Consider the following dynamic problem of a firm. The firm's revenue, at any given date t + s is given by Pt+sqt+s; Pt+s is the price of the good it sells and qt+s is its output. The firm faces wage costs γwt + sqt + s and other costs, which can be represented by (b/2)q2t+s; as well as production adjustment costs, δ2(qt+s - qt+s-1)2.
The firm's profit at date t + s can be expressed as
Πt+s = Pt+sqt+s - γwt+sqt+s - (α/2)q2t+s - (δ/2)(qt+s - qt+s-1)2
where γ > 0, α > 0, δ > 0 are positive constants.
Given, at date t, the previous period's output qt-1 the firm at date t chooses its production level qt as well as makes a contingent plan for future output {qt+s}∞s=1 to maximize its expected discounted profits
Ets=0∑∞βsΠt+s
Additional information regarding the price and wage sequences {Pt+s}∞s=0 and {wt+s}∞s=0 are given below.
Given this set-up, please answer the following questions.
This question has a very similar set-up to the one I laid out in a handout on linear-quadratic models. We will add a new twist here. Price, at any given date t + s is given by:
Pt+s = A - Dqt+s + zt+s (1)
where A, D are positive constants and zt+s, a demand shifter, is an AR(1) process of the form:
zt+s = vzt+s-1 + εt+s
where εt+s is a mean zero iid process, and 0 ≤ v < 1: Similarly,
wt+s = σwt+s-1 + θt+s
where θt+s is a mean zero iid process, uncorrelated with εt+s, and 0 ≤ σ < 1. Assume the firm acts as a monopolist; that is, it understands the impact its production decision qt+s has on Pt+s.
a. Carefully write out the firm's profit-maximization first-order conditions. Find the firm's optimal level of output for any date t: (You can express this in terms of qt-1, wt, and zt).
b. How do current and future production and current and future prices depend on a positive innovation in εt?
What I have in mind is the following: Suppose εt = Δ > 0. How will this positive shock to demand impact on current and future output and prices?
Please be as specific as you can and please do not forget this is a math-econ class (in other words, do your best to flesh out the story in economic terms).
c. Repeat b, only now assuming θt = Δ > 0. Explain any differences in the responses.
d. In either b or c, will the shock terms have persistent effects on prices and/or output if there is no persistence in the processes they are associated with (i.e., if σ or v equal zero)? Explain.