Week 8 Homework Feedback: Chiara Tombolini

Assignment: Dynamic Programming & Value Function Iteration with Population Growth
Week: 8
Date: Week 8 Assessment

⚠️ Overall Assessment

Result: ⚠️ Partial <50% Correct

This submission demonstrates good understanding of Value Function Iteration methodology. The code correctly implements the critical (1+n) factor in the consumption matrix (line 70), performs VFI accurately, and extracts policy functions. However, the submission includes a Solow model simulation (lines 20-27) which is Week 7 material, not Week 8. Additionally, the submission is missing several critical components: consumption path simulation, capital path simulation, convergence history storage, parameter experiments, and comprehensive interpretation. The VFI implementation is correct but incomplete. 6/15 tasks fully correct (40% < 50% but > 0%) with good technical foundation but missing many required components.

Task-by-Task Check

✅ Task 1: Correct Consumption Matrix with (1+n) Factor

Status: ✅ Correct

  • Line 8: n = 0.02; — Population growth parameter defined
  • Line 70: c = y + (1 - delta)*k - (1 + n)*kp; — CORRECT
  • Correctly implements the (1+n) factor for population growth

✅ Task 2: Value Function Iteration Implementation

Status: ✅ Correct

  • Lines 60-89: Proper VFI loop with convergence tracking
  • Line 78: Correct maximization: max(util + beta*V)
  • Proper convergence criterion: max(abs(Vnew - V)) < tol
  • Well-implemented algorithm

❌ Task 3: Convergence History Storage

Status: ❌ Missing

  • No V_history storage during VFI iterations
  • Convergence is tracked but not stored for plotting
  • Missing convergence plot

✅ Task 4: Policy Function Extraction

Status: ✅ Correct

  • Line 78: Extracts policy indices during VFI: [Vnew(i), idx] = max(...)
  • Line 79: Maps to policy function: policy_k(i) = kgrid(idx)
  • Correctly implemented

⚠️ Task 5: Policy Function Plot with 45° Line

Status: ⚠️ Partial

  • Lines 96-100: Policy function plot
  • Missing: 45° line for steady-state visualization
  • Proper formatting and labels
  • Saved to Figures/ directory (lines 106-107)

❌ Task 6: Capital Path Simulation

Status: ❌ Missing

  • No capital path simulation using the policy function
  • Only Solow model simulation (lines 20-27), which is wrong assignment

❌ Task 7: Consumption Path Simulation

Status: ❌ Missing

  • No consumption path simulation using the policy function
  • Only Solow model consumption (lines 23, 27), which is wrong assignment

❌ Task 8: Capital Path Plot

Status: ❌ Missing

  • No capital path plot from VFI policy function
  • Only Solow model plot (lines 30-38), which is wrong assignment

❌ Task 9: Consumption Path Plot

Status: ❌ Missing

  • No consumption path plot from VFI policy function
  • Only Solow model plot (lines 30-38), which is wrong assignment

❌ Task 10: Parameter Experiments

Status: ❌ Missing

  • No parameter experiments
  • Uses fixed parameters throughout

❌ Task 11: Policy Functions for Different Calibrations

Status: ❌ Missing

  • Only one policy function plotted
  • No parameter experiments

❌ Task 12: Capital Paths for Different Calibrations

Status: ❌ Missing

  • No capital paths plotted
  • No parameter experiments

❌ Task 13: Value Function Convergence Plot

Status: ❌ Missing

  • No convergence plot
  • V_history is not stored during VFI

✅ Task 14: Figure Saving to Figures/ Directory

Status: ✅ Correct

  • Lines 41-42, 102-103, 106-107: All figures saved using saveas and savefig with proper paths
  • Professional figure management
  • Creates Figures/ directory structure

⚠️ Task 15: Interpretation Comments

Status: ⚠️ Partial

  • Lines 130-146: Steady-state calculations and results display
  • Missing: Economic interpretation of VFI results
  • Missing: Discussion of population growth effects
  • Missing: Discussion of parameter effects
  • Missing: Comprehensive economic interpretation

Task Summary: 6/15 tasks fully correct (40% < 50% but > 0%)

Grade: ⚠️ (6/15 = 40% < 50% correct, but > 0%)

Technical Implementation

Strengths:

  1. Correct (1+n) Implementation: Critical factor correctly included in consumption matrix
  2. Good VFI Implementation: Proper algorithm with convergence tracking
  3. Clean Code Structure: Well-organized and readable
  4. Proper Figure Saving: All figures saved to Figures/ directory

