Consider the following dynamic problem of a firm. The firm‘s revenue, at any given date t + s is given by P_t+s q_t+s, P_t+s is the price of the good it sells and q_t+s is its output. The firm faces wage costs γω_t+sq_t+s and other costs, which can be represented by b/2q²_t+s, as well as production adjustment costs,

δ/2 (q_t+s - q_t+s-1)².

The firm‘s profit at date t + s can be expressed as

Π_t+s = P_{t + s}q_t+s - γω_t+sq_t+s - α/2q²_t+s - δ/2(q_t+s - q_{t + s -1})².

where γ > 0, α > 0, δ > 0 are positive constants.

Given, at date t, the previous period‘s output q_t-1 the firm at date t chooses its production level q_t as well as makes a contingent plan for future output

(q_t+s}_{s = 1}^∞ to maximixe its expected discounted profits

E_t ∑_s=0^∞βΠ_t+s

Additional information regarding the price and wage sequences (P_t+s}_s=0^∞ and (ω_t+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:

P_t+s = A - Dq_t+s + z_t+s

where A, D are positive constants and Z_t+s, a demand shifter, is an AR(1) process of the form:

Z_t+s = vz_t+s-1 + ∈_t+s

where ∈_t+s is a mean xero iid process, and 0 ≤ v ≤ 1. Similarly,

ω_t+s = σω_{t+s- 1} + θ_t+s

where θ_t+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 q_t+s has on P_t+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 q_t-1, ω_t, and z_t).

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 θ_t = 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.