Homework

General case: technology and demographic change

Part 1. Consider a production function \(A_tF(K_t,L_t)=A_tK_t^\alpha L_t^{1-\alpha}\). With a productivity growth rate of g.This allows us some conveniences for the stationary model. Therefore, the capital motion law is:

\[K_{t+1}=(1-\delta)K_t + sA_tF(K_t,L_t)\]

Using \(\frac{A_{t+1}-A_t}{A_t}=g\) and \(\frac{L_{t+1}-L_t}{L_t}=n\) and considering \(k_t=\frac{K_t}{A_tL_t}\)

1. Derive a theoretical function for \(k_{t+1}=g(k_{t})\) where \(g(k_{t})\) also depends on \(n\) and \(g\)

2. Derive a theoretical function for the steady state of capital such tha \(k^*=h(k^*)\) whereh\(k^*\) also depends on \(n\) and \(g\)

3. Write a function in Julia to determine \(k_{t+1}=g(k_{t})\) for any grid of \(k_t\) and any set of coefficients (use the ones in the lectures)

4. Write a function in Julia to determine the steady state of \(k^*=h(k^*)\).

5. Simulate in Julia \(k_{t+1}=g(k_{t})\) and TEST it converges to the value predicted by \(h(k^*)\).

6. Using \(k_{t+1}=g(k_{t})\) in 3, simulate convergence over time for different values of initial capital \(k_0\)

7. Using \(k_{t+1}=g(k_{t})\) in 3, simulate convergence over time for different values of initial capital \(k_0\) assesing the implications of different values of \(n\), more specifically, \(n=0, n=0.1, n=1\). Explain the intution