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{"parentUuid":"bcbccad8-8621-4607-89c4-8c58802483f2","isSidechain":false,"promptId":"19439f9c-406b-4964-a0f5-368ea65ce5a4","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_015tLmW5tGqWaDz5sXXQVfHb","type":"tool_result","content":"# F5-conf-04 — Value-function iteration against a closed-form policy (optimal growth with log utility)\n\n## Scientific question\nA planner maximises Σ_t β^t ln c_t subject to y_{t+1} = (y_t − c_t)^α, 0 < c_t < y_t, with full depreciation and productivity normalised to one. The Bellman equation is V(y) = max_{0<c<y} { ln c + β V((y − c)^α) }. This model has a known closed-form solution for the optimal consumption policy c(y) and the value function V(y). The estimands are: the closed-form policy at three evaluation points of y; the policy obtained by value-function iteration (VFI) on the prescribed grid at the same points; the maximum absolute discrepancy between them over the evaluation points; the closed-form and numerical value function at y = 1; the savings rate implied by the closed form; and the contraction modulus of the Bellman operator in the sup norm.\n\n## Inputs\n- `data/model.csv` — columns `parameter`, `value`, `description`: `alpha`, `beta`, the grid specification (`y_min`, `y_max`, `grid_points`, `grid_spacing` = log, i.e. points equally spaced in ln y), the VFI stopping tolerance `vfi_tolerance` (sup-norm change between successive value functions), and the three evaluation points `eval_y_1`, `eval_y_2`, `eval_y_3`. Theoretical parameters, not observed data.\n\n## Information-availability rules\n- Use only `data/model.csv` and solve the model on the prescribed grid with the prescribed parameters; if any deviation from the prescribed design is necessary, record it in `data_issues`.\n- The numerical policy must come from the VFI, not from the closed form; the closed form must be derived (e.g. guess-and-verify on the Bellman equation) and stated in the report.\n\n## Required outputs\n- `out/results.json` with the keys:\n\n| key | type | unit | meaning |\n|---|---|---|---|\n| `task_id` | string | — | \"F5-conf-04\" |\n| `c_closed_y05`, `c_closed_y1`, `c_closed_y2` | number | units of output | closed-form optimal consumption at y = 0.5, 1.0, 2.0 |\n| `c_num_y05`, `c_num_y1`, `c_num_y2` | number | units of output | VFI consumption policy at y = 0.5, 1.0, 2.0 (interpolated on the grid if the point is not a grid node) |\n| `max_abs_error_policy` | number | units of output | max over the three evaluation points of \\|c_num − c_closed\\| |\n| `value_closed_y1` | number | utils | closed-form V(1) |\n| `value_num_y1` | number | utils | VFI value function at y = 1 |\n| `savings_rate_closed_form` | number | — | closed-form (y − c)/y (the baseline quantity) |\n| `contraction_modulus` | number | — | modulus of the Bellman operator as a contraction in the sup norm |\n| `grid_points_used` | integer | — | number of grid points actually used |\n| `n_iterations` | integer | — | VFI iterations until the stopping rule was met |\n| `final_sup_change` | number | utils | sup-norm change at the last iteration |\n| `feasibility` | string | — | \"complete\", \"partial\" or \"infeasible\" |\n| `feasibility_reason` | string or null | — | explanation if not complete |\n| `data_issues` | array of {\"issue\",\"handling\"} | — | data problems or design deviations and how handled (empty list if none) |\n| `n_input_rows` | object | — | {\"model.csv\": rows read} |\n| `n_analysed` | integer or null | — | number of grid points |\n| `random_seed` | integer or null | — | seed used, if any |\n\n- `out/report.md` with the sections: Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues. State explicitly the assumptions of the closed form (log utility, Cobb–Douglas technology, full depreciation, interior solution), why the Bellman operator is a contraction and with what modulus, and the scope of the numerical validation (which grid, which parameter point, what the discretisation error depends on).\n- `out/analysis.py` — running `~/wp29/venv/bin/python out/analysis.py` from the workspace root must regenerate `out/results.json` from `data/` only, deterministically.\n- `out/claims.json` — a list of the central claims of the report: [{\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}]. Each claim's status must be justified by the listed results keys.\n\n## Constraints\n- No network access. Python: `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels available). Do not modify `data/`.\n- This task is designed to be completed in well under 20 minutes of work; the VFI on the prescribed grid runs in seconds when vectorised.\ntotal 56\ndrwxr-xr-x  9 simon-pierreboucher  staff   288 oct.  30 00:39 .\ndrwxr-xr-x  7 simon-pierreboucher  staff   224 oct.  30 00:38 ..