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Introduction to Computable General Equilibrium Model (CGE) Dhazn Gillig & Bruce A. McCarl Department of Agricultural Economics Texas A&M University 1 Course Outline Overview of CGE An Introduction to the Structure of CGE An Introduction to GAMS Casting CGE models into GAMS Data for CGE Models & Calibration Incorporating a trade & a basic CGE application Evaluating impacts of policy changes and casting nested functions & a trade in GAMS Mixed Complementary Problems (MCP) 2 This Week’s Road Map Fundamental relationships in a simple CGE model Incorporating taxes Including demand for products and factors Interpretation of results Comparative analysis 3 Fundamental Structure CGE models typically involve determination of economy wide levels of 1. commodity prices based on consumer demand and production possibilities/costs 2. factor prices based on supply and production possibilities/product prices 3. factor usage 4. production levels 5. income to households and resultant demand 6. Government balance 4 Fundamental Structure Basic relationships for a simple CGE 1. Supply–Demand identities for factors & products 2. Zero profit condition for producing Industries 3. Factor demand by producers 4. Product demand by households 5. Income balance constraint for households 6. Government balance 7. Trade balance closed economy 2x2x2 5 Fundamental Structure Basic characteristics of CGE model solutions: A set of non-zero prices, consumption levels, production levels, and factor usages constitutes an economic equilibrium solution (also called a Walrasian equilibrium) and a solution to a CGE of the situation if 1. Total market demand equals total market supply for each and every factor and output. 2. Prices are set so that equilibrium profits of firms are zero with all rents accruing to factors. 3. Household income equals household expenditures. 4. Government transfer payments to consumers equals taxes revenues. 6 Edgeworth Box Pure Exchange Economy Q2 IB IA Q2 A OA OB ω Q1 Q1 A 7 Edgeworth Box Pure Exchange Economy Q1B Q2 IB IA Z Q2A Q2B Q2 A ω OA Max U A s.t P1Q1A + P2Q2A ≤ P1Q1A + P2Q2 A ∂L ∂Q1A ∂U A ∂Q1A OB ∂U A ∂L ∂U A => − λP1 = 0 and => − λP2 = 0 ∂Q1A ∂Q2A ∂Q2A ∂U A P1 = / ∂Q2A P2 Q1 Q1A Q1 A MRSA = MRSB = P1/P2 Max U B s.t P1Q1B + P2Q2B ≤ P1Q1B + P2Q2 B ∂L ∂Q1B ∂U B ∂Q1B ∂U B ∂L ∂U B − λP = => − λP2 = 0 and 0 1 ∂Q1B ∂Q2B ∂Q2B ∂U B P1 = / ∂Q2B P2 8 => Edgeworth Box Pure Exchange Economy Implications: Q1 A + Q1B = Q1 A + Q1B = Q1 Q2 A + Q2 B − Q2 A + Q2 B = Q2 Feasible allocation (Q1 A − Q1 A ) + (Q1B − Q1B ) = 0 (Q2 A − Q2 A ) + (Q2 B − Q2 B ) = 0 Z1 = (Q1 A − Q1 A ) + (Q1B − Q1B ) = 0 Excess demand Z 2 = (Q2 A − Q2 A ) + (Q2 B − Q2 B ) = 0 Walras’ Law P1Q1 A + P2Q2 A = P1Q1 A + P2Q2 A For any price vector P, PZ(P) = 0; i.e., the value of the excess demand, is identically zero, (Varian, page 317) P1Q1B + P2Q2 B = P1Q1B + P2Q2 B P1 (Q1 A − Q1 A ) + P2 (Q2 A − Q2 A ) = 0 P1 (Q1B − Q1B ) + P2 (Q2 B − Q2 B ) = 0 Budget constraints P1 (Q1 A − Q1 A ) + P2 (Q2 A − Q2 A ) + P1 (Q1B − Q1B ) + P2 (Q2 B − Q2 B ) = 0 P1 Z 1 + P2 Z 2 = 0 Walras’ Law 9 Fundamental Structure Basic characteristics of CGE model solutions: A set of non-zero prices, consumption levels, production levels, and factor usages constitutes an economic equilibrium solution (also called a Walrasian equilibrium) and a solution to a CGE of the situation if 1. Total market demand equals total market supply for each and every factor and output. 2. Prices are set so that equilibrium profits of firms are zero with all rents accruing to factors. 3. Household incomes equal household expenditures. 4. Government transfer payments to consumers + consumptions equal taxes revenues. 