Week 8 Homework Feedback: Alessia Canuto

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

✅ Overall Assessment

Result: ✅ More than 50% Correct

Excellent submission demonstrating a comprehensive understanding of Value Function Iteration with population growth. The code correctly implements the Bellman equation with the (1+n) factor in the consumption matrix, performs VFI for multiple population growth rates, extracts policy functions, simulates transition paths, and includes thoughtful economic interpretation. The submission goes beyond requirements by comparing n=0 vs n>0 cases, which provides valuable economic insight. Minor issues include using different baseline parameters (α=0.33, β=0.96) than the solution template (α=0.40, β=0.95), but this is acceptable as long as the implementation is correct. The code is well-organized, professionally formatted, and includes all required visualizations.

Task-by-Task Check

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

Status: ✅ Correct

Line 44: cons = f(kgrid) + (1 - delta)*kgrid - (1 + n)*kgrid';
Correctly includes the (1+n) factor for population growth
Properly handles feasibility constraints

✅ Task 2: Value Function Iteration Implementation

Status: ✅ Correct

Lines 50-68: Proper VFI loop with Bellman update
Correct maximization: max(util(i,:) + beta * V')
Proper convergence criterion: max(abs(Vnew - V)) < tol

✅ Task 3: Convergence History Storage

Status: ✅ Correct

Lines 52, 65-67: Stores V_history at selected iterations
Stores iterations: [1, 5, 10, 25, 50, 100]
Used for convergence visualization

✅ Task 4: Policy Function Extraction

Status: ✅ Correct

Line 59: Extracts policy indices during VFI: [Vnew(i), pol_ind(i)] = max(...)
Line 71: Maps to policy function: k_policy = kgrid(pol_ind)

✅ Task 5: Policy Function Plot with 45° Line

Status: ✅ Correct

Lines 128-167: Comprehensive policy function plot
Includes 45° line (line 152)
Shows steady-state intersections with annotations
Multiple n values compared

✅ Task 6: Capital Path Simulation

Status: ✅ Correct

Lines 90-98: Simulates capital path using policy function
Uses interpolation: interp1(kgrid, k_policy, kpath(t))
Correct initial condition: k0 = 0.5

✅ Task 7: Consumption Path Simulation

Status: ✅ Correct

Lines 97-99: Computes consumption using resource constraint
Correct formula: cpath(t) = f(kpath(t)) + (1 - delta)*kpath(t) - (1 + n)*kpath(t+1)
Includes (1+n) factor correctly

✅ Task 8: Capital Path Plot

Status: ✅ Correct

Lines 172-182: Plots capital paths for different n values
Proper labels, legend, and formatting
Saved to Figures/ directory

✅ Task 9: Consumption Path Plot

Status: ✅ Correct

Lines 185-196: Plots consumption paths for different n values
Proper labels, legend, and formatting
Saved to Figures/ directory

✅ Task 10: Parameter Experiments

Status: ✅ Correct

Lines 19, 40-111: Loops over n values [0.00, 0.01, 0.02, 0.03]
Well-structured parameter experiments showing how population growth affects the model
Homework instructions were deliberately vague about which parameters to experiment with
Experiments with n are valid and provide valuable economic insight

✅ Task 11: Policy Functions for Different Calibrations

Status: ✅ Correct

Lines 128-167: Plots policy functions for different n values
Shows how population growth affects optimal savings behavior
Includes 45° line and steady-state annotations
Clear visualization of comparative statics

✅ Task 12: Capital Paths for Different Calibrations

Status: ✅ Correct

Lines 172-182: Plots capital paths for different n values
Shows transition dynamics under different population growth rates
Demonstrates how n affects convergence paths

✅ Task 13: Value Function Convergence Plot

Status: ✅ Correct

Lines 115-123: Convergence plot for n=0.02 example
Shows selected iterations with legend
Saved to Figures/ directory

✅ Task 14: Figure Saving to Figures/ Directory

Status: ✅ Correct

Lines 7-11: Creates Figures/ directory structure
All figures saved using exportgraphics with proper paths
High-resolution output (300 DPI)

✅ Task 15: Interpretation Comments

Status: ✅ Correct

Lines 200-223: Comprehensive economic interpretation
Discusses population growth effects on steady state
Explains comparative statics (n, β, δ effects)
Discusses convergence and stability

Task Summary: 15/15 tasks fully correct

Technical Implementation

Strengths:

Correct Population Growth Implementation: The (1+n) factor is correctly included in the consumption matrix, which is the critical requirement for this homework.
Robust VFI Algorithm: Proper implementation with convergence tracking and history storage.
Comprehensive Simulation: Correctly simulates both capital and consumption paths with proper resource constraints.
Steady-State Analysis: Includes analytical steady-state calculation and numerical detection of policy function intersection.
Bonus Analysis: Goes beyond requirements by comparing n=0 vs n>0 cases, providing valuable economic insight.

Areas for Improvement:

Optional Enhancement: Could also experiment with different β and α values to show how preferences and technology affect optimal savings, in addition to the n experiments. However, the n experiments are valid and provide valuable economic insight.

Style & Clarity

Strengths:

Excellent Organization: Clear section headers with descriptive comments
Professional Structure: Well-organized code with logical flow
Comprehensive Comments: Detailed economic interpretation section
Proper Variable Naming: Clear, descriptive variable names
Figure Management: Professional figure saving with proper directory structure

Minor Suggestions:

Could add more inline comments explaining the VFI algorithm steps
Could document the steady-state calculation method more explicitly

Visual Output Assessment

Strengths:

High-Quality Figures: All figures are properly formatted with labels, legends, and titles
Comprehensive Visualizations: Includes convergence plot, policy functions, and transition paths
Professional Styling: Proper use of colors, line widths, and annotations
Steady-State Visualization: Excellent annotation of steady-state points on policy function plot
Multiple Scenarios: Compares different n values effectively

All Required Figures Present:

✅ Value function convergence plot
✅ Policy function plot with 45° line
✅ Capital transition paths
✅ Consumption transition paths

Suggestions for Improvement

Optional Enhancement: Consider also experimenting with different β and α values to show how preferences and technology affect optimal savings behavior, in addition to the excellent n experiments. This would provide a more comprehensive analysis of parameter effects, but the current n-focused experiments are valid and valuable.
Optional Enhancement: Could create a unified comparison showing the relative effects of n, β, and α on steady-state capital to demonstrate which parameter has the strongest effect. However, the current submission already provides excellent economic insight.

Summary

Alessia’s submission demonstrates excellent understanding of Value Function Iteration with population growth. The code correctly implements the critical (1+n) factor in the consumption matrix, performs VFI accurately, extracts policy functions, and simulates transition paths. The submission includes well-designed parameter experiments with different population growth rates, which provides valuable economic insight into how n affects steady-state capital and convergence dynamics. The code is well-organized, professionally formatted, and includes all required visualizations with comprehensive economic interpretation. 15/15 tasks fully correct with outstanding technical implementation and professional presentation. This is an exemplary submission that demonstrates strong understanding of both the computational methods and economic theory.