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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.\nmodel.csv","is_error":false}]},"uuid":"c395fbcc-485c-4121-a3e7-416f341bb8d3","timestamp":"2026-10-30T04:37:34.770Z","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). 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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.\nmodel.csv","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"695b4b09-fd10-4c9f-a1c1-e09a38d4f569","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0126","sessionId":"5967f11a-4eb7-4b3f-a0f4-6ff31ff9fe75","version":"2.1.314","gitBranch":"HEAD"}14{"type":"api-request-blob","sessionId":"5967f11a-4eb7-4b3f-a0f4-6ff31ff9fe75","hash":"d51dcbabdcf631a5edf1162af05bcf1169b859fb61e78ebb0a045a1aef5386e2","message":{"role":"assistant","content":[{"type":"tool_use","id":"toolu_013NLvE2SB7cYvQcjTHWyLfe","name":"Bash","input":{"command":"cat TASK.md; ls -R data | head -50","description":"Read task and list 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-p out; cat > out/analysis.py <<'EOF'\n\"\"\"F5-conf-04: VFI vs closed form for log-utility Cobb-Douglas growth (full depreciation).\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\n\"\"\"\nimport json, os\nimport numpy as np\nimport pandas as pd\n\nROOT = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))\ndf = pd.read_csv(os.path.join(ROOT, \"data\", \"model.csv\"))\np = dict(zip(df[\"parameter\"], df[\"value\"]))\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\"]))\ntol = float(p[\"vfi_tolerance\"])\nassert str(p[\"grid_spacing\"]).strip() == \"log\"\nevals = [float(p[\"eval_y_1\"]), float(p[\"eval_y_2\"]), float(p[\"eval_y_3\"])]\n\n# ---------- closed form (guess V = A + B ln y, verified in report) ----------\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)\n\n# ---------- VFI: discrete choice of next-period output y' on the grid ----------\nlny = np.linspace(np.log(y_min), np.log(y_max), N)\ny = np.exp(lny)\nk_needed = y ** (1.0 / alpha) # savings needed to reach y' = grid[j]\nC = y[:, None] - k_needed[None, :] # consumption for (state i, choice j)\nfeasible = C > 0\nU = np.full_like(C, -np.inf)\nU[feasible] = np.log(C[feasible])\n\nV = np.zeros(N)\nn_it = 0\nwhile True:\n RHS = U + beta * V[None, :]\n Vn = RHS.max(axis=1)\n sup = float(np.max(np.abs(Vn - V)))\n V = Vn\n n_it += 1\n if sup < tol:\n break\npol_j = (U + beta * V[None, :]).argmax(axis=1)\nc_grid = C[np.arange(N), pol_j]\n\n# check whether the choice ever hits grid bounds\nhits_lower = bool(np.any(pol_j == 0)); hits_upper = bool(np.any(pol_j == N - 1))\n\n# interpolate (linear in ln y) at evaluation points\ninterp = lambda f, x: float(np.interp(np.log(x), lny, f))\nc_num = [interp(c_grid, e) for e in evals]\nc_cl = [float(c_closed(e)) for e in evals]\nmax_err = float(max(abs(a - b) for a, b in zip(c_num, c_cl)))\nv_num_1 = interp(V, 1.0)\nv_cl_1 = float(V_closed(1.0))\n\n# whole-grid diagnostics (not required keys)\nsup_policy_err_grid = float(np.max(np.abs(c_grid - c_closed(y))))\nsup_value_err_grid = float(np.max(np.abs(V - V_closed(y))))\n\nissues = []\nfor e in evals:\n if not np.any(np.isclose(y, e, rtol=0, atol=1e-12)):\n issues.append({\"issue\": f\"evaluation point y={e} is not a grid node\",\n \"handling\": \"VFI policy/value linearly interpolated in ln y between adjacent grid nodes\"})\nissues.append({\"issue\": \"VFI choice variable not specified in model.csv\",\n \"handling\": \"next-period output y' chosen from the same 400-point log grid (discrete choice, no off-grid interpolation of V); V initialised at 0\"})\n\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], 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not a grid node\",\n \"handling\": \"VFI