# F5-dev-01 — Cobb–Douglas consumer: closed-form demands with numerical validation

# Question

A consumer with u(x1,x2) = α ln x1 + (1−α) ln x2 chooses x1, x2 ≥ 0 subject to p1x1 + p2x2 ≤ m. We derive the Marshallian demands, the indirect utility and the budget residual in closed form, evaluate them at the parameters in data/parameters.csv, and validate them by an independent numerical maximisation of u that does not use the closed form.

# Data and exclusions

data/parameters.csv (SHA-256 a04798ad…4722, matches bundle.json) has 4 rows (n_input_rows): α = 0.35, p1 = 2.5, p2 = 4.0, m = 120. No missing values, duplicates or out-of-range values; data_issues is empty and no rows were excluded (out/exclusions.csv), so all 4 rows are used (n_analysed = 4). Audit details: out/state/data_audit.md.

# Method

Assumptions under which the closed form is the unique optimum: 0 < α < 1, p1 > 0, p2 > 0, m > 0 (all hold here, checked in the script). Then (i) u is strictly increasing in both goods on R²₊₊, so the budget constraint binds; (ii) u is strictly concave (sum of strictly concave logs with positive weights), so any optimum is unique; (iii) ln x → −∞ as x → 0⁺, so the corner solutions are never optimal and the solution is interior (the non-negativity constraints are slack); (iv) the budget set is compact, so an optimum exists.

Derivation. Lagrangian L = α ln x1 + (1−α) ln x2 + λ(m − p1x1 − p2x2). FOCs: α/x1 = λp1, (1−α)/x2 = λp2. Hence p1x1 = α/λ, p2x2 = (1−α)/λ; summing and using the binding budget gives λ = 1/m. Therefore x1* = αm/p1, x2* = (1−α)m/p2, v(p,m) = α ln(αm/p1) + (1−α) ln((1−α)m/p2). Implications: MRS = (α/x1)/((1−α)/x2) = p1/p2 at the optimum, and the expenditure share of good 1 is α.

Numerical validation (independent of the closed form). Implemented in out/analysis.py, using only u, the prices and the income: (a) grid search over 2,000,001 points of x1 ∈ (0, m/p1) on the budget line x2 = (m − p1x1)/p2; (b) bounded Brent minimisation of −u on the same line (xatol 1e-12); (c) 2-D SLSQP on −u with the inequality budget constraint and positivity bounds, started at (1,1), which tests that the budget binds rather than imposing it. indirect_utility_numerical is the largest utility found across the three. Scope: the validation covers only the single parameter point in data/parameters.csv (α = 0.35, p1 = 2.5, p2 = 4, m = 120); the general formula is established by the derivation, not by the numerics. No randomness is used (random_seed = null).

# Results

• Demand for good 1: x1_demand = 16.8 units; demand for good 2: x2_demand = 19.5 units.
• Indirect utility: indirect_utility = 2.9182520129 utils.
• Budget residual: budget_identity_residual = 0.0 currency units.
• Tangency: mrs_at_optimum = 0.625 = price_ratio = 0.625.
• Expenditure share of good 1: expenditure_share_good1 = 0.35 (= α).
• Numerical maximum: indirect_utility_numerical = 2.9182520129; utility_gap_numerical_minus_closed_form = 0.0. Individual methods: numerical_utility_grid = 2.91825201286344, numerical_utility_brent = 2.91825201286353, numerical_utility_slsqp = 2.91825201286353. Numerical maximisers agree with the closed-form demands to within max_abs_demand_gap_x1 = 9.6e-6 (set by grid spacing) and max_abs_demand_gap_x2 = 4.4e-7; SLSQP's budget slack is numerical_slsqp_budget_slack = 2.8e-14 (budget binds).
• feasibility = "complete".

# Uncertainty

This is a deterministic theoretical exercise with given parameters; there is no sampling uncertainty. The only uncertainty is numerical: floating-point rounding (residual exactly 0.0 here) and optimiser/grid tolerance. Because utility is flat near the optimum, demand discrepancies of order 1e-6–1e-5 (max_abs_demand_gap_x1, max_abs_demand_gap_x2) translate into utility discrepancies at or below ~1e-13 (numerical_utility_grid vs indirect_utility).

# Limitations and open issues

• The numerical validation covers one parameter point only; it does not by itself prove the formula for other parameters (the analytic derivation does, under the stated assumptions).
• The closed form is not valid outside 0 < α < 1, p1, p2, m > 0; the script stops with an error in that case rather than reporting values.
• The reported indirect_utility_numerical is the maximum over three methods; the Brent and closed-form values coincide to the last printed digit, so the gap reported is 0.0 at double precision.
• Extra diagnostic keys (prefixed numerical_, utility_gap_…, max_abs_demand_gap_…) are included in out/results.json beyond the required ones.