Week 8 Homework Feedback: Ignazio Nunzi

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 with proper VFI implementation, convergence tracking, policy function extraction, and simulation. The code structure is clean and well-organized. However, there is a critical error: the consumption matrix is missing the (1+n) factor required for population growth. Line 12 computes cons = f + (1-delta)*kgrid - kgrid', which is the formula for the model without population growth (n=0). Additionally, no population growth parameter (n) is defined anywhere in the code. As a result, the entire VFI computation, policy functions, and simulations are for the wrong model (n=0 instead of n>0). Despite this critical error, the submission shows solid technical competence in VFI implementation and proper visualization. 6/15 tasks correct for the right model (40% < 50% but > 0%).

Task-by-Task Check

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

Status: ❌ CRITICAL ERROR

  • Line 12: cons = f + (1-delta)*kgrid - kgrid';
  • WRONG: This is the formula for n=0 (no population growth)
  • CORRECT: Should be cons = f + (1-delta)*kgrid - (1+n)*kgrid';
  • Missing: No population growth parameter n is defined in the code
  • This error affects all subsequent computations

✅ Task 2: Value Function Iteration Implementation

Status: ✅ Correct Algorithm (Wrong Model)

  • Lines 19-37: Proper VFI loop with convergence tracking
  • Correct maximization: max(util(i,:) + beta * V')
  • Proper convergence criterion: diff > 1e-6
  • BUT: Solving the wrong model (n=0 instead of n>0)

✅ Task 3: Convergence History Storage

Status: ✅ Correct

  • Line 22: Initializes Vstore = []
  • Line 36: Stores V each iteration: Vstore(:, iter) = V
  • Well-implemented convergence tracking

✅ Task 4: Policy Function Extraction

Status: ✅ Correct (Wrong Model)

  • Line 30: Extracts policy indices during VFI: [Vnew(i), pol_ind(i)] = max(...)
  • Line 42: Maps to policy function: k_policy = kgrid(pol_ind)
  • BUT: Policy function is for n=0 model, not n>0

✅ Task 5: Policy Function Plot with 45° Line

Status: ✅ Correct Visualization (Wrong Model)

  • Lines 50-54: Policy function plot with 45° line
  • Proper formatting, labels, and legend
  • Saved to Figures/ directory
  • BUT: Shows policy for n=0 model, not n>0

✅ Task 6: Capital Path Simulation

Status: ✅ Correct Algorithm (Wrong Model)

  • Lines 66-72: Simulates capital path using policy function
  • Uses interpolation correctly: interp1(kgrid, k_policy, kpath(t))
  • BUT: Uses policy function from n=0 model

❌ Task 7: Consumption Path Simulation

Status: ❌ Missing

  • No consumption path is computed or plotted
  • Only capital path is simulated

✅ Task 8: Capital Path Plot

Status: ✅ Correct (Wrong Model)

  • Lines 74-78: Plots capital path
  • Proper labels and formatting
  • Saved to Figures/ directory
  • BUT: Path is for n=0 model

❌ Task 9: Consumption Path Plot

Status: ❌ Missing

  • No consumption path plot

❌ 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

  • Only one capital path plotted
  • No parameter experiments

✅ Task 13: Value Function Convergence Plot

Status: ✅ Correct

  • Lines 58-63: Convergence plot showing V_history
  • Proper formatting and saved to Figures/ directory
  • Well-implemented visualization

✅ Task 14: Figure Saving to Figures/ Directory

Status: ✅ Correct

  • Lines 56, 63: All figures saved using exportgraphics with proper paths
  • High-resolution output (300 DPI)
  • Professional figure management

⚠️ Task 15: Interpretation Comments

Status: ⚠️ Partial

  • Lines 80-87: Brief interpretation of beta effects
  • Good economic intuition about patience and savings
  • Missing: Discussion of population growth effects (because n is not implemented)
  • Missing: Discussion of other parameter effects

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

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

Technical Implementation

Critical Error:

  1. Missing (1+n) Factor: The consumption matrix does not include the (1+n) factor required for population growth. This is the most critical requirement of the homework. The entire submission solves the n=0 model instead of the n>0 model.

  2. Missing Population Growth Parameter: No variable n is defined anywhere in the code. The homework requires n>0 (typically n=0.01 or n=0.02).

Strengths:

  1. Good VFI Implementation: Proper algorithm with convergence tracking
  2. Clean Code Structure: Well-organized and readable
  3. Proper Visualization: Good convergence plot and policy function plot
  4. Correct Simulation: Capital path simulation is properly implemented

Issues:

  1. Wrong Economic Model: Entire submission is for n=0 model, not n>0 as required
  2. Missing Consumption Path: No consumption simulation or plot
  3. Missing Parameter Experiments: No experiments with different parameters
  4. Limited Interpretation: Only brief comments on beta effects

Style & Clarity

Strengths:

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

Areas for Improvement:

  1. Missing Population Growth: Need to define n and include (1+n) factor
  2. Add Consumption Path: Should simulate and plot consumption
  3. Add Parameter Experiments: Should test different parameter values
  4. Expand Interpretation: More detailed economic discussion

Visual Output Assessment

Strengths:

  1. Good Convergence Plot: Clear visualization of VFI convergence
  2. Proper Policy Function Plot: Includes 45° line for steady-state visualization
  3. Professional Quality: High-resolution figures with proper formatting

Missing:

  1. Consumption Path Plot: Not included
  2. Parameter Experiment Plots: Not included
  3. Value Function Plot: Not included (only convergence plot)

Suggestions for Improvement

  1. Fix Consumption Matrix (CRITICAL): Add population growth parameter and correct the consumption matrix: 
    n = 0.01;  % Population growth rate
% Line 12 should be:
cons = f + (1-delta)*kgrid - (1+n)*kgrid';  % CORRECT formula
  2. Add Consumption Path: Simulate and plot consumption: 
    cpath = zeros(T,1);
for t = 1:T-1
    cpath(t) = kpath(t)^alpha + (1-delta)*kpath(t) - (1+n)*kpath(t+1);
end
% Plot consumption path

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

  4. Add Value Function Plot: Include a plot of the final value function V(k)

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

Summary

Ignazio’s submission demonstrates good technical competence in VFI implementation. The code structure is clean, the VFI algorithm is correct, convergence tracking is well-implemented, and visualization is professional. However, there is a critical error: the consumption matrix is missing the (1+n) factor required for population growth, and no population growth parameter (n) is defined. As a result, the entire submission solves the n=0 model instead of the n>0 model as required. Additionally, the submission is missing consumption path simulation, parameter experiments, and comprehensive interpretation. 6/15 tasks fully correct for the right model (40% < 50% but > 0%). The submission shows solid understanding of VFI methodology and would be significantly improved once the (1+n) factor is added and the missing components are included.