Problem 3: Investment with Adjustment Costs Consider a firm that lives for 3 periods and maximizes expected discounted profits. Their discount rate is constant and equal to R1, where R=1+r (and r being risk-free interest rate). Per period profits are given by: t=ptKtItc(It) where initial capital level K1 is given, and output price pt is stochastic and i.i.d. The price of investment is normalized to 1. Function c(It) is the capital adjustment cost with properties that cI(It),cII(It)>0 (i.e. it is increasing and convex). 1. Consider the capital choice problem of the firm: max{I1,I2,I3}Et=13(R1)t1[ptKtItc(It)] subject to the law of motion of capital: It=Kt+1(1)Kt. Find an expression for the marginal q (expected value of an additional unit of capital purchased in period 1 as a function of expected future marginal contributions to profits). (Hint: Don't forget about Envelope condition!) 2. What happens to investment in period 1I1 if period-3 price p3 doubles for sure (compared to p1 )? Why