Dale Jorgenson

Samuel W. Morris University Professor

Econometrics -- Preface

This volume contains my econometric studies of producer behavior. The volume includes a self-contained presentation of duality in the theory of production, statistical methods for estimation and inference in systems of nonlinear simultaneous equations, and econometric models based on flexible functional forms. The innovations embodied in these models-duality, simultaneity, and flexibility-have become standard in modeling producer behavior.

The centerpiece of the volume is a suite of econometric models generated from the dual formulation of the theory of producer behavior. The companion volume, Aggregate Consumer Behavior, provides a parallel treatment of my econometric studies of consumer behavior. These flexible representations of technology and preferences serve as building blocks for the general equilibrium models presented in my earlier volumes, Econometric General Equilibrium Modeling and Energy, the Environment, and Economic Growth.

In chapter 1, I survey econometric methods for modeling producer behavior. The goal of empirical research is to determine the nature of substitution among inputs, the character of differences in technology, and the role of economies of scale. Econometric methodology based on duality in the theory of production has generated an extensive body of empirical work. I summarize studies of substitution, technical change, and economies of scale that draw on this methodology.

The traditional approach to econometric modeling begins with additive and homogeneous production functions. Demand and supply functions are derived from the conditions for producer equilibrium. However, the constraints imposed by additivity and homogeneity frustrate the objective of characterizing technology empirically. For example, the production function originated by Charles Cobb and Paul Douglas (1928) requires that elasticities of substitution among all inputs must be equal to unity.

The constant elasticity of substitution (CES) production function introduced by Kenneth Arrow, Hollis Chenery , Bagicha Minhas , and Robert Solow (1961) achieves flexibility by treating the elasticity of substitution as an down parameter. However, the CES production function retains additivity and homogeneity and imposes stringent limitations on patterns of substitution. Daniel McFadden (1963) and Hirofumi Uzawa (1962) have shown, essentially, that elasticities of substitution among all inputs must be the same.

The innovations in econometric methodology for modeling producer behavior, summarized in this volume, stem from the dual formulation of production theory originated by Harold Hotelling (1932). Lawrence Lau and I give a self-contained presentation of duality in production theory in chapters 5 and 6. Technology is characterized by a price or cost function that is dual to the production function. Demand and supply functions are generated without imposing arbitrary restrictions on the underlying technology.

The responses of demands and supplies to changes in prices, technology, and economies of scale characterize the behavior of producers. For example, measures of substitution are specified in terms of the impacts of price changes on demands and supplies. Similarly, measures of technical change are specified in terms of the impacts of changes in technology. A judicious choice of these measures results in a flexible approach to econometric modeling.

In chapter 1, I outline the generation of transcendental logarithmic or translog price and cost functions. I define the share elasticity as the impact on the share of an input in the value of output of a proportional change in the price of an input. If a share elasticity is positive, the corresponding value share increases with the input price. If a share elasticity is negative, the value share decreases with the price. Finally, if a share elasticity is zero, the value share is independent of the price, as in the Cobb-Douglas production function.

Similarly, the bias of technical change is the impact of a change in technology on the input value share. If the bias of some technical change is positive, the corresponding value share increases with a change in the level of technology and we say that the technical change is input-using. If the bias is negative, the corresponding value share decreases with a change in the level of technology and the technical change is input-saving. Finally, if the bias is zero, the value share is independent of technology; in this case we say that the technical change is neutral.

An important feature of models of production based on the translog price function is that the rate of technical change is endogenous, but does not affect future production possibilities. The biases of technical change can be used to derive the implications of changes in input prices for the rate of technical change. If the bias is positive, the rate of technical change decreases with the input price. If the bias is negative, the rate of technical change increases with the input price. Finally, if the bias is zero, so that technical change is neutral, the rate of technical change is independent of the price.

The description of technology is completed by the deceleration of technical change. This is defined as the negative of the rate of change of the rate of technical change. If the deceleration is positive, negative, or zero, the rate of technical change is decreasing, increasing, or independent of the level of technology.

