1# F5-dev-01 — Cobb–Douglas consumer: closed-form demands with numerical validation23## Scientific question4A consumer with utility u(x1, x2) = α ln x1 + (1 − α) ln x2 chooses non-negative quantities of two goods subject to the budget constraint p1 x1 + p2 x2 ≤ m. The estimands are the Marshallian demands x1*(p, m) and x2*(p, m), the indirect utility v(p, m) = u(x1*, x2*), and the budget identity residual p1 x1* + p2 x2* − m at the optimum. Derive the closed-form solution, state the assumptions under which it is the unique optimum, and validate it numerically by an independent constrained maximisation (grid search or a numerical optimiser on the budget line).56## Inputs7- `data/parameters.csv` — columns `parameter`, `value`, `description`; one row per parameter: `alpha` (exponent on good 1, dimensionless), `p1`, `p2` (prices, currency units per unit of good), `m` (income, currency units). These are given parameters of a theoretical exercise, not observed data.89## Information-availability rules10- Use only `data/parameters.csv`. No other input exists.11- The closed-form expressions must be derived from the stated utility function; the numerical validation must not use the closed form (it must maximise utility directly).1213## Required outputs14- `out/results.json` with the keys:1516| key | type | unit | meaning |17|---|---|---|---|18| `task_id` | string | — | "F5-dev-01" |19| `x1_demand` | number | units of good 1 | closed-form Marshallian demand for good 1 |20| `x2_demand` | number | units of good 2 | closed-form Marshallian demand for good 2 |21| `indirect_utility` | number | utils | u evaluated at the closed-form demands |22| `budget_identity_residual` | number | currency units | p1·x1_demand + p2·x2_demand − m (should be numerically zero) |23| `mrs_at_optimum` | number | — | marginal rate of substitution (∂u/∂x1)/(∂u/∂x2) at the closed-form demands |24| `price_ratio` | number | — | p1/p2 |25| `expenditure_share_good1` | number | — | p1·x1_demand / m |26| `indirect_utility_numerical` | number | utils | maximum utility found by the independent numerical maximisation on the budget line (the baseline check) |27| `feasibility` | string | — | "complete", "partial" or "infeasible" |28| `feasibility_reason` | string or null | — | explanation if not complete |29| `data_issues` | array of {"issue","handling"} | — | data problems detected and how handled (empty list if none) |30| `n_input_rows` | object | — | {"parameters.csv": rows read} |31| `n_analysed` | integer or null | — | number of parameter rows used |32| `random_seed` | integer or null | — | seed used, if any |3334- `out/report.md` with the sections: Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues. The Method section must state explicitly the assumptions under which the closed form holds (e.g. interior solution, strict monotonicity/concavity, parameter ranges) and the scope of the numerical validation (which parameter point it covers).35- `out/analysis.py` — running `~/wp29/venv/bin/python out/analysis.py` from the workspace root must regenerate `out/results.json` from `data/` only, deterministically. It must contain both the closed-form computation and the numerical validation.36- `out/claims.json` — a list of the central claims of the report: [{"id": "C1", "text": "...", "results_keys": ["..."], "status": "supported"|"unsupported"|"not_evaluated", "caveat": "..."}].3738## Constraints39- No network access. Python: `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels available). Do not modify `data/`.40- This task is designed to be completed in well under 15 minutes of work.41