# Contract (P0)

Question. Cobb–Douglas consumer, u = α ln x1 + (1−α) ln x2, budget p1x1 + p2x2 ≤ m. Derive Marshallian demands, indirect utility, budget residual; validate numerically by direct constrained maximisation that does not use the closed form.

Estimands. x1*(p,m), x2*(p,m), v(p,m)=u(x*), residual p1x1*+p2x2*−m, MRS at x*, p1/p2, expenditure share of good 1, and numerically maximised utility. Meaningful under: 0<α<1, p1,p2>0, m>0 (then u strictly increasing & strictly concave on R²₊₊, ln→−∞ at the boundary so the optimum is interior, budget binds, 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 (Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues); out/analysis.py; out/claims.json.

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

Invalidity. Missing/non-numeric parameters, α∉(0,1), non-positive prices or income; numerical optimum disagreeing with closed form.