# F5-conf-04 — VFI vs closed-form policy, optimal growth with log utility

## Question
A planner maximises Σ β^t ln c_t subject to y_{t+1} = (y_t − c_t)^α (full depreciation, productivity 1). Does value-function iteration (VFI) on the prescribed grid reproduce the known closed-form policy and value function, and how large is the discrepancy? Parameters (data/model.csv): α = 0.36, β = 0.95; 400-point grid equally spaced in ln y on [0.05, 5]; stopping tolerance 1e-8; evaluation points y = 0.5, 1, 2.

## Data and exclusions
Only `data/model.csv` is used (10 parameter rows, `n_input_rows`; SHA-256 recorded in `out/state/data_audit.md`). Nothing is excluded (`out/exclusions.csv`: 10 → 10). No deviation from the prescribed design was necessary (`data_issues` = []). The grid has `grid_points_used` = 400 nodes (`n_analysed` = 400). No randomness is used (`random_seed` = null).

## Method
**Closed form (guess-and-verify).** Assumptions: log utility, Cobb–Douglas technology y' = k^α with 0<α<1, full depreciation, productivity 1, 0<β<1, interior solution 0<c<y. Guess V(y) = A + B ln y. The Bellman maximand 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: B = 1 + αβB ⇒ B = 1/(1 − αβ), hence

- c(y) = (1 − αβ) y, savings rate (y − c)/y = αβ,
- A = [ln(1 − αβ) + (αβ/(1 − αβ)) ln(αβ)] / (1 − β), V(y) = A + ln y /(1 − αβ).

The constant matches on both sides, so the guess is verified; the policy is interior for all y>0. (An independent numerical maximisation of the Bellman right-hand side at y = 0.3, 1, 3 reproduced this policy and value; `out/state/checks.md`, check 4.)

**Contraction.** The Bellman operator T satisfies Blackwell's sufficient conditions, monotonicity and discounting (T(V + a) = TV + βa). So it is a contraction in the sup norm with modulus β: ‖TV − TW‖∞ ≤ β‖V − W‖∞ (`contraction_modulus` = 0.95). Log utility is unbounded on (0, ∞). On the compact grid, however, every node has feasible choices with bounded log consumption, so the discretised operator acts on bounded functions on a finite set and the argument applies directly.

**VFI.** This is discrete VFI on the prescribed grid. The choice variable is next-period output y' restricted to grid nodes, so consumption is c = y − y'^{1/α} (infeasible choices get −∞). The initial guess is V₀ = 0. Iteration stops once the sup-norm change drops below 1e-8. The numerical policy comes from the argmax of the converged Bellman operator; the closed form is never used to build it. None of the evaluation points is a grid node, so the policy and V are interpolated linearly in ln y between nodes. Everything is generated by `out/analysis.py`.

## Results
| quantity | closed form | VFI |
|---|---|---|
| c(0.5) | 0.329 (`c_closed_y05`) | 0.328818 (`c_num_y05`) |
| c(1.0) | 0.658 (`c_closed_y1`) | 0.658994 (`c_num_y1`) |
| c(2.0) | 1.316 (`c_closed_y2`) | 1.309625 (`c_num_y2`) |
| V(1) | −19.524412 (`value_closed_y1`) | −19.524756 (`value_num_y1`) |

- Maximum absolute policy discrepancy over the three evaluation points: 0.006375 (`max_abs_error_policy`), reached at y = 2. At y = 0.5 the discrepancy is 1.8e-4 and at y = 1 it is 9.9e-4 (`diagnostics.abs_error_policy_each_eval_point`). Relative to c, the error is at most about 0.5%.
- The value at y = 1 differs by about 3.4e-4 utils, i.e. about 0.002% of |V(1)| (`value_num_y1`, `value_closed_y1`).
- Closed-form savings rate αβ = 0.342 (`savings_rate_closed_form`, also `baseline_savings_rate_closed_form`). The VFI-implied savings rates at the three points are 0.342, 0.341 and 0.345 (`out/state/checks.md`, check 12).
- VFI converged after 361 iterations (`n_iterations`) with a final sup-norm change of 9.997e-9 (`final_sup_change`).
- Contraction modulus β = 0.95 (`contraction_modulus`).

## Uncertainty
There is no sampling uncertainty because the model is deterministic. The numerical error has two sources:
1. **Iteration truncation.** By the contraction property, ‖V_n − V*_grid‖∞ ≤ β/(1−β)·‖V_n − V_{n−1}‖∞ ≈ 1.9e-7 (`diagnostics.vfi_truncation_bound_beta_over_1mbeta_times_last_change`). This is negligible next to the discretisation error.
2. **Discretisation.** It dominates. Because y' is restricted to nodes (log step 0.01154, `diagnostics.log_grid_step`), the chosen investment k = y'^{1/α} can be off by up to about half a step in y'. That maps into a consumption error of roughly (1/α)·k·(Δln y/2), so the error grows with y. This explains why the largest error is at y = 2 and why the largest error over all grid nodes is 0.0251 (`diagnostics.max_abs_error_policy_all_grid_nodes`, near y = 5). Interpolating between nodes in ln y adds a further, smaller error. The maximum value-function error over all nodes is 4.8e-4 (`diagnostics.max_abs_error_value_all_grid_nodes`).

As a sensitivity check, the maximum error at the evaluation points is 0.0059 with 200 nodes and 0.0014 with 800 nodes (`diagnostics.sensitivity_max_abs_error_policy_by_grid_points`), against 0.0064 with the prescribed 400. The error shrinks with refinement overall but is **not monotone**: with a discrete choice grid, the error at a given point depends on how the nodes line up with the true optimum. These alternative grids are diagnostics only; every headline number uses the prescribed 400-node grid.

## Limitations and open issues
- The validation covers one parameter point (α = 0.36, β = 0.95), one grid (400 log-spaced nodes on [0.05, 5]) and three evaluation points. It does not establish accuracy for other parameters, grids or regions such as y near 0.
- The closed form depends on log utility, Cobb–Douglas technology, full depreciation and an interior solution. With partial depreciation or CRRA ≠ 1 utility, no such closed form exists and this benchmark does not apply.
- The discrete-choice VFI gives a step-function policy. Its accuracy is limited by the grid spacing, not by the tolerance. Continuous optimisation with interpolated V would reduce the error but was not used, in order to respect the prescribed grid design.
- The closed-form next-period output for y in [0.05, 5] lies within [0.231, 1.213], and the VFI policy never chose a boundary node, so the grid bounds do not bind (`diagnostics.closed_form_yprime_range_on_grid`, `diagnostics.policy_nodes_at_lower_bound`, `diagnostics.policy_nodes_at_upper_bound`).