Monetary Policy, Inflation, and the Business Cycle Chapter 2 A ...

1.0.1 Optimal Consumption and Labor Supply
The optimality conditions implied by the maximization of (1) subject to (2)
are given by:
U n;t
U c;t
= W t
P t
(4)
Q t = E t
Uc;t+1
U c;t
P t
P t+1
for t = 0; 1; 2; :::.
The previous optimality conditions can be derived using a simple variational
argument. Let us …rst consider the impact on utility of a small
departure, in period t, from the household’s optimal plan. That departure
consists of an increase in consumption dC t and and increase in hours dN t ,
while keeping the remaining variables unchanged (including consumption and
hours in other periods). If the household was following an optimal plan to
begin with, it must be the case that
U c;t dC t + U n;t .dN t = 0
for any pair (dC t ; dN t ) satisfying the budget constraint, i.e.
(5)
P t dC t = W t .dN t
for otherwise it would be possible to raise utility by increasing (or decreasing)
consumption and hours, thus contradicting the assumption that the household
is on an optimal plan. Note that by combining both equations we obtain
the optimality condition (4).
Similarly, we can consider the impact on expected utility as of time t of
a reallocation of consumption between periods t and t + 1, while keeping
consumption in any period other than t and t + 1, and hours worked (in all
periods) unchanged. If the household is optimizing it must be the case that
for any pair (dC t ; dC t+1 ) satisfying
U c;t dC t + E t fU c;t+1 .dC t+1 g = 0
P t+1 dC t+1 =
P t
Q t
dC t
3