\n-rw-r--r--  1 simon-pierreboucher  staff  5064 oct.  30 00:38 analysis.py\n-rw-r--r--  1 simon-pierreboucher  staff  1941 oct.  30 00:39 claims.json\n-rw-r--r--  1 simon-pierreboucher  staff   141 oct.  30 00:38 exclusions.csv\n-rw-r--r--  1 simon-pierreboucher  staff  5566 oct.  30 00:39 report.md\n-rw-r--r--  1 simon-pierreboucher  staff  1670 oct.  30 00:38 results.json\ndrwxr-xr-x  2 simon-pierreboucher  staff    64 oct.  30 00:39 review\ndrwxr-xr-x  8 simon-pierreboucher  staff   256 oct.  30 00:39 state\n{\n  \"task_id\": \"F5-conf-04\",\n  \"c_closed_y05\": 0.329,\n  \"c_closed_y1\": 0.658,\n  \"c_closed_y2\": 1.316,\n  \"c_num_y05\": 0.32881774872035086,\n  \"c_num_y1\": 0.6589935200017671,\n  \"c_num_y2\": 1.309625047452993,\n  \"max_abs_error_policy\": 0.006374952547007151,\n  \"value_closed_y1\": -19.524412221722393,\n  \"value_num_y1\": -19.52475624982261,\n  \"savings_rate_closed_form\": 0.34199999999999997,\n  \"contraction_modulus\": 0.95,\n  \"grid_points_used\": 400,\n  \"n_iterations\": 361,\n  \"final_sup_change\": 9.996650618404601e-09,\n  \"feasibility\": \"complete\",\n  \"feasibility_reason\": null,\n  \"data_issues\": [\n    {\n      \"issue\": \"Evaluation points [0.5, 1.0, 2.0] are not grid nodes of the log-spaced grid\",\n      \"handling\": \"VFI policy and value interpolated linearly in ln(y) between adjacent grid nodes\"\n    },\n    {\n      \"issue\": \"Design choice: the choice variable is next-period output y' restricted to the same grid (discrete-choice VFI); consumption c = y - y'^(1/alpha)\",\n      \"handling\": \"No deviation from prescribed grid/parameters; discretisation error of order of the grid spacing is reported, not corrected\"\n    }\n  ],\n  \"n_input_rows\": {\n    \"model.csv\": 10\n  },\n  \"n_analysed\": 400,\n  \"random_seed\": null,\n  \"diag_grid_log_step\": 0.011541779914757022,\n  \"diag_max_abs_policy_error_on_grid\": 0.02512152290513381,\n  \"diag_max_abs_value_error_on_grid\": 0.0004835155258184898,\n  \"diag_policy_hits_lower_bound\": 0,\n  \"diag_policy_hits_upper_bound\": 0,\n  \"diag_value_error_y1\": -0.0003440281002156098,\n  \"diag_closed_form_A\": -19.524412221722393,\n  \"diag_closed_form_B\": 1.519756838905775,\n  \"diag_vfi_error_bound_beta_over_1mbeta_times_final_change\": 1.8993636174968724e-07\n}[\n{\"id\":\"C1\",\"text\":\"The closed-form policy is c(y)=(1-alpha*beta)y, giving 0.329, 0.658, 1.316 at y=0.5,1,2, with a savings rate of alpha*beta=0.342.\",\"results_keys\":[\"c_closed_y05\",\"c_closed_y1\",\"c_closed_y2\",\"savings_rate_closed_form\"],\"status\":\"supported\",\"caveat\":\"Requires log utility, Cobb-Douglas technology, full depreciation and an interior solution.\"},\n{\"id\":\"C2\",\"text\":\"Discrete-choice VFI on the prescribed 400-point log grid reproduces the closed-form policy at the three evaluation points within 0.0064 (about 0.5% of consumption).\",\"results_keys\":[\"c_num_y05\",\"c_num_y1\",\"c_num_y2\",\"max_abs_error_policy\",\"grid_points_used\"],\"status\":\"supported\",\"caveat\":\"Error is mostly discretisation plus log-linear interpolation; it holds for this grid and parameter point only.\"},\n{\"id\":\"C3\",\"text\":\"The VFI value at y=1 matches the closed-form V(1)=-19.5244 to within about 3.4e-4 utils.\",\"results_keys\":[\"value_closed_y1\",\"value_num_y1\",\"diag_value_error_y1\"],\"status\":\"supported\",\"caveat\":\"Interpolated in ln y; y=1 is not a grid node.\"},\n{\"id\":\"C4\",\"text\":\"The Bellman operator is a sup-norm contraction with modulus beta=0.95, and VFI met the 1e-8 stopping rule in 361 iterations, so the iteration error is bounded by about 1.9e-7 utils.\",\"results_keys\":[\"contraction_modulus\",\"n_iterations\",\"final_sup_change\",\"diag_vfi_error_bound_beta_over_1mbeta_times_final_change\"],\"status\":\"supported\",\"caveat\":\"The contraction argument uses bounded utility on the compact grid; on the full domain, log utility needs a weighted norm.\"},\n{\"id\":\"C5\",\"text\":\"Discretisation, not iteration, dominates the numerical error: the maximum policy error over all grid nodes is 0.025 and grows with y.\",\"results_keys\":[\"diag_max_abs_policy_error_on_grid\",\"diag_grid_log_step\",\"final_sup_change\"],\"status\":\"supported\",\"caveat\":\"The claim that the error grows with y comes from the step-function structure; only the maximum is stored in the results.\"}\n]","is_error":false}]},"uuid":"769eba04-7f08-49c2-b7e1-32e3808aa4a4","timestamp":"2026-10-30T04:39:58.908Z","toolUseResult":{"stdout":"# F5-conf-04 — Value-function iteration against a closed-form policy (optimal growth with log utility)\n\n## Scientific question\nA planner maximises Σ_t β^t ln c_t subject to y_{t+1} = (y_t − c_t)^α, 0 < c_t < y_t, with full depreciation and productivity normalised to one. The Bellman equation is V(y) = max_{0<c<y} { ln c + β V((y − c)^α) }. This model has a known closed-form solution for the optimal consumption policy c(y) and the value function V(y). The estimands are: the closed-form policy at three evaluation points of y; the policy obtained by value-function iteration (VFI) on the prescribed grid at the same points; the maximum absolute discrepancy between them over the evaluation points; the closed-form and numerical value function at y = 1; the savings rate implied by the closed form; and the contraction modulus of the Bellman operator in the sup norm.