1. (20 points)
a. Consider Taylor who has 20 hours in her day, and can earn a wage of $15 per
hour chosen to work (and not consumed as leisure 𝑙). Taylor can spend her
income on a composite consumption good 𝑐 that costs $10 per unit. She also
receives $80 from her supportive parents every day.
i. Write the equation of Taylor’s budget constraint.
For Taylor to be able to “afford” the bundle (𝑙, 𝑐), Taylor’s expenditure
on the consumption good must equal the sum of Taylor’s labor income
and non-labor income.
10𝑐 = (20 − 𝑙)15 + 80
or
(20 − 𝑙)15 + 80
𝑐=
10
Note that the slope of the budget constraint (with 𝑐 on the vertical axis)
!"
is − !#.
ii. Suppose Taylor’s utility function is 𝑈(𝑙, 𝑐) = 𝑙𝑐. How much labor will
Taylor supply? Show your work.
Note: you can assume that the solution is in the interior, and that the
second order condition is satisfied at the solution to the FOC.
Given the strictly convex indifference curves, we know the optimum is
at a point where the indifference curve is tangent to the budget
constraint. Equate the two slopes
𝑀𝑈$ 15
− =−
𝑀𝑈% 10
𝑐 3
→ =
𝑙 2
, 3
→𝑐= 𝑙
2
Substitute this for 𝑐 in the budget constraint to get
3
10 4 𝑙5 = (20 − 𝑙)15 + 80
2
→ 15𝑙 = 300 − 15𝑙 + 80
380
→ 𝑙∗ =
30
'(
Thus, the labor supplied by Taylor is 20 − 𝑙 ∗ = 20 − '
= 7.33 hours.
b. Ignore the information in part (a). A consumer lives for two periods and
consumes a single good 𝑥 that costs $1/unit. Their consumption in periods 1
and 2 is labelled 𝑥! and 𝑥) respectively. They earn an income of $100 in period
1 and $150 in period 2. The interest rate for borrowing and lending is 50%.
Suppose this consumer’s utility function is 𝑈(𝑥! , 𝑥) ) = 𝑥!#.' + 0.5𝑥)#.'
i. Write down this consumer’s constrained optimization problem, that
says what they are trying to maximize/minimize, with respect to what
variables, subject to what constraints.
Budget constraint: the present value of the consumer’s income stream
should equal the present value of their consumption expenditures.
150 𝑥)
100 + = 𝑥! +
1 + 0.5 1 + 0.5
→ 𝑥) = 1.5(200 − 𝑥! )
Note, the slope of this line, when 𝑥) is on the vertical axis, is −1.5.
a. Consider Taylor who has 20 hours in her day, and can earn a wage of $15 per
hour chosen to work (and not consumed as leisure 𝑙). Taylor can spend her
income on a composite consumption good 𝑐 that costs $10 per unit. She also
receives $80 from her supportive parents every day.
i. Write the equation of Taylor’s budget constraint.
For Taylor to be able to “afford” the bundle (𝑙, 𝑐), Taylor’s expenditure
on the consumption good must equal the sum of Taylor’s labor income
and non-labor income.
10𝑐 = (20 − 𝑙)15 + 80
or
(20 − 𝑙)15 + 80
𝑐=
10
Note that the slope of the budget constraint (with 𝑐 on the vertical axis)
!"
is − !#.
ii. Suppose Taylor’s utility function is 𝑈(𝑙, 𝑐) = 𝑙𝑐. How much labor will
Taylor supply? Show your work.
Note: you can assume that the solution is in the interior, and that the
second order condition is satisfied at the solution to the FOC.
Given the strictly convex indifference curves, we know the optimum is
at a point where the indifference curve is tangent to the budget
constraint. Equate the two slopes
𝑀𝑈$ 15
− =−
𝑀𝑈% 10
𝑐 3
→ =
𝑙 2
, 3
→𝑐= 𝑙
2
Substitute this for 𝑐 in the budget constraint to get
3
10 4 𝑙5 = (20 − 𝑙)15 + 80
2
→ 15𝑙 = 300 − 15𝑙 + 80
380
→ 𝑙∗ =
30
'(
Thus, the labor supplied by Taylor is 20 − 𝑙 ∗ = 20 − '
= 7.33 hours.
b. Ignore the information in part (a). A consumer lives for two periods and
consumes a single good 𝑥 that costs $1/unit. Their consumption in periods 1
and 2 is labelled 𝑥! and 𝑥) respectively. They earn an income of $100 in period
1 and $150 in period 2. The interest rate for borrowing and lending is 50%.
Suppose this consumer’s utility function is 𝑈(𝑥! , 𝑥) ) = 𝑥!#.' + 0.5𝑥)#.'
i. Write down this consumer’s constrained optimization problem, that
says what they are trying to maximize/minimize, with respect to what
variables, subject to what constraints.
Budget constraint: the present value of the consumer’s income stream
should equal the present value of their consumption expenditures.
150 𝑥)
100 + = 𝑥! +
1 + 0.5 1 + 0.5
→ 𝑥) = 1.5(200 − 𝑥! )
Note, the slope of this line, when 𝑥) is on the vertical axis, is −1.5.
