COMPUTABLE GENERAL EQUILIBRIUM MODELS: SOME POLICY DEDUCTIONS

Mustapha Muktar, Ph.D

Department of Economics

Bayero University, Kano-Nigeria

Introduction

Equilibrium can be seen as a situation whereby the forces determining a state are in balance so that there is no tendency for the variables of a system to change. Equilibrium can either be static or dynamic. General equilibrium analysis was developed by French economist Kon Walras, and he developed a general equilibrium model based on interdependencies that exist in an Economy, since then a lot of general equilibrium models were developed and they include the Leontiet input-output model, Walras – Cassel model, Arrow- Debreu model, Zeutham model, John Von - Neumann model, Paratian general equilibrium model and etc. General equilibrium analysis is a disaggregative structure of macroeconomic theory, since all economic units are interdependent and an equilibrium point is revolved simultaneously by all.

Computable general equilibrium (CGE) modeling is an attempt to use general equilibrium theory a san operational tool for emphatically oriented analyses of resources allocation issues in market economies (Bergam 2003). Following this the introduction paper is arrived at highling the concept of computable general equilibrium model and to appreciate the practical application of the models to economic analysis.            

Computable general equilibrium modeling general equilibrium refers to an analytic approach were an economy is regarded as a complete system of interdependent components (different market, industries, households and etc.) and all discussions are taken according to fully optimizing behaviour. Economic shocks affecting away one of these components may produce repercussions through out the whole system. Assessing the effects of the shocks can be means of simulation i.e. by measuring the repercussions that are triggered by shocking the system in various ways. The models are called “Computable” in the sense that they should produce numerical results that are applicable to particular situation in particular countries (Rumter 1999).

C.G.E models are constructed in order to capture the multi sectoral nature of the chain reaction in the economy resulting from a change from the economy within the general equilibrium frame work, the repercussions in the economy are taking place through changed relative prices, to which agents adjust their production and consumption. A common feature of C.G.E models is that quantities and relative prices are endogenously determined within the model (Unemo 1993).

Computation of C.G.E. models.

There are two main ways of computing C.G.E. models, namely the non linear programming approach was not used widely, and it is based on the idea that the solution to a C.G.E. model can be deduced form the solution to an optimizing problem. Although this method has been applied successfully in solving C.G.E. models, the experience of most researchers in C.G.E. suggests that the derivative approach is more convenient and flexible, and therefore and more suitable for practical policy analysis and forecasting Rumter (1999).

The derivative approach to solving C.G.E. models otherwise known as Johansen/Euler method was first used by Johansen in 1960 it involves restricting the model so that it is linear in its derivative and solving for perturbations of endogenous variables due to exogenous changes matrix inversion. However the development of piecewise linear approximations by Dixon etal (1984) has come to be applied to models with non linear derivatives.

To introduce the main idea of this method we have to consider a model in which equilibrium is described by a vector, V, of length n satisfying a system of equations of the form f (V) = 0.--------------------------------------------------------------- (1).

Where f is a vector function of length m we further assume that f (is twice continuously) differentiable and that the number of variables, n, exceeds the number equations, m, by (n-m), The components of vector V typically represent demand for and supply commodities or factors, policy variables, technological coefficients and other economic variables. The function f imposes relations such as demands equals supplies and prices equals costs, while preferences and technologies are represented by differentiable a utility and production functions in system (1) finally let’s assume that we know an initial solution to the model V¹, which implies that we have known vector V¹ satisfying F (V¹)= 0---------------------------------------------- (2).

Now re-writing equation (1) as F (V₁,V₂) = 0 ---------------------------------------- (3) The Euler method has been satisfied where V₁ is a vector of length m containing all endogenous variables and V₂ is the vector of length n – m of exogenous variables. By differentiation equation (3) totally the deviations dV₁ and dV₂ from the known solution V₁ must, to an approximation, satisfy.

                        F₁(V₁)dV₁ + F₂ (V₁)dV₂ = 0 -------------------------------------------------(4)

Where F₁ and F₂ are matrices of partial derivations of F evaluated at V₁, Now solving for the impact of movements in the exogenous variables, dV₂, on the endogenous variables dV₁, on the endogenous variables dV₁, in the vicinity of V₁, i.e. calculating a one-step Euler or Johansen approximation, gives

           

dV1 = B (V1)dV2------------------------------------------------------------(5)

where B(V¹) = - F₁ ^-1 (V¹) F₂ (V₁) -----------------------------------------------(6)

provided that we can evaluate B (V¹) which directly hinges on the inveritability or equivalency non- singularity of F₁ (V¹)3. The MX (n -m) matrix B(V¹) show the partial derivatives, evaluated at V¹, of the endogenous variables V₁ with respect to the exogenous variable V₂. In other words B(.) is the Jacobian matrix of the implicit solution function G⁴. Therefore for a given change in exogenous variables equation 5 provides only a first order approximation to the effects on the endogenous variables of the solution function G is highly non linear i.e. if the higher order terms in the associated Tailors series approximation are not negligible, then serious approximation errors may arise requiring further refinements to be done, one way out of this problem is to compute multi-step Johansen/Euler solutions.

