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

# Question

Planner 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. Compare the closed-form policy/value with value-function iteration (VFI) on the prescribed grid at y = 0.5, 1, 2.

# Data and exclusions

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], VFI tolerance 1e-8 (sup-norm change), evaluation points 0.5, 1.0, 2.0. Nothing excluded. Design choices recorded in data_issues: (i) evaluation points are not grid nodes → linear interpolation in y of the VFI policy and value; (ii) next-period output y' is chosen on the same grid (discrete-choice VFI), infeasible choices (c ≤ 0) get −∞. Every node has feasible choices, so all 400 points are used.

# Method

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). Matching 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. Matching constants: A = [ln(1 − αβ) + (αβ/(1 − αβ)) ln(αβ)]/(1 − β) = −19.5244, B = 1.5198. Assumptions: log utility, Cobb–Douglas technology y' = k^α with TFP 1, full depreciation, interior solution (0 < c < y, guaranteed by Inada of ln and 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.

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 until sup|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.

Contraction. On the finite grid the return ln c is bounded on the feasible set, and the Bellman operator T satisfies Blackwell's sufficient conditions (monotonicity; discounting T(V + a) = TV + βa), so T is a contraction in the sup norm with modulus β = 0.95. (On the continuous unbounded domain, log utility is unbounded; there the contraction argument needs a weighted norm or restriction to a compact invariant set such as [y_min, y_max], which contains the steady state y* = (αβ)^{α/(1−α)} ≈ 0.547.)

# Results

• max |c_num − c_closed| over evaluation points: 0.00641 (≈0.5% of c at y = 2).
• V(1): closed form −19.52441, VFI −19.52478 (difference 3.7e-4).
• Savings rate (closed form): αβ = 0.342. Contraction modulus: β = 0.95.
• VFI converged in 361 iterations; final sup change 9.997e-9. Grid points used: 400.
• 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 the log-spaced grid of y' becomes coarse in levels).

# Uncertainty

There is no sampling uncertainty (deterministic model, no data, no randomness). Numerical error has two parts: (a) iteration error, bounded by β·tol/(1 − β) = 1.9e-7 in sup norm on V — negligible; (b) discretisation error from restricting y' to the grid and interpolating between nodes, which dominates (policy errors 1e-4 to 6e-3 at the evaluation points). The policy error scales with the gap between adjacent grid nodes in levels (log step 0.0115, so gaps grow ∝ y), which is why the error rises with y.

# Limitations and open issues

• 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 do not automatically transfer to other parameters or grids.
• Discrete choice produces a step-function policy; linear interpolation at off-grid points adds error. Continuous choice (e.g. golden-section search with interpolated V) or finer grids would reduce the error, but were not used to stay with the prescribed design.
• 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) is well inside the grid.