In the system of demand and supply functions generated from the translog price function the share elasticities and biases of technical change are unknown parameters. The dependent variables are the value shares of all inputs and the rate of technical change. All the dependent variables are functions of the same independent variables, namely, prices and the level of technology, These functions are nonlinear in the variables; the functions may also be nonlinear in the parameters. Finally, the parameters may be subject to nonlinear constraints arising from the theory of production. Additional constraints arise from restrictions on technology such as additivity and homogeneity.

Myopic decision rules for econometric models of producer behavior can be derived by treating the price of capital input as the rental price of capital services. Production decisions depend only on current prices, including the price of investment goods. More details about myopic decision rules are given in my paper, "Technology and Decision Rules in the Theory of Investment Behavior," in the companion volume, Tax Policy and the Cost of Capital. These decision rules greatly facilitate the implementation of the econometric models.

The constraints on the system of demand and supply functions implied by the theory of production are

1. Homogeneity. The value shares and the rate of technical change are homogeneous of degree zero in the input prices. 2. Product exhaustion. The sum of the value shares is equal to unity. 3. Symmetry. The matrix of share elasticities , biases of technical change, and the deceleration of technical change must be symmetric. 4. Nonnegativity . The value shares must be nonnegative. 5. Monotonicity . The matrix of share elasticities must be nonpositive definite.

Statistical methods for estimating the unknown parameters of systems of demand and supply functions depend on the character of the data set. For cross-section observations on individual producing units, prices can be treated as exogenous variables. Unknown parameters can be estimated by the nonlinear multivariate regression techniques introduced by Robert Jennrich (1969) and Edmond Malinvaud (1970, 1980). These techniques deal with nonlinearities in the parameters, nonlinearities in the variables, or both.

For time series observations on industry groups, the prices that determine demands and supplies must be treated as endogenous variables. Unknown parameters can be estimated by techniques for non-linear simultaneous equations. In chapter 7 Jean-Jacques Laffont and I present the method of nonlinear three-stage least squares for estimation of parameters in systems of demand and supply functions. These methods deal with simultaneity, as well as nonlinearities in the parameters and the variables.

The theory of production can be tested statistically by deriving constraints on the parameters of a system of demand and supply functions implied by the theory. Additional constraints, for example, the constraints implied by additivity and homogeneity, can also be tested. Jennrich and Malinvaud have introduced methods for statistical inference in nonlinear multivariate regression models. Ronald Gallant and I present methods for statistical inference in systems of nonlinear simultaneous equations in chapter 11.

I conclude chapter 1 by considering flexible representations of technology for econometric general equilibrium modeling. I also describe the use of panel data techniques for modeling technical change and economies of scale simultaneously. Finally, I outline methods for constructing dynamic models of production that incorporate internal costs of adjustment. The optimal production plan at each point of time depends on the initial level of "quasi-fixed" inputs, such as capital inputs, as well as expectations about future prices of outputs and inputs.

In chapter 2, I survey empirical studies of depreciation, an important special topic in econometric modeling of producer behavior. The measurement of depreciation requires modeling substitution among different vintages of capital inputs, corresponding to units of capital accumulated at different points of time. In principle each vintage could be treated as a separate input in an econometric model of production. However, the number of parameters would increase with the square of the number of inputs in a flexible functional form like the translog , rendering this approach infeasible.

The key simplifying assumption in the vintage model of capital is that different vintages are perfect substitutes in production. The services of these different vintages are proportional to initial investments with constants of proportionality given by relative efficiencies in production. The prices of different vintages of capital inputs are proportional to the relative efficiencies. Empirical research on the vintage model of capital reduces to modeling the relative efficiencies. Although the assumption of perfect substitutes is restrictive, the vintage approach has become the method of choice for modeling substitution among capital inputs.

