1.0 Introduction

1.1 Introduction

Partial equilibrium analysis illustrates results for one market at a time. However, there often exist market interactions and thus market feedbacks. As Nicholson suggests, pricing outcomes in one market usually have effects in other markets, and these effects, in turn, create ripples throughout the economy, perhaps even to the extent of affecting the price-quantity equilibrium in the original market. To represent this complex set of economic relationships, it is necessary to go beyond partial equilibrium analysis and construct a model that permits viewing many markets simultaneously. The general equilibrium model is a framework for analyzing linkages between markets and thus interactions between industries, factor resources and institutions.

de Melo and Tarr argue that inter-industry linkages are best captured in a general equilibrium framework. Although partial equilibrium may yield accurate estimates for particular sectors, estimates of aggregate costs of regional policies across sectors, for example, require a general equilibrium model to account for region-wide budget and resource constraints.

In the past, implementation of general equilibrium analysis was constrained by inadequate data and computational resources. Currently, however, the existence of large-capacity computer technology has made possible applications of such models to actual market situations. By recommending general equilibrium analysis, we do not mean that econometric estimates representing different sectors have little value. Rather, the two approaches should be viewed as complementary because it is neither feasible nor desirable to estimate, as a system of simultaneous equations, the full set of conditions describing a multisector economy model
(de Melo and Tarr). In many cases, general equilibrium analysis borrows parameter estimates from partial equilibrium econometric studies.

1.2 General equilibrium economic models

Several approaches have been used to represent the regional macroeconomy interactions among sectors and, hence, the analysis of impacts of alternative policies. Most general equilibrium procedures are broadly categorized into fixed-price (multiplier) impact analysis and the endogenous price, quantity and income computable general equilibrium (CGE) methods. This section provides an overview and comparison of these model types and their variants.

Input-output analysis, attributed to Leontief, has been used for assessing the impact of a change in the demand conditions for a given sector of the economy. The basic relationship in these models is represented by

, (1.1)

where , the amount of sector i’s output required for the production of sector j’s output, is assumed to be proportional to sector j’s output , and is the relevant input-output coefficient. Summing over sectors and adding final demand to equation (1.1) produces the
I-O model:

, (1.2)

which is also assumed to hold in first-difference form (depicting changes in the variables). An increase in final demand in a particular sector by, say, D will initially increase production for that sector, which in turn raises the intermediate demand for all sectors. To produce these intermediate inputs, however, more intermediate inputs are required. Although sectoral outputs keep on rising in several rounds, these increases become smaller and smaller such that their total always has a limit (Sadoulet and de Janvry). Equation (1.2) is often written in matrix notation:

, (1.3)

where X is the vector of outputs, F is the vector of final demands, A is the matrix of input-output coefficients, and I is the identity matrix (with ones on the diagonal and zeros elsewhere). The matrix represents a multiplier used to calculate overall changes in sectoral outputs caused by changes in final demand. For a more complete discussion of input-output see the web text chapter by William A. Schaffer.

Input-output analysis hinges on the crucial assumption that sectoral production is completely demand-driven, implying that there is always excess capacity in all sectors that is capable of meeting increased demand with no price increase. Because this assumption is likely to be unrealistic, input-output models are more useful as guidelines to potential induced linkage effects, and as indicators of likely bottlenecks that may occur in a growing economy, than as predictive models (Sadoulet and de Janvry).

Further, I-O models assume a constant returns to scale production function with no substitution among the different inputs. Prices are also assumed constant, which is not a major problem as substitution among factors is expected to be induced only by nonexistent relative price movements.

Extension of the I-O model to a social accounting matrix (SAM) framework is performed by partitioning the accounts into endogenous and exogenous accounts and assuming that the column coefficients of the exogenous accounts are all constant. According to Sadoulet and de Janvry, endogenous accounts are those for which changes in the level of expenditure directly follow any change in income, while exogenous accounts are those for which we assume that the expenditures are set independently of income. In determining exogenous accounts, it is common practice to pick one or more among the government, capital, and the rest of the world accounts based on macroeconomic theory and the objectives of the study.

Although I-O and SAM models have typically been used for impact analyses, they do not consider the special case where productive capacity of a sector is curtailed or eliminated (Seung, et al.). This concern has led to the emergence of mixed exogenous/endogenous I-O models where the production capacity of a sector is exogenously reduced (Petkovich and Ching). To examine the impacts of timber production potentials on income distribution, Marcouiller, Schreiner and Lewis demonstrated an application of a SAM version of the mixed exogenous/endogenous model, the supply-determined SAM (SDSAM) model, to the analysis of forest products.

However, these mixed exogenous/endogenous models, though relatively easy to implement, have limitations similar to fixed-price models. These are fixity of prices and no factor substitution in production and no commodity substitution in consumption. Seung et al. contend that, by these restrictive assumptions, the SDSAM model lacks microtheoretic foundation. Thus, such models are internally inconsistent because outputs for some sectors are forced to be fixed and final demands for the same sectors are assumed endogenous.

To circumvent the limitations posed by the SDSAM model, regional economists have turned to using the more theoretically sound computable general equilibrium (CGE) models as a tool for policy and impact analyses. In CGE analysis, output in all sectors is endogenously determined and prices are assumed sufficiently flexible to clear the commodity and factor markets. An empirical comparison of the SDSAM and CGE approaches by Seung et al. indicates that, compared to the CGE model, the SDSAM model tends to overestimate the policy impacts and to estimate production decreases in sectors where production may not change or may increase. The authors conclude that a regional CGE model is theoretically more sound than mixed exogenous/endogenous fixed price models for impact analyses where productive capacity of sectors is curtailed or eliminated.

