An Applied Model: The CGE Mini Model

Abstract

In this chapter a CGE model (the CGE mini model) is presented. The model is simple enough to be presented in a few pages and yet complicated enough to demonstrate the application of the general CGE structure. In short, the focus of this chapter is to provide examples of structural adjustment in an open economy. The numerical applications of this chapter will be an examination of the sensitivity of the model to systematic variation in key variables of the adjustment process. Here we emphasise the effect of changes (government intervention) in the fixed rate of real exchange and growth in the capital stock.

Appendices

Appendix 1: The Mathematical Equations of the Model

5.1.1 Prices

5.1.1.1 Definition of Domestic Import Prices

$$ p_j^M = p_j^{WM } \times ER \times (1 + t{m_j} + p{r_j}) $$

(5.1)

$p_j^{WM }$ is the world market price of imports, ER is the real exchange rate, tm _j is the tariff rate on imports, and pr _j is the import premium rate. Note, that the world market price of imports $p_j^{WM }$ and the tariff rates are fixed. Depending on the exchange rate, the domestic import price $p_j^M$ is flexible or fixed.

5.1.1.2 Definition of Domestic Export Prices

$$ p_j^E = p_j^{WE} \times (1 + t{e_j})\times ER $$

(5.2)

$p_j^E$ is the domestic price of exports, $p_j^{WE }$ is the world market price of exports, te _j are the export duty rates, and ER is the real exchange rate. Note, the world market price of exports $p_j^{WE }$ and the duty rates are fixed. Depending on the exchange rate, the domestic export price $p_j^E$ is flexible or fixed.

5.1.1.3 Value of Domestic Sales

$$ {P_i}\times {x_i} = p_j^Z\times x_j^Z + p_j^M\times {M_j} $$

(5.3)

p _i is the price of composite commodities, x _i is the composite commodity supply, $p_j^Z$ is the domestic price, $x_j^Z$ are the domestic sales, $p_j^M$ is the domestic price of imports, and M _j is imports by sector.

5.1.1.4 Value of Domestic Output (Market Value)

$$ p_j^Z \times {Z_j} = p_j^Z \times x_j^Z + p_j^E \times {E_j} $$

(5.4)

$p_j^Z$ is the average output price by sector, Z _j is the domestic output by sector, $x_j^Z$ are domestic sales, $p_j^E$ is the domestic price of exports, and E _j is exports by sector.

5.1.1.5 Definition of Activity Prices

$$ p_j^Z\times \left( {1-ITA{X_j}} \right) = PV{A_j}+{\Sigma_j},{a_{ij }}\times {p_i} $$

(5.5)

$p_j^Z$ is the average output price by sector, ITAX _j is the indirect tax rate, PVA _j is the value added price by sector, a _ij are the input–output coefficients, and p _i is the price of composite commodities.

5.1.1.6 Definition of Capital Commodity Price

$$ p_j^{\mathrm{ K}}={\Sigma_i},{p_i}\times {c_{ij }} $$

(5.6)

$p_j^{\mathrm{ K}}$ is the rate of capital rent by sector, p _i is the price of composite commodities, and c _ij is the capital composition matrix.

5.1.1.7 Definition of General Price Level

$$ {p_{\mathrm{ index}}} = {\Sigma_i},pwt{s_i} \times {p_i} $$

(5.7)

p _index is the general price level, pwts _i are the CPI weights, and p _i is the price of the composite commodity.

5.1.2 Output and the Factors of Production

5.1.2.1 Production Function (Cobb-Douglas)

$$ {Z_j}=\mathrm{A}{{\mathrm{D}}_j}\,\,{\Pi_{lc }}{L_{j,lc }}\,{\alpha_{j,lc }}\,\,{K_j}^{{\left( {1-{\Sigma_{lc }},\,\,{\alpha_{j,lc }}} \right)}} $$

(5.8)

Z _j is the domestic output by sector, AD _j is the production function shift parameter, α _j,lc is the labour share parameter, L _j,lc is the employment by sector and labour category (lc), and K _j is the capital stock by sector.