\n\n## Inputs\n- `data/model.csv` — columns `parameter`, `value`, `description`: `alpha`, `beta`, the grid specification (`y_min`, `y_max`, `grid_points`, `grid_spacing` = log, i.e. points equally spaced in ln y), the VFI stopping tolerance `vfi_tolerance` (sup-norm change between successive value functions), and the three evaluation points `eval_y_1`, `eval_y_2`, `eval_y_3`. Theoretical parameters, not observed data.\n\n## Information-availability rules\n- Use only `data/model.csv` and solve the model on the prescribed grid with the prescribed parameters; if any deviation from the prescribed design is necessary, record it in `data_issues`.\n- The numerical policy must come from the VFI, not from the closed form; the closed form must be derived (e.g. guess-and-verify on the Bellman equation) and stated in the report.\n\n## Required outputs\n- `out/results.json` with the keys:\n\n| key | type | unit | meaning |\n|---|---|---|---|\n| `task_id` | string | — | \"F5-conf-04\" |\n| `c_closed_y05`, `c_closed_y1`, `c_closed_y2` | number | units of output | closed-form optimal consumption at y = 0.5, 1.0, 2.0 |\n| `c_num_y05`, `c_num_y1`, `c_num_y2` | number | units of output | VFI consumption policy at y = 0.5, 1.0, 2.0 (interpolated on the grid if the point is not a grid node) |\n| `max_abs_error_policy` | number | units of output | max over the three evaluation points of \\|c_num − c_closed\\| |\n| `value_closed_y1` | number | utils | closed-form V(1) |\n| `value_num_y1` | number | utils | VFI value function at y = 1 |\n| `savings_rate_closed_form` | number | — | closed-form (y − c)/y (the baseline quantity) |\n| `contraction_modulus` | number | — | modulus of the Bellman operator as a contraction in the sup norm |\n| `grid_points_used` | integer | — | number of grid points actually used |\n| `n_iterations` | integer | — | VFI iterations until the stopping rule was met |\n| `final_sup_change` | number | utils | sup-norm change at the last iteration |\n| `feasibility` | string | — | \"complete\", \"partial\" or \"infeasible\" |\n| `feasibility_reason` | string or null | — | explanation if not complete |\n| `data_issues` | array of {\"issue\",\"handling\"} | — | data problems or design deviations and how handled (empty list if none) |\n| `n_input_rows` | object | — | {\"model.csv\": rows read} |\n| `n_analysed` | integer or null | — | number of grid points |\n| `random_seed` | integer or null | — | seed used, if any |\n\n- `out/report.md` with the sections: Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues. State explicitly the assumptions of the closed form (log utility, Cobb–Douglas technology, full depreciation, interior solution), why the Bellman operator is a contraction and with what modulus, and the scope of the numerical validation (which grid, which parameter point, what the discretisation error depends on).\n- `out/analysis.py` — running `~/wp29/venv/bin/python out/analysis.py` from the workspace root must regenerate `out/results.json` from `data/` only, deterministically.\n- `out/claims.json` — a list of the central claims of the report: [{\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}]. Each claim's status must be justified by the listed results keys.\n\n## Constraints\n- No network access. Python: `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels available). Do not modify `data/`.\n- This task is designed to be completed in well under 20 minutes of work; the VFI on the prescribed grid runs in seconds when vectorised.\ntotal 56\ndrwxr-xr-x  9 simon-pierreboucher  staff   288 oct.  30 00:39 .\ndrwxr-xr-x  7 simon-pierreboucher  staff   224 oct.  30 00:38 ..\n-rw-r--r--  1 simon-pierreboucher  staff  5064 oct.  30 00:38 analysis.py\n-rw-r--r--  1 simon-pierreboucher  staff  1941 oct.  30 00:39 claims.json\n-rw-r--r--  1 simon-pierreboucher  staff   141 oct.  30 00:38 exclusions.csv\n-rw-r--r--  1 simon-pierreboucher  staff  5566 oct.  30 00:39 report.md\n-rw-r--r--  1 simon-pierreboucher  staff  1670 oct.  30 00:38 results.json\ndrwxr-xr-x  2 simon-pierreboucher  staff    64 oct.  30 00:39 review\ndrwxr-xr-x  8 simon-pierreboucher  staff   256 oct.  30 00:39 state\n{\n  \"task_id\": \"F5-conf-04\",\n  \"c_closed_y05\": 0.329,\n  \"c_closed_y1\": 0.658,\n  \"c_closed_y2\": 1.316,\n  \"c_num_y05\": 0.32881774872035086,\n  \"c_num_y1\": 0.6589935200017671,\n  \"c_num_y2\": 1.309625047452993,\n  \"max_abs_error_policy\": 0.006374952547007151,\n  \"value_closed_y1\": -19.524412221722393,\n  \"value_num_y1\": -19.52475624982261,\n  \"savings_rate_closed_form\": 0.34199999999999997,\n  \"contraction_modulus\": 0.95,\n  \"grid_points_used\": 400,\n  \"n_iterations\": 361,\n  \"final_sup_change\": 9.996650618404601e-09,\n  \"feasibility\": \"complete\",\n  \"feasibility_reason\": null,\n  \"data_issues\": [\n    {\n      \"issue\": \"Evaluation points [0.5, 1.0, 2.0] are not grid nodes of the log-spaced grid\",\n      \"handling\": \"VFI policy and value interpolated linearly in ln(y) between adjacent grid nodes\"\n    },\n    {\n      \"issue\": \"Design choice: the choice variable is next-period output y' restricted to the same grid (discrete-choice VFI); consumption c = y - y'^(1/alpha)\",\n      \"handling\": \"No deviation from prescribed grid/parameters; discretisation error of order of the grid spacing is reported, not corrected\"\n    }\n  ],\n  \"n_input_rows\": {\n    \"model.csv\": 10\n  },\n  \"n_analysed\": 400,\n  \"random_seed\": null,\n  \"diag_grid_log_step\": 0.011541779914757022,\n  \"diag_max_abs_policy_error_on_grid\": 0.02512152290513381,\n  \"diag_max_abs_value_error_on_grid\": 0.0004835155258184898,\n  \"diag_policy_hits_lower_bound\": 0,\n  \"diag_policy_hits_upper_bound\": 0,\n  \"diag_value_error_y1\": -0.0003440281002156098,\n  \"diag_closed_form_A\": -19.524412221722393,\n  \"diag_closed_form_B\": 1.519756838905775,\n  \"diag_vfi_error_bound_beta_over_1mbeta_times_final_change\": 1.8993636174968724e-07\n}[\n{\"id\":\"C1\",\"text\":\"The closed-form policy is c(y)=(1-alpha*beta)y, giving 0.329, 0.658, 1.316 at y=0.5,1,2, with a savings rate of alpha*beta=0.342.