general-equilibrium-model-builder（一般均衡模型构建器）

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:

1. Number of goods ($L$): How many commodities in the economy?
2. Number of consumers ($I$): How many agents?
3. Preferences: What utility functions? (Cobb-Douglas, CES, Leontief, quasilinear)
4. Endowments: What is each agent's initial endowment vector?
5. 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:

1. Utility Maximization: For each $i$, $x^{*i}$ solves consumer $i$'s problem at prices $p^*$
2. 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:

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

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

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

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

Mathematical Background

Assumptions for Existence

Standard assumptions ensuring equilibrium existence:

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

Properties of Excess Demand

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

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

Numerical Solution Strategy

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

Best Practices

1. Always verify market clearing after solving
2. Check Walras' Law holds numerically ($p \cdot z \approx 0$)
3. Verify Pareto efficiency by checking MRS equality across consumers
4. Use multiple initial guesses if solver doesn't converge
5. 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