1# F5-conf-01 — Competitive equilibrium of a 2×2 Cobb–Douglas exchange economy23## Question4Find the Walrasian equilibrium relative price p = p1/p2 (good 2 is the numeraire, p2 = 1) and the equilibrium allocation for agents A and B with u_i = α_i ln x1 + (1−α_i) ln x2. Check market clearing and Walras' law with numbers, and check the closed-form price against an independent numerical root of the aggregate excess demand for good 1.56## Data and exclusions7- `data/endowments.csv` (2 rows): A = (8, 2), B = (4, 8). No missing values or duplicates, and every endowment is strictly positive.8- `data/parameters.csv` (4 rows): α_A = 0.4, α_B = 0.6, and the recorded aggregates 14 (good 1) and 10 (good 2). Row counts are in `n_input_rows`.9- **Data issue (`data_issues`):** the recorded aggregate endowment of good 1 is 14 (`recorded_aggregate_endowment_good1`). The individual endowments add up to 8 + 4 = 12. Good 2 is consistent (2 + 8 = 10). By definition the aggregate endowment is the sum of individual endowments, and `endowments.csv` is the primary data. I therefore used 12 (`aggregate_endowment_good1`) and 10 (`aggregate_endowment_good2`) in the market-clearing condition. With the recorded value of 14, good 1 could not clear: aggregate excess demand would be −2 (`excess_demand_good1_if_recorded_aggregate_used`).10- No agents were excluded. `out/exclusions.csv` shows 2 agents before and 2 after, so `n_analysed` = 2.1112## Method13- Cobb–Douglas demands: x1_i = α_i m_i / p and x2_i = (1−α_i) m_i, where income is m_i = p·e1_i + e2_i.14- Setting the good-1 market to clear, Σ α_i (p e1_i + e2_i) = p Σ e1_i, gives the closed form15 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) = 5.6/6.4.16- Assumptions behind the closed form:17 - 0 < α_i < 1.18 - Endowments are strictly positive, so incomes are positive and the optimal demands are interior. Under log utility there are no corner solutions.19 - Prices are normalised so that p2 = 1. Only relative prices are determined.20 - With Cobb–Douglas preferences the goods are gross substitutes, so the equilibrium is unique.21- Numerical validation: `scipy.optimize.brentq` finds the root of z1(p) = Σ_i x1_i(p) − W1 on p ∈ [1e-6, 1e6] (xtol 1e-14). It evaluates the demand functions and never uses the closed-form expression. z1 changes sign once on this bracket, so the root found is the unique positive equilibrium. Scope of the check: it confirms the algebra and the arithmetic for these inputs only. It does not test the economic model.22- Script: `out/analysis.py`. It is deterministic and uses no random numbers (`random_seed` = null).2324## Results25- Closed-form price: p1/p2 = 0.875 (`price_ratio_p1_over_p2`).26- Numerical root: 0.875 (`price_ratio_numerical`). The absolute difference is 1.1e-16 (`abs_diff_closed_form_vs_numerical`), which is machine precision.27- Allocation:28 - Agent A: x1_A = 4.1143 (`x1_A`), x2_A = 5.4 (`x2_A`).29 - Agent B: x1_B = 7.8857 (`x1_B`), x2_B = 4.6 (`x2_B`).30 - Totals are 12 and 10, equal to the aggregate endowments.31- Excess demand for good 1 at p*: 0.0 (`excess_demand_good1`). For good 2: 0.0 (`excess_demand_good2`).32- Walras' law residual p·z1 + z2: 0.0 (`walras_law_residual`).33- Feasibility: `feasibility` = "complete".3435## Uncertainty36The inputs are theoretical parameters, not sample data, so there is no statistical uncertainty. The only error is floating-point error, about 1e-16 (`abs_diff_closed_form_vs_numerical`). The single substantive ambiguity is which aggregate endowment of good 1 is correct (12 or 14), discussed above.3738## Limitations and open issues39- The results depend on resolving the 14 vs 12 inconsistency in favour of the individual endowments. If the individual endowments were the wrong figures instead (for example, if some agent's good-1 endowment should be 2 units higher), the price and the allocation would be different. The data give no basis for deciding which agent such a correction would apply to. Because of that, and because an aggregate must equal the sum of its parts, I took the primary file as authoritative.40- The numerical validation only checks consistency for these inputs. It is not a general proof.41