is given by Equation 330, since it allows us to test for quasifixity and capacity utilization. Decomposing the Divisia index of the quasifixed factor in Equation 330, we get the following equation fdlnx, = ,2O,'gdlnyr +73,y .;dlnz,+ (s, f (, f)dln w) (332) r=l k=l I=1 Using the definitions of 73, O,' and /u it is easy to show that VVC 73k'k 0, 1 dn VC I In zan d k 730 ,' dlnzk :,' kl(( ~ k aInzk =,'d Inzk = C1 d ln zk k= In zk k= Substituting now this term back into Equation 332, it transforms the inputdemand system into 'g n S fdnx,: d ny,.+ ,, dlz, +( f(8,,f)dlnw,) (333) 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 333 over i and using the previous result that ,' = 1. The most important result is that the summation of Equation 333 over i for a specific quasifixed input gives us an estimate of the elasticity of variable cost with respect to the level of that quasifixed factor. This estimate, ge,.( = (VC/ zk)(zk /VC), provides a tool to test for quasifixity of input k. Specifically, a testable hypothesis for quasifixity is Ho : (aVC / z) + vk = 0, where vk is the exante market rental price of the quasifixed input. If H0 holds then it implies that the quasifixed input is at its full equilibrium level and should not be included in the righthand side of the demand equation. Given that we have an elasticity estimate we need to transform the null hypothesis H VC z VC into Ho: Ho: VC Zk + vk = 0, where the first term in the parenthesis is the zk VC zk estimate from the input demand estimation and is being multiplied by (VC / z ) at each data point at the sample. If the null holds at some data point then the quasifixed input k is at its full equilibrium level and the model is misspecified, while deviations from H0 show that the input is quasifixed. A one way ttest 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) ttest, providing the whole path of changes between full static equilibrium and shortrun 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 quasifixity through a Hausman test for specification error in a system of simultaneous equations, where their system consisted of a restricted cost function, shortrun demand for variable inputs and longrun 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 31, and the longrun demands for the quasifixed inputs can be obtained. This would be a simultaneousequations error specification test. However, the estimation of Equation 333 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 quasifixed 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 quasifixity. However, we provide directions for the estimation methods for such systems in the following chapter. CHAPTER 4 ESTIMATION METHODS 4.1 Choice of Estimation Method In Chapter 3, a model for the decisions of a multiproduct firm over a period of time was presented. While this formulation seems to be restrictive for real applications, it should be noted that it can be transformed to reflect different situations. For instance, the one firm could represent one sector of the whole economy, such as agriculture. Further, if one was considering the inputdemand 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 inputdemand and outputsupply 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 timespecific, fixedeffects and firmspecific, randomeffects 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 crossequations restrictions. For convenience, we reproduce the systems of equations x,, = ,Y,, + ,Zk, + ~ ,Dw,, + r, (41) r=l k=1 k=l an = aDp,, a,, ,s Dw,, ,rk Dzk, + (42) 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 41 and 42 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 41 and 42 is that we have assumed constant technology for the firm, but this can be resolved by adding a constant term in both systems. Laitinen (1980, page 118) suggested that these terms would represent systematic changes in the firm's technology (Hicks neutral technical change). Before we proceed into the estimation method for the joint system of Equations 41 and 42, we need to impose the addingup 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 addingup property in the input demand system, which creates the problem of a singular variance matrix of disturbances in the system of Equations 41. 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 41) 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 (MoorePenrose) 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 inputdemand and output supply equations (Equations 41 and 42, respectively) by employing one of the previously mentioned methods for the inputdemand system and then to provide a joint method of estimation for both systems. However, the nonlinear crossequations 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 quasifixed inputs aVC / azk = vk + ,k used in the parameterization of the inputdemand system (Equation 41) serves that purpose, since the summation of zk over i addsup to a constant and thus the last equation can be dropped. The homogeneity property of the inputdemand system (Equation 41) in input prices and outputsupply system (Equation 42) 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 addingup 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 inputdemand 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 quasifixed inputs