EC356 Problem Set 1
Fall 2025 Max Levine
Due: September 16, 2025 by 5pm on Blackboard
Note: You have to submit an electronic version of your homework as a
PDF on blackboard on the due date by 5pm. This can be either written in
a word processor or handwritten and scanned. Make sure your write-up is
legible and clearly structured.
Important: Late submissions will not be accepted. Make sure
you use backups and leave enough time for mishaps (scanner
not working, your computer breaks, ...).
Problem 1: Solve the following maximization problems:
(a) max f(x, y) = 5x− x2 − y2
(b) max f(x, y) = 1− x2 − y2 + 2x+ 4y − xy
(c) max g(x, y) = −x2 − y2 subject to 3x− 5y = 1
(d) max h(x, y) = x0.5y0.5 subject to 2x+ 2y = 10
Problem 2: Suppose there is a labor market with two workers, Bianca
and Caroline. Bianca’s individual labor supply is hB = −40 + 3w, and
Caroline’s individual labor supply is hC = 30 + 0.1w, where w is the hourly
wage (in USD) and h is hours worked per week.
(a) What is the aggregate labor supply ha(w) in this market? Create a figure
where you draw Bianca and Caroline’s inverse labor supply functions, as
well as the inverse aggregate labor supply function (that is, with wage
on the y-axis and hours on the x-axis).
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(b) Suppose labor demand in this market is given as: hd = 400−10w. What
is the equilibrium wage and the equilibrium number of total hours worked
in this market? At this equilibrium, how many hours does Bianca and
how many hours does Caroline work? What is the total pay Bianca and
Caroline get (the total pay of an individual is individual hours worked
times the hourly wage)?
(c) Let ηa =
dha
dw
w
ha
be the aggregate labor supply elasticity at the market
equilibrium. Similarly, let ηi =
dhi
dw
w
hi
be the individual labor supply elas-
ticity for individual i. Calculate the elasticities ηa, ηB, ηC . Whose labor
supply is more elastic, Bianca’s or Caroline’s? How do the individual
labor supply elasticities compare to the aggregate?
(d) Suppose there is a demand shock in this labor market and demand in-
creases to hd = 600− 10w. What is the new equilibrium wage, and how
much do Bianca and Caroline work in the new equilibrium? Compare
the change in the number of hours worked for Bianca and Caroline with
the labor supply elasticities in (c)–how are they related?
Problem 3: Consider the utility function u(C,L) = C− 110000 · (H−L)
1+1e
1+ 1e
.
Sometimes it is easier to rewrite the utility function first in terms of hours
worked (rather than leisure). If we substitute h = H − L, we can use the
equivalent utility function u˜(C, h) = C − 110000 · h
1+1e
1+ 1e
. The worker’s labor
choice problem is then:
max
C,h
u˜(C, h) subject to C = wh+G
and the solution for h will be the labor supply curve.
(a) Set up the Lagrangian for this problem and solve for the optimal level of
consumption C∗(w) and hours worked (the labor supply function) L∗(w).
Hint: the utility function may look complicated, but the solution is very
simple if you do things correctly (if you are having trouble, double-check
the rules for derivatives).
(b) Suppose w = 10 and e = 0.3. How much does the person work per week?
Calculate the elasticity of labor supply at that point.
(c) Calculate the elasticity of labor supply in general (that is, without making
any assumptions about w or e). Which terms does elasticity depend on?
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(d) Suppose that wages increase by 10%. If e = 0.5, how much (in percentage
terms) does the number of hours increase?
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