policy/value linearly interpolated in ln y between adjacent grid nodes\"\n },\n {\n \"issue\": \"VFI choice variable not specified in model.csv\",\n \"handling\": \"next-period output y' chosen from the same 400-point log grid (discrete choice, no off-grid interpolation of V); V initialised at 0\"\n }\n ],\n \"n_input_rows\": {\n \"model.csv\": 10\n },\n \"n_analysed\": 400,\n \"random_seed\": null,\n \"diagnostics\": {\n \"grid_log_step\": 0.011541779914757022,\n \"sup_abs_policy_error_on_grid\": 0.02512152290513381,\n \"sup_abs_value_error_on_grid\": 0.0004835155258184898,\n \"value_error_bound_from_tolerance\": 1.8999999999999982e-07,\n \"policy_hits_lower_grid_bound\": false,\n \"policy_hits_upper_grid_bound\": false,\n \"closed_form_A\": -19.524412221722393,\n \"closed_form_B\": 1.519756838905775,\n \"steady_state_y\": 0.5468770327603131\n 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-p out; cat > out/analysis.py <<'EOF'\n\"\"\"F5-conf-04: VFI vs closed form for log-utility Cobb-Douglas growth (full depreciation).\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\n\"\"\"\nimport json, os\nimport numpy as np\nimport pandas as pd\n\nROOT = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))\ndf = pd.read_csv(os.path.join(ROOT, \"data\", \"model.csv\"))\np = dict(zip(df[\"parameter\"], df[\"value\"]))\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\"]))\ntol = float(p[\"vfi_tolerance\"])\nassert str(p[\"grid_spacing\"]).strip() == \"log\"\nevals = [float(p[\"eval_y_1\"]), float(p[\"eval_y_2\"]), float(p[\"eval_y_3\"])]\n\n# ---------- closed form (guess V = A + B ln y, verified in report) ----------\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)\n\n# ---------- VFI: discrete choice of next-period output y' on the grid ----------\nlny = np.linspace(np.log(y_min), np.log(y_max), N)\ny = np.exp(lny)\nk_needed = y ** (1.0 / alpha) # savings needed to reach y' = grid[j]\nC = y[:, None] - k_needed[None, :] # consumption for (state i, choice j)\nfeasible = C > 0\nU = np.full_like(C, -np.inf)\nU[feasible] = np.log(C[feasible])\n\nV = np.zeros(N)\nn_it = 0\nwhile True:\n RHS = U + beta * V[None, :]\n Vn = RHS.max(axis=1)\n sup = float(np.max(np.abs(Vn - V)))\n V = Vn\n n_it += 1\n if sup < tol:\n break\npol_j = (U + beta * V[None, :]).argmax(axis=1)\nc_grid = C[np.arange(N), pol_j]\n\n# check whether the choice ever hits grid bounds\nhits_lower = bool(np.any(pol_j == 0)); hits_upper = bool(np.any(pol_j == N - 1))\n\n# interpolate (linear in ln y) at evaluation points\ninterp = lambda f, x: float(np.interp(np.log(x), lny, f))\nc_num = [interp(c_grid, e) for e in evals]\nc_cl = [float(c_closed(e)) for e in evals]\nmax_err = float(max(abs(a - b) for a, b in zip(c_num, c_cl)))\nv_num_1 = interp(V, 1.0)\nv_cl_1 = float(V_closed(1.0))\n\n# whole-grid diagnostics (not required keys)\nsup_policy_err_grid = float(np.max(np.abs(c_grid - c_closed(y))))\nsup_value_err_grid = float(np.max(np.abs(V - V_closed(y))))\n\nissues = []\nfor e in evals:\n if not np.any(np.isclose(y, e, rtol=0, atol=1e-12)):\n issues.append({\"issue\": f\"evaluation point y={e} is not a grid node\",\n \"handling\": \"VFI policy/value linearly interpolated in ln y between adjacent grid nodes\"})\nissues.append({\"issue\": \"VFI choice variable not specified in model.csv\",\n \"handling\": \"next-period output y' chosen from the same 400-point log grid (discrete choice, no off-grid interpolation of V); V initialised at 0\"})\n\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\": v_cl_1, \"value_num_y1\": v_num_1,\n \"savings_rate_closed_form\": float(ab),\n \"contraction_modulus\": float(beta),\n \"grid_points_used\": int(N),\n \"n_iterations\": int(n_it),\n \"final_sup_change\": sup,\n \"feasibility\": \"complete\", \"feasibility_reason\": None,\n \"data_issues\": issues,\n \"n_input_rows\": {\"model.csv\": int(len(df))},\n \"n_analysed\": int(N),\n \"random_seed\": None,\n \"diagnostics\": {\n \"grid_log_step\": float(lny[1] - lny[0]),\n \"sup_abs_policy_error_on_grid\": sup_policy_err_grid,\n \"sup_abs_value_error_on_grid\": sup_value_err_grid,\n \"value_error_bound_from_tolerance\": float(beta * tol / (1 - beta)),\n \"policy_hits_lower_grid_bound\": hits_lower,\n \"policy_hits_upper_grid_bound\": hits_upper,\n \"closed_form_A\": float(A), \"closed_form_B\": float(B),\n \"steady_state_y\": float(ab ** (alpha / (1 - alpha))),\n },\n}\nwith open(os.path.join(ROOT, \"out\", \"results.json\"), \"w\") as f:\n json.dump(res, f, indent=2)\nprint(json.dumps(res, indent=2))\nEOF\n~/wp29/venv/bin/python out/analysis.py","description":"Write and run VFI 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> out/report.md <<'EOF'\n# F5-conf-04 — VFI vs closed-form policy, log-utility optimal growth\n\n## Question\nPlanner maximises Σ β^t ln c_t s.t. y_{t+1} = (y_t − c_t)^α (productivity 1, full depreciation). Compare the closed-form policy/value function with the value-function-iteration (VFI) solution on the prescribed grid, at y ∈ {0.5, 1, 2}, and report V(1), the closed-form savings rate, and the contraction modulus of the Bellman operator.\n\n## Data and exclusions\n`data/model.csv` (10 rows, all used): α = 0.36, β = 0.95, grid of 400 points equally spaced in ln y on [0.05, 5], VFI tolerance 1e-8 (sup-norm change), evaluation points 0.5, 1.0, 2.0. Nothing excluded. Design decisions (also in `data_issues`):\n- none of the three evaluation points is a grid node, so the VFI policy and value are interpolated linearly in ln y between the two nearest nodes;\n- the file does not say what the choice variable is. I chose next-period output y′ from the same grid (discrete choice), so c = y − (y′)^{1/α}. V starts at 0.\n\n## Method\n**Closed form (guess and verify).** Assumptions: log utility, Cobb–Douglas technology y′ = k^α with productivity 1, full depreciation, and an interior solution (0 < c < y; the Inada property of ln guarantees this). Guess V(y) = A + B ln y. The Bellman RHS is ln c + β[A + Bα ln(y − c)]. The FOC 1/c = αβB/(y − c) gives c = y/(1 + αβB). Matching the coefficients on ln y gives B = 1 + αβB, so B = 1/(1 − αβ). Matching the constants gives A = [ln(1 − αβ) + (αβ/(1 − αβ)) ln(αβ)]/(1 − β). Therefore\n- c(y) = (1 − αβ) y, with savings rate (y − c)/y = αβ = 0.342;\n- V(y) = A + ln y/(1 − αβ), with A = −19.5244 and B = 1.51976.\n\n**Contraction.** The Bellman operator T is monotone (v ≤ w ⇒ Tv ≤ Tw) and discounts (T(v + a) = Tv + βa). By Blackwell's sufficient conditions, T is a contraction in the sup norm with modulus β = 0.95. One caveat: ln is unbounded on (0, ∞), so the theorem applies directly to the discretised problem, which is bounded on a compact grid. For the continuous problem, a weighted-norm or bounded-state argument is needed (for example, restricting y to an invariant compact set). The tolerance gives an error bound for the fixed point: ‖V_n − V*‖ ≤ β·tol/(1 − β) = 1.9e-7.\n\n**VFI.** All 400 × 400 (state, choice) pairs are evaluated in one vectorised step, with infeasible pairs (c ≤ 0) set to −∞. Iteration continues until the sup-norm change is below 1e-8. The policy is the argmax at the fixed point. The optimal y′ never hits either grid bound.\n\n## Results\n| y | c closed form | c VFI | abs diff |\n|---|---|---|---|\n| 0.5 | 0.32900 | 0.32882 | 0.00018 |\n| 1.0 | 0.65800 | 0.65899 | 0.00099 |\n| 2.0 | 1.31600 | 1.30963 | 0.00637 |\n\n- Max abs policy error over the evaluation points: **0.00637**, at y = 2, about 0.5% of c.