and four outputs. Then, the homogeneity and symmetry imposed inputdemand 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 (23) 21 22 "2 0 0 (wi*)Y l"~ e wz) y,, y2, 2, 3 ~ 72] Y~ P ( P (wi) (w 2w3) 0 0 0 , ( ] p0 0 0 p2 (w,10)(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 quasifixed inputs to save space, and we have dropped one inputdemand 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 addingup, linear symmetry and homogeneity restrictions, the inputdemand and outputsupply 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 41 and 42 are written in matrix form as x, = Oy, + Kz, + Dw, +, = Nv, + i = ,...,n1 (43) y, = Ap, + Cw, + Fz + = Mq, + s"*, r = 1,.... m (44) which is subject to the following restrictions Homogeneity Ai,, + Ci, = 0, in output supply (45) Di, = 0, in input demand (46) Linear Symmetry Conditions A = A', in output supply (47) D = D', in input demand (48) Nonlinear Symmetry C = A K', in input demand and output supply (49) The addingup property of the inputdemand 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 inputdemand 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 Lnl 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 43 and 44, 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 43) are stochastically independent of those of the supply system (Equation 44), 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 outputsupply 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 43) but also in the supply system (Equation 44). Therefore, in spite of independence of the disturbances of the two systems, a joint method of estimation of Equations 43 and 44 is more appropriate in order to impose these restrictions on the marginal shares. Further, in the supply system 44 the parameters are nonlinear if we impose symmetry and homogeneity. Let us denote the variance across equations in the inputdemand system and output supply system, as E(, ,*") = ',,,,, and E(e,c,) = c n,_) (410) 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 =) ( 411) However, we choose not to force the offdiagonal elements of the covariance matrix to be zero. Bronsard and SalvasBronsard, (1984) suggested to test for the exogeneity of y, in the inputdemand system, by estimating the joint system one time with Equation 411 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, (412) 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 firstorder 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 411 then the offdiagonal elements in Equation 413 would be zero. In this case one could use a two step estimator, where in the first step Q*i and Q are estimated from each system separately (impose homogeneity in each system at this step) and then use those as an initial estimator of x, where now impose the linear and nonlinear cross equations restrictions. To apply the nonlinear symmetry constraints, it is convenient to regard the elements of M and N as functions of a vector u that contains only the free parameters in the joint system. Further, we choose to substitute the nonlinear symmetry restrictions, C = A K' at the objective (likelihood function). Another, equivalent way would be to include it as a constraint and maximize the constrained loglikelihood 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 firstorder conditions with respect to the i'h element of vector p/, are given4 by aM IFMq, 9L y y, Mq, q, = ,Nv,, =0 (414) x,, x,Nv, ON where = K' N K D In Appendix A. 1, the analytical derivatives of OM /ap, and ON/ap, are provided for a system that consists of three variable inputs, two quasifixed 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 (415) 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 414, and E the square and symmetric matrix, whose (ij)element is E given by Equation 415. The iterative procedure that Laitinen (1980) suggests, consists of the following steps: Compute i using Equation 413 with M, N evaluated at the given vector u . Use Equations 414 and 415 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 + Eco. Get /t, . Then from Equation 412, the concentrated loglikelihood function at the optimum becomes Lm T(m n l)ln2re n I (416) 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 fixedeffects 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 inputdemand system to be independent of those in the outputsupply system, then there is no need to follow the structure of equations as initially presented in Equations 43 and 44, and was followed until Equation 416. Instead, we could consider that the two systems of (n1) 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 outputsupply 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 (418) ,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 (, (419) 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 (420) Y(,;", 0 ... ... X;, ; u ;it This can be written in a compact form as Y, = X,, +U,, (421) where Y,, is a G xl vector, X,, is a G x K matrix of exogenous variables and f is a K x vector with K = I K, and Kg is the number of regressors in the g' equation g=I including a constant; U,, is a G x 1 vector of the error terms. Since we want to impose symmetry and some coefficients appear in at least two equations, then we redefine f as the complete coefficient vector, which is nonlinear and does not