Textbook Problem

Most economies have a goal of maximizing the average consumption per period. Assume that during each year, an economy saves the same (to be determined) percentage S of its production. During a year in which the beginning capital level is K, a quantity K^1/2 of capital is produced. If the economy saves a percentage S of its capital, then during the current year it consumes (1 − S)K units of capital and, through savings, adds (SK)^1/2 units of capital. Also, during any year, 10% of all capital present at the beginning of the year depreciates or wears out.

1. a. What annual savings percentage S maximizes the long-run average consumption level? Assume that year 50 represents the long run, so that the objective is the consumption level in year 50. You can assume the initial capital is 1 (for some appropriate measurement unit).
2. b. Use SolverTable to see how the optimal value of S depends on the annual depreciation rate.