I have presented a vintage model of capital in "The Economic Theory of Replacement and Depreciation," chapter 5 of the companion volume, Tax Policy and the Cost of Capital. This model, originally formulated by Hotelling (1925), is characterized by price-quantity duality. Capital goods decline in efficiency with age, requiring replacement investments to maintain productive capacity The price of a capital good also falls with age, reflecting both the current decline and the present value of future declines in efficiency Depreciation is the decrease in the value of a capital good with age.

Laurits Christensen and I have presented a vintage accounting system for prices and quantities of capital goods in our paper, "Measuring the Performance of the Private Sector of the U.S. Economy, 1929-1969," chapter 5 in the companion volume, Postwar U.S. Economic Growth. This accounting system provides an internally consistent framework for measuring depreciation and capital stock. We have extended this framework to encompass income, product, and wealth data for the econometric general equilibrium models described below.

In the model of capital goods prices introduced by Robert Hall (1971) the relative efficiencies of a capital good are expressed as functions of age and calendar time. The unknown parameters of the model can be estimated from observations on the prices of capital goods of different vintages. This model can be generalized to capital goods with different varieties that are perfect substitutes in production. Relative efficiencies are represented as functions of the technical characteristics of each variety.

Providing an illustration of modeling the relative efficiencies of different vintages of capital, Charles Hulten and Frank Wykoff (1981b) have constructed econometric models for the prices of eight categories of capital goods. Making use of the asset classification scheme of the Bureau of Economic Analysis (1987) capital stock study, Kun-Young Yun and I (1991b) have derived economic depreciation rates for thirty-five asset categories. These estimates of depreciation have been incorporated into price and quantity indices of capital services for thirty-five industries in "Productivity and Economic Growth," chapter 1 of the companion volume, International Comparisons of Economic Growth. The research of Hulten and Wykoff has been successfully exploited by the Bureau of Economic Analysis in measuring depreciation in the U.S. national accounts, as described by Barbara Fraumeni (1997).

Ellen Dulberger (1989) has employed speed of processing and main memory as technical characteristics of different varieties of computer processors. The Bureau of Economic Analysis (1986) has introduced price indices for computers based on this model of relative efficiencies into the U.S. National Income and Product Accounts. Kevin Stiroh and I (1999) have derived price and quantity indices for the capital services of computers in our paper, "Information technology and Economic Growth."

In chapter 3, Lawrence Lau and I present an economic theory of agricultural household behavior. This theory relates household consumption and production decisions to the prices of outputs, variable inputs, and consumption goods. Additional determinants of these decisions include stocks of quasi-fixed inputs, household wealth, and the composition of the household. Extensive empirical studies of agricultural households based on this approach were published in Resource Use in Agriculture, Applications of the Profit Function to Selected Countries, a special issue of Food Research Institute Studies, edited by Pan Yotopoulos and Lau (1978).

Our theory of agricultural household behavior expresses household welfare as a function of the utility functions of individual household members. An important simplifying assumption is that the utility functions for all individuals are identical, except for proportional transformations of units of measurement. These transformations are equivalence scales that depend on the characteristics of the individual such as age and sex. Daniel Slesnick and I have used this approach to modeling household behavior in our paper, "Aggregate Consumer Behavior and Household Equivalence Scales," chapter 5 in the companion volume, Measuring Social Welfare.

The objective of the agricultural household is to maximize house-hold welfare, subject to the technology of the enterprise, the constraints on the time available to household members, and the total expenditure of the household. The household takes the profits of the agricultural enterprise and non-agricultural income as given in making consumption decisions. It maximizes welfare with respect to leisure, consumption of goods produced within the agricultural enterprise, and purchased consumption goods.

Given competitive markets for agricultural inputs, including hired labor, production decisions depend only on technology and are independent of preferences. Lau and I represent the technology of the agricultural enterprise in terms of outputs, variable inputs such as labor, materials, energy, and quasi-fixed inputs such as land and reproducible capital. We derive a profit function that is dual to the agricultural production function. This gives the maximized value of profit of the enterprise as a function of the prices of the outputs and the variable inputs and the quantities of the quasi-fixed inputs.

