For each of these questions, you should be able to set up the consumer's

problem, derive and interpret the first order conditions, solve for optimal

consumption, and discuss any implications. You should also be able to ex-

plain the rational expectations assumption and how it is used.

The essential structure of this question is a variant of the consumption-

savings problem, where future income is uncertain (and can take on many

possible values).

Let the per period utility function be quadratic:

u

(

c

) =

c

a

2

c

2

:

The

consumer discounts the future at rate

. You can assume that

=

r:

Future income, is uncertain, but evolves according to:

y

0

=

+

y

+

0

;

where

,

>

0, and

0

is a random variable with mean zero. Assume that

parameter values are such that the present value of the consumer's lifetime

income is positive for all possible realizations of

0

. The consumer has rational

expectations.

The possible questions are:

1. Suppose there is a tax that reduces the consumer's income in the rst

period (i.e. after-tax income in the rst period equals

y

t

, where

t

is

the tax). The tax is known to be temporary, and will not be paid in

the second period.

2. Suppose there is a tax that reduces the consumer's income in both

periods (i.e. after-tax income in the rst period equals

y

t

, and in

the second period will be

y

0

t

, where

t

is the tax). It is known that

the tax will have to be paid in both periods.