contain any duplicates, apart from the nonlinear terms (see example in Appendix A.2). Further, we redefine X,, = [xI',,X...x, x,,,]', where the k'h element of x ,, is redefined to contain the observations on the variable in the g'' equation which corresponds to the kth coefficient in 8 If the latter does not occur in the gth equation, then the k'' element of x,, is set to zero. A pooled model would then consist of regressing Equation 421 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 421 as a twoway error component model: U,, = a, + v,, i=l ...,N, t= ,...,T (422) where a* denotes all the unobserved, omitted variables from Equation 421, which are specific to each firm and are time invariant; a, denotes all unobserved, omitted variables from Equation 421 that are period individualinvariant variables. That is, variables that are the same for all crosssectional 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 timeinvariant variables or firminvariant variables whose omission could bias the estimates in a typical crosssection or timeseries study that reveals the advantages of a panel. The way we treat a, and a,, it then differentiates between fixedeffects and randomeffects models. Specifically, if we treat a, as fixed parameters to be estimated as coefficients of firmspecific dummies in the sample, then we follow a fixedeffects approach. Instead, if we assume that a, are random variables that are drawn from a distribution we have a randomeffects model. The same arguments are true for a,. Randomeffects models are considered in Section 4.3. So suppose that we formulate a fixedeffects model for N multiproduct firms, where the effects of omitted, unobserved, firmspecific variables are treated as fixed constants over time. Then, Equation 421 becomes ,, =a, +X,, + U, i = 1,...,, t=, ..., T (423) In this formulation a7 represents the firmspecific 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 firmspecific, fixedeffects model in this study for two reasons. Our sample consists of T > fixed and N + large and a fixedeffects 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 422 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 randomeffects model in the next section. A more appropriate fixedeffects model would be to consider timespecific 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 (424) \ t=l T where i, is a vector of ones, and we impose a, = 0 to avoid the dummy variable trap, t=1 since /f contains an overall constant in each equation. Further, we make the following assumptions Assumption 4.1: The error terms of Equation 424 are independent and identically distributed as U,, IIN,; (0(,,, ,,) (425) Assumption 4.2: X, and U,, are uncorrelated (426) 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 (427) 0 ifi j,t s 01", ... 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 424 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, (428) 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 (429) Y17 ,,I_ U,,, And let V =[U,,,..., U,,](;7, with U, = vecV,, i= ,...,N (430) Then making use of Equation 425, the (GT xl) vectors U, are distributed as IIN(0, ), with variance matrix E(U,, U) = 1,. 1,, = Q (431) Y, O 0 0 Q= =17, 0 I. Y E O C  The model as presented in Equation 428 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 loglikelihood associated with the linear model (Equation 428), 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, )= 27r2 012 e1u ') (432) The loglikelihood function for firm i is then, 1 11 L,= GTln 2r ln n (U/,''U,) (433) 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 loglikelihood function can be written as 1 T L, = GT n27 In I  U,,'U,, (434) 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 428, but nonlinear in the parameters, under the Assumptions 4.1 and 4.2. Then the gradient vector, and the N information matrix for = L, are given by 1=1 Ac NT 1I T? 8N 7 = 1 1+ U"UU' ,= _I I X,, ,__ (Y,X,,_f) (435) aX1 2 2 ,=1 ,=1Ph Ph S X  1= h (436) 0 NT 2 2 To prove this proposition notice that for a given vector /, we can differentiate Equation 434 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 435. Further, we can solve for 1,, in the above firstorder condition, to get 1 = .UU', = iv;' (437) 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 434 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 (438) Summing this gradient vector over all individuals it gives Equation 435. 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 438 would give us the GLS estimator, GS = XI:x'X, X^ZY, (439) In order now to find the Hessian we need to take the secondorder derivatives. We begin with the covariance matrix 02L, T 2 02 NT 2 ' 2 and so = (440) aC I gy 2 9 ,9l 1 2 Taking the secondorder derivative of the coefficient vector, after some algebra we obtain the following expression Yfi,, N)',, (_,l, (441) af a) Since E(Y, X,,f) = 0, the expectation of the above expression gives the information matrix of the parameters 11 ,, (442) Given that the information matrix, which is defined as I = E(D2 (/, u)), is block diagonal (Heymans and Magnus, 1979), and combining Equation 442 with Equation 440 we get the expression in Equation 436. 