\",\"results_keys\":[\"c_closed_y05\",\"c_closed_y1\",\"c_closed_y2\",\"savings_rate_closed_form\"],\"status\":\"supported\",\"caveat\":\"Requires log utility, Cobb-Douglas technology, full depreciation and an interior solution.\"},\n{\"id\":\"C2\",\"text\":\"Discrete-choice VFI on the prescribed 400-point log grid reproduces the closed-form policy at the three evaluation points within 0.0064 (about 0.5% of consumption).\",\"results_keys\":[\"c_num_y05\",\"c_num_y1\",\"c_num_y2\",\"max_abs_error_policy\",\"grid_points_used\"],\"status\":\"supported\",\"caveat\":\"Error is mostly discretisation plus log-linear interpolation; it holds for this grid and parameter point only.\"},\n{\"id\":\"C3\",\"text\":\"The VFI value at y=1 matches the closed-form V(1)=-19.5244 to within about 3.4e-4 utils.\",\"results_keys\":[\"value_closed_y1\",\"value_num_y1\",\"diag_value_error_y1\"],\"status\":\"supported\",\"caveat\":\"Interpolated in ln y; y=1 is not a grid node.\"},\n{\"id\":\"C4\",\"text\":\"The Bellman operator is a sup-norm contraction with modulus beta=0.95, and VFI met the 1e-8 stopping rule in 361 iterations, so the iteration error is bounded by about 1.9e-7 utils.\",\"results_keys\":[\"contraction_modulus\",\"n_iterations\",\"final_sup_change\",\"diag_vfi_error_bound_beta_over_1mbeta_times_final_change\"],\"status\":\"supported\",\"caveat\":\"The contraction argument uses bounded utility on the compact grid; on the full domain, log utility needs a weighted norm.\"},\n{\"id\":\"C5\",\"text\":\"Discretisation, not iteration, dominates the numerical error: the maximum policy error over all grid nodes is 0.025 and grows with y.\",\"results_keys\":[\"diag_max_abs_policy_error_on_grid\",\"diag_grid_log_step\",\"final_sup_change\"],\"status\":\"supported\",\"caveat\":\"The claim that the error grows with y comes from the step-function structure; 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{"parentUuid":"1988e6a2-4ff8-4a44-bb7b-3c5917b6dc65","isSidechain":false,"promptId":"19439f9c-406b-4964-a0f5-368ea65ce5a4","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01R6NGXscKGrUUxT9rWBpjma","type":"tool_result","content":"parameter,value,description\nalpha,0.36,capital elasticity in y_{t+1} = k_{t+1}^alpha\nbeta,0.95,discount factor\ny_min,0.05,lower bound of the output grid\ny_max,5.0,upper bound of the output grid\ngrid_points,400,number of grid points for y (log-spaced between y_min and y_max)\ngrid_spacing,log,grid points equally spaced in ln(y)\nvfi_tolerance,1e-08,stop value-function iteration when the sup-norm change is below this\neval_y_1,0.5,evaluation point for the policy (output level)\neval_y_2,1.0,evaluation point for the policy (output level)\neval_y_3,2.0,evaluation point for the policy (output level)\n\"\"\"F5-conf-04: VFI vs closed form, optimal growth with log utility.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\nReads only data/model.csv; deterministic (no randomness).\"\"\"\nimport json, os, csv\nimport numpy as np\n\nROOT = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))\nDATA = os.path.join(ROOT, \"data\", \"model.csv\")\nOUT = os.path.join(ROOT, \"out\")\n\nwith open(DATA, newline=\"\") as f:\n    rows = list(csv.DictReader(f))\np = {r[\"parameter\"]: r[\"value\"] for r in rows}\nalpha = float(p[\"alpha\"]); beta = float(p[\"beta\"])\ny_min = float(p[\"y_min\"]); y_max = float(p[\"y_max\"]); N = int(float(p[\"grid_points\"]))\nassert p[\"grid_spacing\"].strip() == \"log\"\ntol = float(p[\"vfi_tolerance\"])\neval_y = [float(p[\"eval_y_1\"]), float(p[\"eval_y_2\"]), float(p[\"eval_y_3\"])]\n\n# ---------------- closed form (guess-and-verify V = A + B ln y) ----------------\nab = alpha * beta\nB = 1.0 / (1.0 - ab)\nA = (np.log(1.0 - ab) + ab / (1.0 - ab) * np.log(ab)) / (1.0 - beta)\nc_closed = lambda y: (1.0 - ab) * np.asarray(y)\nV_closed = lambda y: A + B * np.log(y)\nsavings_rate = ab\n\n# ---------------- VFI on prescribed grid (discrete choice of y' on grid) --------\ngrid = np.exp(np.linspace(np.log(y_min), np.log(y_max), N))\nk_needed = grid ** (1.0 / alpha)                 # capital needed to reach y' = grid[j]\nC = grid[:, None] - k_needed[None, :]            # c(i, j) = y_i - y'_j^(1/alpha)\nfeasible = C > 0\nU = np.full_like(C, -np.inf)\nU[feasible] = np.log(C[feasible])\nassert feasible.any(axis=1).all(), \"some grid point has no feasible choice\"\n\nV = np.zeros(N)\nn_iter = 0\nsup_change = np.inf\nmax_iter = 100000\nwhile sup_change >= tol and n_iter < max_iter:\n    W = U + beta * V[None, :]\n    V_new = W.max(axis=1)\n    sup_change = float(np.max(np.abs(V_new - V)))\n    V = V_new\n    n_iter += 1\nconverged = sup_change < tol\npol_j = (U + beta * V[None, :]).argmax(axis=1)\nc_pol = grid - k_needed[pol_j]\n\n# boundary diagnostics: does the optimal choice ever hit grid edges?