Understanding investment behavior is important to understanding the cyclical

behavior of the economy. Here we start with a simple production function, similar in

spirit to that of question 1 only now labour's contribution is indexed at one.

Y = AKa

In the above, Y is output, K is the capital stock anda is capital's share in output.

a) Derive an expression for the desired capital stock ( K d ) with Y on the right-hand

side. In addition show that the elasticity of K d with respect to Y is one.

b) Based on your equation for K d , derive an equation for gross investment (I) and

discuss how it is related to both Y and the user cost of capital (uc). Now suppose

that the marginal product of capital for the firm is taxed at a rate (). Defining

the real rate of interest as r, depreciation as d and Pk is the price of capital, show

how the existence of the tax affects the user cost of capital (uc) and your

investment equation.

c) You are given the following information: Y = 100; r = 5%; d = 12.5%;t = 30%;

a = 0.25and Pk = 1. If the current capital stock ( Kt ) is 102, what is the level of

investment?

d) You will now study the role of depreciation and obsolescence.

1) Start by assuming that d rises to 15%. Calculate the effect that this has on

investment. You will note that there are two competing effects. In the first

term in your investment equation, a rise in depreciation lowers the desired

demand for capital since the user cost has risen. That said, the rise in

depreciation also tends to increase actual investment spending, as more of

the capital stock needs to be replaced. This is shown in the second term of

the investment equation.

2) Working with these two concepts, assume that new investments in capital are

much longer lived than the existing capital stock (K). You can think more

powerful computers with better, longer-lasting batteries. In this case, we can

imagine that d in the user cost term falls to 10%, while the d in the capital

stock term rises to 15%. Based on this information, find the effect on

investment.