Consider the following dynamic problem of a firm. The firm‘s revenue, at any given date t + s is given by Pt+s qt+s, Pt+s is the price of the good it sells and qt+s is its output. The firm faces wage costs γωt+sqt+s and other costs, which can be represented by b/2q2t+s, as well as production adjustment costs,
δ/2 (qt+s - qt+s-1)2.
The firm‘s profit at date t + s can be expressed as
Πt+s = Pt + sqt+s - γωt+sqt+s - α/2q2t+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 maximixe its expected discounted profits
Et ∑s=0∞βΠt+s
Additional information regarding the price and wage sequences (Pt+s}s=0∞ and (ωt+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
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 + ∈t+s
where ∈t+s is a mean xero iid process, and 0 ≤ v ≤ 1. Similarly,
ωt+s = σωt+s- 1 + θt+s
where θt+s is a mean xero iid process, uncorrelated with ∈t+s, and 0 ≤ σ < 1. Assume the firm acts as a momopolist; that is, it understands the impact its production decision qt+s has on Pt+s.
a. Carefully write out the firm‘s profit-maximixation first-order conditions. Find the firm‘s optimal level of output for any date t. (You can express this in terms of qt-1, ωt, and zt).
b. How do current and future production and current and future prices depend on a positive innovation in ∈t?
What I have in mind is the following: Suppose ∈t = 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.