In chapter 4 Christensen, Lau and I, present an econometric model of production that embodies three innovations. The model utilizes translog functional forms, statistical methods for nonlinear systems of simultaneous equations, and duality in production theory. In this model the economy supplies outputs of consumption and investment goods and demands inputs of capital and labor services. Price and quantity data for the inputs and outputs of the U.S. private domestic economy are taken from the system of U.S. national accounts that Christensen and I have constructed for the period 1929-1969.

An increase in the output of investment goods requires foregoing a part of the output of consumption goods, so that adjusting the rate of investment is costly. However, costs of adjustment are fully reflected in the market price of investment goods. The cost of capital input is a function of this price, so that costs of adjustment are external to the production process. In models of production with internal costs of adjustment, like those presented in section 1.7 of chapter 1, the cost of capital input must be inferred from the shadow value of the adjustment costs. Further details are given in my paper, "Technology and Decision Rules in the Theory of Investment Behavior," in the companion volume, Tax Policy and the Cost Of Capital.

Our first objective is to develop tests on the theory of Production that do not employ additivity and homogeneity as part of the maintained hypothesis. For this purpose we generate an econometric model of aggregate producer behavior from the translog production possibility frontier. The dependent variables are ratios of the values of investment goods and labor services to the value of capital services. The independent variables are logarithms of the quantities of outputs of investment and consumption goods, quantities of inputs of capital and labor services, and the level of technology.

Under constant returns to scale our model of aggregate producer behavior implies the existence of a price possibility frontier, defined by the set of prices consistent with zero profits. The price possibility frontier and the system of demand and supply functions are dual to the production possibility frontier and the necessary conditions for producer equilibrium. An econometric model generated from the translog price possibility frontier has the same dependent variables. The independent variables are logarithms of the prices of investment and consumption goods, logarithms of the prices of capital and labor services, and the level of technology

The translog production and price possibility frontiers correspond to two distinct representations of technology. We have estimated the unknown parameters of both models by the method of nonlinear three-stage least squares presented in chapter 7. We have tested hypotheses implied by the theory of production for both models, using the test statistics presented in chapter 10. Results for both models are consistent with the validity of an extensive set of restrictions implied by the theory.

Our second objective is to test the additivity and homogeneity restrictions that underlie the constant elasticity of the substitution production function. We employ the same data and econometric methodology as in our tests of the theory of production. The constraints implied by additivity and homogeneity conflict sharply with the empirical evidence. We further simplify the technology by requiring that the elasticity of substitution between capital and labor inputs is equal to unity, as in the Cobb-Douglas production function. Conditional on additivity and homogeneity, this is also strongly rejected by our tests.

Our overall conclusion is that flexible representations of technology are appropriate for dynamic general equilibrium modeling at the aggregate level. A representation incorporating additivity and homogeneity is much less satisfactory. Yun and I have employed the translog price function in our dynamic general equilibrium model of the impact of U.S. tax policy. We have presented the model in our paper, "The Efficiency of Capital Allocation," chapter 10 in the companion volume, Tax Policy and the Cost of Capital. We have used this model in analyzing the impact of U.S. tax reforms in our paper, "Tax Policy and Capital Allocation," chapter 11 in the same volume.

Our dynamic general equilibrium model of tax policy also incorporates a flexible representation of preferences. This is based on the translog indirect utility function presented in my paper with Christensen and Lau in the companion volume, Aggregate Consumer Behavior. Equilibrium in the tax model is characterized by an intertemporal price system that clears markets for consumption and investment goods and for capital and labor services. The price of investment goods reflects the present value of capital services and links the pre-sent to the future. Capital has been accumulated through previous investments, linking the present to the past.

In chapters 8, 9, and 10, Klaus Conrad and I have applied the econo -metric methodology presented in chapter 1 to aggregate data for the Federal Republic of Germany. These data are presented in our 1975 book, Measuring Performance in the Private Economy of the Federal Reptlb -lit of Germany , 2950-7973. The economy supplies investment and consumption goods and demands capital and labor services. An additional feature of this model is that the rate of technical change is endogenous and depends on the same independent variables as the demands and supplies. We utilize translog price and production functions to generate systems of nonlinear simultaneous equations that describe aggregate producer behavior.