\nhits_lower = int(np.sum(pol_j == 0)); hits_upper = int(np.sum(pol_j == N - 1))\n\n# interpolation at evaluation points: linear in ln y (grid is uniform in ln y)\nlg = np.log(grid)\ninterp = lambda f, y: float(np.interp(np.log(y), lg, f))\nc_num = [interp(c_pol, y) for y in eval_y]\nc_cl = [float(c_closed(y)) for y in eval_y]\non_grid = [bool(np.any(np.isclose(grid, y, rtol=1e-12, atol=0))) for y in eval_y]\nmax_err = float(max(abs(a - b) for a, b in zip(c_num, c_cl)))\n\n# whole-grid diagnostics (for report only)\ngrid_policy_err = float(np.max(np.abs(c_pol - c_closed(grid))))\ngrid_value_err = float(np.max(np.abs(V - V_closed(grid))))\n\ndata_issues = []\nif not all(on_grid):\n    data_issues.append({\"issue\": \"Evaluation points %s are not grid nodes of the log-spaced grid\" %\n                        [y for y, o in zip(eval_y, on_grid) if not o],\n                        \"handling\": \"VFI policy and value interpolated linearly in ln(y) between adjacent grid nodes\"})\ndata_issues.append({\"issue\": \"Design choice: the choice variable is next-period output y' restricted to the same grid (discrete-choice VFI); consumption c = y - y'^(1/alpha)\",\n                    \"handling\": \"No deviation from prescribed grid/parameters; discretisation error of order of the grid spacing is reported, not corrected\"})\n\nfeas = \"complete\" if converged else \"partial\"\nres = {\n    \"task_id\": \"F5-conf-04\",\n    \"c_closed_y05\": c_cl[0], \"c_closed_y1\": c_cl[1], \"c_closed_y2\": c_cl[2],\n    \"c_num_y05\": c_num[0], \"c_num_y1\": c_num[1], \"c_num_y2\": c_num[2],\n    \"max_abs_error_policy\": max_err,\n    \"value_closed_y1\": float(V_closed(1.0)),\n    \"value_num_y1\": interp(V, 1.0),\n    \"savings_rate_closed_form\": float(savings_rate),\n    \"contraction_modulus\": float(beta),\n    \"grid_points_used\": int(N),\n    \"n_iterations\": int(n_iter),\n    \"final_sup_change\": float(sup_change),\n    \"feasibility\": feas,\n    \"feasibility_reason\": None if converged else \"VFI did not reach tolerance within max_iter\",\n    \"data_issues\": data_issues,\n    \"n_input_rows\": {\"model.csv\": len(rows)},\n    \"n_analysed\": int(N),\n    \"random_seed\": None,\n    # supplementary diagnostics\n    \"diag_grid_log_step\": float(lg[1] - lg[0]),\n    \"diag_max_abs_policy_error_on_grid\": grid_policy_err,\n    \"diag_max_abs_value_error_on_grid\": grid_value_err,\n    \"diag_policy_hits_lower_bound\": hits_lower,\n    \"diag_policy_hits_upper_bound\": hits_upper,\n    \"diag_value_error_y1\": interp(V, 1.0) - float(V_closed(1.0)),\n    \"diag_closed_form_A\": float(A), \"diag_closed_form_B\": float(B),\n    \"diag_vfi_error_bound_beta_over_1mbeta_times_final_change\": float(beta / (1 - beta) * sup_change),\n}\nos.makedirs(OUT, exist_ok=True)\nwith open(os.path.join(OUT, \"results.json\"), \"w\") as f:\n    json.dump(res, f, indent=2)\nwith open(os.path.join(OUT, \"exclusions.csv\"), \"w\") as f:\n    f.write(\"step,rule,n_before,n_after,n_removed,rationale\\n\")\n    f.write(\"1,none (all parameter rows used),%d,%d,0,model.csv contains only parameters; nothing excluded\\n\" % (len(rows), len(rows)))\nprint(json.dumps({k: v for k, v in res.items() if k != \"data_issues\"}, indent=1))\n# F5-conf-04 — VFI vs closed-form policy, optimal growth with log utility\n\n## Question\nA planner maximises Σ β^t ln c_t subject to y_{t+1} = (y_t − c_t)^α (Cobb–Douglas, full depreciation, productivity 1). We compare the closed-form solution of V(y) = max_{0<c<y} {ln c + β V((y−c)^α)} with value-function iteration (VFI) on the prescribed grid, at y = 0.5, 1, 2, and at V(1).\n\n## Data and exclusions\nOnly `data/model.csv` (10 parameter rows, `n_input_rows`) is used: α = 0.36, β = 0.95, a grid of 400 points equally spaced in ln y on [0.05, 5] (`grid_points_used`, `n_analysed`), VFI tolerance 1e-8, evaluation points 0.5, 1.0, 2.0. Nothing was excluded (`out/exclusions.csv`, 10 → 10). The SHA-256 matches the bundle manifest (`out/state/data_audit.md`). None of the evaluation points is a grid node; see `data_issues`.\n\n## Method\n**Closed form (guess and verify).** Guess V(y) = A + B ln y. The right-hand side is ln c + β(A + Bα ln(y−c)). The first-order condition 1/c = αβB/(y−c) gives c = y/(1+αβB). Matching the ln y coefficients gives B = 1 + αβB, so B = 1/(1−αβ). Then c(y) = (1−αβ) y and the savings rate is (y−c)/y = αβ. Matching the constants gives A = [ln(1−αβ) + (αβ/(1−αβ)) ln(αβ)]/(1−β). With these parameters, B = 1.5198 (`diag_closed_form_B`) and A = −19.5244 (`diag_closed_form_A`). This rests on four assumptions: log utility; Cobb–Douglas technology y' = k^α with 0<α<1; full depreciation, with productivity normalised to 1; and an interior solution (0 < c < y, guaranteed because ln c → −∞ at c→0 and ln(y−c) → −∞ at c→y). Both functions are concave, so the first-order condition is sufficient. As a numerical check, brute-force maximisation of the Bellman right-hand side at y = 1 recovers V(1) and c = 0.658 to about 1e-13 (`out/state/checks.md`, item 2).