\n- V(1): closed form −19.52441, VFI −19.52476. The difference of 3.4e-4 is far larger than the tolerance bound, so it comes from discretisation.\n- Savings rate (closed form): 0.342. Contraction modulus: 0.95.\n- VFI converged in 361 iterations, with final sup change 9.997e-9.\n- Over the whole grid: sup |c_VFI − c| = 0.0251 (at the upper end of the grid), and sup |V_VFI − V| = 4.8e-4.\n\n## Uncertainty\nThere is no sampling uncertainty: the model is deterministic and the parameters are given. The numerical error has two parts:\n1. **Iteration error.** It is bounded by 1.9e-7 in V, which is negligible.\n2. **Discretisation error.** This dominates. With discrete choice of y′ on a log grid with step h = 0.01154, the chosen ln y′ can be off by up to h/2. The implied savings k = y′^{1/α} is then off by a relative amount of up to about h/(2α) ≈ 1.6%. So the policy error grows roughly in proportion to k = αβy, which is why the error is largest at y = 2. Linear interpolation between nodes adds a smaller error; the true policy is linear in y but not in ln y. The error shrinks as h → 0, i.e. with more grid points, or with continuous choice using interpolation of V.\n\n## Limitations and open issues\n- I validated only one parameter point (α = 0.36, β = 0.95) and one grid (400 log-spaced points on [0.05, 5]). I did not study grid refinement or other parameter values.\n- The result depends on my choice of discrete-choice design (choosing y′ on the grid). A continuous-choice or endogenous-grid method would produce different, probably smaller, errors.\n- The closed form relies on the special case of log utility, Cobb–Douglas technology and full depreciation. Outside that case the comparison does not extend.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"Closed-form policy c(y) = (1-alpha*beta) y gives c = 0.329, 0.658, 1.316 at y = 0.5, 1, 2; savings rate 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, full depreciation, interior solution.\"},\n {\"id\":\"C2\",\"text\":\"VFI on the prescribed 400-point log grid reproduces the closed-form policy at the evaluation points within 0.0064 (about 0.5%).\",\"results_keys\":[\"c_num_y05\",\"c_num_y1\",\"c_num_y2\",\"max_abs_error_policy\",\"grid_points_used\"],\"status\":\"supported\",\"caveat\":\"Discrete choice of y' on the grid; error is discretisation-driven and grows with y.\"},\n {\"id\":\"C3\",\"text\":\"VFI value at y=1 (-19.5248) matches the closed form (-19.5244) to about 3.4e-4.\",\"results_keys\":[\"value_num_y1\",\"value_closed_y1\"],\"status\":\"supported\",\"caveat\":\"Gap dominated by discretisation, not by iteration tolerance.\"},\n {\"id\":\"C4\",\"text\":\"The Bellman operator is a sup-norm contraction with modulus beta = 0.95 (Blackwell); VFI converged in 361 iterations to a sup change below 1e-8.\",\"results_keys\":[\"contraction_modulus\",\"n_iterations\",\"final_sup_change\"],\"status\":\"supported\",\"caveat\":\"Blackwell applies directly to the bounded discretised problem; unbounded log utility in the continuous problem needs a weighted-norm argument.\"}\n]\nEOF\n~/wp29/venv/bin/python out/analysis.py >/dev/null && ~/wp29/venv/bin/python -c \"import json;json.load(open('out/claims.json'));print('ok')\"","description":"Write report and claims, verify