Consider the following dynamic problem of a firm. The firm's revenue, at any given date t + s is given by P_t+sq_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 γw_t + _sq_t + s and other costs, which can be represented by (b/2)q²_t+s; as well as production adjustment costs, δ2(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+sq_t+s - γw_t+sq_t+s - (α/2)q²_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 maximize its expected discounted profits

Et_s=0∑^∞β^sΠ_t+s

Additional information regarding the price and wage sequences {P_t+s}^∞_s=0 and {w_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                                          (1)

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 + ε_t+s

where ε_t+s is a mean zero iid process, and 0 ≤ v < 1: Similarly,

w_t+s = σw_t+s-1 + θ_t+s

where θ_t+s is a mean zero iid process, uncorrelated with ε_t+s, and 0 ≤ σ < 1. Assume the firm acts as a monopolist; 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-maximization 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, w_t, and z_t).

b. How do current and future production and current and future prices depend on a positive innovation in ε_t?

What I have in mind is the following: Suppose ε_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.