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SKILL.md
General Equilibrium Model Builder
Purpose
This skill helps economists build, analyze, and numerically solve Walrasian General Equilibrium (GE) models. It covers both the theoretical foundations (existence, uniqueness, welfare theorems) and computational implementation in Julia for finding equilibrium prices and allocations.
Current Scope: Pure exchange economies (no production). Future versions will extend to production economies, Arrow-Debreu with uncertainty, and dynamic models.
When to Use
Theory Development: Formalizing a pure exchange GE model for a paper
Teaching: Creating examples for microeconomic theory courses
Computation: Numerically solving for equilibrium prices and allocations
Welfare Analysis: Evaluating Pareto efficiency and social welfare
Comparative Statics: Analyzing how equilibrium changes with parameters
Instructions
Step 1: Understand the Economic Environment
Before generating any model, ask the user:
Number of goods ($L$): How many commodities in the economy?
Number of consumers ($I$): How many agents?
Preferences: What utility functions? (Cobb-Douglas, CES, Leontief, quasilinear)
Endowments: What is each agent's initial endowment vector?
Output format: Theory derivation, Julia code, or both?
Step 2: Set Up the Theoretical Framework
A pure exchange economy $\mathcal{E}$ is characterized by:
Equivalently, the excess demand function $z(p) = \sum_{i=1}^{I} [x^i(p) - \omega^i]$ satisfies $z(p^*) = 0$.
Step 4: State Key Theoretical Results
Include the following theorems as appropriate:
Theorem (Walras' Law): For any price vector $p$:
$$p \cdot z(p) = 0$$
Interpretation: The value of excess demand is always zero (budget constraints bind).
Theorem (First Welfare Theorem): Every Walrasian equilibrium allocation is Pareto efficient.
Theorem (Second Welfare Theorem): Under convexity assumptions, any Pareto efficient allocation can be supported as a Walrasian equilibrium with appropriate lump-sum transfers.
Theorem (Existence - Debreu, 1959): Under standard assumptions (continuity, strict convexity, strict monotonicity of preferences, strictly positive endowments), a Walrasian equilibrium exists.
Step 5: Generate Julia Code for Computation
Use Julia with the following structure:
# ============================================
# General Equilibrium Solver in Julia
# Pure Exchange Economy
# ============================================
using LinearAlgebra
using NLsolve
using Plots
# Define the economy structure
struct PureExchangeEconomy
n_goods::Int # Number of goods (L)
n_consumers::Int # Number of consumers (I)
endowments::Matrix{Float64} # I × L matrix of endowments
utility_params::Vector{Any} # Parameters for utility functions
utility_type::Symbol # :cobb_douglas, :ces, :leontief
end
# Cobb-Douglas utility: u(x) = ∏ x_l^α_l
function utility_cobb_douglas(x, α)
return prod(x .^ α)
end
# Marshallian demand for Cobb-Douglas preferences
function demand_cobb_douglas(p, wealth, α)
# x_l = (α_l / sum(α)) * (wealth / p_l)
α_normalized = α / sum(α)
return α_normalized .* wealth ./ p
end
# Excess demand function
function excess_demand(p, economy::PureExchangeEconomy)
z = zeros(economy.n_goods)
for i in 1:economy.n_consumers
ω_i = economy.endowments[i, :]
wealth_i = dot(p, ω_i)
if economy.utility_type == :cobb_douglas
α_i = economy.utility_params[i]
x_i = demand_cobb_douglas(p, wealth_i, α_i)
else
error("Unsupported utility_type: $(economy.utility_type). Only :cobb_douglas is currently implemented.")
end
z += x_i - ω_i
end
return z
end
# Solve for equilibrium prices (normalize p_1 = 1)
# Uses log-price parameterization to ensure prices remain strictly positive
function solve_equilibrium(economy::PureExchangeEconomy)
# Initial guess in log-space (log of ones = zeros)
p0 = zeros(economy.n_goods - 1)
# Excess demand for goods 2 to L (Walras' Law implies good 1 clears)
# Reparameterize using log-prices: x = log(p_rest), so p_rest = exp(x)
function excess_demand_reduced!(F, x)
p_rest = exp.(x) # Exponentiate to get positive prices
p = vcat(1.0, p_rest) # Numeraire p_1 = 1
z = excess_demand(p, economy)
F .= z[2:end]
end
# Solve z(p) = 0 in log-space
result = nlsolve(excess_demand_reduced!, p0, autodiff=:forward)
if converged(result)
p_rest_star = exp.(result.zero) # Convert back from log-space
p_star = vcat(1.0, p_rest_star)
return p_star
else
error("Equilibrium solver did not converge")
end
end
# Compute equilibrium allocations
function equilibrium_allocations(p_star, economy::PureExchangeEconomy)
allocations = zeros(economy.n_consumers, economy.n_goods)
for i in 1:economy.n_consumers
ω_i = economy.endowments[i, :]
wealth_i = dot(p_star, ω_i)
if economy.utility_type == :cobb_douglas
α_i = economy.utility_params[i]
allocations[i, :] = demand_cobb_douglas(p_star, wealth_i, α_i)
else
throw(ArgumentError("Unsupported utility_type: $(economy.utility_type) in equilibrium_allocations. Only :cobb_douglas is currently implemented."))
end
end
return allocations
end
# Check Pareto efficiency via MRS equality
function check_pareto_efficiency(allocations, economy::PureExchangeEconomy)
# Currently only supports 2-good economies
if economy.n_goods != 2
throw(ArgumentError("check_pareto_efficiency currently only supports 2-good economies. Got economy.n_goods = $(economy.n_goods)."))
end
if economy.utility_type == :cobb_douglas
# MRS_{12} = (α_1/α_2) * (x_2/x_1) should be equal for all consumers
epsilon = 1e-12 # Small threshold for near-zero detection
mrs_values = []
for i in 1:economy.n_consumers
α_i = economy.utility_params[i]
x_i = allocations[i, :]
# Guard against division by zero: check both x_i[1] and α_i[2]
if abs(x_i[1]) < epsilon || abs(α_i[2]) < epsilon
# Handle corner case: set sentinel value for zero/near-zero consumption
push!(mrs_values, Inf)
else
mrs_i = (α_i[1] / α_i[2]) * (x_i[2] / x_i[1])
push!(mrs_values, mrs_i)
end
end
return mrs_values
else
throw(ArgumentError("Unsupported utility type: $(economy.utility_type) in check_pareto_efficiency. Only :cobb_douglas is currently implemented."))
end
end
