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

# Question

Find the Walrasian equilibrium relative price p = p1/p2 (with good 2 as numeraire, p2 = 1) and the equilibrium allocation for agents A and B with u_i = α_i ln x1 + (1−α_i) ln x2. Then check market clearing and Walras' law numerically, and validate the price with an independent numerical root-finder.

# Data and exclusions

• data/endowments.csv has 2 rows: A = (8, 2), B = (4, 8). data/parameters.csv has 4 rows: α_A = 0.4, α_B = 0.6, plus summary aggregates (n_input_rows). No rows were excluded, so 2 agents were analysed (n_analysed; see out/exclusions.csv).
• Data issue (data_issues): the summary sheet gives an aggregate endowment of good 1 of 14. The individual endowments sum to 8 + 4 = 12. I used 12 (aggregate_endowment_good1), because the individual endowments are the primary data. Market clearing also has to match the agents' budget constraints, which are built from those endowments. If I had used 14, the clearing condition would not match the budgets and Walras' law would fail. For good 2 the two sources agree at 10 (aggregate_endowment_good2). For reference, at the equilibrium price total demand for good 1 is 2 units below the summary figure of 14 (sensitivity_excess_demand_good1_vs_summary_E1).

# Method

Assumptions:

• 0 < α_i < 1 and all endowments strictly positive. Together these give unique, interior Marshallian demands x1_i = α_i w_i / p and x2_i = (1−α_i) w_i, with wealth w_i = p e1_i + e2_i.
• Prices are normalised with p2 = 1.

Setting aggregate excess demand for good 1 to zero gives the closed form p = Σ_i α_i e2_i / Σ_i (1−α_i) e1_i. Cobb–Douglas preferences satisfy gross substitutes, so this equilibrium is unique.

Independent validation: scipy.optimize.brentq finds the root of the excess demand of good 1, z1(p) = Σ_i x1_i(p) − E1. The function is built directly from the demand functions, and the closed form is not used. The search bracket is [1e-6, 1], expanded if needed. The validation checks this one economy only; it does not cover other parameter values.

No randomness is involved (random_seed = null).

# Results

• Equilibrium price p1/p2 = 0.875 (price_ratio_p1_over_p2). The numerical root is also 0.875 (price_ratio_numerical), and the two differ by 1.1e-16 (abs_diff_closed_vs_numerical).
• Wealth at equilibrium: A = 9.0 (wealth_A), B = 11.5 (wealth_B).
• Allocation:
  • A gets x1_A = 4.1143 (x1_A) and x2_A = 5.4 (x2_A).
  • B gets x1_B = 7.8857 (x1_B) and x2_B = 4.6 (x2_B).
• Aggregate excess demand for good 1 is 0.0 (excess_demand_good1). For good 2 it is also 0.0 (excess_demand_good2).
• The Walras' law residual p·z1 + z2 is 0.0 (walras_law_residual).
• Further checks are in out/state/checks.md: both budget constraints hold, and MRS_A = MRS_B = 0.875, so the allocation is Pareto efficient.
• Status: feasibility = "complete".

# Uncertainty

This is a deterministic theoretical exercise, so there is no statistical uncertainty. The only remaining error is floating-point rounding, of order 1e-16 (abs_diff_closed_vs_numerical). The main real uncertainty comes from the inputs: if the summary aggregate of 14 for good 1 were correct, the individual endowments would be wrong. The results depend on the individual endowments being correct.

# Limitations and open issues

• The two input files disagree on the aggregate endowment of good 1 (14 vs 12). I resolved this in favour of the individual endowments, but I could not confirm which source is right.
• The closed form holds only under the assumptions above: interior solution, strictly positive endowments, Cobb–Douglas preferences with α in (0,1), and p2 = 1.
• The numerical validation covers this single parameter set only.