\n\n**Contraction.** The Bellman operator T is monotone, and it discounts: T(V + a) = TV + βa. By Blackwell's sufficient conditions it is therefore a contraction in the sup norm with modulus β = 0.95 (`contraction_modulus`). One caveat: ln is unbounded, so on all of (0, ∞) the argument needs a weighted norm. On the compact grid used here, every per-period utility is finite and the bounded-function argument applies directly.\n\n**VFI.** We use discrete-choice VFI on the prescribed grid. For each grid point y_i, the choice is next-period output y'_j on the same grid, with consumption c_ij = y_i − (y'_j)^{1/α}. Infeasible pairs (c ≤ 0) get −∞. Iteration starts at V_0 = 0 and stops when ‖V_{n+1} − V_n‖_∞ < 1e-8. The closed form is never used inside the VFI. The policy c(y_i) comes from the argmax. Values at the off-grid evaluation points are linear interpolations in ln y between neighbouring nodes. No randomness is involved (`random_seed` = null).\n\n## Results\n| y | closed form c | VFI c |\n|---|---|---|\n| 0.5 | 0.329 (`c_closed_y05`) | 0.32882 (`c_num_y05`) |\n| 1.0 | 0.658 (`c_closed_y1`) | 0.65899 (`c_num_y1`) |\n| 2.0 | 1.316 (`c_closed_y2`) | 1.30963 (`c_num_y2`) |\n\n- The largest absolute policy discrepancy across the three points is 0.00637 (`max_abs_error_policy`), at y = 2. That is about 0.5% of consumption.\n- V(1) is −19.52441 from the closed form (`value_closed_y1`) and −19.52476 from VFI (`value_num_y1`), a gap of −3.4e-4 utils (`diag_value_error_y1`).\n- The closed-form savings rate is αβ = 0.342 (`savings_rate_closed_form`). This is the baseline.\n- VFI met the stopping rule after 361 iterations (`n_iterations`), with a final sup change of 9.997e-9 (`final_sup_change`). Status is `feasibility` = \"complete\".\n\n## Uncertainty\nThe model is deterministic, so there is no sampling uncertainty. The numerical error has two parts.\n1. **Iteration error.** It is bounded by β/(1−β)·‖V_n − V_{n−1}‖ = 1.9e-7 utils (`diag_vfi_error_bound_beta_over_1mbeta_times_final_change`). This is negligible.\n2. **Discretisation and interpolation error.** This part dominates. It depends on the log grid step of 0.01154 (`diag_grid_log_step`). Because y' is restricted to grid nodes, the implied capital k = y'^{1/α} moves in relative steps of about 0.0115/α ≈ 3.2%. The policy is therefore a step function, and its error is of the order of the grid spacing. Over all 400 nodes, the maximum policy error is 0.0251 (`diag_max_abs_policy_error_on_grid`), largest at high y, where absolute steps are bigger. The maximum value error is 4.8e-4 utils (`diag_max_abs_value_error_on_grid`).\n\nThe value function converges much more tightly than the policy. This is expected, because the objective is flat near its optimum.\n\n## Limitations and open issues\n- **Scope.** The validation covers a single parameter point (α = 0.36, β = 0.95), one grid (400 log-spaced points on [0.05, 5]) and one tolerance. It does not show accuracy for other parameters, grids, or for models without a closed form, such as partial depreciation or CRRA utility other than log.\n- **Grid truncation.** The bounds were checked: the optimal y' never hits a grid bound (`diag_policy_hits_lower_bound` = `diag_policy_hits_upper_bound` = 0).\n- **Off-grid evaluation.** The evaluation points 0.5, 1, 2 are not grid nodes, so the VFI numbers include interpolation error on top of the discrete-choice error (recorded in `data_issues`). A continuous-choice VFI would reduce the policy error. We did not use one, to stay within the prescribed design.\n- The discrete-choice formulation is a modelling choice made within the prescribed design, not a change to it. It is documented in `data_issues`.\nstep,rule,n_before,n_after,n_removed,rationale\n1,none (all parameter rows used),10,10,0,model.csv contains only parameters; nothing excluded\nchecks.md\ncontract.md\ndata_audit.md\ngate_receipts\nlog.md\nphases.json","is_error":false}]},"uuid":"7f75e6b4-76a2-417a-8e6e-b901434938fe","timestamp":"2026-10-30T04:40:00.523Z","toolUseResult":{"stdout":"parameter,value,description\nalpha,0.36,capital elasticity in y_{t+1} = k_{t+1}^alpha\nbeta,0.95,discount factor\ny_min,0.05,lower bound of the output grid\ny_max,5.0,upper bound of the output grid\ngrid_points,400,number of grid points for y (log-spaced between y_min and y_max)\ngrid_spacing,log,grid points equally spaced in ln(y)\nvfi_tolerance,1e-08,stop value-function iteration when the sup-norm change is below this\neval_y_1,0.5,evaluation point for the policy (output level)\neval_y_2,1.0,evaluation point for the policy (output level)\neval_y_3,2.0,evaluation point for the policy (output level)\n\"\"\"F5-conf-04: VFI vs closed form, optimal growth with log utility.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\nReads only data/model.csv; deterministic (no randomness).