In chapter 8, we derive constraints on the parameters of econometric models of producer behavior implied by the theory of production. Since the price and production functions provide two distinct representations of technology, we present tests of these constraints for both. The theory of production is consistent with the results of both sets of tests, corroborating and extending the findings of chapter 4. We test inequality restrictions price and production tested in chapter 4.

In chapter 10, Conrad and I have tested and rejected restrictions on technology associated with the additivity and homogeneity implied by the constant elasticity of substitution production function. These findings also corroborate and extend those of chapter 4. In chapter 9 we represent technical change by commodity augmentation factors that are analogous to the equivalence scales of chapter 3. We find that technical change is factor-augmenting, so that inputs of capital and labor services can be transformed into efficiency units, while investment and consumption goods outputs can be represented in natural units.

In chapter 12, Fraumeni and I present econometric models for each of thirty-five industrial sectors of the U.S. economy. These models are based on a translog price function for each sector. The price of output is a function of the prices of the primary factors of production-capital and labor services-prices of inputs of energy and materials, and time as an index of technology. An important feature of these models is that the rate of technical change is endogenous, but does not affect future production possibilities.

The econometric model of producer behavior for each of the thirty-five industries consists of a system of nonlinear simultaneous equations. The equations give the value shares of capital, labor, energy, and materials (KWEM) inputs and the rate of technical change as functions of relative prices and time. Price and quantity data for the inputs and outputs of each industry are taken from the system of national accounts presented in my 1980 paper, "Accounting for Capital," while the rate of technical change is an index number constructed from these data. This paper extends the vintage accounting system I had developed with Christensen to include both sectoral and aggregate production accounts.

The parameters of the system of input demand equations for each industry are estimated by the method of nonlinear three-stage least squares presented in chapter 7. These parameters included the share elasticities that describe substitution and the biases that describe technical change. We have estimated these parameters from time-series data for each industry. The industry-level data for the U.S. are described in my 1980 paper with Fraumeni , "The Role of Capital in U.S. Economic Growth, 1948-1976." In 1987 we published updated sectoral and aggregate production accounts in our book with Frank Gollop , Productivity and U.S. Economic Growth. The results are sum- marized in chapter 1 of the companion volume, Postwar U.S. Economic Growth.

As before, we describe substitution patterns by share elasticities , giving the impact of a proportional price change on the share of an input in the value of an input. As an illustration, the share elasticity of capital with respect to the price of labor is zero if the elasticity of substitution between the two inputs is equal to unity, since the share of capital is constant. We describe patterns of technical change by biases, giving the impact of a change in technology on the input value share. For example, we say that technical change is capital-using if the capital share increases with time, holding input prices constant.

The empirical findings on patterns of substitution and technical change reveal striking similarities among industries. In general, share elasticities are nonnegative, so that shares increase with proportional input price changes and elasticities of substitution are greater than unity. The elasticities of the shares of capital with respect to the price of labor are nonnegative for thirty-three of the thirty-five industries. Elasticities of substitution between capital and labor are greater than unity for these industries.

Similarly, elasticities of the shares of capital with respect to the price of energy are nonnegative for thirty-four industries and elasticities with respect to the price of materials are nonnegative for all thirty-five industries. The share elasticities of labor with respect to the price of materials are nonnegative for all thirty-five industries. However, the share elasticities of labor with respect to the price of energy are non-negative for only nineteen of the thirty-five industries. Finally, the share elasticities of energy with respect to the price of materials are nonnegative for thirty of the thirty-five industries.

A classification of industries by patterns of the biases of technical change is given in table 1.1 of chapter 1. The most common pattern is capital-using, labor-using, energy-using, and materials-saving technical change. The economic interpretation is that changes in technology conserve material inputs or increase value added through inputs of capital, labor, and energy. This occurs for nineteen of the thirty-five industries. Technical change is capital-using for twenty-five of the thirty-five industries, labor-using for thirty-one, energy-using for twenty-nine, and materials-using for only two.