5.1.2.2 First Order Condition for Profit Maximum

$$ P_{lc}^{\mathrm{ L}} \times {{\mathrm{ W}}_{\mathrm{ dist}}} \times {L_{j,lc }} = x_j^Z \times PV{A_j} \times {\alpha_{j,lc }} $$

(5.9)

$P_{lc}^{\mathrm{ L}}$ is the average wage rate by labour category (lc), W_dist are the wage proportionality factors, L _j,lc denote the employment by sector and labour category, and PVA _j is the value added price by sector.

5.1.2.3 Labour Market Equilibrium

$$ {\Sigma_j},{L_{j,lc }}\le\ {L_{lc }} $$

(5.10)

L _j,lc denote the employment by sector and labour category, and L _lc is the labour supply by labour category (lc).

5.1.2.4 CET Function: Exports (Domestic Output)

$$ {Z_j}=\mathrm{ A}{{\mathrm{ T}}_j}{{\left[ {{\gamma_j}E_j^{{{\varphi_j}}}+(1-{\gamma_j})x_j^{{Z{\varphi_j}}}} \right]}^{{1/{\varphi_j}}}} $$

(5.11)

Z _j is the domestic output by sector, AT _j is the CET function shift parameter, GAMMA is the CET function share parameter, E _j is exports by sector, ϕ _j is the CET function exponent, and $x_j^Z$ are the domestic sales. This function applies to commodities that are both sold domestically and exported. The equation above reflects the assumption of imperfect transformability between domestic sales and exports.

5.1.2.5 Export Supply

$$ \frac{{{E_j}}}{{x_j^Z}}=\frac{{p_j^E}}{{p_j^Z}}\times \frac{{1-{\gamma_j}}}{{{\gamma_j}}}{^{{\frac{1}{{{\varphi_j}-1}}}}} $$

(5.12)

$p_j^E$ is the domestic price of exports, and $p_j^Z$ is the domestic price.

5.1.2.6 CES Function: Composite Commodity Aggregation Function

$$ {x_i}=\mathrm{A}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{\mathrm{C}}}_j}{{\left[ {{\delta_j}\,M_j^{{-{\rho_j}}}+(1-{\delta_j}){x_j}^{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{Z}}-{\rho_j}}}} \right]}^{{-1/{\rho_j}}}} $$

(5.13)

x _i is the composite commodity supply, AC _j is the Armington function shift parameter, ${\delta_j}$ is the Armington function share parameter, M _j is imports, ρ _j is the Armington function exponent, and $x_j^Z$ are the domestic sales. This function applies to commodities that are both produced and sold domestically and imported, i.e., composite commodities. The equation above reflects the assumption of imperfect substitutability between imports and domestic produced commodities sold domestically.

5.1.2.7 Cost Minimisation of Composite Good

$$ \frac{{{M_j}}}{{x_j^Z}}=\frac{{p_j^Z}}{{p_j^M}} \times {{\frac{{{\delta_j}}}{{1-{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{\delta}}_j}}}}^{{\frac{1}{{1+{\rho_j}}}}}} $$

(5.14)

$p_j^Z$ is the domestic prices, and $p_j^M$ is the domestic price of imports.

5.1.2.8 Domestic Sales for Non-traded Sectors

A first step toward more realism has been taken by introducing non-tradable commodities. Non-tradable commodities are commodities that are not subject to international trade. In general, most service as well as housing and construction fit this category.

$$ x_j^Z = {Z_j} $$

(5.15)

$x_j^Z$ are the domestic sales, and Z _j is the domestic output by sector.

5.1.2.9 Composite Commodity Aggregation for Non-traded Sectors

$$ {x_i}=x_j^Z $$

(5.16)

x _i is the composite commodity supply, and $x_j^Z$ are domestic sales.

5.1.3 Demand

5.1.3.1 Total Intermediate Uses

$$ {x_{ij }} = {\Sigma_j},{a_{ij }}\times {Z_j} $$

(5.17)

x _ij are the intermediate uses, a _ij is the input–output coefficients, and Z _j is the domestic output by sector. The sector balances of intermediate inputs (inter-industry matrix) form the basis of the input–output table. The input–output matrix is derived from the inter-industry matrix, by dividing each element in a column by the row sum of the corresponding row. The Leontief matrix is obtained from the input–output matrix by subtracting it from an n by n identity matrix. This changes the sign of all off-diagonal elements and makes all diagonal elements into their complements to one. Theoretically, the input coefficients are in physical terms. Empirically, the coefficients are in monetary terms. As long as we assume that prices are constant, the input coefficients should be the same either in physical or monetary terms.

