# 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 (Cobb–Douglas, productivity 1, full depreciation). We derive the closed-form policy and value, 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

Single input data/model.csv (10 rows; SHA-256 098123…195b, matching the bundle manifest): 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), evaluation points 0.5, 1.0, 2.0 (n_input_rows). No rows excluded (out/exclusions.csv); data_issues is empty — no deviation from the prescribed design was needed. Grid points used: 400 (grid_points_used, n_analysed). None of the evaluation points is a grid node (diag_eval_points_on_grid_node), so the VFI policy is interpolated, as the task allows.

# Method

Closed form (guess-and-verify). Guess V(y) = A + B ln y. The first-order condition of max_c {ln c + β(A + Bα ln(y − c))} gives 1/c = αβB/(y − c), i.e. 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 savings rate (y − c)/y = αβ. Matching constants gives A = [ln(1 − αβ) + (αβ/(1 − αβ)) ln(αβ)]/(1 − β), so V(1) = A. The verification was also done numerically: maximising the right-hand side with this V at y = 0.5, 1, 2 reproduces c = (1 − αβ)y and T V = V (out/state/checks.md). Assumptions: log utility; Cobb–Douglas technology with productivity 1; full depreciation; interior solution (0 < c < y, guaranteed by the Inada property of ln); 0 < α < 1, 0 < β < 1.

VFI. On the log grid, the choice variable is next-period output y' restricted to the grid; consumption is c = y − y'^{1/α} (infeasible pairs get −∞). Starting from V₀ = 0, V_{n+1}(y_i) = max_j {ln c_ij + β V_n(y_j)} is iterated (vectorised 400×400) until the sup-norm change is below 1e-8. The policy is the argmax; it never uses the closed form. Policy values at the evaluation points are linearly interpolated in y between neighbouring grid nodes; V(1) is linearly interpolated in ln y (the coordinate in which the grid is uniform). No randomness (random_seed = null).

Contraction. On the discretised (finite, bounded) problem the Bellman operator 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 (contraction_modulus = 0.95). The state space is effectively compact: under the optimal policy y' = (αβy)^α stays in [0.231, 1.213] ⊂ [0.05, 5] (diag_closed_yprime_min, diag_closed_yprime_max), and on the grid ln c is bounded, so the bounded-returns theory applies. (On the unbounded domain with log utility one would need a weighted-norm argument; this is not required here.)

# Results

• Maximum absolute policy discrepancy over the three points: 0.00641 units of output (max_abs_error_policy), attained at y = 2; the largest relative discrepancy is 0.49 % (diag_policy_max_rel_error_eval).
• Value at y = 1 differs by 3.4e-4 utils (diag_value_abs_error_y1); the sup-norm value error over the grid is 4.8e-4 (diag_value_sup_error_grid).
• Closed-form savings rate αβ = 0.342 (savings_rate_closed_form).
• VFI converged in 361 iterations (n_iterations) with final sup-norm change 9.997e-9 (final_sup_change) < 1e-8.
• Contraction modulus β = 0.95 (contraction_modulus).
• The VFI policy never hits the edge of the grid (diag_policy_edge_hits = 0).

# Uncertainty

The problem is deterministic; there is no sampling uncertainty and no confidence interval is meaningful. Numerical error has two sources:

1. Stopping rule. By the contraction property, ‖V_n − V*grid‖ ≤ β/(1 − β)·‖V_n − V{n−1}‖ ≈ 1.9e-7 utils (diag_vfi_tolerance_error_bound) — negligible.
2. Discretisation, which dominates: the choice of y' is restricted to grid nodes with log-step 0.01154 (diag_grid_log_step), so capital k = y'^{1/α} can be off by up to about step/(2α) ≈ 1.6 % in relative terms, i.e. an error in c of up to about 0.83 % of c; the observed relative errors (≤ 0.49 %) are within that bound. Linear interpolation between nodes adds a smaller component. The error grows in absolute terms with y because c and k scale with y (sup over the grid: 0.025 at y = 5, diag_policy_sup_error_grid), and shrinks proportionally with the grid step (more points or a narrower range); it also depends on α (through 1/α) and on β only through the tolerance bound.

# Limitations and open issues

• The validation covers only the prescribed parameter point (α = 0.36, β = 0.95) and grid (400 log-spaced points on [0.05, 5]); no other grids or parameters were tested, as the task does not require it.
• The closed form relies on log utility, Cobb–Douglas technology, full depreciation and an interior solution; with partial depreciation or CRRA ≠ 1 no closed form of this kind exists and the comparison would not apply.
• The policy discrepancy (max_abs_error_policy) reflects the discrete-choice VFI on this grid; it is not an error in the closed form. A continuous-choice VFI with interpolated V would reduce it but is outside the prescribed design.
• The contraction statement with modulus β is for the bounded, discretised problem; the unbounded continuous problem with log utility requires a weighted sup-norm or a direct verification argument, which we do not develop here.