Critical Issues:

  1. Wrong Assignment Section: Lines 20-27 implement Solow model with constant saving rate, which is Week 7 material, not Week 8
  2. Missing Simulation: No capital and consumption path simulation using the VFI policy function
  3. Missing Convergence History: No V_history storage for convergence plot
  4. Missing Parameter Experiments: No experiments with different parameter values
  5. Missing 45° Line: Policy function plot doesn’t include 45° line for steady-state visualization

Style & Clarity

Strengths:

  1. Clean Code: Well-organized with clear structure
  2. Good Comments: Helpful comments explaining key steps
  3. Proper Formatting: Clean, readable code

Areas for Improvement:

  1. Remove Solow Model: The Solow model section (lines 20-27) is not part of Week 8 homework
  2. Add Simulation: Implement capital and consumption path simulation using the VFI policy function
  3. Add Convergence Tracking: Store V_history during VFI
  4. Add Parameter Experiments: Test different parameter values
  5. Add 45° Line: Include 45° line in policy function plot
  6. Expand Interpretation: More detailed economic discussion

Visual Output Assessment

Strengths:

  1. Good Value Function Plot: Clear visualization of value function
  2. Good Policy Function Plot: Clear visualization (but missing 45° line)
  3. Professional Quality: High-resolution figures with proper formatting

Issues:

  1. Solow Model Plot: Includes Solow model dynamics plot, which is wrong assignment
  2. Missing 45° Line: Policy function plot doesn’t include 45° line
  3. Missing Transition Plots: No capital and consumption path plots from VFI
  4. Missing Convergence Plot: Not included
  5. Missing Parameter Experiment Plots: Not included

Suggestions for Improvement

  1. Remove Solow Model Section (IMPORTANT): The Solow model simulation (lines 20-27) is Week 7 material, not Week 8. Week 8 requires VFI with optimal savings, not constant saving rate.

  2. Add Capital Path Simulation: Simulate capital path using the VFI policy function: 
    T = 50;
kpath = zeros(T,1);
kpath(1) = 0.5;
for t = 1:T-1
    kpath(t+1) = interp1(kgrid, policy_k, kpath(t));
end
  3. Add Consumption Path Simulation: Compute consumption along the simulated path: 
    cpath = zeros(T,1);
for t = 1:T-1
    cpath(t) = kpath(t)^alpha + (1-delta)*kpath(t) - (1+n)*kpath(t+1);
end
% Final period
k_next_T = interp1(kgrid, policy_k, kpath(T));
cpath(T) = kpath(T)^alpha + (1-delta)*kpath(T) - (1+n)*k_next_T;
  4. Add 45° Line to Policy Plot: Include 45° line for steady-state visualization: 
    plot(kgrid, policy_k, 'LineWidth', 2); hold on;
plot(kgrid, kgrid, '--k', 'LineWidth', 1);  % 45° line
legend('Policy', '45° line', 'Location', 'NorthWest');
  5. Add Convergence History Storage: Store V_history during VFI for convergence plot: 
    V_history = [];
for iter = 1:maxIter
    % ... VFI update ...
    V_history(:, end+1) = V;  % Store each iteration
end
% Plot convergence
figure; plot(kgrid, V_history); title('Value Function Convergence');

  6. Add Parameter Experiments: Test different β, α, or n values to show how they affect optimal savings behavior

  7. Add Transition Path Plots: Plot capital and consumption paths from the VFI simulation

  8. Expand Interpretation Comments: Add more detailed discussion of:
    • How population growth affects steady state
    • Comparison of policy functions for different parameters
    • Economic intuition behind the results

Summary

Chiara’s submission demonstrates good understanding of Value Function Iteration methodology. The code correctly implements the critical (1+n) factor in the consumption matrix, performs VFI accurately, and extracts policy functions. However, the submission includes a Solow model simulation which is Week 7 material, not Week 8. Additionally, the submission is missing several critical components: consumption path simulation, capital path simulation, convergence history storage, parameter experiments, and comprehensive interpretation. The VFI implementation is correct but incomplete. 6/15 tasks fully correct (40% < 50% but > 0%) with good technical foundation but missing many required components. The main improvements needed are to remove the Solow model section, add capital and consumption path simulation using the VFI policy function, add convergence tracking, include parameter experiments, add the 45° line to the policy plot, and expand the interpretation comments.