# Contract (P0)

**Question.** Deterministic optimal growth, log utility, y' = (y-c)^alpha, full depreciation, A=1. Compare the
closed-form policy/value (guess-and-verify) with value-function iteration (VFI) on the grid prescribed in data/model.csv.

**Estimands.** c_closed(y) = (1-alpha*beta) y at y=0.5,1,2; VFI policy at those points (linear interpolation between grid
nodes); max |c_num - c_closed| over the three points; V_closed(1) = A0 (since ln 1 = 0) and V_num(1);
closed-form savings rate alpha*beta; contraction modulus of the Bellman operator in sup norm (= beta, by Blackwell,
on the bounded discretised problem); grid points used, VFI iterations, final sup-norm change.
Assumptions: log utility, Cobb-Douglas technology with A=1, full depreciation, interior solution, 0<alpha<1, 0<beta<1.

**Outputs.** out/results.json (keys: task_id, c_closed_y05/_y1/_y2, c_num_y05/_y1/_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 (state/, exclusions.csv).

**Information rules.** Only data/model.csv; prescribed grid (400 log-spaced points on [0.05,5]) and parameters;
numerical policy from VFI only; deviations go to data_issues.

**Invalidity conditions.** Missing/inconsistent parameters; optimal next-period output leaving the grid (truncation
binding); VFI not converging; using the closed form inside the VFI.