Consider the following dynamic problem of a ?rm

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Consider the following dynamic problem of a ?rm. The ?rm?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 ?rm faces wage costs wt+sqt+s and other costs, which can be represented by b

2 q2t+s; as well as production adjustment costs,

? 2 (qt+s ?qt+s?1)

2 :

The ?rm?s pro?t at date t+s can be expressed as

?t+s = Pt+sqt+s ? wt+sqt+s ? ?

2 q2t+s ?

?

2 (qt+s ?qt+s?1)

2

where > 0;? > 0;? > 0 are positive constants.

Given, at date t, the previous period?s output qt?1 the ?rm at date t chooses its production level qt as well as makes a contingent plan for future output fqt+sg

1 s=1 to maximize its expected discounted pro?ts

Et

1X s=0

?s?t+s

Additional information regarding the price and wage sequences fPt+sg 1 s=0 and

fwt+sg 1 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 = ?zt+s?1 + ?t+s

where ?t+s is a mean zero iid process, and 0 ? ? < 1: Similarly,

wt+s = ?wt+s?1 +?t+s

where ?t+s is a mean zero iid process, uncorrelated with ?t+s, and 0 ? ? < 1: Assume the ?rm acts as a monopolist; that is, it understands the impact its production decision qt+s has on Pt+s.

a. Carefully write out the ?rm?s pro?t-maximization ?rst-order conditions. Find the ?rm?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 ?t?

What I have in mind is the following: Suppose ?t = ? > 0. How will this positive shock to demand impact on current and future output and prices?

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Please be as speci?c as you can and please do not forget this is a math-econ class (in other words, do your best to ?esh out the story in economic terms).

c. Repeat b, only now assuming ?t = ? > 0. Explain any di¤erences in the responses.

d. In either b or c, will the shock terms have persistent e¤ects on prices and/or output if there is no persistence in the processes they are associated with (i.e., if ? or ? equal zero)? Explain.

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