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 GaussNewton 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 GaussNewton iterative procedure consists of the following steps: Get initial consistent estimates of the vector f, using the GLS estimator presented in Equation 439 by disregarding the linear and nonlinear symmetry conditions. Impose though homogeneity and obtain the relevant estimates. Compute i' using Equation 437 at the given vector /. Use Equations 435 and 442 to evaluate co and I1 . Let A/8 = Ip1co. 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 + Il1 and go back to step 2. Continue until convergence. Get /, 1,,. To avoid potential confusion between the Scoring method and the multivariate GaussNewton 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 GaussNewton 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 loglikelihood function is concentrated with respect to i,, at step 2 in the procedure, then it obvious that 1 the inverse term in the last equation is Z,I 'Z,' = I,, while the last term of this equation is equal to Z,YX = C . From Proposition 4.1 we have that the loglikelihood 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 loglikelihood 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, (443) 2 2 The previous method of estimation accommodates unbalanced paneldata 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 timespecific or firm specific, randomeffects 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 individualspecific 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 421 for the joint inputdemand and output supply system as Y, = X,, + ,, (444) e,, = a, + u,, (445) In this formulation we assume that a, are firmspecific, randomeffects, and E,, are random errors. Further the coefficient vector / has no duplicates and includes an overall intercept. The matrix of exogenous variables is assumed to have the form X, = [x',,,..., xI;,, ] Concerning the distributional form of the random variables we assume Y ifi=j,t=s u,, IIN(; (0G,, ), that is E(u,,)= 0Gx and E(u,u ;,)= 0 ifi j,t=s (447) 0 ifi j,t s X,,, a, and e,, are uncorrelated (448) 1 Fi _a _a L *** Wl IG Then we have = '. and a = '. From Equations 446448 it is easily shown that 5 The error terms in this section are not related to any disturbances in the previous sections. Z1,u+ ifi=j,t=s E(E,,) = 0(,, and E(,,E',) = if i = j, t s (449) 0 ififj,t s As before, we stack the observations by time to get Y, = X,p + i u, a, + u,, = X,, + s, (450) 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,, (451) 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,) (452) 1 1 where A, = J, and B, = I, J, are symmetric and idempotent matrices. T T Following Magnus (1982) the loglikelihood for the i'h individual is given by GT 1 1 ) L,= In27r 1ln (Y,)'Q (YX ) (453) 2 2 2 Defining Y, = ,, + TS, we have that Q = JA, 0 + B, 9 Y,, (454) Using the property that A, and B, are symmetric and idempotent matrices, Magnus (1982) shows in his lemma 2.1 that = 1, I 1 (455) 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 455 we can rewrite the log likelihood as GT 1 1 1 T 1 (456) L, =2 In2 2n'(71)n2 e, e' (I' )e,, (456) 2 2 2 2 ( ts=\ For given covariance matrices, Y,, and E,, we take the first and second order conditions of Equation 456 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 (457) 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 secondorder 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 timespecific error component and not a firmspecific, to get 88 1 N1 N L V,(IT A,)V,', L = ) V,(TArlr)V,' (459) (T()NA, T(T1)N, An iterative procedure could be employed as in the previous section for the estimates that maximize the loglikelihood function. Further, to prevent the solution to converge towards a local maximum, Magnus (1982) suggests ensuring that E, and 1a are positive semidefinite. CHAPTER 5 APPLICATION TO U.S. BANKING INDUSTRY 5.1 Introduction One of the objectives of this study is to utilize the differential production model as means of estimation of the input demand, output supply and efficiency measures of US banks. To examine the robustness of the differential model results and to highlight the differences in the description of the technology that are induced by fitting the differential model, a comparison is provided against a commonly used in the literature parametric specification (translog). The discussion of the results focuses on three aspects of technology: concavity, returns to scale and input substitution as measured by the Allen Uzawa elasticities of substitution. The differential model is based on the total differentiation of the firstorder derivatives of any arbitrary cost or profit function given a technological constraint. As it was shown in Chapter 2. this provides an inputdemand and outputsupply system of equations for the multiproductmultifactor 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 secondorder 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 secondorder 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 51) 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 19851992, 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 RiegleNeal 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 GrammLeachBliley 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 51 illustrates the number of banks for the period 19902000 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 