# Contract (P0)

Question. Consumer with u(x1,x2)=α ln x1 + (1−α) ln x2, budget p1x1+p2x2 ≤ m, x ≥ 0. Derive Marshallian demands, indirect utility, budget residual; validate with an independent numerical 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, numerical max utility on the budget line. Meaningful under: 0<α<1, p1,p2>0, m>0 (then u strictly increasing and strictly concave on R²₊₊, Inada at boundary → unique interior optimum, budget binds).

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; plus protocol files out/exclusions.csv, out/state/*.

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

Invalidity / impossibility. Missing/duplicate/non-numeric parameters; α∉(0,1), non-positive prices or income (closed form invalid or not unique); numerical optimum disagreeing with closed form.