Economics Mathematics

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?

1

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.

2