Partridge and Rickman argue that fixed-price regional models are limiting cases of the more general Walrasian general equilibrium system. In fixed-price models, which are characterized by perfectly elastic supply, the total change in the regional economy is always predicted to be proportionate to the exogenous change. The Walrasian general equilibrium procedure, which is grounded in neoclassical theory, specifies less than elastic supply with equilibration of demand and supply achieved through flexible prices. In these models, the total response in an economy to an exogenous change is not necessarily proportionate and depends upon the various elasticities of demand and supply.

2.0 Overview of CGE Analysis

The CGE framework offers an alternative for regional analysis. It encompasses both the I-O and SAM frameworks by making demand and supply of commodities and factors dependent on prices. A CGE model simulates the working of a market economy in which prices and quantities adjust to clear all markets. It specifies the behavior of optimizing consumers and producers while including the government as an agent and capturing all transactions in circular flow of income (Robinson, Kilkenny and Hanson).

In the Walrasian neoclassical general equilibrium approach, the main equations are derived from constrained optimization of the neoclassical production and consumption functions. Producers are assumed to choose their level of operation so as to maximize profits or minimize costs using constant returns to scale production technology. Production factors – labor, capital and land – are all paid in accordance with their respective marginal productivities. Consumers are assumed to choose their purchases to maximize their utility subject to budget constraints. At equilibrium, the model solution provides a set of prices that clears all commodity and factor markets and makes all the individual agent optimizations feasible and mutually consistent (Bandara).

CGE analysis has been applied to a wide range of policy issues, which include, among others, income distribution, trade policy, development strategy, taxes, long-term growth and structural change in both developed and less developed countries (LDCs). Dixon and Parmenter associate the proliferation of these models in LDCs with two major conditions. First, growing realization that CGE models, unlike a number of other types of economic models, allow the simulation of policy alternatives in a way which is readily understood and perceived to be both relevant and useful by policy makers. Second, vast progress in the development of user friendly, readily transferable high capacity computer software, which has greatly increased researchers’ ability to handle models with considerable detail.

2.1 CGE analysis at national and regional levels

Most CGE models have been used to capture the effects of policies and economic shocks at the national level. Application of the technique to regions (such as states) is more recent. Examples of regional applications in Oklahoma include Koh, Lee, Budiyanti, and Amera.

Regional CGE models differ from their national counterparts in several respects. Most of these differences stem from the fact that regions are relatively more open economies compared to nations. Because of regional openness, commodity trade and resource migration are more important in regional CGE models. For example, regional households and entrepreneurs would not invest within the region if other regions offered higher rates of return. Thus, while national CGE models require that savings be equal to investment, regional CGE models permit excess savings to flow out of the region and vice-versa. This is not to say that regional policymakers cannot influence rates of return to investments but that control over major components of monetary policy is mainly determined at the national level.

In general, CGE models require considerable data, which, in most cases, is difficult to obtain. This problem is more severe at the regional level, where data in most cases is virtually non-existent. In fact, one of the possible reasons for the relatively slow start of regional CGE modeling is the paucity of regional data, in addition to unresolved theoretical issues of regional specification (Partridge and Rickman). Most of the limitations of regional CGE models are also inherent in alternative empirical regional modeling, such as I-O, SAM, and econometric.

Although regional CGE models have grown in popularity in recent years as an alternative method for examining regional economies and policy issues, their contribution has yet to be assessed. Partridge and Rickman present an extensive review of literature related to regional CGE modeling and conclude that regional CGE models, though still with unclear conclusions on issues of quantitative accuracy, represent a significant advancement in regional economic analysis. For details on the current state of the art of regional CGE modeling, readers are referred to Partridge and Rickman.

The greater openness of regional economies suggests some desired divergence in structure between national and regional CGE models. In spite of the differences between national and regional CGE models discussed above, the general formulation used in most studies is basically the same. While some studies have been designed to capture the added complexity, others have relied on the specifications common to the national CGE literature.

Most empirical applications of CGE models have been developed on the simplifying assumption of constant returns to scale production technology and perfectly competitive market structures. This has made these models fail to adequately represent industries with declining unit cost structures. Recently, de Melo and Tarr used the theory of duality to develop and apply a production modeling technique that accommodates imperfect competition in the U.S. auto and steel industries. Tembo has suggested and demonstrated an application of this technique to regional economies. Vargas and Schreiner show an application to monopsony markets in the regional timber industry.

The purpose of this chapter is to present and illustrate application of the salient features of the regional CGE model and to provide a step-by-step example of their empirical implementation. In this endeavor, the more traditional perfectly competitive constant returns to scale version of the CGE model is presented first. This is then followed by a variation that accommodates imperfect competition (see section 5.0).

2.2 Data and data organization

CGE models are very data intensive. Thus, the first step in implementation of a CGE model is identification and organization of data into a social accounting matrix (SAM). The SAM is a square matrix representing a series of accounts which describe flows between agents of commodity and factor markets and institutions. It is a double-entry book-keeping system capable of tracing monetary flows through debits and credits and constructed in such a way that expenditures (columns) and receipts (rows) balance. Benjamin distinguishes two objectives for the SAM: 1) to organize information about the economic and social structure of a country, region in a country, city or any other geographic unit of analysis; and 2) to provide a "fixed point" basis for the creation of a plausible model.