10 Fundamental Structure Basic relationships for a simple CGE 1. Supply–Demand identities for factors & products 2. Zero profit condition for producing Industries 3. Factor demand by producers 4. Product demand by households 5. Income balance constraint for households 6. Government balance 11 Fundamental Structure Basic notations: Sectors denoted by j Households denoted by h Factors denoted by L and K and are owned by households Qj is the production of goods by the jth sector Pj is the price of goods produced by the jth sector aj1,j is the amount of goods used from sector j1 when producing one unit of goods in the jth sector Lj is the usage of labor by the jth sector WL is the price of labor (the wage rate) Kj is the usage of capital in the jth sector WK is the price of capital 12 Fundamental Structure 1. Supply-Demand identities for factors & products a. Factor market: Total demand is less than or equal to total supply in every factor market or the excess demand in the factor market is less than or equal to zero. ∑ L j − ∑ Lh ≤ 0 j h ∑Kj −∑Kh ≤ 0 j h Total supply is the sum across the household endowments 13 Fundamental Structure 1. Supply-Demand identities (con’t) b. Product or output market: Total demand in every output market including consumer and intermediate production usage is less than or equal to total supply in that market or the excess demand in each output market is less than or equal to zero. ∑ X jh h + ∑ a j,j1Q j1 Consumer consumption (ff) − Qj ≤ 0 ∀j j1 Intermediate usage Total production (ff) 14 Fundamental Structure 2. Zero profits in each sector Pj Q j = ∑ Pj1 a j1, j Q j + WL L j + WK K j ∀j j1 Revenues Costs Note that: CRS + Perfect Competition Unit price = unit cost 15 Fundamental Structure 3. Household income identity This relationship implies that household exhausts its income. Incomeh ≥ WL L h + WK K h Note that: Household consumptions (Xj) are a function of price and income. Maximize U(Xj) s.t. ∑ Pj X j ≤ WL L + WK K = Income j 16 Complementarity Recall: Walras’ Law: For any price vector P, PZ(P) = 0; i.e., the value of the excess demand, is identically zero, (Varian, page 317) Namely, if total demand is less than total supply for the factor/commodity markets then the price in that market must be zero; otherwise, prices will be nonzero only if supply equals demand This implies PRICE OR EXCESS DEMAND = 0. This leads to the following complementary relationships. 17 Complementarity (con’t) In a CGE model, a set of prices P and quantities Q are defined as variables such that D = S (Walras’ Law) (Qs-Qd)P = 0 (Pd-P)Qd (Ps-P)Qs = 0 = 0 Implications: Each equation must be binding or an associated complementary variable must be zero. IF P > 0 then Qs = Qd IF Qd > 0 then Pd = P, IF Qs > 0 then Ps = P, and This is similar to KT conditions of the following optimization model. 18 Complementarity (con’t) Max 6Qd-0.15Qd 2 − Qs − 0.1Qs 2 s.t Qd − Qs ≤ 0 Qd , Qs ≥ 0 Pd = 6 - 0.3*Qd Ps = 1 + 0.2*Qs + where P is the dual variable associated with the first constraint. 19 Complementarity (con’t) 1. 0 ≤ WL ⊥ ∑ L j − ∑ Lh ≤ 0 0 ≤ WK ⊥ ∑K j − ∑Kh ≤ 0 j j representing complementary relationship h h Factor prices must be zero if factors are not all used up. Non zero prices exist if factors all are consumed. 2. 0 ≤ Pj ⊥ ∑ X jh + h ∑ a j,j1Q j1 − Qj ≤ 0 ∀j j1 Product prices must be zero if products are not all consumed. Non zero prices exist if products all are consumed. 20 Complementarity (con’t) 3. 0 ≤ Q j ⊥ Pj Q j ≤ ∑ Pj1a j1, jQ j + WL L j + WK K j j1 Firm profits must equal zero and a non-zero production level is achieved. Firm profits can be less than costs without the firm producing. 4. 0 ≤ Incomeh ⊥ Incomeh ≥ WL Lh + WK K h Household incomes must be non-zero if expenditures exhaust incomes. 21 Complementarity (con’t) 5. Factor demand by producers identity (functional forms: CES, Cobb Douglas, Leontief?) 0 ≤ Lj ⊥ 0≤ Kj ⊥ δ jW K 1 Lj = Q j δ j + (1 − δ j ) (1 − δ )W φj j L (1 − δ )W 1 j L K j = Q j δ j φj δ jWK (1−σ j ) (1−σ j ) σ j /(1−σ j ) + (1 − δ j ) σ j /(1−σ j ) 22 Complementarity (con’t) 6. Product demand by households identity (functional forms: CES, Cobb Douglas, LES ?) 