# F5-conf-01 — Competitive equilibrium of a 2×2 Cobb–Douglas exchange economy

# Question

Find the Walrasian equilibrium price p = p1/p2 (good 2 is the numeraire, p2 = 1) and allocation for two agents with u_i = α_i ln x1 + (1−α_i) ln x2. Check market clearing and Walras' law, and confirm the closed-form price with an independent numerical root-finder.

# Data and exclusions

• data/endowments.csv (2 rows): A = (8, 2), B = (4, 8).
• data/parameters.csv (4 rows): α_A = 0.4, α_B = 0.6, sheet aggregates good1 = 14, good2 = 10.
• Data issue: the summary sheet gives an aggregate endowment of good 1 of 14, but the individual endowments sum to 12. For good 2 the sheet value (10) matches the sum. I used the individual endowments, which TASK.md calls the primary data, so the aggregates are 12 and 10. In an exchange economy the aggregate endowment is by definition the sum of the individual endowments. If 14 were imposed, budget-balanced demands could not clear both markets. As a check, solving z1 = 0 with E1 = 14 gives p = 0.667, and at that price the good-2 market has an excess demand of −1.333. The sheet value of 14 was therefore not used. No rows were excluded, and both agents are analysed.

# Method

• Demands: with income w_i = p·e1_i + e2_i, x1_i = α_i w_i / p and x2_i = (1−α_i) w_i.
• Closed form: setting z1(p) = Σ_i x1_i − E1 = 0 gives p* = Σ_i α_i e2_i / Σ_i (1−α_i) e1_i = (0.4·2 + 0.6·8)/(0.6·8 + 0.4·4) = 5.6/6.4.
• Assumptions: strictly positive endowments (so every income is positive), α_i ∈ (0,1) (so the solution is interior and unique, with all consumptions > 0), and normalisation p2 = 1. The code checks all of these.
• Numerical validation: Brent's method (scipy.optimize.brentq) on the aggregate excess-demand function z1(p) itself, over the bracket [1e-6, 1] (the upper end is doubled until the sign changes). It does not evaluate the closed-form formula.
• Checks: z1 and z2 at p*, and the Walras residual p·z1 + z2.

# Results

The allocations add up to the aggregate endowments (12, 10). Each agent spends exactly their income: A's is 9 and B's is 11.5.

# Uncertainty

This is a deterministic theoretical exercise, so there is no sampling uncertainty. The only numerical error is floating-point error: the closed-form and numerical prices differ by about 1e-16, and the residuals are 0 to machine precision. No random seed is used.

# Limitations and open issues

• The results depend on resolving the good-1 aggregate inconsistency (14 vs 12) in favour of the individual endowments. If the per-agent endowments were the ones in error instead (for example, one agent's good-1 endowment should be 2 units higher), the equilibrium would be different. The data cannot tell which is right.
• The numerical validation only confirms the root of z1 inside the bracket. Uniqueness comes from the theory (Cobb–Douglas preferences give gross substitutes), not from a global numerical search.
• The closed form holds only for interior Cobb–Douglas solutions with positive endowments. Boundary cases are not covered.