Week 8 Homework Feedback: Davide Magnini

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

✅ Overall Assessment

Result: ✅ More than 50% Correct

The submission demonstrates good understanding of VFI and includes well-implemented parameter experiments with different β and α values. The consumption matrix correctly includes the (1+n) factor (line 27), and the code structure is clear. The submission includes valuable bonus work comparing n=0 vs n>0 cases. However, there are some missing elements: the value function convergence plot is not created (no V_history storage), and interpretation comments on how population growth affects steady state are minimal. The parameter experiments are correctly implemented, and the simulation of capital and consumption paths is accurate.

Task-by-Task Check

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

Status: ✅ Correct

Lines 27-28: cons = f + (1-delta)*kgrid - (1+n) * kgrid'; — CORRECT
Includes comment explaining population growth requirement
The consumption matrix correctly includes the (1+n) factor for population growth
Code structure computes n=0.01 first, then later compares with n=0, which is fine

✅ Task 2: Value Function Iteration Implementation

Status: ✅ Correct

Lines 34-44: Proper VFI loop with Bellman update
Correct maximization: max(util(i,:) + beta*V')
Proper convergence criterion: max(abs(Vnew - V)) < 1e-6

❌ Task 3: Convergence History Storage

Status: ❌ Missing

No storage of V_history during iterations
Cannot create convergence plot

✅ Task 4: Policy Function Extraction

Status: ✅ Correct

Line 39: Extracts policy indices during VFI: [Vnew(i), pol_ind(i)] = max(...)
Line 45: Maps to policy function: policy = kgrid(pol_ind)

✅ Task 5: Policy Function Plot with 45° Line

Status: ✅ Correct

Lines 52-81: Policy function plot with 45° line (line 65)
Shows multiple (β, α) combinations
Proper labels and legend

✅ Task 6: Capital Path Simulation

Status: ✅ Correct

Lines 92-102: Simulates capital path using policy function
Uses interpolation: interp1(kgrid, T_policy{:,j}, kpath{t,j})
Correct initial condition: k0 = 0.5

✅ Task 7: Consumption Path Simulation

Status: ✅ Correct

Lines 120-133: Computes consumption using resource constraint
Line 128: Correct formula: cpath{t,j} = k_now^alpha + (1 - delta)*k_now - (1+n) * k_next
Includes (1+n) factor correctly

✅ Task 8: Capital Path Plot

Status: ✅ Correct

Lines 104-114: Plots capital paths for different (β, α) combinations
Proper labels, legend, and formatting
Saved to Figures/ directory

✅ Task 9: Consumption Path Plot

Status: ✅ Correct

Lines 135-145: Plots consumption paths for different (β, α) combinations
Proper labels, legend, and formatting
Saved to Figures/ directory

✅ Task 10: Parameter Experiments (β and α)

Status: ✅ Correct

Lines 6, 22-50: Loops over β ∈ {0.9, 0.95, 0.98} and α ∈ {0.25, 0.30, 0.40}
Stores results in tables for comparison
Well-structured implementation

✅ Task 11: Policy Functions for Different Calibrations

Status: ✅ Correct

Lines 52-81: Plots policy functions for all (β, α) combinations
Includes 45° line for steady-state visualization
Bonus: Also compares n=0 vs n=0.01 (lines 193-230)

✅ Task 12: Capital Paths for Different Calibrations

Status: ✅ Correct

Lines 104-114: Plots capital paths for all (β, α) combinations
Shows transition dynamics for each calibration

❌ Task 13: Value Function Convergence Plot

Status: ❌ Missing

No convergence plot created
V_history is never stored during VFI iterations

✅ Task 14: Figure Saving to Figures/ Directory

Status: ✅ Correct

Line 83: Creates Figures/ directory
Lines 84, 114, 145, 230: All figures saved to Figures/ directory
Uses PDF format with high resolution

⚠️ Task 15: Interpretation Comments

Status: ⚠️ Partial

Lines 86-90: Basic interpretation of β and α effects
Lines 116-118: Comments on capital path convergence
Missing: Interpretation of how population growth (n) affects steady state and policy functions
The n=0 vs n>0 comparison (lines 232-234) has minimal interpretation

Task Summary: 11/15 tasks fully correct, 1 task incorrect (convergence plot), 1 task partial (interpretation), 2 tasks need clarification

Grade: ✅ (11/15 = 73% > 50% correct)

Technical Implementation

Strengths:

Correct Parameter Experiments: Well-implemented loops over β and α values as required.
Bonus n Comparison: Includes valuable comparison between n=0 and n>0 cases.
Proper Simulation: Correctly simulates capital and consumption paths with proper resource constraints.
Table Organization: Uses tables to store results, which is clean and organized.

Critical Issues:

Missing Convergence Plot: The value function convergence plot is a required deliverable but is completely missing. Need to store V_history during VFI iterations.
Code Structure Confusion: The code computes results for n=0.01 first (lines 16-91), then recomputes for n=0 (lines 149-231). While both implementations are correct, the structure is confusing. It would be clearer to compute n=0 first as baseline, then n>0.

Areas for Improvement:

Interpretation: Missing detailed comments on how population growth affects the model, especially the steady-state comparative statics.
Convergence Tracking: Need to add V_history storage to enable convergence visualization.

Style & Clarity

Strengths:

Good Organization: Clear sections with comments
Table Usage: Effective use of tables for storing results
Proper Variable Naming: Clear variable names

Areas for Improvement:

Code Structure: The flow is confusing - computes n=0.01 first, then n=0. Consider reorganizing.
Comments: Could add more detailed comments explaining the economic intuition, especially for the n comparison.
Interpretation Section: The interpretation comments are minimal. Need more detailed discussion of parameter effects.

Visual Output Assessment

Strengths:

Comprehensive Plots: Includes policy functions, capital paths, and consumption paths
Multiple Calibrations: Shows results for different (β, α) combinations
Bonus Comparison: Includes n=0 vs n>0 comparison plots
Proper Formatting: Figures have labels, legends, and titles

Missing:

Value Function Convergence Plot: This is a required deliverable but is completely absent.

Suggestions for Improvement

Add Convergence Plot (CRITICAL): Store V_history during VFI iterations and create a convergence plot showing how the value function evolves. Add code like:

V_history = [];
store_iter = [1, 5, 10, 25, 50, 100];
while diff > 1e-6
    % ... VFI update ...
    if ismember(iter, store_iter)
        V_history(:,end+1) = V;
    end
end
% Then plot convergence

Add Population Growth Interpretation: Include detailed comments explaining:
- How higher n reduces steady-state capital per worker (dilution effect)
- The economic intuition: population growth requires more capital to equip new workers
- Comparison of policy functions: n>0 shifts policy function downward relative to n=0
Reorganize Code Structure: Consider computing n=0 case first as baseline, then n>0 cases. This makes the comparison clearer.
Expand Interpretation Comments: Add more detailed discussion of:
- How β affects savings behavior (patience → higher k*)
- How α affects steady state (higher productivity → higher k*)
- How these effects interact with population growth

Summary

Davide’s submission demonstrates good understanding of VFI and includes well-implemented parameter experiments with different β and α values. The code correctly includes the (1+n) factor in the consumption matrix. The submission includes valuable bonus work comparing n=0 vs n>0 cases. However, there are two key missing elements: (1) the value function convergence plot (required deliverable), and (2) detailed interpretation comments on how population growth affects the model. 11/15 tasks fully correct (73% > 50%), with the main gaps being the missing convergence plot and incomplete interpretation. The submission would benefit from adding convergence tracking and more detailed economic interpretation, especially regarding population growth effects.