Math Problem, Macro & Micro economics profit maximizing firms

Project description
1. (Analytical question on Profit Maximizing Firms with Nominal Price Stickiness) Consider a profit maximization problem in which the firms choose current price Pi,0 at a given period 0.They face a stochastic probability of adjusting prices: the probability of not being able to adjust price in period t is denoted as qt . Further assume that q1 = q2 = q3 = 1 and for t > 3, qt = 0. Assume stochastic discount factor ?t = 1 for simplicity. Given this environment, do the following ?t = 1 for simplicity. Given this environment, do the following
(a) Set up the maximization problem for the price adjusting firms, in terms of the the period real profit Rt. What are the choice variables for these firms?
(b) Rewrite the above problem as a minimization problem using the secnd-order approximation used in the textbook. Define variables as pi,0 = lnPi,0 and pt = lnPt.
(c) Solve for the optimal price in (b). Economically interpret the result.
(d) Suppose that probability of nonadjustment changes to q1 = q2 = 1 and for t > 2, qt = 0. How does the change affect the above result in (c)? Explain.

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