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:

$$\mathcal{E} = \left{ (u^i, \omega^i)_{i=1}^{I} \right}$$

where:

$u^i: \mathbb{R}^L_+ \to \mathbb{R}$ is consumer $i$'s utility function
$\omega^i \in \mathbb{R}^L_+$ is consumer $i$'s endowment vector
$L$ is the number of goods
$I$ is the number of consumers

Consumer's Problem: Given prices $p \in \mathbb{R}^L_{++}$, consumer $i$ solves:

$$\max_{x^i \in \mathbb{R}^L_+} u^i(x^i) \quad \text{s.t.} \quad p \cdot x^i \leq p \cdot \omega^i$$

The solution yields the Marshallian demand $x^i(p, p \cdot \omega^i)$.

Step 3: Define Walrasian Equilibrium

Definition (Walrasian Equilibrium): An allocation $(x^{*1}, \ldots, x^{I})$ and price vector $p^ \in \mathbb{R}^L_{++}$ constitute a Walrasian equilibrium if:

Utility Maximization: For each $i$, $x^{i}$ solves consumer $i$'s problem at prices $p^$
Market Clearing: $\sum_{i=1}^{I} x^{*i} = \sum_{i=1}^{I} \omega^i$

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:

julia

# ============================================
# 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

Step 6: Provide Complete Example

julia

# ============================================
# Example: 2×2 Pure Exchange Economy
# ============================================

# Two consumers, two goods
# Consumer 1: u(x,y) = x^0.6 * y^0.4, endowment (4, 1)
# Consumer 2: u(x,y) = x^0.3 * y^0.7, endowment (1, 4)

economy = PureExchangeEconomy(
    2,                           # 2 goods
    2,                           # 2 consumers
    [4.0 1.0; 1.0 4.0],         # Endowment matrix
    [[0.6, 0.4], [0.3, 0.7]],   # Cobb-Douglas parameters
    :cobb_douglas
)

# Solve for equilibrium
p_star = solve_equilibrium(economy)
println("Equilibrium prices: p = ", p_star)

# Compute allocations
x_star = equilibrium_allocations(p_star, economy)
println("Consumer 1 allocation: ", x_star[1, :])
println("Consumer 2 allocation: ", x_star[2, :])

# Verify market clearing
total_endowment = sum(economy.endowments, dims=1)
total_allocation = sum(x_star, dims=1)
println("Market clearing check: ", isapprox(total_endowment, total_allocation))

# Check Pareto efficiency (MRS equality)
mrs = check_pareto_efficiency(x_star, economy)
println("MRS values (should be equal): ", mrs)

Step 7: Visualize with Edgeworth Box

julia

# ============================================
# Edgeworth Box Visualization
# ============================================

function plot_edgeworth_box(economy::PureExchangeEconomy, p_star, x_star)
    # Total endowment defines box dimensions
    ω_total = vec(sum(economy.endowments, dims=1))
    
    # Create plot
    plt = plot(
        xlim=(0, ω_total[1]),
        ylim=(0, ω_total[2]),
        xlabel="Good 1",
        ylabel="Good 2",
        title="Edgeworth Box",
        legend=:topright,
        aspect_ratio=:equal
    )
    
    # Plot endowment point
    ω1 = economy.endowments[1, :]
    scatter!([ω1[1]], [ω1[2]], label="Endowment", markersize=8, color=:red)
    
    # Plot equilibrium allocation
    scatter!([x_star[1, 1]], [x_star[1, 2]], label="Equilibrium", markersize=8, color=:green)
    
    # Plot budget line through endowment
    # p_1 * x_1 + p_2 * x_2 = p_1 * ω_1 + p_2 * ω_2
    wealth1 = dot(p_star, ω1)
    x1_range = range(0, ω_total[1], length=100)
    x2_budget = (wealth1 .- p_star[1] .* x1_range) ./ p_star[2]
    plot!(x1_range, x2_budget, label="Budget line", color=:blue, linewidth=2)
    
    # Plot contract curve (locus of Pareto efficient allocations)
    # For Cobb-Douglas, contract curve: x_2^1 / x_1^1 = (α_2^1/α_1^1) / (α_2^2/α_1^2) * (ω_2 - x_2^1) / (ω_1 - x_1^1)
    
    return plt
end

# Generate the plot
plt = plot_edgeworth_box(economy, p_star, x_star)
savefig(plt, "edgeworth_box.png")

Example Prompts

Users might invoke this skill with prompts like:

"Set up a 2-good, 3-consumer pure exchange economy with CES preferences"
"Derive the Walrasian equilibrium conditions for a Cobb-Douglas economy"
"Write Julia code to solve for equilibrium prices in my exchange economy"
"Prove the First Welfare Theorem for a pure exchange economy"
"Plot an Edgeworth box showing the contract curve and equilibrium"
"Compute comparative statics: how does equilibrium change if endowments shift?"