\"\"\"\nimport json, os, csv\nimport numpy as np\n\nROOT = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))\nDATA = os.path.join(ROOT, \"data\", \"model.csv\")\nOUT = os.path.join(ROOT, \"out\")\n\nwith open(DATA, newline=\"\") as f:\n    rows = list(csv.DictReader(f))\np = {r[\"parameter\"]: r[\"value\"] for r in rows}\nalpha = float(p[\"alpha\"]); beta = float(p[\"beta\"])\ny_min = float(p[\"y_min\"]); y_max = float(p[\"y_max\"]); N = int(float(p[\"grid_points\"]))\nassert p[\"grid_spacing\"].strip() == \"log\"\ntol = float(p[\"vfi_tolerance\"])\neval_y = [float(p[\"eval_y_1\"]), float(p[\"eval_y_2\"]), float(p[\"eval_y_3\"])]\n\n# ---------------- closed form (guess-and-verify V = A + B ln y) ----------------\nab = alpha * beta\nB = 1.0 / (1.0 - ab)\nA = (np.log(1.0 - ab) + ab / (1.0 - ab) * np.log(ab)) / (1.0 - beta)\nc_closed = lambda y: (1.0 - ab) * np.asarray(y)\nV_closed = lambda y: A + B * np.log(y)\nsavings_rate = ab\n\n# ---------------- VFI on prescribed grid (discrete choice of y' on grid) --------\ngrid = np.exp(np.linspace(np.log(y_min), np.log(y_max), N))\nk_needed = grid ** (1.0 / alpha)                 # capital needed to reach y' = grid[j]\nC = grid[:, None] - k_needed[None, :]            # c(i, j) = y_i - y'_j^(1/alpha)\nfeasible = C > 0\nU = np.full_like(C, -np.inf)\nU[feasible] = np.log(C[feasible])\nassert feasible.any(axis=1).all(), \"some grid point has no feasible choice\"\n\nV = np.zeros(N)\nn_iter = 0\nsup_change = np.inf\nmax_iter = 100000\nwhile sup_change >= tol and n_iter < max_iter:\n    W = U + beta * V[None, :]\n    V_new = W.max(axis=1)\n    sup_change = float(np.max(np.abs(V_new - V)))\n    V = V_new\n    n_iter += 1\nconverged = sup_change < tol\npol_j = (U + beta * V[None, :]).argmax(axis=1)\nc_pol = grid - k_needed[pol_j]\n\n# boundary diagnostics: does the optimal choice ever hit grid edges?\nhits_lower = int(np.sum(pol_j == 0)); hits_upper = int(np.sum(pol_j == N - 1))\n\n# interpolation at evaluation points: linear in ln y (grid is uniform in ln y)\nlg = np.log(grid)\ninterp = lambda f, y: float(np.interp(np.log(y), lg, f))\nc_num = [interp(c_pol, y) for y in eval_y]\nc_cl = [float(c_closed(y)) for y in eval_y]\non_grid = [bool(np.any(np.isclose(grid, y, rtol=1e-12, atol=0))) for y in eval_y]\nmax_err = float(max(abs(a - b) for a, b in zip(c_num, c_cl)))\n\n# whole-grid diagnostics (for report only)\ngrid_policy_err = float(np.max(np.abs(c_pol - c_closed(grid))))\ngrid_value_err = float(np.max(np.abs(V - V_closed(grid))))\n\ndata_issues = []\nif not all(on_grid):\n    data_issues.append({\"issue\": \"Evaluation points %s are not grid nodes of the log-spaced grid\" %\n                        [y for y, o in zip(eval_y, on_grid) if not o],\n                        \"handling\": \"VFI policy and value interpolated linearly in ln(y) between adjacent grid nodes\"})\ndata_issues.append({\"issue\": \"Design choice: the choice variable is next-period output y' restricted to the same grid (discrete-choice VFI); consumption c = y - y'^(1/alpha)\",\n                    \"handling\": \"No deviation from prescribed grid/parameters; discretisation error of order of the grid spacing is reported, not corrected\"})\n\nfeas = \"complete\" if converged else \"partial\"\nres = {\n    \"task_id\": \"F5-conf-04\",\n    \"c_closed_y05\": c_cl[0], \"c_closed_y1\": c_cl[1], \"c_closed_y2\": c_cl[2],\n    \"c_num_y05\": c_num[0], \"c_num_y1\": c_num[1], \"c_num_y2\": c_num[2],\n    \"max_abs_error_policy\": max_err,\n    \"value_closed_y1\": float(V_closed(1.0)),\n    \"value_num_y1\": interp(V, 1.0),\n    \"savings_rate_closed_form\": float(savings_rate),\n    \"contraction_modulus\": float(beta),\n    \"grid_points_used\": int(N),\n    \"n_iterations\": int(n_iter),\n    \"final_sup_change\": float(sup_change),\n    \"feasibility\": feas,\n    \"feasibility_reason\": None if converged else \"VFI did not reach tolerance within max_iter\",\n    \"data_issues\": data_issues,\n    \"n_input_rows\": {\"model.csv\": len(rows)},\n    \"n_analysed\": int(N),\n    \"random_seed\": None,\n    # supplementary diagnostics\n    \"diag_grid_log_step\": float(lg[1] - lg[0]),\n    \"diag_max_abs_policy_error_on_grid\": grid_policy_err,\n    \"diag_max_abs_value_error_on_grid\": grid_value_err,\n    \"diag_policy_hits_lower_bound\": hits_lower,\n    \"diag_policy_hits_upper_bound\": hits_upper,\n    \"diag_value_error_y1\": interp(V, 1.0) - float(V_closed(1.0)),\n    \"diag_closed_form_A\": float(A), \"diag_closed_form_B\": float(B),\n    \"diag_vfi_error_bound_beta_over_1mbeta_times_final_change\": float(beta / (1 - beta) * sup_change),\n}\nos.makedirs(OUT, exist_ok=True)\nwith open(os.path.join(OUT, \"results.json\"), \"w\") as f:\n    json.dump(res, f, indent=2)\nwith open(os.path.join(OUT, \"exclusions.csv\"), \"w\") as f:\n    f.write(\"step,rule,n_before,n_after,n_removed,rationale\\n\")\n    f.write(\"1,none (all parameter rows used),%d,%d,0,model.csv contains only parameters; nothing excluded\\n\" % (len(rows), len(rows)))\nprint(json.dumps({k: v for k, v in res.items() if k != \"data_issues\"}, indent=1))\n# F5-conf-04 — VFI vs closed-form policy, optimal growth with log utility\n\n## Question\nA planner maximises Σ β^t ln c_t subject to y_{t+1} = (y_t − c_t)^α (Cobb–Douglas, full depreciation, productivity 1). We compare the closed-form solution of V(y) = max_{0<c<y} {ln c + β V((y−c)^α)} with value-function iteration (VFI) on the prescribed grid, at y = 0.5, 1, 2, and at V(1).