Regionalized economic datasets that can serve as a basis for regional CGE models are now available. This section describes the types of data required for building regional CGE models. These data needs include regional social accounts and parameters required for incorporating economic relationships among industries, in production and factor usage, among institutions, and in the generation of regional economic output. Each is addressed in-turn in the following sections.

2.2.1 Social accounting matrices

The base data upon which a regional CGE model is constructed relies on a static accounting for economic transactions taking place in a base year and specific to the region under examination. Input-output (I-O) tables provide one data framework but lack the comprehensive accounting of income flows. Base data on these income flows are necessary to address labor components, production structures, and government interaction necessary to conduct policy analysis. A more comprehensive accounting structure for regional economies is provided through an I-O extension known as a social accounting matrix (or SAM.) SAM extensions were initially developed during the late 1960's and early 1970's as a result of general dissatisfaction with the manner in which income flows were treated. A good overview of SAM development and analytical background for the interested reader can be found in Pyatt and Round and Hewings and Madden. SAMs as a basis for CGE models is addressed in Isard et al.

What is a SAM?

Like input-output accounts, social accounting matrices provide a comprehensive accounting structure of regional market-based productive activities and utilize similar double-counting book-keeping entries. Unlike input-output, however, social accounts focus on the household as the relevant unit of analysis and provide a comprehensive, and additional, set of accounts that track how household income is generated and distributed. Where input-output tables are focused on industries and their respective relationships with regional output, SAMs extend this into a more complete range of market mechanisms associated with generating household income. The relevant focus thus shifts from how regional output is produced to also address how regional income is generated and distributed. This comprehensive element is particularly important in regional CGE models that focus on both production processes and the economics of household factor supply, commodity demand, and government interaction.

How are SAM’s useful for policy analysis?

Social accounting matrices have been employed in a wide array of situations arising in policy development to address key issues of economic structure and impact assessment. A good overview of SAM applications in policy analysis was written by Erik Thorbecke and found in the recent text by Isard et al. (pages 317-331.) Basically, SAMs are useful in assessments that require a more comprehensive accounting of circular flows of an economy.

Particularly useful for addressing issues of income distribution, SAMs have been widely employed in assessing development effectiveness in attaining equity-based outcomes of policy. Applications, however, are not limited to assessing redistributive income policies. This is particularly true in the United States as national and state level policies that support the redistribution of income to the poor are largely out of favor. Increasingly, welfare reform legislation has emphasized the role of private markets to provide for individual welfare. SAMs have been employed to assess the relative impacts of alternative market-based changes on the distribution of income within regions. Thus SAMs will continue to be relevant tools to address a wide array of policy situations and development issues.

How is a regional/state SAM constructed?

SAMs can be constructed in a variety of ways. The manner in which a SAM is specified is typically driven by the problem being addressed. A thorough assessment of the various types of SAM structures is beyond the scope of this chapter. Rather, for this discussion a generic SAM structure will be discussed illustrated by a modest empirical SAM constructed for the Oklahoma economy.

Data elements for constructing a SAM. An illustrative SAM framework is provided in Figure 2.1. From an input-output perspective, the rows and columns that correspond to industry and commodity are the focus. Whereas input-output is limited to this industrial perspective, social accounting matrices extend the dataset to more fully capture income distribution resulting from returns to primary factors of production (land, labor, and capital.) In this way, the circular flow of goods and services to households from firms and the corresponding factor market flows to firms from households are captured.

In the SAM, row totals and column totals are equal thus representing a regional economy in equilibrium. For example, total industry output just equals the outlay used in its production. Institutional income (to households for example) just equals the outlay required for the use of institutionally-owned land, labor, and capital in the factor markets. In general, total income equals total cost of inputs. SAM accounts are constructed to balance outputs with inputs.

Data sources for SAM building. Once again, the specific data requirements for constructing a regional SAM vary depending on the type of problems being addressed. However, some generalizations can be made. In addition to standard input-output data (industry production, interindustry transactions, final demands, factors of production and imports/exports), typical SAMs require additional data on total factor payments, total household income (by income category), total government expenditures and receipts (including intergovernmental transactions), institutional income distribution, and transfer payments (both to households and to production sectors.) SAMs are typically built as static snapshots of a region thus, data elements will need to be generally consistent in temporal and geographic specificity.

2.2.2 Using IMPLAN to construct a SAM.

For purposes of illustration, discussion will center on a readily available dataset for the initial regional static equilibrium. A good example of this base economic equilibrium data is found in the county-level files available from the Minnesota IMPLAN Group (or MIG.) This consultancy group develops relational datasets built from secondary data available at the national, state, and county-level from the BEA REIS, BLS ES202, County Business Patterns and other sources. Specifically, this group first gathers data at the national level, converts it to a standardized format, derives national input-output tables and national tables for deflators, margins and regional purchase coefficients. State level data is gathered and controlled totaled to the national. County level data is gathered and controlled totaled to each state. County or regional-level input-output tables are derived using various data elements employed in the model development software embedded within IMPLAN Pro.

