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

# Question

A planner maximises Σ β^t ln c_t subject to y_{t+1} = (y_t − c_t)^α (Cobb–Douglas, productivity 1, full depreciation). We derive the closed-form policy and value function, solve the Bellman equation numerically by value-function iteration (VFI) on the prescribed grid, and compare the two at y = 0.5, 1, 2 (policy) and y = 1 (value).

# Data and exclusions

The only input is data/model.csv (10 rows; n_input_rows): α = 0.36, β = 0.95, grid of 400 points (grid_points_used) log-spaced on [0.05, 5], VFI tolerance 1e-8 (sup norm), evaluation points 0.5, 1.0, 2.0. These are model parameters, not observations. Nothing was excluded (out/exclusions.csv); all 400 grid points are analysed (n_analysed). No deviation from the prescribed design was needed (data_issues = []). The evaluation points are not grid nodes (diag_eval_points_on_grid all false; log step diag_grid_log_step ≈ 0.01154), so the VFI policy and value are linearly interpolated in ln y between the neighbouring nodes, as the task allows. No randomness is used (random_seed = null).

# Method

Closed form (guess and verify). Guess V(y) = A + B ln y. The first-order condition of max_c {ln c + βA + βBα ln(y − c)} gives 1/c = αβB/(y − c), so c = y/(1 + αβB). Matching the coefficient on ln y gives B = 1 + αβB ⇒ B = 1/(1 − αβ), hence c(y) = (1 − αβ) y and the savings rate is (y − c)/y = αβ. Matching constants gives A = [ln(1 − αβ) + (αβ/(1 − αβ)) ln(αβ)] / (1 − β). Assumptions: log utility, Cobb–Douglas technology y' = k^α with 0 < α < 1 and unit productivity, full depreciation, deterministic environment, 0 < β < 1, and an interior solution (0 < c < y, which holds since 0 < αβ < 1). I checked independently that this V is a fixed point of the Bellman operator at y = 0.5, 1, 2 by numerical maximisation (see out/state/checks.md, check 3).

VFI. State: y on the 400-point log grid. Choice: next-period output y'_j on the same grid, which requires capital k = (y'_j)^{1/α} and leaves c = y_i − (y'j)^{1/α}; choices with c ≤ 0 are infeasible (utility −∞). Starting from V₀ = 0, iterate V{n+1}(y_i) = max_j {ln c_ij + β V_n(y'j)} until sup|V{n+1} − V_n| < 1e-8. The policy is c at the argmax. The numerical policy never uses the closed form. No optimal choice lies at either grid bound (diag_choice_at_lower_bound = 0, diag_choice_at_upper_bound = 0), so the grid does not truncate the solution.

Contraction. The Bellman operator T satisfies Blackwell's sufficient conditions on bounded functions: monotonicity (V ≤ W ⇒ TV ≤ TW) and discounting (T(V + a) = TV + βa). Hence ‖TV − TW‖∞ ≤ β‖V − W‖∞, modulus β = 0.95 (contraction_modulus). (Log utility is unbounded on (0, ∞); on the compact discretised grid all values are bounded, so the argument applies exactly to the discrete operator that is iterated.)

# Results

• Maximum absolute policy discrepancy over the three points: 0.006375 (max_abs_error_policy), attained at y = 2 (≈ 0.5 % of c); relative errors at y = 0.5 and 1 are below 0.2 %.
• Value at y = 1 differs by 3.4e-4 utils (diag_value_abs_error_y1).
• Closed-form savings rate αβ = 0.342 (savings_rate_closed_form), i.e. c = 0.658 y.
• Contraction modulus 0.95 (contraction_modulus). VFI met the stopping rule after 361 iterations (n_iterations) with final sup change 9.997e-9 (final_sup_change).
• Over the whole grid, max policy error is 0.0251 (diag_policy_max_abs_error_on_grid) and max value error 4.8e-4 (diag_value_max_abs_error_on_grid).

# Uncertainty

The problem is deterministic; there is no sampling uncertainty. The numerical error has two sources: (i) iteration error: by the contraction property, ‖V_n − V*‖∞ ≤ β/(1 − β)·‖V_n − V_{n−1}‖∞ ≤ 1.9e-7 (diag_value_tolerance_bound), negligible; (ii) discretisation error: the choice y' is restricted to grid nodes and results at off-grid points are linearly interpolated. This dominates: the 3.4e-4 value gap at y = 1 is far larger than the 1.9e-7 iteration bound. The policy error is roughly proportional to the change in capital between adjacent choice nodes, Δk ≈ (k/α)·Δln y', which grows with y; this explains why the error is largest at y = 2 and at the top of the grid.

# Limitations and open issues

• The validation holds only for the single parameter point (α = 0.36, β = 0.95) and the single grid (400 log-spaced points on [0.05, 5], tolerance 1e-8). Discretisation error depends on the grid density (log step ≈ 0.0115), on α (through k = y'^{1/α}), on the level of y, and on the interpolation rule for off-grid evaluation points. A finer grid or a continuous choice of c would reduce it; these were not run because the task prescribes the grid.
• The VFI policy is a step function between nodes, so pointwise errors at individual nodes can be as large as diag_policy_max_abs_error_on_grid, which is larger than the error at the three evaluation points.
• The closed form needs log utility, Cobb–Douglas technology, full depreciation and an interior solution; it does not extend to partial depreciation or CRRA utility with σ ≠ 1.
• Feasibility: complete (feasibility); no part of the task was left undone.