The transactions may be valued at either the price received by the producer, producer’s value, or at the price paid by the consumer, purchaser’s value. The difference between these values is that transport margins, net indirect commodity taxes, i.e., indirect taxes less subsides, and trade margins are added to the basic producer’s values in the national accounts. Since the demand components are computed at purchaser’s values, production and imports are converted to these values too.

5.1.3.2 Inventory Investment

$$ DS{T_j} = DST{R_j} \times {Z_j} $$

(5.18)

DST _j is inventory investment by sector, DSTR _j is the ratio of inventory investment to gross output, and Z _j is the domestic output by sector.

5.1.3.3 Private Consumption Behaviour

$$ {P_j}\times C{D_j} = {\Sigma_h},CLE{S_{j,h }} \times (1-MP{S_h})\times Y{H_h}\times (1-HTA{X_h}) $$

(5.19)

p _j are the price of composite commodities, CD _j is the final demand for private consumption, CLES _j,h are the private consumption shares, MPS _h is the marginal propensity to save by household type, YH _h is the total income by household type, and HTAX _h is the income tax rate by household type

5.1.3.4 Private GDP

$$ Y={\Sigma_h}\ Y{H_h} $$

(5.20)

Y is private GDP, YH _h is the total income by household type.

5.1.3.5 Total Income Accruing to Labour

$$ Y{H_h} = {\Sigma_{lc }},{P_{lc}}^{\mathrm{ L}} \times {L_{lc }} + REMIT\times ER $$

(5.21)

YH _h is the total income by household type, ${P_{lc}}^{\mathrm{ L}}$ is the average wage rate by labour category, L _lc is the labour supply by labour category, REMIT is the net remittances from abroad, and ER is the real exchange rate.

5.1.3.6 Total Income Accruing to Capital

$$ \begin{gathered} Y{H_h} = {\Sigma_j},\,\,PV{A_j}\times {Z_j}-DEPRECIA-{\Sigma_{lc }},{P_{lc}}^{\mathrm{L}} \times {L_{lc }}\ \hfill \\ \quad \quad \quad \quad \quad +FBOR\times ER+YPR \hfill \\ \end{gathered} $$

(5.22)

YH _h is the total income by household type, PVA _j is value added price by sector, Z _j is the domestic output by sector, DEPRECIA is total depreciation expenditure, ${P_{lc}}^{\mathrm{ L}}$ is the average wage rate by labour category, L _lc is the labour supply by labour category, FBOR is the net flow of foreign borrowing, ER is the real exchange rate, and YPR is total premium income accruing to capitalists.

5.1.4 Saving and Income

5.1.4.1 Household Savings

$$ HSAV={\Sigma_h},MP{S_h} \times Y{H_h} \times (1-HTA{X_h}) $$

(5.23)

HSAV are the total household savings, MPS _h is the marginal propensity to save by household type h, YH _h is the total income by household type, and HTAX _h is the income tax rate by household type.

5.1.4.2 Government Revenue

$$ GR=TARIFF-NETSUB+INDTAX+TOTHTAX $$

(5.24)

GR is the government revenue, TARIFF is the tariff revenue, NETSUB is the export duty revenue, INDTAX is the indirect tax revenue, TOTHTAX is the household tax revenue.

5.1.4.3 Government Savings

$$ GR={\Sigma_j},{p_j}\times G{D_j} + GOVSAV $$

(5.25)

GR is the government revenue, p _j are the price of composite commodities, GD _j is the final demand for government consumption, and GOVSAV are government savings. It is an essential assumption for a real equilibrium model that the government must balance its budget.

5.1.4.4 Government Consumption Shares

$$ G{D_j} = GLE{S_j} \times GDTOT $$

(5.26)

GD _j is the final demand for government consumption, GLES _j is the government consumption shares, and GDTOT is the total volume of government consumption.

5.1.4.5 Tariff Revenue

$$ TARIFF={\Sigma_j},T{M_j} \times {M_j} \times p_j^{WM } \times ER $$

(5.27)

TARIFF is the tariff revenue, TM _j are the tariff rates on imports, M _j are imports, $p_j^{WM }$ are world market price of imports, ER is the real exchange rate.