Requirements

Software

Julia 1.9+

Packages

julia

using Pkg
Pkg.add(["NLsolve", "LinearAlgebra", "Plots", "ForwardDiff"])

Package	Purpose
`NLsolve`	Nonlinear equation solver for excess demand = 0
`LinearAlgebra`	Vector/matrix operations
`Plots`	Visualization (Edgeworth box, etc.)
`ForwardDiff`	Automatic differentiation for Jacobians

Mathematical Background

Assumptions for Existence

Standard assumptions ensuring equilibrium existence:

Continuity: Each $u^i$ is continuous
Strict Monotonicity: $x \gg y \Rightarrow u^i(x) > u^i(y)$
Strict Convexity: $u^i$ is strictly quasiconcave
Positive Endowments: $\omega^i \gg 0$ for all $i$

Properties of Excess Demand

Under standard assumptions, $z(p)$ satisfies:

Continuity: $z$ is continuous
Homogeneity of degree 0: $z(\lambda p) = z(p)$ for all $\lambda > 0$
Walras' Law: $p \cdot z(p) = 0$
Boundary behavior: If $p_l \to 0$, then $z_l(p) \to +\infty$

Numerical Solution Strategy

Normalize prices: Set $p_1 = 1$ (numeraire)
Reduce dimension: Solve $z_2(p) = \cdots = z_L(p) = 0$ (Walras' Law gives $z_1 = 0$)
Use Newton's method:
```
NLsolve.jl
```
with autodiff for Jacobian
Handle boundaries: Ensure $p_l > 0$ during iteration

Best Practices

Always verify market clearing after solving
Check Walras' Law holds numerically ($p \cdot z \approx 0$)
Verify Pareto efficiency by checking MRS equality across consumers
Use multiple initial guesses if solver doesn't converge
Normalize prices to avoid indeterminacy (homogeneity of degree 0)

Common Pitfalls

❌ Forgetting that prices are only determined up to a scalar (must normalize)
❌ Not checking for corner solutions (zero consumption of some good)
❌ Ignoring numerical precision issues near boundaries
❌ Assuming uniqueness without verifying (multiple equilibria are possible)
❌ Confusing Marshallian (uncompensated) and Hicksian (compensated) demands

Extensions (Future Versions)

Production economies: Firms with profit maximization
Arrow-Debreu securities: Contingent claims and uncertainty
Overlapping generations (OLG): Dynamic GE with generational overlap
Computable GE (CGE): Calibrated models for policy analysis
Incomplete markets: When not all contingencies can be traded

References

Textbooks

Mas-Colell, Whinston, and Green (1995). Microeconomic Theory. Oxford University Press. Chapters 15-17.
Debreu, G. (1959). Theory of Value. Yale University Press.
Varian, H. (1992). Microeconomic Analysis. 3rd Edition. Chapters 17-18.

Computational Resources

QuantEcon Julia lectures: https://julia.quantecon.org/
Judd, K. (1998). Numerical Methods in Economics. MIT Press.

Key Papers

Arrow, K. J., & Debreu, G. (1954). Existence of an equilibrium for a competitive economy. Econometrica, 22(3), 265-290.
Scarf, H. (1967). The approximation of fixed points of a continuous mapping. SIAM Journal on Applied Mathematics, 15(5), 1328-1343.

Changelog

v1.0.0

Initial release: Pure exchange economies with Cobb-Douglas preferences
Julia implementation with NLsolve
Edgeworth box visualization
Theoretical framework and welfare theorems

general-equilibrium-model-builder

NPX Install

Tags

SKILL.md Content

General Equilibrium Model Builder

Purpose

When to Use

Instructions

Step 1: Understand the Economic Environment

Step 2: Set Up the Theoretical Framework

Step 3: Define Walrasian Equilibrium

Step 4: State Key Theoretical Results

Step 5: Generate Julia Code for Computation

Step 6: Provide Complete Example

Step 7: Visualize with Edgeworth Box

Example Prompts

Requirements

Software

Packages

Mathematical Background

Assumptions for Existence

Properties of Excess Demand

Numerical Solution Strategy

Best Practices

Common Pitfalls

Extensions (Future Versions)

References

Textbooks

Computational Resources

Key Papers

Changelog

v1.0.0