We have emphasized that rates of technical change are endogenous in our econometric models of producer behavior. These rates depend on prices of inputs and the level of technology. If the bias of technical change is capital-using, then an increase in the price of a capital input reduces the rate of technical change. Since this is typical of the pat-terns we have described, an increase in the price of capital inputs reduces the rate of technical change. Similarly, increases in the prices of labor and energy inputs typically depress the rate of technical change, while an increase in the price of materials inputs raises the rate of technical change.

Over extended periods of time, energy prices have fallen relative to the prices of other inputs, elevating rates of technical change at the industry level. However, prices of labor inputs have risen relative to other input prices, depressing these rates of technical change. The substantial increases in energy prices after 1973 have had the effect of reducing sectoral rates of technical change, decreasing the aggregate rate of technical change, and diminishing the rate of growth of the U.S. economy.

In chapter 1 of the companion volume, Energy, the Environment, and Economic Growth, Peter Wilcoxen and I present a dynamic general equilibrium model of the U.S. economy with a flexible representation of technology for each of thirty-five industries. For this purpose we have employed econometric models of producer behavior for these industries based on translog price functions. We have used our dynamic general equilibrium model to analyze the economic impact of alternative energy, environmental, and tax policies. Mun Ho and I use this model to analyze the impact of trade policies in chapters 8, 9 and 10 of the same volume.

Our dynamic general equilibrium model of the U.S. economy also incorporates a flexible representation of preferences. This is presented in my paper with Lau and Thomas Stoker, "Transcendental Logarithmic Model of Aggregate Consumer Behavior," chapter 8 of the companion volume, Aggregate Consumer Behavior. The model is based on exact aggregation over systems of household demand functions derived from the translog indirect utility function. Equilibrium is characterized by an intertemporal price system that clears markets for the outputs of all thirty-five industries as well as for capital and labor services. The price of investment goods is forward-looking and depends on future capital service prices, while the stock of capital is backward-looking and depends on past investments.

In chapter 14, I consider the relationship between energy prices and rates of technical change in greater detail. I present an econometric model of producer behavior for thirty-five U.S. industries, based on data for the period 1958-1979. I divide energy inputs between electricity and non-electrical energy, so that the model for each sector includes the shares of five inputs-capital and labor services, electricity and non-electrical energy, and materials. The shares are functions of the prices of these inputs, as well as time is an index of technology. Finally, the model includes an endogenous rate of technical change, also a function of the five input prices and time.

The patterns of substitution for the models presented in chapter 14 are similar to those for the models of chapter 12. Technical change is electricity-using for twenty-three of the thirty-five industries and non-electrical energy-using for twenty-eight of the thirty-five industries. An increase in the price of electricity reduces the rate of technical change for twenty-three industries and reduces this rate for the remaining twelve industries. An increase in the price of non-electrical energy reduces the rate of technical change for twenty-eight industries and reduces the rate for the remaining seven.

Historically, the price of electricity has fallen relative to the price of non-electrical energy over extended periods of time. Prices of both types of energy have fallen relative to prices of capital and labor ser-vices and materials inputs. Electrification associated with the positive bias of technical change for electricity has raised rates of technical change in a wide range of industries. However, greater use of non-electrical energy has increased rates of technical change in an even broader range. Jumps in the prices of both forms of energy after 1973 have had a depressing effect on rates of technical change at the indus -try level, slowing the aggregate rate of technical change and the growth rate of the U.S. economy.

In chapter 13, Masahiro Kuroda, Kanji Yoshioka, and I present econometric models of producer behavior for thirty industries of the Japanese economy. These models are based on the translog price function introduced in chapter 12. We implement this model for price and quantity data for inputs and outputs of Japanese industries, as well as rates of technical change for these industries. The data are also employed in my paper with Kuroda and Mieko Nishimizu , "Japan-U.S. Industry-Level Productivity Comparisons, 1960-1979," chapter 7 in the companion volume, International Comparisons of Economic Growth.

