# Contract (P0) — F5-dev-01

Question. A consumer maximises u(x1,x2) = α ln x1 + (1−α) ln x2 s.t. p1x1 + p2x2 ≤ m, x ≥ 0. Derive the closed-form Marshallian demands, evaluate them at the given parameters, and validate numerically by maximising u directly on the budget line (without using the closed form).

Estimands (at the single parameter point in data/parameters.csv): x1*(p,m), x2*(p,m), v(p,m)=u(x*), budget residual p1x1*+p2x2*−m, MRS at x*, p1/p2, expenditure share of good 1, and the numerically found maximum utility.

Assumptions for meaningfulness/uniqueness. 0<α<1, p1>0, p2>0, m>0. Then u is strictly increasing (budget binds), strictly concave on R²₊₊, and ln x → −∞ as x→0 (interior solution); the feasible set is compact and convex, so the optimum exists and is unique.

Outputs. out/results.json (keys: task_id, x1_demand, x2_demand, indirect_utility, budget_identity_residual, mrs_at_optimum, price_ratio, expenditure_share_good1, indirect_utility_numerical, feasibility, feasibility_reason, data_issues, n_input_rows, n_analysed, random_seed); out/report.md (sections Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues); out/analysis.py; out/claims.json; plus protocol files (state/, exclusions.csv).

Information rules. Only data/parameters.csv. Numerical validation must not use the closed form.

Invalidity conditions. Missing/non-numeric parameters, α∉(0,1), non-positive prices or income (closed form would not be the unique interior optimum / problem ill-posed).