labor economics question

Jane is trying to figure out how to allocate her time between leisure and work, both in the
summer and in the winter. Jane is a full time student, so that she has only 5 hours per day
to allocate between leisure and work in each season. She earns $8 per hour in the summer
as a tennis instructor, but only $4 per hour in the winter working at a skating rink.
Consumption increases Jane’s utility, but the more she spends on consumption, the less
additional utility she receives from further spending on consumption. The following function
captures how Jane’s marginal utility from consumption in the summer falls as she consumes
more in the summer:
MUSum
C (CSum) = 4 −
1
4
CSum
The rate at which additional utility per dollar of consumption declines with further consumption
is the same in winter as it is in summer:
MUW in
C (CW in) = 4 −
1
4
CW in
In the summer, Jane likes to go to the beach, but the more hours per day she spends
enjoying leisure in the summer, the less additional utility she receives per hour (she starts
to get sunburned). Thus, the following function captures how Jane’s marginal utility from
leisure falls as she enjoys more hours of leisure in the summer:
MUSum
L (LSum) = 12 − 4LSum
In the winter, Jane likes to go skiing. Skiing makes her tired, so she also derives less
additional utility per hour of skiing the more she has already skied that day. Thus, the
following function captures how Jane’s marginal utility from leisure falls as she enjoys more
hours of leisure in the winter:
MUW in
L (LW in) = 12 − 4LW in
Suppose that at first Jane cannot save or borrow across seasons. She can only spend in
each season what she earns in that season.
1A) Write an equation for the budget constraint that Jane faces in the summer (in terms
of CSum and LSum). Write a second equation for the budget constraint that Jane faces in
the winter (in terms of CW in and LW in).
1B) Use an optimality condition for utility maximization to relate the optimal level of
leisure in summer L
∗
Sum to the optimal level of consumption in the summer C
∗
Sum. Provide
intuition for why these conditions should hold at the optimal values of consumption
and leisure. Rearrange the equation (show your work!) to obtain the following expression:
1
L
∗
Sum =
1
2C
∗
Sum − 5. Do the same for L
∗
W in and C
∗
W in to obtain: L
∗
W in =
1
4C
∗
W in − 1.
1C) Use the equations from parts 1A and 1B to solve for the optimal level of consumption
in summer. Do the same for winter. How much will Jane consume in the summer? How
much will she consume in the winter?
1D) How much will Jane work in the summer? How much will she work in the winter?
1E) If you have done problems 1A-1D correctly, you should have found that Jane works
more (and enjoys less leisure) in the winter than in the summer, even though she earns a
higher wage in the summer. How can you explain this finding?
Now suppose that Jane can save or borrow across seasons at an interest rate of 0. In other
words, a dollar earned but not consumed during the summer will allow an extra dollar of
spending during the winter, and vice versa.
1F) Rather than having a separate budget constraint for each season, the ability to save
and borrow means that Jane now has one budget constraint for the entire year. She can
choose any combination of LSum, LW in, CSum, and CW in such that the total dollars of
consumption she purchases over the course of the year is equal to her total yearly earnings.
Write an equation for her year-long budget constraint in terms of LSum, LW in, CSum, and
CW in.
1G) Argue that even when earnings can be saved or borrowed across seasons, the two
optimality conditions from problem 1B will still continue to hold. Substitute these into
the budget constraint from 1F and rearrange (show your work!) to obtain the following
expression relating C
∗
Sum to C
∗
W in: 104 = 5C
∗
Sum + 2C
∗
W in.
1H) Argue that the ability to save and borrow implies that, at the (new) optimal values of
consumption in each season (denoted C
∗
Sum and C
∗
W in), the marginal utility from additional
consumption must be the same: MUSum
C
(C
∗
Sum) = MUW in
C
(C
∗
W in). Use this condition to
relate C
∗
Sum to C
∗
W in.
1I) Substitute the condition obtained in 1H into the re-written budget constraint from 1G.
How much consumption will Jane enjoy in the winter now that she can save? How much
will she enjoy in the summer?
1J) How much will Jane work in the winter? How much will she work in the summer?
1K) If you have done parts 1F-1J correctly, you should have found that once she can save
and borrow, Jane increases her work hours in the summer, and decreases them in the winter,
to the point that she now works more in the summer than she does in the winter. How can
you explain this reversal?


Having issues with 1H and questions after