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

# Question

Agents A and B have u_i = α_i ln x1 + (1−α_i) ln x2 and endowments (e1_i, e2_i). Good 2 is numeraire (p2 = 1). We find the Walrasian relative price p = p1/p2 and the equilibrium allocation, check market clearing and Walras' law, and confirm the closed-form price by solving z1(p) = 0 numerically.

# 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, plus summary-sheet aggregates (n_input_rows).
• Data issue (data_issues): the summary sheet gives aggregate_endowment_good1 = 14 (aggregate_endowment_good1_summary_sheet). The individual endowments add up to 8 + 4 = 12. For good 2 the sheet agrees with the rows (10). The task calls the individual endowments the primary data, and they also set each agent's wealth. So I took the aggregates as row sums: aggregate_endowment_good1 = 12 and aggregate_endowment_good2 = 10. Using 14 would contradict the budget constraints. At the equilibrium allocation it would show a spurious excess demand of −2 (excess_demand_good1_vs_summary_sheet_aggregate), and no price could clear that market.
• No rows were excluded. Both agents are analysed (n_analysed = 2; out/exclusions.csv). SHA-256 hashes are in out/state/data_audit.md.

# Method

• Demands (p2 = 1, wealth w_i = p e1_i + e2_i): x1_i = α_i w_i / p and x2_i = (1−α_i) w_i.
• Closed form. Market clearing for good 1 gives Σ α_i (p e1_i + e2_i) = p Σ e1_i. Solving: p* = Σ α_i e2_i / Σ (1−α_i) e1_i = (0.4·2 + 0.6·8)/(0.6·8 + 0.4·4) = 5.6/6.4.
• Assumptions:
  • Cobb–Douglas with 0 < α_i < 1, so demands are interior and strictly positive.
  • Endowments are strictly positive, so wealth is positive at any p > 0.
  • Price normalisation p2 = 1.
  • Agents are price-takers.
  • With these preferences the equilibrium is unique, because z1(p) is strictly decreasing in p.
• Independent numerical validation: Brent's method (scipy.optimize.brentq) on the aggregate excess demand z1(p) built from the individual demand functions, over the bracket [1e−6, 1e6]. The sign change was checked. The closed-form expression is not used in this step.
• Diagnostics:
  • Walras' law, p·z1 + z2, evaluated at p* and also on a 41-point logarithmic grid of prices from 0.01 to 100.
  • Budget identities.
• The method is deterministic and uses no randomness (random_seed = null).

# Results

• Closed-form price: price_ratio_p1_over_p2 = 0.875. Numerical root: price_ratio_numerical = 0.875. The absolute difference is 1.1e−16 (price_abs_diff_closed_vs_numerical).
• Allocation:
  • Agent A: x1_A ≈ 4.1143 (= 144/35), x2_A = 5.4.
  • Agent B: x1_B ≈ 7.8857 (= 276/35), x2_B = 4.6.
  • Wealth: wealth_A = 9 and wealth_B = 11.5.
• Market clearing: excess_demand_good1 = 0 (and excess_demand_good2 = 0). The allocations add up to the aggregates 12 and 10.
• Walras' law: walras_law_residual = 0 at p*. The largest absolute residual on the price grid is 5.7e−14 (walras_law_residual_max_abs_on_price_grid), consistent with an identity up to floating-point rounding.
• feasibility = "complete".

# Uncertainty

This is a deterministic theoretical calculation with no sampling uncertainty. The only numerical error is floating-point rounding, of order 1e−16 for the price (price_abs_diff_closed_vs_numerical) and 1e−13 for the Walras check (walras_law_residual_max_abs_on_price_grid). The main substantive uncertainty is the conflict between the inputs for the good-1 aggregate (see Limitations).

# Limitations and open issues

• The results depend on resolving the aggregate conflict (14 vs 12) in favour of the individual endowments. If the individual endowments were the ones in error (for example if some agent actually holds 2 more units of good 1), the equilibrium would be different, and the data cannot tell us which agent. I did not compute alternative scenarios because that would mean inventing an endowment.
• Scope of the numerical validation: it confirms the closed-form price for this one parameter set only, by root-finding z1. It does not prove the general formula. Uniqueness comes from the analytical monotonicity of z1 under Cobb–Douglas, not from a numerical search over several roots.
• The closed form assumes an interior solution, strictly positive endowments and p2 = 1. It does not apply to corner preferences (α ∈ {0, 1}) or zero wealth.