5.1.4.6 Indirect Taxes on Domestic Production

$$ INDTAX={\Sigma_j},ITA{X_j} \times p_j^Z \times {Z_j} $$

(5.28)

INDTAX is the indirect tax revenue, ITAX _j is the indirect tax rates, $p_j^Z$ is the average output price by sector, and Z _j is the domestic output by sector.

5.1.4.7 Export Duties

$$ NETSUB={\Sigma_j},t{e_j} \times {E_j} \times p_j^{WE } \times ER $$

(5.29)

NETSUB is export duty revenue, te _j are export duty rates, E _j are exports by sector, $p_j^{WE }$ is the world market price of exports, ER is the real exchange rate.

5.1.4.8 Total Import Premium Income

$$ YPR={\Sigma_j},p_j^{WM } \times {M_j} \times ER\times pr $$

(5.30)

YPR is the total premium income accruing to capitalists, $p_j^{WM }$ is the world market price of imports, M _j are imports, ER is the real exchange rate, and pr is the import premium.

5.1.4.9 Total Household Taxes Collected by Government

$$ TOTHTAX={\Sigma_h},HTA{X_h} \times Y{H_h} $$

(5.31)

TOTHTAX is the household tax revenue, HTAX _h is the income tax rate by household type h, YH _h is the total income by household type h.

5.1.5 Capital Formation

5.1.5.1 Depreciation Expenditure

$$ DEPRECIA={\Sigma_j},\ DEP{R_j} \times p_j^{\mathrm{ K}} \times {K_j} $$

(5.32)

DEPRECIA is the total depreciation expenditure, DEPR _j is the depreciation rate, K _j is the capital stock by sector, $p_j^{\mathrm{ K}}$ is the rate of domestic capital rent by sector, ER is the exchange rate. As the capital stock gets older, the quasi-rent in the Marshallian sense falls and eventually becomes zero. The economic decision is then taken to scrap the capital object as obsolete.

5.1.5.2 Total Savings

$$ SAVINGS=HSAV+GOVSAV+DEPRECIA+FSAV\times ER $$

(5.33)

SAVINGS are total savings, HSAV are total household savings, GOVSAV are government savings, DEPRECIA is total depreciation expenditure, FSAV are foreign savings. Thus, the sum of domestic and foreign savings in domestic currency.

5.1.5.3 Domestic Investment by Sector of Destination

In the CGE mini-model domestic investment by sector of destination is given by:

$$ p_j^{\mathrm{ K}} \times I_j^{\mathrm{ D}} = K{I^{\mathrm{ o}}}_j \times INVEST - K{I^{\mathrm{ o}}}_j \times {\Sigma_j},DS{T_j} \times {p_j} $$

(5.34)

Thus, $p_j^{\mathrm{ K}}$ is rate of capital rent by sector, $I_j^{\mathrm{ D}}$ is volume of investment by sector of destination, $K{I^{\mathrm{ o}}}_j$ are the shares of investment by sector of destination, INVEST is the total investment, DST _j is inventory investment by sector, p _j is the price of composite goods. The sector share parameters for investment are assumed fixed. Total investment is determined by savings in the economy (saving determined investment).

The sector capital stocks K _j are fixed within periods. However, they change over time given aggregate growth of the capital stock and the sector allocation of investment. Sector share parameters of investment by sector of destination $K{I^{\mathrm{ o}}}_j$ are assumed to be fixed. For information, the numerical values of the sector share parameters of investment are in these applications arbitrary assumed to be: 0.13 for agriculture, 0.29 for industry, and 0.58 for services. The sum is equal to one. However, the sector allocation of investment is here assumed to be adjusted over time (endogenously) to equate rental rates $p_j^{\mathrm{ K}}$ in the industrial sectors by the terminal year.