In table 13.3 we compare patterns of substitution between U.S. and Japanese industries. These are broadly similar. In figure 13.2 we com-pare patterns of technical change for the two countries. The bias of technical change is labor-using for all thirty industries in Japan , while the bias is material-saving for twenty-eight of the thirty industries. The bias is energy-using for twenty-six of these industries and capital-saving for twenty-two. Capital-saving bias Predominates in Japan , while capital-using bias predominates in the U.S. For both countries an increase in the price of energy results in a reduction in the rate of technical change and a slowdown in economic growth.

In chapter 15, Kuroda, Hikaru Sakuramoto , Yoshioka, and I present bilateral econometric models of producer behavior for twenty-eight Japanese and U.S. industries. We treat data on production patterns for Japan and the U.S. as separate sets of observations. However, we assume that econometric models of producer behavior for the two countries have common parameters. The point of departure for the econometric model is a bilateral translog production function. Output depends on a dummy variable--one for the U.S. and zero for Japan -that allows for differences in technology between the two countries, as well as inputs and time as an index of the level of technology.

We combine data for the U.S. and Japan to estimate the parameters that describe substitution and technical change. These data are employed for bilateral productivity comparisons in my paper with Kuroda and Nishimizu . For the same input prices and level of technology production is more capital-intensive and intermediate input-intensive in Japanese industries and more labor-intensive in U.S. industries. Rates of technical change are higher for Japanese industries and lower for U.S. industries. Not surprisingly, the technology gap between Japan and the U.S. is gradually closing, as Kuroda, Nishimizu , and I have shown.

One of the key innovations in the econometric models of production presented in this volume is the application of duality in production theory. The starting point of the theory of production is the set of production possibilities, containing all the production plans available to the producing unit. The production function gives the maximum net output of any commodity as a function of the net outputs of all other commodities. The theory of marginal productivity completes the theory of production; this may be identified with the gradient of the production function.

In chapter 5, Lau and I present equivalent specifications of the set of production possibilities, the production function, and the marginal productivities. We can take a characterization of any one of the three as a starting point of the theory of production and derive the proper-ties of the other two. Marginal productivities provide the vehicle for generating econometric models of production in chapters 4, 8, 9, and 10 at the aggregate level and in chapter 15 at the sectoral level. The theory of production provides the links between these econometric models and representations of technology in terms of the production function and the set of production possibilities.

The second objective of chapter 5 is to characterize the set of feasible production plans from the point of view of economic behavior. For this purpose Lau and I present a theory of supply that parallels the theory of marginal productivity. We develop equivalent specifications of the profit function, the net supplies, and the set of price and profit possibilities. Any one of the three can be taken as the starting point for the theory of production. The net supplies provide the vehicle for generating econometric models of production in chapters 4, 8, 9, and 10 and chapters 12, 13, and 14. The theory of production provides links between these econometric models and the alternative representations of technology we consider in chapter 5.

The final objective of chapter 5 is to link the theory of supply with the theory of marginal productivity. Lau and I demonstrate the equivalence of technological ands behavioral viewpoints of the theory of production. For this purpose we employ equivalent specifications of the production and profit functions. The theory also implies equivalent specifications of marginal productivities and net supplies and of the sets of production and price possibilities. The econometric models of production considered in this volume can be linked to any of these six alternative specifications of the theory of production.

Sets of production and price possibilities are not employed in econometric modeling. However, Kenneth Hoffman and I have utilized these specifications of technology in our paper, "Economic and Technological Models for Evaluation of Energy Policy" included in the companion volume, Econometric General Equilibrium Modeling. We present a linear activity analysis model of the U.S. energy sector. This model is based on information from detailed engineering studies of technologies that are not currently available, but could be implemented under alternative technology policies. The activity analysis model of the energy sector is linked to econometric models for the non-energy sectors of the U.S. economy to provide a complete representation of technology.

