The Standard CGE Model (stdcge)#
This document maps each equation in cge_core/models/standard/model.py to
Hosoe, Gasawa & Hashimoto (2010), Chapters 5–6, and to the GAMS Model Library
file stdcge.gms (SEQ=276).
The bundled example has goods \(i \in \{\mathrm{BRD},\mathrm{MLK}\}\) and factors \(h \in \{\mathrm{CAP},\mathrm{LAB}\}\).
Production and factors#
Code |
Equation |
Meaning |
|---|---|---|
|
\(Y_i=b_i\prod_h F_{h,i}^{\beta_{h,i}}\) |
Cobb-Douglas composite-factor production |
|
\(F_{h,i}=\dfrac{\beta_{h,i}p_i^yY_i}{p_h^f}\) |
Factor demand |
|
\(X_{i,j}=a^x_{i,j}Z_j\) |
Leontief intermediate demand |
|
\(Y_i=a_i^yZ_i\) |
Leontief composite-factor demand |
|
\(p_j^z=a_j^yp_j^y+\sum_i a^x_{i,j}p_i^q\) |
Unit cost / zero-profit price |
Taxes#
Code |
Equation |
Meaning |
|---|---|---|
|
\(T^d=\tau^d\sum_h p_h^fFF_h\) |
Direct-tax revenue |
|
\(T_i^z=\tau_i^zp_i^zZ_i\) |
Production-tax revenue |
|
\(T_i^m=\tau_i^mp_i^mM_i\) |
Import-tariff revenue |
Final demand and saving#
Code |
Equation |
Meaning |
|---|---|---|
|
\(X_i^p=\dfrac{\alpha_i}{p_i^q}\left(\sum_h p_h^fFF_h-S^p-T^d\right)\) |
Household demand |
|
\(X_i^g=\dfrac{\mu_i}{p_i^q}\left(T^d+\sum_jT_j^z+\sum_jT_j^m-S^g\right)\) |
Government demand |
|
\(X_i^v=\dfrac{\lambda_i}{p_i^q}\left(S^p+S^g+\varepsilon S^f\right)\) |
Investment demand |
|
\(S^p=ss^p\sum_h p_h^fFF_h\) |
Private saving |
|
\(S^g=ss^g\left(T^d+\sum_jT_j^z+\sum_jT_j^m\right)\) |
Government saving |
Trade: Armington, CET, and the rest of the world#
Code |
Equation |
Meaning |
|---|---|---|
|
\(Q_i=\gamma_i\left[\delta_i^mM_i^{\eta_i}+\delta_i^dD_i^{\eta_i}\right]^{1/\eta_i}\) |
Armington composite |
|
\(M_i=\left[\dfrac{\gamma_i^{\eta_i}\delta_i^mp_i^q}{(1+\tau_i^m)p_i^m}\right]^{1/(1-\eta_i)}Q_i\) |
Import demand |
|
\(D_i=\left[\dfrac{\gamma_i^{\eta_i}\delta_i^dp_i^q}{p_i^d}\right]^{1/(1-\eta_i)}Q_i\) |
Domestic demand for the Armington composite |
|
\(Z_i=\theta_i\left[\xi_i^eE_i^{\phi_i}+\xi_i^dD_i^{\phi_i}\right]^{1/\phi_i}\) |
CET transformation |
|
\(E_i=\left[\dfrac{\theta_i^{\phi_i}\xi_i^e(1+\tau_i^z)p_i^z}{p_i^e}\right]^{1/(1-\phi_i)}Z_i\) |
Export supply |
|
\(D_i=\left[\dfrac{\theta_i^{\phi_i}\xi_i^d(1+\tau_i^z)p_i^z}{p_i^d}\right]^{1/(1-\phi_i)}Z_i\) |
Domestic supply |
|
\(p_i^e=\varepsilon p_i^{We}\) |
Export price |
|
\(p_i^m=\varepsilon p_i^{Wm}\) |
Import price |
|
\(\sum_i p_i^{We}E_i+S^f=\sum_i p_i^{Wm}M_i\) |
Balance of payments |
Market clearing#
Code |
Equation |
Meaning |
|---|---|---|
|
\(Q_i=X_i^p+X_i^g+X_i^v+\sum_jX_{i,j}\) |
Composite-good market |
|
\(\sum_iF_{h,i}=FF_h\) |
Factor market |
Objective#
The model maximizes the Cobb-Douglas household-utility index
The household-demand equations already embody utility maximisation, so the constraint system pins down the equilibrium once the model is square. The objective gives the NLP solver a well-defined optimization problem and its solution value is the utility level used by the model’s welfare reporting.
Closure and degrees of freedom#
Counting the scalar expanded system:
variables: 48; one is fixed as the numeraire (
pf['LAB']), leaving 47 free;equality constraints: 48.
Therefore,
The raw system is over-determined by one equation because Walras’ law makes one market-clearing condition dependent on the others plus the agents’ budget constraints.
Dropping one admissible market-clearing equation with
cge.model_drop_redundant("eqpf", "LAB")
gives a square system with zero degrees of freedom. The dropped factor market still clears at the solution and is checked by the test suite.
The standard model declares eqpf and eqpqd as admissible redundant
market-clearing families. One scalar market-clearing condition is dropped in
a solve; arbitrary behavioural equations are not valid closure choices.
Calibration#
All behavioural parameters are recovered from the SAM so that the base instance reproduces the benchmark:
expenditure and factor shares: \(\alpha_i\), \(\beta_{h,i}\), \(\mu_i\), and \(\lambda_i\);
Leontief coefficients: \(a^x_{i,j}\) and \(a_i^y\);
CES/CET parameters: \(\delta_i^m\), \(\delta_i^d\), \(\gamma_i\), \(\xi_i^e\), \(\xi_i^d\), and \(\theta_i\);
substitution/transformation parameters derived from the fixed benchmark elasticities \(\sigma_i=\psi_i=2\);
tax and saving rates: \(\tau^d\), \(\tau_i^z\), \(\tau_i^m\), \(ss^p\), and \(ss^g\).
At the base equilibrium all prices equal 1. The bundled standard SAM yields
Numerical lower bounds#
The model reproduces the reference implementation’s explicit numerical lower bounds to keep divisions and fractional powers away from zero.
All positive stdcge quantities, prices, the exchange rate, private saving,
government saving, and direct tax use a lower bound of \(10^{-5}\).
Production-tax and import-tariff revenues (Tz, Tm) may be zero.
The simple model uses \(10^{-3}\) for its positive variables.