In this exercise, treat the real interest rate r as identical to the the rental rate on capital.
A. We will now consider the responsiveness—or elasticity—of savings and borrowing behavior with respect to changes in the interest rate (and other prices). Suppose that tastes over consumption now and in the future are homothetic, and further suppose that production frontiers (that use labor and capital as inputs) are homothetic.
(a) Can you tell whether the interest rate elasticity of savings (or capital supply) is positive or negative for someone who earns income now but not in the future?
(b) Can you tell whether the interest rate elasticity of borrowing (or capital demand) is positive or negative for someone who earns no income now but will earn income in the future?
(c) Is the interest rate elasticity of demand for capital by firms positive or negative?
(d) Is the wage elasticity of demand for capital by firms positive or negative?
(e) Is the output price elasticity of demand for capital positive or negative?
B. Suppose that inter temporal tastes over consumption are Cobb-Douglas. Furthermore, suppose that production technologies (which take capital and labor as inputs) have decreasing returns to scale and are Cobb-Douglas.
(a) Suppose that your income this period is e1 and your income in the future is e2. Set up your inter temporal utility maximization problem and derive your demand for consumption c1 now.
(b) Suppose all your income occurs now (i.e. e2 = 0). What is your savings (or capital supply) function, and what is the interest rate elasticity of savings?
(c) Suppose instead that all your income happens next period (i.e. e1 = 0). What is the interest rate elasticity of borrowing (or capital demand)?
(d) Next, derive the interest rate elasticity of capital demand by firms. Is it positive or negative?
(e) Repeat this for the wage elasticity of capital demand as well as the output price elasticity of capital demand for firms.