The linear activity analysis model does not require that the marginal products corresponding to a given production plan and the net supplies corresponding to a given price system are unique. How- ever, uniqueness of the marginal products and the net supplies is essential for econometric modeling. In chapter 6 Lau and I consider the implications of differentiability of the production and profit functions or, equivalently, uniqueness of the marginal products and net supplies. By strengthening convexity assumptions for the production and profit functions we are able to develop the theory of production in terms of properties of differentiable convex functions and their gradients.

A second key innovation in the econometric models of production we present in this volume is the application of methods for systems of nonlinear simultaneous equations. In chapter 7 Laffont and I present the method of nonlinear three-stage least squares for estimation of the parameters of these models. Gallant and I provide the corresponding methods for statistical inference in chapter 11. These methods were greatly extended by Lars Hansen (1982) and became the basis for the Generalized Method of Moments that is now the standard approach to estimation and inference in macroeconometric modeling. In chapter 7, Laffont and I consider two lines of attack on efficient estimation of systems of nonlinear simultaneous equations. The non-linear three-stage least squares estimator is obtained by minimizing a weighted sum of squared residuals, where the weights depend on a set of instrumental variables. The first step in constructing this estimator is to linearize the system of nonlinear simultaneous equations. Arnold Zellner and Henri Theil's (1962) method of three-stage least squares is applied to the linearized model. This process is reiterated until the weighted sum of squared residuals is a minimum.

An alternative approach to efficient estimation of systems of non-linear simultaneous equations is an extension of the method of efficient instrumental variables for linear systems developed in my papers with James Brundy (1971, 1973). Laffont and I show that efficient instrumental variables and minimum distance estimators achieve the same efficiency. Both are less efficient than the fill information maximum likelihood estimator for systems of nonlinear simultaneous equations; however, the maximum likelihood estimator is computationally burdensome, and requires estimating all the equations of the model at the same time.

In chapter 11, Gallant and I present statistics for testing hypotheses about the parameters of systems of nonlinear simultaneous equations. We consider statistics based on likelihood ratio and Wald approaches, applied to the nonlinear three-stage least-squares estimator. The likely- hood ratio approach involves a comparison of values of the minimized criterion function with and without the constraints implied by the hypothesis to be tested. Both approaches can be used to generate confidence intervals and regions for the unknown parameters. These methods are available in many econometric software packages.

The econometric models of producer behavior presented in this volume have been incorporated into the dynamic general equilibrium models. These general equilibrium models also include econometric models of consumer behavior presented in the companion volume, Aggregate Consumer Behavior. The advantage of the econometric approach to general equilibrium modeling is that responses of production, and consumption decisions to changes in energy prices, environmental controls, trade restrictions, and tax policies can be derived from historical experience. This experience is an indispensable guide to economic policy making.

Implementation of the econometric approach to general equilibrium modeling has necessitated innovations in economic theory and econometric method. Duality has been especially critical in generating econometric models that provide flexible representations of technology and preferences. These models have required the development of new statistical methods for systems of nonlinear simultaneous equations. Duality, simultaneity, and flexibility in econometric modeling have led to a burgeoning empirical literature, characterizing technology and preferences in a wide range of empirical settings.

I would like to thank June Wynn of the Department of Economics at Harvard University for her excellent work in assembling the manuscripts for this volume in machine-readable form. Renate D'Arcangelo of the Editorial Office of the Division of Engineering and Applied Science at Harvard edited the manuscripts, proofread the machine-readable versions and prepared them for typesetting. Warren Hrung , then a senior at Harvard College , checked the references and proof-read successive versions of the typescript. William Richardson and his associates provided the index. Gary Bisbee of Chiron Incorporated typeset the manuscript and provided camera-ready copy for publication. The staff of the MIT Press, especially Terry Vaughn, Victoria Richardson, Lindsey Kistler , and Michael Sims, has been very helpful at every stage of the project. Financial support was provided by the Program on Technology and Economic Policy of the Kennedy School of Government at Harvard. As always, the author retains sole responsibility for any remaining deficiencies in the volume.