0 ≤ X jh ⊥ X jh = α j (Income h ) Pj σ ∑ (α (P ) ) 1−σ j j j 23 Incorporating Taxes Basic characteristics of CGE model solutions: A set of non-zero prices, consumption levels, production levels, and factor usages constitutes an economic equilibrium solution (also called a Walrasian equilibrium) and a solution to a CGE of the situation if 1. Total market demand equals total market supply for each and every factor and output. 2. Prices are set so that equilibrium profits of firms are zero with all rents accruing to factors. 3. Household income equals household expenditures. 4. Government transfer payments to consumers equal taxes revenues. 24 Incorporating Taxes (con’t) Fundamental structure 1. Supply-Demand identities 2. Zero profits in each sector 3. Household income-expenditure balance 4. Government tax revenue balance This relationship implies that the government satisfies its budget constraint when total revenue is positive. 25 Incorporating Taxes (con’t) Assumptions: 1. Total tax revenues are a. redistributed to households in the form of transfer payments (TRh) at the rate (sh) => TRh = sh R OR b. expended on government purchase of goods (GPj) 2. Government purchases are proportional to tax revenues at the rate (sj) => GPj = s j R ∑ sh + ∑ s j = 1 h j 3. th percent is imposed on household income tfj percent is imposed on factor f in sector j Fh is a total deduction imposed on household 26 Incorporating Taxes (con’t) Modification 1. The market balance : include government purchases transformed to be in a quantity unit ∑ X jh + ∑ a j, j1Q j1 − Q j + s j R / Pj ≤0 ∀j j1 h 2. The zero profit condition : include a tax effect as a cost of doing business Pj Q j ≤ ∑ Pj1 a j1, j Q j + 1 + tlj WL L j + 1 + tkjWK K j ∀j j1 27 Incorporating Taxes (con’t) 3. The household income equation : include tax effects to reflect tax incidence and tax revenue redistribution ( ) (1 − t h ) WL L h + WK K h − Fh + sh R ≤ Incomeh 4. Add the government tax revenue balance : apply the household income tax (th) to gross revenue less deductions, and the factor tax (tfj) R ≤ ∑ t h (W L Lh + W K K h − Fh ) h + tljW L L j + tkjW K K j 28 Incorporating Taxes (con’t) 5. Complementary 0≤ R ⊥ R ≤ ∑ t h (W L Lh + W K K h − Fh ) h + tljW L L j + t kjW K K j the government satisfies its budget constraint when total revenue is positive. 29 Including Demand for Products and Factors Specification of product and factor demand functions should be consistent with theory but analytically tractable. (1). Theoretical approach ¾ choosing functional forms that satisfy : classical restrictions i.e. nonnegative, continuous, and homogenous degree zero in price : Walras’s law => the value of market excess demands equals zero at all prices (2). Analytically tractable ¾ choosing functional forms that are easy to evaluate e.g. Leontief, CD, CES, LES, etc. 30 Including Demand for Products and Factors Factor demand derived from CES function : Production function σ j /(σ j −1) Q j = φ j (δ j L j (σ j −1) /σ j + (1 − δ j ) K j (σ j −1) /σ j ) : Factor demand δ jW K 1 Lj = Q j δ j + (1 − δ j ) (1 − δ )W φj j L (1 − δ )W 1 j L K j = Q j δ j φj δ jWK (1−σ j ) (1−σ j ) σ j /(1−σ j ) + (1 − δ j ) σ j /(1−σ j ) Note that: σ → 0 then CES tends to Leontief σ → 1 then CES tends to Cobb - Douglas 31 Including Demand for Products and factors Household product demand from CES Utility function Maximize Utility U = ∑ α j ( ) (X j ) (σ −1) / σ 1/σ σ /(σ −1) s.t ∑ Pj X j ≤ WL L + WK K ≡ Income j Yields Demand Curve X j = α P jσ (Income ) 1−σ ∑ (α j (P j ) j j ) 32 Numerical Example Symbol Brief Description σ h α jh Elasticity of substitution in household CES Lh, K h Household endowments of factors aj1,j φ j δ j σ j Use of goods in sector1 when producing in sector j Consumption share in household CES Scale parameter in CES production function Distribution parameter in CES production Elasticity of production factor substitution sh Household share