5.1.5.4 Investment by Sector of Origin

The request for the volume of investment by sector of destination $I_j^{\mathrm{ D}}$ (the sector capital accumulation) is translated into a demand for investment commodities by sector of origin $I_i^{\mathrm{ S}}$ (producing sectors of capital commodities), thus investment by sector of origin:

$$ I_i^{\mathrm{ S}} = {\Sigma_j},IMA{T_{ij }}\times I_j^{\mathrm{ D}} $$

(5.35)

$I_i^{\mathrm{ S}}$ is the final demand for productive investment, IMAT _IJ is the capital composition matrix, and $I_j^{\mathrm{ D}}$ is the volume of domestic investment by sector of destination. In accordance with the production structure, as represented by the input–output model, the investment by sector of origin $I_i^{\mathrm{ S}}$ is also known as final demand for productive investment. The summation of the capital composition matrix IMAT _IJ is, as the sector share parameters of investment, equal to one. Following this application, the two sectors producing capital commodities are industry (the dominating sector), and a small fraction from services.

5.1.5.5 Balance of Payments

$$ {\Sigma_j},\ p_j^{WM } \times {M_j} = {\Sigma_j},p_j^{WE } \times {E_j} + FSAV+REMIT+FBOR $$

(5.36)

$p_j^{WM }$ is the world market price of imports, M _j are imports, $p_j^{WE }$ is the world market price of exports, E _j are exports by sector, FSAV are foreign savings, REMIT are net remittances from abroad, and FBOR is the net flow of foreign borrowing. In the experiments in this book the exchange rate is fixed and the net flow of foreign borrowing is unfixed. Following this specification, the trade deficit is free to vary.

5.1.6 Market Equilibrium

5.1.6.1 Commodity Market Equilibrium

$$ {x_i} = {x_{ij }} + \mathrm{ C}{D_j} + G{D_j} + I_i^{\mathrm{ S}} + DS{T_j} $$

(5.37)

x _i are the composite commodity supply, x _ij are intermediates uses, CD _j is the final demand for private consumption, GD _j is the final demand for government consumption, $I_i^{\mathrm{ S}}$ is the final demand for productive investment, and DST _j is the inventory investment by sector.

5.1.6.2 Objective Function

$$ OMEGA={\Pi_j}\ C{D_j}{{}^{CLESj,h }} $$

(5.38)

OMEGA is the objective function variable, CLES _j,h is the private consumption shares, and CD _j is the final demand for private consumption.

For full specification of the numerical input in the original input version of the model, see the computer program of the CGE mini-model. The CGE mini-model is a minor version of an equilibrium model that originally comes from Chenery, Lewis, de Melo, and Robinson in their work on designing an equilibrium development model for Korea. The model illustrates the basic use of CGE models. See further: Chenery et al. (1986). The model is included in the GAMS model library (korcge.gms). The reader can reach the GAMS homepage at www.gams.com.

Appendix 2: Some Parameters Assignments of the Model

PARAMETER ASSIGNMENTS

$$ \mathbf{INCOME}\mathbf{TAX}\mathbf{RATE}\mathbf{BY}\mathbf{LABOUR}=\mathbf{0}.\mathbf{08910} $$

$$ \mathbf{INCOME}\mathbf{TAX}\mathbf{RATE}\mathbf{BY}\mathbf{CAPITALIST}=\mathbf{0}.\mathbf{08910} $$

LABOUR SHARE PARAMETER IN THE PRODUCTION FUNCTION

	LABOUR1	LABOUR2	LABOUR3
Agriculture	0.38258	0.06740	0.00000
Industry	0.00000	0.53476	0.00000
Services	0.00000	0.16234	0.42326

INPUT–OUTPUT COEFFICIENTS

	Agriculture	Industry	Services
Agriculture	0.12591	0.19834	0.01407
Industry	0.10353	0.35524	0.18954
Services	0.02358	0.11608	0.08390

CAPITAL COMPOSITION MATRIX

	Agriculture	Industry	Services
Agriculture	0.00000	0.00000	0.00000
Industry	0.93076	0.93774	0.93080
Services	0.06924	0.06226	0.06920

WAGE PROPORTIONALITY FACTORS

	LABOUR1	LABOUR2	LABOUR3
Agriculture	1.00000	0.52780	0.00000
Industry	0.00000	1.21879	0.00000
Services	0.00000	1.11541	1.00000

PRIVATE CONSUMPTION SHARES

	LAB-HH	CAP-HH
Agriculture	0.47000	0.47000
Industry	0.31999	0.31999
Services	0.21001	0.21001