Over the course of development, the Minnesota IMPLAN Group has endeavored to adapt, expand, and extend datasets into more comprehensive accounting structures and regional modeling approaches. For example, a set of social accounts has been added to the county-level IMPLAN datasets. These accounts are available for use both in assessing inter-institutional transactions and in regional CGE modeling. The latter application has been under development for the past few years. Notable discussions of these developments can be found in Robinson and Sullivan, McCollum, and Alward.

Specific data incorporated into the IMPLAN SAM begins with standardized elements of the National Income and Product Accounts (NIPA.) Household transfer payments and distributional breakdowns come from the Census of Population, BEA REIS dataset and the BLS Consumer Expenditure Survey. Government data requirements originate from the Annual Survey of State and Local Government Expenditures. This data source provides state and local revenues and expenditures by detailed category.

Generating a SAM from an IMPLAN model is rather straightforward given general knowledge of software and dataset operations. The SAMs generated from IMPLAN are not, however, without drawback. One key drawback of using the IMPLAN system to generate a social accounting matrix is the rather rigid categorization scheme used in dataset and model construction. For example, due to the manner in which the dataset was developed, value added remains in rather nebulous categories that match published secondary data sources. Instead of value added being separated into returns to land, labor, and capital, value added in IMPLAN is reported in categories that include employee compensation, other property type income, proprietary income, and indirect business taxes. One ad hoc method of conversion is to simply use employee compensation as a proxy for labor returns (which neglects proprietary income), other property type income as a proxy for land returns, and proprietary income as a proxy for capital returns (actually more a mixture of labor and capital returns.) Although there exist procedures for disaggregating total value added into more standard categories of factor return, these methods tend to be data intensive and complex.

The aggregate SAM for Oklahoma

The number of sectors represented in the SAM and, hence, the number of markets in the CGE model depends to a large extent on the purpose of the study. Budiyanti, for example, aggregated the Oklahoma 1991 SAM to 14 industrial sectors of market goods, two sectors of non-market goods, three value-added sectors (capital, labor, and land), and three institutions (enterprises, households and government). The labor sector was further sub-divided into five skill levels. The household sector was also divided into low-, medium- and high-income classes. Government was represented by a state/local level and a federal level. Amera’s 1993 Oklahoma SAM has 30 industrial sectors, three factor sectors, three household sectors, two government sectors, one enterprise sector, one investment sector, and a rest-of-the-world sector. For illustration purposes in this chapter, a highly aggregated (four-industrial sector) version of Amera’s SAM is used as the data source (Table 2.1). This SAM also aggregates the household and government sectors into one sector each.

2.3 Determining parameter values

Once the economic agents are identified and their optimizing behavior specified by algebraic equations, the parameters in those equations must be evaluated. Data on endogenous and exogenous variables obtained at a snapshot point in time are typically used for this purpose. This process is referred to as calibration. Calibration or benchmarking determines the values of the normalizing (or free) parameters so as to replicate the observed flow values incorporated in the SAM (de Melo and Tarr). This process assumes that all equations describing market equilibriums in the system (model) are met in the benchmark period.

When dealing with flexible functional forms, such as the constant elasticity of substitution (CES) or the constant elasticity of transformation (CET), it is necessary to supplement the calibration process with these exogenously determined elasticities. Other parameters obtained from literature (econometric studies) include income elasticities, migration elasticities, and price elasticities of export demand. These parameters are used to illustrate the calibration process of the various components of the regional CGE model.

The calibration process starts with choice of units. Because in CGE analysis only relative prices matter, all prices and factor rents are normalized to unity in the initial equilibrium. With prices normalized to one, then the flow "values" in the SAM (Table 2.1) may be interpreted as a physical index of quantity in the commodity (industry) and factor markets (click here for further explanation of normalized prices). Once all the parameters are specified, the model is solved to
reproduce the benchmark data. The solution obtained with the benchmark data is referred to as the "replication" equilibrium, assuming the benchmark represents an equilibrium outcome, given existing exogenous conditions (Partridge and Rickman). In addition to providing a check on the accuracy of the calibration, the replication also shows that the complete circular flows of income and expenditures are balanced, which is referred to as microconsistency of the data. Counterfactual equilibria are obtained by introducing shocks to exogenous variables, changes in market conditions, or changes in any policy variable and rerunning the model. The general algebraic modeling system (GAMS) software is used for solving the regional CGE model. The following sections in this paper outline the general features of a regional CGE model and demonstrate the calibration and solution processes under both perfect and imperfect competition.

In a market economy there is generally a large number of homogeneous goods and services, which include not only consumption items but also factors used in production. Each of these goods and services has a market price, determined by the forces of supply and demand. Every market is assumed to clear at this set of prices. The perfectly competitive model further assumes zero transactions cost, a large number of price taking market participants (consumers and suppliers), and existence of perfect information, all of which support the law of one price (Nicholson).

Under these conditions, computable general equilibrium (CGE) models are similar to multimarket models, in which agents’ decisions are price responsive and markets reconcile supply and demand. Because they also encompass macroeconomic components, such as investment and savings, balance of payments and government budget, they are best chosen for policy analysis when the socioeconomic structure, prices, and macroeconomic phenomena all prove important (Sadoulet and de Janvry). CGE models have been built to simulate the economic and social impacts of various scenarios. Examples of alternative scenarios include foreign trade shocks, changes in economic policies, and changes in domestic economic and social structure.

