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

# Question

Two agents, A and B, have preferences u_i = α_i ln x1 + (1−α_i) ln x2. Find the Walrasian relative price p = p1/p2 (good 2 is the numeraire, so p2 = 1) and the equilibrium allocation. Check that the good-1 market clears and that Walras' law holds, and confirm the closed-form price by solving numerically for the root of the excess-demand function.

# Data and exclusions

• endowments.csv (2 rows): A = (8, 2), B = (4, 8).
• parameters.csv (4 rows): α_A = 0.4, α_B = 0.6. It also gives the summary sheet's aggregates: good 1 = 14, good 2 = 10.
• Data issue: the individual endowments of good 1 add up to 8 + 4 = 12, not 14. The good-2 figure is consistent (2 + 8 = 10). In a pure exchange economy, aggregate supply is by definition the sum of the agents' endowments, and the task names endowments.csv as the primary data. I therefore used 12 as the aggregate of good 1 and treated 14 as a transcription error in the summary sheet. No rows were excluded; both agents are analysed.
• Sensitivity (diagnostic only, out/diagnostics.json): if 14 is put into the good-1 market-clearing condition, the root is p = 0.6667. At that price the excess demand for good 2 is −1.333 and the Walras residual p·z1 + z2 is −1.333, not zero. This is because 2 units of good 1 that nobody owns would be "supplied". That outcome cannot be a competitive equilibrium, which supports the choice of 12.

# Method

Assumptions behind the closed form. Preferences are strictly monotone Cobb–Douglas with 0 < α_i < 1. Endowments are strictly positive, so wealth m_i = p·e1_i + e2_i > 0 at every p > 0. Demands are therefore interior: x1_i = α_i m_i / p and x2_i = (1−α_i) m_i. Prices are normalised by p2 = 1.

Closed form. Good-1 market clearing, p·(e1_A + e1_B) = α_A m_A + α_B m_B, gives

p* = (α_A e2_A + α_B e2_B) / ((1−α_A) e1_A + (1−α_B) e1_B) = (0.8 + 4.8) / (4.8 + 1.6) = 5.6 / 6.4 = 0.875.

Numerical validation. I found the root of the good-1 aggregate excess demand z1(p) directly, using Brent's method (scipy.optimize.brentq) on [1e−6, 1e6] with xtol = 1e−14. This does not evaluate the closed-form expression. I also checked Walras' law, p·z1(p) + z2(p) = 0, on a log grid of 61 prices from 1e−3 to 1e3. The computation is deterministic, so no random seed is used.

# Results

Both markets clear: 4.1143 + 7.8857 = 12 and 5.4 + 4.6 = 10. Across the whole price grid, the largest absolute Walras residual is 9.1e−13, which is floating-point rounding. This confirms that Walras' law holds as an identity, not only at p*.

# Uncertainty

The model is a deterministic theoretical one, so there is no sampling uncertainty. The only numerical uncertainty is floating-point error, about 1e−16 for the price. Excess demand is strictly decreasing in p, which makes the equilibrium unique, so the root-finder could not have converged to a different equilibrium. The substantive uncertainty is the data inconsistency: if the summary-sheet aggregate of 14 were correct, at least one individual endowment would have to be wrong, and the equilibrium would change. With the given individual endowments, using 14 gives no consistent equilibrium (see the sensitivity check).

# Limitations and open issues

• The good-1 aggregate inconsistency (14 vs 12) was resolved in favour of the individual endowments. I cannot tell from the inputs which figure is the true one.
• The closed form relies on interior Cobb–Douglas demands. It would not apply if α_i were 0 or 1, or if an endowment were zero.
• The numerical validation only checks the given parameter values and the bracket [1e−6, 1e6]. It is not a general proof.