of tax disbursements sj Government goods purchase dependence on revenues th Household tax level t fj Tax on factor f in sector j Fh Household tax exemptions 33 Parameter Specification Example of simple 2x2x2 CGE (Shoven and Whalley 1984) Production Parameters Sector (j) Food Non-Food φ j 1.5 2.0 δ j 0.6 0.7 tfj σ j 2.0 0.5 Labor Capital 0.0 0.0 0.0 0.0 Consumer Parameters Household (h) σ Farmer Non-Farmer h sh 0.75 0.6 1.5 0.4 Farmer Non-Farmer α hj Food Non-Food 0.3 0.7 0.5 0.5 Government Food Non-Food 0.0 0.0 th 0.00 0.00 Fh 0.0 0.0 Endowments Labor Capital 60 0 0 25 34 Equilibrium Results 1. Total demand for each output exactly matches the amount produced X jh = α j (Income h ) Pj σ ∑ (α (P ) ) 1−σ j j j σ j /(σ j −1) Q j = φ j (δ j L j (σ j −1) /σ j + (1 − δ j ) K j (σ j −1) /σ j ) Recall: ∑ X jh h + X ∑ a j,j1Q j1 j1 − Qj ≤ 0 ∀j 35 Equilibrium Results (con’t) 2. Producer revenues equal consumer expenditures ∑ Pj X jh h Pj Q j 36 Equilibrium Results (con’t) 3. Labor and capital are exhausted δ jW K 1 Lj = Q j δ j + (1 − δ j ) (1 − δ )W φj j L (1 − δ )W 1 j L K j = Q j δ j φj δ jWK (1−σ j ) (1−σ j ) σ j /(1−σ j ) + (1 − δ j ) σ j /(1−σ j ) 37 Equilibrium Results (con’t) 4. Unit cost = selling price => zero profits wl + rk From model solutions 38 Equilibrium Results (con’t) 5. Consumer factor incomes equal producer factor costs WL L h + WK K h WL = 1.00 WK = 1.37 WL L j + WK K j 39 Equilibrium Results (con’t) 6. Household expenditures exhaust their incomes ∑ Pj X j j WL L h + WK K h 40 Introducing Shocks (1). Imposing a 50% tax on a capital used in a food sector Production Parameters Sector (j) Food Non-Food φ j 1.5 2.0 δ j 0.6 0.7 σ tfj tj j 2.0 0.5 Labor Capital 0.0 0.5 0.0 0.0 0.0 0.0 δ j (1 + t kj )W K 1 Lj = Q j δ j + (1 − δ j ) (1 − δ j )W L φj (1 − δ )W 1 j L Kj = Q j δ j δ j (1 + t kj )WK φj (1−σ j ) (1−σ j ) σ j /(1−σ j ) + (1 − δ j ) σ j /(1−σ j ) 41 Introducing Shocks 1. The market balance ∑ X jh − Q j + s j R / Pj ≤ 0 h 2. The zero profit condition Pj Q j ≤ (1X + tlj ) WL L j + (1 + tkj ) WK K j 3. The household income equation (1 −Xti ) (WL L h + WK K h − Fh ) + sh R ≤ Incomeh 4. Add the government tax revenue constraint R = ∑Xth (WL Lh + WK K h − Fh ) + XtljWL L j + t kjW K K j h 42 Introducing Shocks (2). At equilibrium, transfer payments = tax revenues TRh = sh R R = ∑ Xt h (W L Lh + W K K h − Fh ) h tljW L L j + t kjW K K j +X 43 Taxrate("capital","Food") = 0.5 ; Comparative Analysis SOLVE CGEModel USING MCP ; Note that: Tax revenue under non-CGE excluding a tax including tax = 1.69 1.37 * 6.212 * 0.5 = $4.265 Factor Factor Tax level Cost Quantity VS. CGE tax revenue =$2.28 44 Comparative Analysis (con’t) 1. Unit cost = selling price => zero profits Relative priceP1/P2 = 1.40/1.09 = 1.28 (no tax) = 1.47/1.01 = 1.45 (with tax) 45 Comparative Analysis 2. Total demand for each output exactly matches the amount produced 46 Comparative Analysis (con’t) 3. Producer revenues equal consumer expenditures 47 Comparative Analysis (con’t) 4. Labor and capital are exhausted 48 Comparative Analysis (con’t) 5. Consumer factor incomes equal producer factor costs 49 Comparative Analysis (con’t) 6. Household expenditures exhaust their incomes 50 Wrap Up Overview of CGE Fundamental relationship of simple CGE model Interpretation of results Incorporating shocks (Taxes) Comparative analysis Next: Using GAMS for CGE Modeling What is GAMS? GAMS IDE Dissecting GAMS Formulation 51 References McCarl, B. A. and D. Gillig. “Notes on Formulating and Solving Computable General Equilibrium Models within GAMS.” Ferris, M. C. and J. S. Pang. “Engineering and Economic Applications of Complementarity Problems.” SIAM Review, 39:669-713, 1997. Shoven, J. B. and J. Whalley. “Applying general equilibrium.” Surveys of Economic Literature, Chapters 3 and 4, 1998. Shoven, J. B. and J. Whalley. “Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey.” J. Economic Literature, 22:1007-1051, 1984. 52