Scrooge has an initial wealth of W4 at the start of every date, and earns capital income on that wealth at the rate of return r. He also earns a labor income of y at every date.
(a) Write down a formula for Scrooge’s total income at date t.
(b) Now suppose that Scrooge always save his entire existing wealth as well as a fraction s of his total income; that is, his capital income plus his labor income. Write down a formula that connects Wl to Wt+1.
(c) Suppose that Scrooge’s labor income grows exogenously at the rate of g from period to period. Find the growth rate in Scrooge’s total wealth in the long run. (Hint: you will need to look at two cases.)
(d) Suppose that the rate of growth of overall per-capita income is 2% per year. Suppose, also, that Scrooge is a pure capital income earner (y = 0), and saves 25% of his income (s = 0.25). Find the rate of return on capital that will allow Scrooge to double his income relative to national income in 20 years.