1
Problem 1.
Monetary Neutrality or Nonneutrality? (25 points)
.
Suppose firms need to use
only labor in order to produce output, and output in period
t
is given by
t
t
t
y
An
!
, where
t
A
is a
technology shock and
t
n
is the number of hours of labor.
Firms choose their labor in a profit
maximizing way every period.
Suppose
the
representative
consumer’s
period-t
utility
function
is
given
by
ln(
) ln
ln
t
t t
t
t
M
B c
l
P
""
, where, as always,
1
tt
ln
!
#
denotes
leisure
,
t
B
denotes a preference
shock, and
/
M
P
is real money.
Note that rather than supposing there are “168 units of time”
available, here we are supposing that there is only “one unit of time” available (hence, both
n
and
l
will be fractions between zero and one).
Finally, assume the labor supply curve is always
upward-sloping (i.e., it never bends backwards), and that the labor tax rate is always zero.
a.
(7 points)
In the diagram below, show the effect on the consumer’s optimal choice of
consumption and leisure in period t due to a
simultaneous rise in
t
A
and rise in
t
B
.
Clearly
label your diagram, and explain precisely any and all effects pertinent here.
consumption
Solution:
With the representative firm maximizing profit, we know that the real wage equals
the marginal product of labor.
In this case (with the linear production technology), the
marginal product of labor is simply
t
A
, which means the slope of the budget line in the above
diagram (which is the real wage) is
t
A
.
The rise in
t
A
means the budget line becomes
steeper, pivoting around the fixed point on the leisure axis.
Next, consider the MRS between consumption and leisure.
We know the MRS between
consumption and leisure (i.e., the slope of the indifference curve) is
/
lc
uu
.
With the given
utility function,
1/
l
ul
!
and
/(
) 1/
c
u
B Bc
c
!
!
.
Note that the B term drops out of the
marginal utility of consumption here (you had to use the chain rule to fully differentiate).
Thus, the MRS is
unaffected
by the preference shifter in this case, meaning the indifference
map is unaffected.