In a regional CGE model, production creates demand for value-added factors and goods and services used as intermediate inputs. Intermediate inputs consist of both imports and locally produced goods and services. Demand for value-added factors interacts with available factor supplies to determine factor prices. Margins, such as taxes and transportation costs, increase factor costs to firms, which in turn increase product prices. Factor rates of return and ownership of factor supplies determine personal income, which in turn influences demand for imports and locally produced goods and services. Equilibrium occurs at prices which equate the demands for goods and services with supplies, and the demands for factors with factor supplies.

Because the CGE model attempts to look at all adjustments simultaneously, it is inherently an extensive formulation. To enhance understanding by students and prospective users of CGE analysis, the model here is split into components and each component is explained separately. The components include commodity markets, factor markets, production systems, institutional agents, and welfare measures.

3.1 Production system

Unlike regional input-output and SAM models, which are based on Leontief technology, neoclassical theory guides specification of production in regional CGE models. In consequence, the CGE model does not represent factor demands as linear functions of output. Instead, factor demands depend on both output and relative prices. The only exception, however, is in relation to treatment of those goods and services that are used as intermediate inputs. The Leontief input-output production function is used to represent production of regional output with fixed proportions of composite primary factors and composite intermediate inputs.

The composite primary factors generally enter the production process in a manner allowing factor substitution. Thus, production is best described as a multi-level or nested production process. Note that all factors in a constant elasticity of substitution (CES) function have the same elasticity of substitution between any pair of factors. To allow for differing elasticities between sets of factors, multi-level or "nested" production function forms are used in CGE, with each level containing a different set of factors and their own corresponding elasticities of substitution. That is, the use of a multi-level structure allows for use of both fixed-coefficients and price responsiveness in the CES form.

3.1.1 Composite value-added and intermediate inputs

The Leontief input-output production function that represents the non-substitutability between intermediate and primary inputs constitutes the first level of the three-level production process characteristic of most CGE models. For a single industry/sector, the Leontief production function is presented as:

(3.1.1)

where X_i is gross output of sector i, VA_i is composite factor (value-added) inputs of industry i and is composite intermediate inputs of industry i. Constants and represent industry i’s input-output coefficients for composite factor inputs and composite intermediate inputs.

By rearranging terms in equation (3.1.1), the (input-output coefficient) parameters of the Leontief production function are calibrated as follows:

, and

(3.1.2)

For each calculation in equation (3.1.2), values of the variables on the right-hand-side (RHS) are given in the SAM. For the agricultural sector in Table 2.1, for example, total output X_i=4,344,160,000 (the column or row total), composite factor inputs =1,713,668,000, and composite intermediate inputs (locally produced plus imports) =2,534,191,000. Therefore, Leontief parameter values are =0.40 and =0.58 (click here for graphic presentation of Leontief production function). Although an industry is an aggregation of many producers, it is treated as a single firm in the CGE framework.

3.1.2 Substitution among primary factors of production

What generally distinguishes a regional CGE production structure from a simple input-output model is that value-added (primary) factor usage is responsive to factor costs, and imports of intermediate goods are price responsive (Partridge and Rickman). At the second level of production, nesting allows different treatment of intermediate goods from that of value-added factors.

Primary inputs and their demands

Cobb-Douglas (CD) or constant-elasticity-of-substitution (CES) functions are commonly specified to represent substitution among primary factors of production in a sector - land, labor, and capital. Here production technology is assumed to possess constant returns to scale (CRS). The CD function implicitly specifies unitary factor substitution elasticities, while the CES is a more general case that allows different from unitary elasticities of substitution. For simplicity, the Cobb-Douglas functional form is used to represent the second level of production:

, (3.1.3)

where, , and are labor, capital, and land inputs for industry i, respectively. Coefficient is the total factor efficiency parameter for composite primary factor inputs in sector i. Parameters , , and are production elasticities (click here for CD production elasticities) and correspond to labor, capital and land, respectively. Constant returns to scale are imposed by assuming that the sum of the elasticities in equation (3.1.3) is equal to unity. Individually, the production parameters are also assumed to have values that lie between zero and one. By substituting and rearranging terms in equations (3.1.2) and (3.1.3), sectoral gross output () can be expressed in the Cobb-Douglas production function form:

, (3.1.4)

or

, where (3.1.5)

Assuming that labor, land, and capital are the only value-added (or primary) inputs in the production of sector i's output X_i, the sector’s profit function is

(3.1.6)

where p _i is profit (click here for example of profits) for sector i, PN_i is net price of output (i.e. output price less cost of intermediate inputs and indirect business taxes), PL is wage rate, PK_i is capital rent (assuming capital is fixed by sector), and PT is land rent.

Assuming all firms in the sector strive to maximize profits, differentiating equation (3.1.6) with respect to each of the inputs and equating the outcome to zero will give the first order conditions. Thus, the first order condition with respect to capital is:

. (3.1.7)

Rearranging terms in equation (3.1.7), the marginal product of capital is equal to the ratio of capital rent to output net price:

. (3.1.8)

Substituting equation (3.1.5) into equation (3.1.7) yields the following:

, (3.1.9)

which translates into:

. (3.1.10)

Rearranging terms in equation (3.1.10) and substituting for using equation (3.1.5), yields an expression for capital’s share parameter in the Cobb-Douglas production function:

, or (3.1.11)