\n\n## Data and exclusions\nOnly `data/model.csv` (10 parameter rows, `n_input_rows`) is used: α = 0.36, β = 0.95, a grid of 400 points equally spaced in ln y on [0.05, 5] (`grid_points_used`, `n_analysed`), VFI tolerance 1e-8, evaluation points 0.5, 1.0, 2.0. Nothing was excluded (`out/exclusions.csv`, 10 → 10). The SHA-256 matches the bundle manifest (`out/state/data_audit.md`). None of the evaluation points is a grid node; see `data_issues`.\n\n## Method\n**Closed form (guess and verify).** Guess V(y) = A + B ln y. The right-hand side is ln c + β(A + Bα ln(y−c)). The first-order condition 1/c = αβB/(y−c) gives c = y/(1+αβB). Matching the ln y coefficients gives B = 1 + αβB, so B = 1/(1−αβ). Then c(y) = (1−αβ) y and the savings rate is (y−c)/y = αβ. Matching the constants gives A = [ln(1−αβ) + (αβ/(1−αβ)) ln(αβ)]/(1−β). With these parameters, B = 1.5198 (`diag_closed_form_B`) and A = −19.5244 (`diag_closed_form_A`). This rests on four assumptions: log utility; Cobb–Douglas technology y' = k^α with 0<α<1; full depreciation, with productivity normalised to 1; and an interior solution (0 < c < y, guaranteed because ln c → −∞ at c→0 and ln(y−c) → −∞ at c→y). Both functions are concave, so the first-order condition is sufficient. As a numerical check, brute-force maximisation of the Bellman right-hand side at y = 1 recovers V(1) and c = 0.658 to about 1e-13 (`out/state/checks.md`, item 2).\n\n**Contraction.** The Bellman operator T is monotone, and it discounts: T(V + a) = TV + βa. By Blackwell's sufficient conditions it is therefore a contraction in the sup norm with modulus β = 0.95 (`contraction_modulus`). One caveat: ln is unbounded, so on all of (0, ∞) the argument needs a weighted norm. On the compact grid used here, every per-period utility is finite and the bounded-function argument applies directly.\n\n**VFI.** We use discrete-choice VFI on the prescribed grid. For each grid point y_i, the choice is next-period output y'_j on the same grid, with consumption c_ij = y_i − (y'_j)^{1/α}. Infeasible pairs (c ≤ 0) get −∞. Iteration starts at V_0 = 0 and stops when ‖V_{n+1} − V_n‖_∞ < 1e-8. The closed form is never used inside the VFI. The policy c(y_i) comes from the argmax. Values at the off-grid evaluation points are linear interpolations in ln y between neighbouring nodes. No randomness is involved (`random_seed` = null).\n\n## Results\n| y | closed form c | VFI c |\n|---|---|---|\n| 0.5 | 0.329 (`c_closed_y05`) | 0.32882 (`c_num_y05`) |\n| 1.0 | 0.658 (`c_closed_y1`) | 0.65899 (`c_num_y1`) |\n| 2.0 | 1.316 (`c_closed_y2`) | 1.30963 (`c_num_y2`) |\n\n- The largest absolute policy discrepancy across the three points is 0.00637 (`max_abs_error_policy`), at y = 2. That is about 0.5% of consumption.\n- V(1) is −19.52441 from the closed form (`value_closed_y1`) and −19.52476 from VFI (`value_num_y1`), a gap of −3.4e-4 utils (`diag_value_error_y1`).\n- The closed-form savings rate is αβ = 0.342 (`savings_rate_closed_form`). This is the baseline.\n- VFI met the stopping rule after 361 iterations (`n_iterations`), with a final sup change of 9.997e-9 (`final_sup_change`). Status is `feasibility` = \"complete\".\n\n## Uncertainty\nThe model is deterministic, so there is no sampling uncertainty. The numerical error has two parts.\n1. **Iteration error.** It is bounded by β/(1−β)·‖V_n − V_{n−1}‖ = 1.9e-7 utils (`diag_vfi_error_bound_beta_over_1mbeta_times_final_change`). This is negligible.\n2. **Discretisation and interpolation error.** This part dominates. It depends on the log grid step of 0.01154 (`diag_grid_log_step`). Because y' is restricted to grid nodes, the implied capital k = y'^{1/α} moves in relative steps of about 0.0115/α ≈ 3.2%. The policy is therefore a step function, and its error is of the order of the grid spacing. Over all 400 nodes, the maximum policy error is 0.0251 (`diag_max_abs_policy_error_on_grid`), largest at high y, where absolute steps are bigger. The maximum value error is 4.8e-4 utils (`diag_max_abs_value_error_on_grid`).\n\nThe value function converges much more tightly than the policy. This is expected, because the objective is flat near its optimum.\n\n## Limitations and open issues\n- **Scope.** The validation covers a single parameter point (α = 0.36, β = 0.95), one grid (400 log-spaced points on [0.05, 5]) and one tolerance. It does not show accuracy for other parameters, grids, or for models without a closed form, such as partial depreciation or CRRA utility other than log.\n- **Grid truncation.** The bounds were checked: the optimal y' never hits a grid bound (`diag_policy_hits_lower_bound` = `diag_policy_hits_upper_bound` = 0).\n- **Off-grid evaluation.** The evaluation points 0.5, 1, 2 are not grid nodes, so the VFI numbers include interpolation error on top of the discrete-choice error (recorded in `data_issues`). A continuous-choice VFI would reduce the policy error. We did not use one, to stay within the prescribed design.\n- The discrete-choice formulation is a modelling choice made within the prescribed design, not a change to it. 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