Only For Question 1, you should submit a Julia code file as well as copying and pasting the code into the PDF.
1. Simulating the basic overlapping generations model
Consider the basic overlapping generations model we studied in lecture. We saw that the capital stock in intensive form kt satisfies the difference equation
kt+1 = s[f'(kt+1)][f(kt) – ktf'(kt)] (1) 1+n
where the savings rate s(r) satisfies
s(r) = (1 + r)1/?-1 + (1 + ?)1/? (2)
(1 + r)1/?-1
The wage and rental rate follow directly from the capital stock:
rt = f'(kt) (3) wt = f(kt) – ktf'(kt) (4)
Let f(k) = ka for a = 0.3,? = 0.1,n = 0.1. For ? ? {1/2,1,2}, simulate a time series for capital from 1 = t = 10 starting from k0 = 0.01. Then calculate corresponding series for the interest rate rt and the savings rate st. There should be three plots, one for each
1
variable, and three curves on each plot, corresponding to ? = 1/2,1, or 2. Hint: for ? = 1, we just have s = 1/(2 + ?).
How does ? affect capital accumulation and the resulting savings and interest rates? Provide economic intuition. (Hint: define the savings rate function in Julia and use a for loop to update the capital stock. You can use a root-finding method, such as fzero, to calculate kt+1 given kt in (1).)