This is equivalent to multiplying capital’s marginal product (see equation 3.1.8 above) by the ratio of capital to output, which is also the formula for elasticity. Therefore, expression (3.1.11) shows that factor shares are equal to production elasticities in a Cobb-Douglas function. Share parameters for labor and land are derived in a similar fashion. In equation (3.1.11), making the subject of the formula yields the conditional demand (i.e. fixed output level) for capital in the industry, given by:

. (3.1.12)

Similarly, conditional demands for labor and land can be expressed as:

, and (3.1.13)

. (3.1.14)

Calibration of the Cobb-Douglas production equation (3.1.5), involves determining and evaluating two sets of parameters – share parameters and the efficiency parameter, where all prices are normalized to one. The numerator and denominator in equation (3.1.11) are provided in the SAM as total capital returns and total value-added, respectively. For the agricultural sector in the above SAM (Table 2.1), capital returns and total value-added are $571,360,000 and $1,713,668,000, respectively. Substituting these values into (3.1.11) yields =0.333. Similarly, =0.253 and =0.414. The efficiency parameter for the Cobb-Douglas production function is calculated by rearranging equation (3.1.5):

. (3.1.15)

Calibration of equation (3.1.15) proceeds by substituting the calibrated factor share parameters and the quantities for the factor variables obtained from the SAM. For the agricultural sector, =7.46. Multiplying by a_oi yields the value . (Click here for graphic presentations of the calibrated production function and factor demands).

Intermediate inputs and their demands

By the Armington assumption (Armington), goods produced in different regions (and possibly countries) are assumed to be imperfect substitutes, usually specified as a constant elasticity of substitution (CES) function. These intermediate goods from different regions combine at the second level of production to form composite intermediate goods that enter the first level of production. The CES function representing the relationship between the two categories of intermediate inputs can be expressed as:

, (3.1.16)

where is the intermediate input efficiency parameter, is the share parameter, represents intermediate goods imported by sector i from sector j in the exporting region, is regionally produced intermediate goods for sector i from sector j, is the elasticity of substitution for industry j, and is the substitution parameter. The value of depends on the degree of substitutability between the two sources of intermediate inputs. If , the two are perfect substitutes. If , they are used in fixed proportions.

The following cost minimization problem is used to derive demand functions for regionally produced and imported intermediate inputs:

Minimize

Subject to: ,

where PM and represent, respectively, prices of imported and regionally produced intermediate inputs from sector j. Solving the first-order conditions of this problem and rearranging terms yields the following expression:

. (3.1.17)

Calibration of this equation requires knowledge of the elasticity of substitution and normalizing the two prices, PM and to one. As stated above, values of elasticities of substitution are obtained from other sources. For the Oklahoma agricultural sector, for example, manufacturing input has a value of 3.55 (Table 3.1). This leaves the share parameter as the only unknown in equation (3.1.17). The value of is calculated by substituting the elasticity of substitution and the base values for imported and regionally produced intermediate inputs (from SAM) in the rearranged form of equation (3.1.17):

. (3.1.18)

From the SAM, the known values for intermediate inputs from manufacturing to agriculture are =159,671,000 and =446,829,000. Thus, from equation (3.1.18), =0.48. The efficiency parameter is computed by rearranging terms in the CES function (equation 3.1.16) and making the relevant substitutions:

. (3.1.19)

Total intermediate inputs from manufacturing to agriculture is, =606,500,000 (see the SAM). Thus, evaluating equation (3.1.19) yields the value =1.931 for the agricultural sector. (Click here for graphic presentation of the substitution between the two sources of intermediate inputs).

3.1.3 Substitution among types of factor inputs

A third level in the nested production process may represent substitution among labor skills within the overall labor input, among classes of land within the overall land input for agriculture, or types of capital inputs within the overall classification of capital. (The SAM presented in Table 2.1 does not show subcategories of primary inputs.) A common procedure is to consider the CES form of production which allows elasticities of substitution to differ among industries but requires the elasticity of substitution among any two subcategories (i.e. labor skills, land classes or types of capital) to be the same. Alternatively, subcategories could be grouped into two parts, such as production labor and all other, with one elasticity of substitution between the two and then two different classes of production labor with a different elasticity of substitution.

The elasticities of substitution for this level of the production process must come from other studies. (Click here for modeling substitution among labor skills). The studies by Koh and Budiyanti classified labor into five skill levels following work by Rose. They then assumed the Cobb-Douglas elasticity of substitution (equal to one) for all combinations of skill levels and for all industries. No sensitivity analysis was completed to test the results of varying these elasticities.

3.1.4 Net output price

Net output price in the competitive model is regional output price minus the unit cost of intermediate inputs and unit value of indirect business tax:

(3.1.20)

where PN_i is commodity i's net price, PX_i is the composite regional output price, a_ji is the amount of the j^th commodity per unit output of the i^th commodity, P_j is the composite purchase price of the j^th commodity, and ibt_i is the indirct business tax per unit value of output. (See section 3.2.4 for explanation of composite regional output price and composite purchase price). The net output price is the per unit value of output available to compensate for primary factor use. Under conditions of constant returns to scale in production, the sum of the marginal value products for all primary factor use should exactly equal the commodity net price (see equation 3.1.6).

3.2 Commodity markets

Commodity trade involves both regional and export markets. Within the region, commodity supplies are obtained from regional sources (regional production sectors) as well as from out-of-region sources (imports). Though differentiated by source, these commodities are bought by industries (intermediate inputs), households and other institutions. Inter-industry commodity flows have been discussed in Section 3.1 as intermediate input demands. In this section, we discuss regional output markets and household commodity demand systems.