rerun"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":3767,"cache_read_input_tokens":4359,"output_tokens":2807,"output_tokens_details":{"thinking_tokens":104},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":3767},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXoYHZ62ArYLaVQUEVSW","requestRef":"d3ffa8a6-e3fe-44b8-aaa3-7acd99849dfd","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"78cf185e-935c-49a1-80ee-f0a008f5a93e","timestamp":"2026-10-30T04:38:24.718Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0126","sessionId":"5967f11a-4eb7-4b3f-a0f4-6ff31ff9fe75","version":"2.1.314","gitBranch":"HEAD"}29{"parentUuid":"78cf185e-935c-49a1-80ee-f0a008f5a93e","isSidechain":false,"promptId":"898d07de-16c6-43ba-8c70-82683f527d4d","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_018r6ZNxdbv46uxp1QtAdLZw","type":"tool_result","content":"ok","is_error":false}]},"uuid":"1062f9c0-6c85-4e1f-b54e-d7ead9f47527","timestamp":"2026-10-30T04:38:25.047Z","toolUseResult":{"stdout":"ok","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"78cf185e-935c-49a1-80ee-f0a008f5a93e","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0126","sessionId":"5967f11a-4eb7-4b3f-a0f4-6ff31ff9fe75","version":"2.1.314","gitBranch":"HEAD"}30{"type":"api-request-blob","sessionId":"5967f11a-4eb7-4b3f-a0f4-6ff31ff9fe75","hash":"a0ca0817a857973c88b30330de1238119d63f8165981153a522b650f9dfdb583","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_018r6ZNxdbv46uxp1QtAdLZw","name":"Bash","input":{"command":"cat > out/report.md <<'EOF'\n# F5-conf-04 — VFI vs closed-form policy, log-utility optimal growth\n\n## Question\nPlanner maximises Σ β^t ln c_t s.t. y_{t+1} = (y_t − c_t)^α (productivity 1, full depreciation). Compare the closed-form policy/value function with the value-function-iteration (VFI) solution on the prescribed grid, at y ∈ {0.5, 1, 2}, and report V(1), the closed-form savings rate, and the contraction modulus of the Bellman operator.\n\n## Data and exclusions\n`data/model.csv` (10 rows, all used): α = 0.36, β = 0.95, grid of 400 points equally spaced in ln y on [0.05, 5], VFI tolerance 1e-8 (sup-norm change), evaluation points 0.5, 1.0, 2.0. Nothing excluded. Design decisions (also in `data_issues`):\n- none of the three evaluation points is a grid node, so the VFI policy and value are interpolated linearly in ln y between the two nearest nodes;\n- the file does not say what the choice variable is. I chose next-period output y′ from the same grid (discrete choice), so c = y − (y′)^{1/α}. V starts at 0.\n\n## Method\n**Closed form (guess and verify).** Assumptions: log utility, Cobb–Douglas technology y′ = k^α with productivity 1, full depreciation, and an interior solution (0 < c < y; the Inada property of ln guarantees this). Guess V(y) = A + B ln y. The Bellman RHS is ln c + β[A + Bα ln(y − c)]. The FOC 1/c = αβB/(y − c) gives c = y/(1 + αβB). Matching the coefficients on ln y gives B = 1 + αβB, so B = 1/(1 − αβ). Matching the constants gives A = [ln(1 − αβ) + (αβ/(1 − αβ)) ln(αβ)]/(1 − β). Therefore\n- c(y) = (1 − αβ) y, with savings rate (y − c)/y = αβ = 0.342;\n- V(y) = A + ln y/(1 − αβ), with A = −19.5244 and B = 1.51976.\n\n**Contraction.** The Bellman operator T is monotone (v ≤ w ⇒ Tv ≤ Tw) and discounts (T(v + a) = Tv + βa). By Blackwell's sufficient conditions, T is a contraction in the sup norm with modulus β = 0.95. One caveat: ln is unbounded on (0, ∞), so the theorem applies directly to the discretised problem, which is bounded on a compact grid. For the continuous problem, a weighted-norm or bounded-state argument is needed (for example, restricting y to an invariant compact set). The tolerance gives an error bound for the fixed point: ‖V_n − V*‖ ≤ β·tol/(1 − β) = 1.9e-7.\n\n**VFI.** All 400 × 400 (state, choice) pairs are evaluated in one vectorised step, with infeasible pairs (c ≤ 0) set to −∞. Iteration continues until the sup-norm change is below 1e-8. The policy is the argmax at the fixed point. The optimal y′ never hits either grid bound.\n\n## Results\n| y | c closed form | c VFI | abs diff |\n|---|---|---|---|\n| 0.5 | 0.32900 | 0.32882 | 0.00018 |\n| 1.0 | 0.65800 | 0.65899 | 0.00099 |\n| 2.0 | 1.31600 | 1.30963 | 0.00637 |\n\n- Max abs policy error over the evaluation points: **0.00637**, at y = 2, about 0.5% of c.