1. Laura wants to buy songs which cost $2 a song. But in order to afford this, she has to babysit for
her neighbor. Her neighbor has agreed to pay her $10 an hour for babysitting. Laura dislikes
babysitting, but has no other way of buying her music. Her utility is given by the following function:
𝑈ሺ𝐶, 𝐿ሻ ൌ minሼ2𝐶, 22 െ 𝐿ሽ
Where 𝐶 is the number of songs, 𝐿 is the number of hours she babysits.
Laura can work a maximum of 22 hours.
a) Draw Laura’s indifference curves. Plot the number of work hours, 𝐿 (𝑥ଵ) on the horizontal
axis and 𝐶 (𝑥ଶ) on the vertical axis. Show the line along which the indifference curves
kink.
b) In which direction are her preferences increasing in this graph?
c) Is there a way to redefine Laura’s utility function so that her indifference curves have the
usual convex (or L-shaped) shape?
d) How many hours will Laura chose to work?
e) In a more general case, suppose the hourly wage is 𝑤 for babysitting, then the labor
supply curve shows the relationship between the optimal number of hours that Laura
chooses to work as a function of the wage 𝑤. Derive Laura’s labor supply function which
shows her choice of hours of work, 𝐿, as a function of wages 𝑤.
f) Does Laura work more or less as wages increase?
2. Construction workers in the town of Cortland, NY have Cobb-Douglas utility for labor and
consumption,
𝑈ሺ𝐿ሻ ൌ ሺ20 െ 𝐿ሻ𝐶,
𝑀𝑅𝑆 ൌ
𝐶
20 െ 𝐿
where 𝐿 is the number of hours of labor supplied in a day and 𝐶 is the dollar amount of
consumption goods purchased with wage income (𝑝 ൌ $1). 𝑅 ൌ 20 െ 𝐿 is the number of hours of
leisure that the worker has during the day.
a) Derive the labor supply function for workers in Cortland.
b) Draw the labor supply curve on a graph with 𝐿 on the horizontal axis and 𝑤 on the vertical
axis.
3. Tim’s utility function labor and consumption is given by the utility function
𝑈்ሺ𝐿ሻ ൌ 10ሺ12 െ 𝐿ሻ 𝐶,
where 𝐿 is the number of hours of labor supplied in a day and 𝐶 is the dollar amount of
consumption goods purchased with wage income (𝑝 ൌ $1). 𝑅 ൌ 12 െ 𝐿 is the number of hours of
leisure that the worker has during the day.
Tim’s friend Sophia has the following preferences for labor and consumption:
𝑈ௌሺ𝐿ሻ ൌ 5ሺ12 െ 𝐿ሻ 𝐶,
a) Derive Tim and Sophia’s labor supply functions.
b) Draw Tim and Sophia’s labor supply functions on the same graph and compare.
The current wage rate is 𝑤ௗ ൌ $8. A minimum wage increase is being proposed by the state
legislature that would increase the hourly wage to 𝑤௪ ൌ $12.
c) How does the labor supply change for Tim and Sophia from the old wage rate to the new
minimum wage?
Suppose the state government were to impose a lump-sum tax (𝑇) on their wage income.
d) What is the highest tax that can be imposed so that Tim is no worse-off than before the
minimum wage was increased?
e) Redo part d) for Sophia and compare your answers for Tim and Sophia
Risk Preferences
4. The utility function over wealth, 𝑤, is given by the function, 𝑈ሺ𝑤ሻ ൌ √𝑤.
I have 𝑤 ൌ $100 which I can either invest in the stock market (Option A) or keep as cash (Option
B).
Option A: If I invest in the stock market, my future wealth is either, 𝑠ଵ ൌ $150 or 𝑠ଶ ൌ $50 with
equal probability, i.e. 𝑝 ൌ 1 െ 𝑝 ൌ 0.5.
Option B: If I keep my wealth as cash, I face no risk, but also get no return, so that my future
stays 𝑤 ൌ $100.
a) What is my expected value of wealth under Option A?
b) What is my expected utility under Option A? Under Option B? Compare.
c) What is the Certainty Equivalent for Option A?