3.2.1 Market outlets for regional output

Each industry in the region produces a composite commodity that can be exported or sold in the regional market. Export markets include other regions within the country and international markets. In CGE analysis, exports and regionally sold products are assumed to be differentiated by market, with the relationship between them represented by a constant elasticity of transformation (CET) function. Price ratios and elasticities of transformation determine the levels of output exported and sold in the region. The substitution possibilities are, thus, represented as

, (3.2.1)

where is industry i’s total output (as defined above), is the output efficiency parameter, is the share parameter, represents sector i’s supply for export, is the sector’s output supply to the regional market, is the elasticity of transformation for industry i, and is the output substitution parameter. The value of depends on the degree of transformability between the two market outlets. If , the two are perfect in their transformation. If , the two markets are not substitutable and further market behavior for each must be specified (see Berck et al. for an alternative to the CET).

Each firm allocates it's output between the regional and export markets so as to maximize revenue, subject to the CET function. Because the production process is assumed the same for each market, revenue maximization may be substituted for profit maximization. Thus, for given regional and export prices, the problem faced by the firm is to:

maximize

subject to: ,

where PE_i and are, respectively, prices of exported and regionally sold commodities from sector i. Solving the first-order conditions and rearranging terms yields the following:

. (3.2.2)

Calibration of this equation requires knowledge of the elasticity of transformation , which is obtained from exogenous sources, and normalizing the two prices, and to one. For the Oklahoma agricultural sector in Table 3.2, . The value of , the only unknown in equation (3.2.2), is calculated by substituting the elasticity of transformation and the benchmark values for exported and regionally sold commodities (from SAM) in the rearranged form of equation (3.2.2):

. (3.2.3)

For the agricultural sector (see SAM in Table 2.1), =1,752,557,000 and =2,591,603,000. Thus, from equation (3.2.3), =0.47. The efficiency parameter is computed by rearranging terms in the CET function (equation 3.2.1) and making the relevant substitutions:

. (3.2.4)

For the agricultural sector, =4,344,160,000 (see the SAM). Thus, evaluating equation (3.2.4) yields =2.01 for the agricultural sector. (Click here for graphic presentation of the output transformation between the regional and export markets shown in the form of a market possibility frontier).

3.2.2 Commodity consumption by households

Regional household income available for commodity expenditure is calculated as gross income minus government taxes, savings and, in this case, payments for labor employed by households. Equation (3.2.5) is an algebraic representation of this relationship:

, (3.2.5)

where is household expenditure, is household disposable (minus government taxes) income, represents household savings, PL is wage rate, and is labor employed directly by households. The subscript h represents household category (low, medium or high income). The current SAM (Table 2.1) shows only total households.

The regional consumption by households is nested in two levels. At the first level, households maximize utility from leisure and consumption of composite market commodities, subject to total time (work plus leisure), household budget constraints and prices. At the second level, they choose optimal combinations of imported and locally produced commodities, which are imperfect substitutes, so as to minimize their cost of purchasing predetermined amounts of market commodities. Substitution between these commodity groups is captured in a CES function. A detailed presentation of each of these levels of the household consumption follows below.

Household commodity demand systems

Several alternative formulations have been used to represent household demand systems in the literature. Examples include the almost ideal demand systems (AIDS) by Deaton and Muellbauer, the Rotterdam model by Theil, and Barten, and the linear expenditure system (LES) by Stone. In general, a theoretically consistent demand system permits imposition of the general restrictions of classical demand theory. These restrictions are a) adding-up: value of total demands equals total expenditure, b) homogeneity: demands are homogeneous of degree zero in total expenditure and prices, c) symmetry: cross-price derivatives of the Hicksian demands are symmetric, and d) negativity: direct substitution effects are negative for the Hicksian demands.

The linear expenditure system is the most commonly used in CGE analysis due, in part, to convention and because it allows representation of subsistence consumption, in addition to satisfying the above restrictions. In this subsection, we provide an overview of the LES demand system and its adaptation to the CGE framework. Readers interested in more detail about the LES and other demand systems are referred to Deaton and Muellbauer.

In the LES, demand equations are assumed to be linear in all prices and incomes and the set of demand functions is expressed in expenditure form:

, (3.2.6)

where is the price of the i^th commodity, is the quantity of the commodity demanded, is the i^th intercept, are the price parameters, is the marginal budget share for the commodity, and y is the household’s income. Empirically, the LES is derived from constrained maximization of the Klein-Rubin (also known as Stone-Geary) utility function, whose general form is

(3.2.7)

where U is the utility level, is level of commodity i, is as defined above, and , if positive, is subsistence minima as perceived by the consumer.