Another computational approach regards the equilibrium condition of CGE models as simultaneous set of non-linear equations and solves such equations for the equilibrium values of the endogenous variables using a fixed point algorithms or an alternatives method of numerical iteration. It is equally possible to classify CGE models on how such models are numerically specified. The best known method involves the deterministic calibration of the model. The procedure entails choosing benchmark equilibrium time period, constructing, a general equilibrium data set of the period, selecting the functional configuration of the model, the generating model parameters from benchmark data set or from independent econometric works. Mansur and Whalley (1984) examine this typology of CGE model and remarked that CGE model Calibration is often faced with problem of unavailability of social accounting practices and of econometric estimates of key model parameters in developing countries. The challenge that is posed by this problem will always be to construct a social accounting matrix to provide methodological framework for CGE models fashioned after deterministic calibration method.

Social Accounting Matrix

A social accounting matrix (SAM) is defined as a numerical representation of economic cycle with emphasis on distributive aspects. As in the complete system of national accounts and in the input-output framework, transactions in a particular year appear in a matrix format showing receipt on the rows and outlays in the columns. Indeed a SAM shows how sectoral value added accrues to production factors and their institution not owners, how these incomes, corrected for net current transaction are spent, and how expenditures on commodities lead to sectoral production and value added with the linkages from this cycle also shown (Keuning and Ruijter, 1988).

SAM is a square matrix whose rows and columns sums must balance as required by budget constraint condition. The conventions of double entry book-keeping guarantee that these will be no linkages or injections into the system and there is no room for any statistical discrepancy”, suggesting that every flow must go to some actor. Hence a SAM is a synthesis of two well known ideas in economics, namely the input output table and the national income accounting. SAM requires a balance of in the accounts of every factor in the economy, such that the income from say, sales in the agricultural sector must equal to its total expenditure on intermediate inputs, labour, Imports and capital services.

Applications of CGE model to economic analysis since the invention of CGE model in 1960 it has been used as a powerful technique for quantitative analysis in economic terms, these includes analyzing the effects on, for instance industries, region the labour market, income and welfare induced by changes in e.g. government policies including taxes trade restrictions, and/or technology. More recently also dynamic CGE models have been put forth, enabling researchers to address also multi period (dynamic) problems by means of numerical general equilibrium analysis. (Rumler 1999). Other areas where CGE models are applied include income distribution, sectoral manpower planning and resources/environmental economics.   

Application of CGE to Environmental Economics

In order to apply CGE model to environmental resources, Bojo etal (1992) presented a modified version of a schematic social accounting matrix the matrix is shown below

Table 1: A Hypothetical Sam with Environmental Account

	Institutions	Factors	Production	Savings investment	Rest of the world	Environment
Institution		NNP				Rent
Factors			NNP
Production	C			GI	X-M
Savings investment	S		D
Rest of the World				X – M
Environment	-ED		Env U	Est

Source: Bojo et al (1992)

In table 3.1, Net National Product (NNP) is the net value added production, net of the depreciation of real capital. The total income consists of NNP plus rent on environmental resources. This income is equal to purchase of consumption goods (C) plus savings (S) minus the damage to households from a degraded environment, (ED). In the product account, it is assumed that there is an imputed value (Env U) on the companies use of environmental resourcwes, including the harvest of forest resources Savings (S) plus the depreciation (D) equal;s gross investment (GI) in real capital plus the foreign net investment (X - M) plus the value of the change in the stock of environmental resources (Est). it is important to note that in this accounting framework, the sum in the first column, that is, (C + S - ED) gives the net “green” national product where environmental damage and changes in environmental stocks have been netted out, this represents the aggregated measure of human well being which is often differentiated from the System of National Accounts SNA by referring to the sum as the Net Welfare Measure (NWM). The sum is equal to the Net National Product (NNP) plus environmental rent.

Alabi in 1998 tried to apply the technique of CGE model to deforestation. Problem in Nigeria and used a static CGE model of an open economy, although the model have certain implicit dynamic features like inclusion of discount rate to account for future valuation of forested land. The model has two types of sectors, the tradable-producing sectors namely forest, agriculture and industry and the non-tradable-producing sectors which includes infrastructure and service. In addition to this there are two other sectors that clear land namely logging and squatting sectors. The model was solved using a special package called the general algebraic modeling system (GAMS) it was developed by team researchers at World Bank.

Criticism of Computable General Equilibrium Models

One of the general Criticism leveled against CGE models are “black boxes” that can not be understood by unintended. While CGE models and the techniques needed to solve them could easily be quite intimidating when first introduced, methodological and computational advances makes this untrue today. Also the development of packaged software to solve complex mathematical or statistical problems has permitted modeless to return their attention to economics.

Conclusion

Computable general equilibrium models are very crucial to the understanding of intersectoral linkages within an economy, they are employed to address, multi-regional and multiperiod problems. They are capable of directly capturing inter industry inter action suggesting forward and backward linkages arising from the provision and purchase of intermediate inputs. The models have become a dominant methodology in the economic analysis of developing countries and some developed economies.

References

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Keuninh S. and Ruijter W. (1998) “Guideline for the construction of social accounting matrix” Review of income and wealth series 34 No. 1

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