\n- V(1): closed form −19.52441, VFI −19.52476. The difference of 3.4e-4 is far larger than the tolerance bound, so it comes from discretisation.\n- Savings rate (closed form): 0.342. Contraction modulus: 0.95.\n- VFI converged in 361 iterations, with final sup change 9.997e-9.\n- Over the whole grid: sup |c_VFI − c| = 0.0251 (at the upper end of the grid), and sup |V_VFI − V| = 4.8e-4.\n\n## Uncertainty\nThere is no sampling uncertainty: the model is deterministic and the parameters are given. The numerical error has two parts:\n1. **Iteration error.** It is bounded by 1.9e-7 in V, which is negligible.\n2. **Discretisation error.** This dominates. With discrete choice of y′ on a log grid with step h = 0.01154, the chosen ln y′ can be off by up to h/2. The implied savings k = y′^{1/α} is then off by a relative amount of up to about h/(2α) ≈ 1.6%. So the policy error grows roughly in proportion to k = αβy, which is why the error is largest at y = 2. Linear interpolation between nodes adds a smaller error; the true policy is linear in y but not in ln y. The error shrinks as h → 0, i.e. with more grid points, or with continuous choice using interpolation of V.\n\n## Limitations and open issues\n- I validated only one parameter point (α = 0.36, β = 0.95) and one grid (400 log-spaced points on [0.05, 5]). I did not study grid refinement or other parameter values.\n- The result depends on my choice of discrete-choice design (choosing y′ on the grid). A continuous-choice or endogenous-grid method would produce different, probably smaller, errors.\n- The closed form relies on the special case of log utility, Cobb–Douglas technology and full depreciation. Outside that case the comparison does not extend.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"Closed-form policy c(y) = (1-alpha*beta) y gives c = 0.329, 0.658, 1.316 at y = 0.5, 1, 2; savings rate 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, full depreciation, interior solution.\"},\n {\"id\":\"C2\",\"text\":\"VFI on the prescribed 400-point log grid reproduces the closed-form policy at the evaluation points within 0.0064 (about 0.5%).\",\"results_keys\":[\"c_num_y05\",\"c_num_y1\",\"c_num_y2\",\"max_abs_error_policy\",\"grid_points_used\"],\"status\":\"supported\",\"caveat\":\"Discrete choice of y' on the grid; error is discretisation-driven and grows with y.\"},\n {\"id\":\"C3\",\"text\":\"VFI value at y=1 (-19.5248) matches the closed form (-19.5244) to about 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finished the task, and the numerical solution closely matches the closed form. The analysis script runs cleanly and regenerates `out/results.json` from `data/model.csv`.\n\n**Closed form.** I derived it by guessing V(y) = A + B ln y and checking it against the Bellman equation:\n- **Policy:** consumption is c(y) = (1 − αβ)y, so the savings rate is αβ = 0.342.\n- **Value function:** V(y) = A + ln y / (1 − αβ), with A = −19.5244.\n- **Contraction modulus:** the Bellman operator shrinks distances by a factor of β = 0.95 (Blackwell's conditions).\n\n**Numerical solution (value-function iteration, 400 points evenly spaced in ln y on [0.05, 5]):**\n\n| y | closed form | numerical | difference |\n|---|---|---|---|\n| 0.5 | 0.329 | 0.32882 | 0.0002 |\n| 1.0 | 0.658 | 0.65899 | 0.0010 |\n| 2.0 | 1.316 | 1.30963 | 0.0064 |\n\n- **Largest policy gap at the three points:** 0.0064, at y = 2, about 0.5% of consumption.\n- **Value at y = 1:** −19.52476 numerically against −19.52441 in closed form.\n- **Convergence:** 361 iterations, with a last change of 1.0e-8.\n\n**Where the error comes from.** Almost all of it comes from the coarseness of the grid, not from stopping the iteration too early. 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