You've been given the following production function:

Q = 50L 2L^2 + 15K 0.5K^2

Let's assume this is a perfectly competitive market, so you have no control over the market price price for wages (w = 5), rental rate (r = 8), and the price for the produced good (p = 9).

(a) Calculate the returns to scale and explain what would happen with a 5% increase in capital (K) and labor (L). Just give an approximate amount.

(b) Assume we are in the short-run and labor (L) is fixed at 10 units. What is the optimal amount of capital we need?

(c) What is the profit in the short-run?

(d) What is the marginal rate of technical substitution (MRT S)? Assume we are now in the long-run, and no input is fixed.

(e) What is the optimal combination of capital and labor? You do not have to numerically solve. Just find one as a function of the other (ie - K in terms of L or L in terms of K)