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