1# F5-conf-04 — VFI vs closed form, optimal growth with log utility23## Question4Planner maximises Σ β^t ln c_t s.t. y_{t+1} = (y_t − c_t)^α (full depreciation, TFP = 1), α = 0.36, β = 0.95.5Compare the closed-form policy/value with value-function iteration (VFI) on the prescribed grid at y = 0.5, 1, 2.67## Data and exclusions8`data/model.csv` (10 rows, theoretical parameters only): α = 0.36, β = 0.95, grid of 400 points equally spaced in ln y on [0.05, 5],9VFI tolerance 1e-8 (sup-norm change), evaluation points 0.5, 1.0, 2.0. Nothing excluded. Design choices recorded in `data_issues`:10(i) evaluation points are not grid nodes → linear interpolation in y of the VFI policy and value; (ii) next-period output y' is chosen11on the same grid (discrete-choice VFI), infeasible choices (c ≤ 0) get −∞. Every node has feasible choices, so all 400 points are used.1213## Method14**Closed form (guess and verify).** Guess V(y) = A + B ln y. FOC of max_c {ln c + βA + βBα ln(y − c)}: 1/c = αβB/(y − c) ⇒ c = y/(1+αβB).15Matching the ln y coefficient: B = 1 + αβB ⇒ B = 1/(1 − αβ), so c(y) = (1 − αβ) y and savings rate (y − c)/y = αβ = 0.342.16Matching constants: A = [ln(1 − αβ) + (αβ/(1 − αβ)) ln(αβ)]/(1 − β) = −19.5244, B = 1.5198.17Assumptions: log utility, Cobb–Douglas technology y' = k^α with TFP 1, full depreciation, interior solution (0 < c < y, guaranteed by Inada of ln18and α ∈ (0,1), β ∈ (0,1)), and the transversality/limit condition β^t V(y_t) → 0, which holds since ln y_t stays bounded along the optimal path.1920**VFI.** V_0 = 0; V_{n+1}(y_i) = max_j {ln(y_i − y_j^{1/α}) + β V_n(y_j)} over grid nodes y_j, vectorised (400×400) and iterated until21sup|V_{n+1} − V_n| < 1e-8. Policy = argmax node; c = y_i − y_j^{1/α}. Policy and value at off-grid points are linearly interpolated.2223**Contraction.** On the finite grid the return ln c is bounded on the feasible set, and the Bellman operator T satisfies Blackwell's24sufficient conditions (monotonicity; discounting T(V + a) = TV + βa), so T is a contraction in the sup norm with modulus β = 0.95.25(On the continuous unbounded domain, log utility is unbounded; there the contraction argument needs a weighted norm or restriction to a26compact invariant set such as [y_min, y_max], which contains the steady state y* = (αβ)^{α/(1−α)} ≈ 0.547.)2728## Results29| y | c closed form | c VFI | abs. error |30|---|---|---|---|31| 0.5 | 0.32900 | 0.32882 | 0.00018 |32| 1.0 | 0.65800 | 0.65898 | 0.00098 |33| 2.0 | 1.31600 | 1.30959 | 0.00641 |3435- max |c_num − c_closed| over evaluation points: **0.00641** (≈0.5% of c at y = 2).36- V(1): closed form −19.52441, VFI −19.52478 (difference 3.7e-4).37- Savings rate (closed form): αβ = 0.342. Contraction modulus: β = 0.95.38- VFI converged in 361 iterations; final sup change 9.997e-9. Grid points used: 400.39- Diagnostics: sup-norm value error over the whole grid 4.8e-4; sup policy error over grid nodes 0.025 (largest at high y, where the40 log-spaced grid of y' becomes coarse in levels).4142## Uncertainty43There is no sampling uncertainty (deterministic model, no data, no randomness). Numerical error has two parts: (a) iteration error,44bounded by β·tol/(1 − β) = 1.9e-7 in sup norm on V — negligible; (b) discretisation error from restricting y' to the grid and interpolating45between nodes, which dominates (policy errors 1e-4 to 6e-3 at the evaluation points). The policy error scales with the gap between46adjacent grid nodes in levels (log step 0.0115, so gaps grow ∝ y), which is why the error rises with y.4748## Limitations and open issues49- Validation is for one parameter point (α = 0.36, β = 0.95) and one grid (400 log-spaced points on [0.05, 5]); conclusions about accuracy do50 not automatically transfer to other parameters or grids.51- Discrete choice produces a step-function policy; linear interpolation at off-grid points adds error. Continuous choice (e.g. golden-section52 search with interpolated V) or finer grids would reduce the error, but were not used to stay with the prescribed design.53- Grid truncation at y_min/y_max can bias behaviour near the bounds; the evaluation points are interior and the ergodic set (around 0.547) is54 well inside the grid.55