Given a fixed amount of household income that can be allocated to consumption, , the household faces the following constrained maximization problem:

Maximize

subject to: ,

where the subscript h represents a particular category of households. Solving the first order conditions of the Lagrangean to this problem produces the following results:

, and (3.2.8)

. (3.2.9)

Rearranging terms in (3.2.8), summing across i, and solving for the Lagrangean multiplier yields

, (3.2.10)

where, as stated above, . Substituting (3.2.10) into (3.2.8) produces an expression for the expenditure on commodity i by household category h:

. (3.2.11)

As expected, the first derivative of equation (3.2.11) with respect to total expenditure is the marginal budget share, . The linear expenditure system (equation 3.2.12) is obtained by dividing equation (3.2.11) by :

. (3.2.12)

To evaluate equation (3.2.12), we need values for and , prices, and total consumption expenditure data from the SAM. Because cannot be directly estimated from empirical data and because cannot be calculated from a one-period data set in the SAM, equation (3.2.12) is often implemented using a simplified version of the Stone-Geary LES. Rearranging equation (3.2.12) gives

. (3.2.13)

If we assume that the average budget share is equal to the marginal budget share, equation (3.2.13) implies the following:

, and (3.2.14)

. (3.2.15)

Because and , the relationship in equation (3.2.15) is guaranteed only if the minimum/subsistence consumption for all commodities. If this is the case, the LES demand function, equation (3.2.12), simplifies to:

. (3.2.16)

Coefficients are calculated from equation (3.2.14) by using the benchmark data in the SAM. This process is accomplished by normalizing the prices to one, which transforms the expenditure results in the SAM to physical quantities. In our example (Table 2.1), total household expenditure on both imported and regionally produced commodities,=$50,665,679,000 and expenditure on agricultural commodities is $328,760,000. Thus, the marginal (equal to the average) budget share for agriculture is 0.0065. (Click here for a graphic presentation of commodity demand).

As you notice, equation (3.2.16) is based on very restrictive and somewhat unrealistic assumptions. It implies that income elasticities of demand are unitary for all commodities. Although the results are not appropriate for dynamic analysis, this assumption does not pose serious problems for comparative static analysis, particularly if expenditure patterns for several household income groups are embodied in the model. For the interested reader, click here for a more general case of the LES demand system, which provides for leisure, household labor supply, and varying commodity income elasticities.

Commodity substitution of imports for domestic product

The second level of household commodity demand involves determination of the minimum cost combination of regional and imported commodities. For each commodity i, substitution between the two sources is captured in the following CES function:

, (3.2.17)

where is the household consumption efficiency parameter, is the share parameter, represents household demand for imports, household demand for regional products, is the elasticity of substitution, and is the substitution parameter. The determination of the domestic (regional) and imported amounts of a fixed total household demand is the same as presented in equations (3.1.16) to (3.1.19). (Click here for a graphic presentation of the substitution relationship between imported and regionally produced commodities as shown in the form of a household indifference curve).

3.2.3 Institutional markets

Governments and capital formation are the two remaining commodity markets represented in the Oklahoma SAM. Quantity demanded is assumed exogenous for each of these markets. However, price is endogenous and, hence, expenditure by governments and for capital formation varies with price. Similar to intermediate commodity inputs and household commodity demands, imported and regionally produced commodities are imperfect substitutes in meeting the composite commodity demands. Exogenous commodity demand for governments (QG_i) and capital formation (QC_i) from the two sources (regional and imported) is given by the following CES function:

(3.2.18)

where QX_i=QG_i + QC_i, QXM_i is quantity imported and QXR_i is quantity domestically produced. All parameters are identified similar to those for equation (3.1.16). The elasticities of substitution are the same as for intermediate inputs and household demand (see Table 3.1). Solution to quantities imported and domestically produced is similar to equations (3.1.16) to (3.1.19).

3.2.4 Commodity prices

Composite purchase price

Commodity purchase prices are a composite of regional and import prices:

(3.2.19)

The composite purchase price (P_i) is the unit value for household consumption goods, intermediate inputs, and institutional purchases. PR_i is the regional purchase price and PM_i is the import price. R_i is the total amount of commodity regionally produced and consumed and M_i is the total amount of commodity imported:

(3.2.20)

(3.2.21)

The right hand side terms are as previously defined.

Composite output price

Commodity output prices are a composite of regional and export prices:

(3.2.22)

The composite output price (PX_i) is the weighted unit value of revenue received from regional and export sales. PR_i is the regional price and PE _i is the export price. R_i is the regional quantity and EXP_i is the export quantity.

3.2.5 Commodity market equilibrium

Total commodity demand is the sum of intermediate demand, institutional demand, and export demand. Total commodity supply is the sum of regional production and imports. Market equilibrium for commodity i is the following:

(3.2.23)

where X_i=regional production, M_i=imports, TV_i=total composite intermediate input demand, TQ_i=total composite household demand, QX_i=total composite exogenous commodity demands (governments plus capital formation), and EXP_i=export demand.

ENDNOTES

1. These are all Ph.D. studies completed at Oklahoma State University. Results of the studies have been published in Koh, Schreiner and Shin; Schreiner, Koh, Lee and Budiyanti; and Amera and Schreiner. Partridge and Rickman give an extensive review of many other regional studies.

2. The major strength of regional SAMs include accounting comprehensiveness. Although still widely used, it is important to note that SAMs have some rather serious theoretical shortcomings when used to model economic change. These modeling caveats have, in part, driven the movement toward developing more flexible modeling systems. As such their use in computable general equilibrium models remains the focus of this chapter.

3. The IMPLAN Pro software and county/state datasets are available from the Minnesota IMPLAN Group (accessed on the world wide web at www.implan.com.) Their mailing address is 1725 Tower Drive West, Suite 140, Stillwater, MN 55082, Voice: 651/439-4421 Fax: 651/439-4813.

4. Application of these elasticities in the CGE framework are discussed in section 3.0.

5. Often, households have been categorized into 'high income', 'medium income', and 'low income'. See, for example, Budiyanti. For simplicity, we assume here that all households are homogeneous (h=1).