It is relatively easy to extend the single-period model of labor supply
presented in Chapter 16 to many periods. Here we look at a simple example.
Suppose that an individual makes his or her labor supply and consumption
decisions over two periods. \(^{14}\) Assume that this person begins period 1
with initial wealth \(W_{0}\) and that he or she has 1 unit of time to devote to
work or leisure in each period. Therefore, the two-period budget constraint is
given by
\(W_{0}=c_{1}+c_{2}-w_{1}\left(1-h_{1}\right)-w_{2}\left(1-h_{2}\right),\) where
the \(w^{\prime}\) s are
the real wage rates prevailing in each period. Here we treat \(w_{2}\) as
uncertain, so utility in period 2 will also be uncertain. If we assume utility
is additive across the two periods, we have \(E\left[U\left(c_{1}, h_{1},
c_{2}, h_{2}\right)\right]=U\left(c_{1}, h_{1}\right)+E\left[U\left(c_{2},
h_{2}\right)\right]\)
a. Show that the first-order conditions for utility maximization in period 1
are the same as those shown in Chapter \(16 ;\) in particular, show
\(\operatorname{MRS}\left(c_{1} \text { for } h_{1}\right)=w_{1}\) Explain how
changes in \(W_{0}\) will affect the actual choices of \(c_{1}\) and \(h_{1}\)
b. Explain why the indirect utility function for the second period can be
written as \(V\left(w_{2}, W^{*}\right),\) where \(W^{*}=W_{0}+\)
\(w_{1}\left(1-h_{1}\right)-c_{1} .\) (Note that because \(w_{2}\) is a random
variable, \(V\) is also random.) c. Use the envelope theorem to show that
optimal choice of \(W^{*}\) requires that the Lagrange multipliers for the
wealth constraint in the two periods obey the condition
\(\lambda_{1}=E\left(\lambda_{2}\right)\) (where \(\lambda_{1}\) is the Lagrange
multiplier for the original problem and \(\lambda_{2}\) is the implied Lagrange
multiplier for the period 2 utility-maximization problem \() .\) That is, the
expected marginal utility of wealth should be the same in the two periods.
Explain this result intuitively.
d. Although the comparative statics of this model will depend on the specific
form of the utility function, discuss in general terms how a governmental
policy that added \(k\) dollars to all period 2 wages might be expected to
affect choices in both periods.
