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Marco Del Negro, Teoria y Politica Monetaria, Fall 1999
Problem set 2
Solutions
Consider the 'price of durable goods' model studied in class, and assume that
the household benefits only from a fraction
of the durable goods she owns.
Specifically, the household's
problem is:
subject to the budget constraint (which holds in each period t):
and the initial condition s-1. (Remember that st-1 and st represent
the amount of durable
goods held by the household at the end of periods t and t+1
respectively, qt is the price of the durable goods in period t,
yt and ct are endowment and consumption at time t
respectively. All variables are expressed in units of the consumption
good) The supply of the durable goods is 1 in each period.
a) Find the first order condition of the agent with respect to st and interpret it
from an economic standpoint.
b) Find the equilibrium values for ct, st, and qt.
a) Let us substitute the constraint in the objective function:
and notice that the term st appears in the sum above only in the two terms:
Taking derivatives, the first order condition is:
where
,
and
which can be rewritten as:
The marginal cost (benefit) from decreasing (increasing) durable goods holdings today is
where qt is the decrease (increase) in consumption, and
is the
decrease (increase) in today's utilty. The marginal benefit (cost) from increasing (decreasing)
durable goods holdings tomorrow comes from the additonal resources that can be spent from consumption,
plus the service in terms of utility that comes from holding the durable good:
Following what we did in class,
we can write the FOC for periods
t+1,..,t+T,
substitute again and again for
qt+1 uc(ct+1,st),...
and obtain the following expression:
Assume that
,
and we obtain:
b) in equilibrium it must be that in each period ct=yt and st=1
the pricing formula then becomes:
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Marco Del Negro
2000-02-04