# Contract (P0)

Question. Brock–Mirman optimal growth with log utility, y' = (y-c)^alpha, full depreciation, A=1. Compare the closed-form policy/value with those obtained by discrete value-function iteration (VFI) on a prescribed log-spaced grid.

Estimands. Closed-form c(y) at y=0.5,1,2; VFI c(y) at the same points (interpolated if off-grid); max |c_num-c_closed| over the 3 points; closed-form V(1) and VFI V(1); closed-form savings rate (alpha*beta); contraction modulus of the Bellman operator in sup norm (beta, via Blackwell, on the bounded/compact grid problem). Meaningful under: log utility, Cobb–Douglas technology, full depreciation, interior optimum, 0<alpha<1, 0<beta<1.

Outputs. out/results.json with keys task_id, c_closed_y05, c_closed_y1, c_closed_y2, c_num_y05, c_num_y1, c_num_y2, max_abs_error_policy, value_closed_y1, value_num_y1, savings_rate_closed_form, contraction_modulus, grid_points_used, n_iterations, final_sup_change, 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/state/*, out/exclusions.csv).

Information rules. Use only data/model.csv; prescribed grid/parameters (deviations -> data_issues). Numerical policy must come from VFI, never from the closed form; closed form derived by guess-and-verify and stated in report.

Invalidating conditions. Grid that cannot contain optimal choices (binding grid bounds), non-convergence of VFI, infeasible consumption, use of closed form inside the VFI.