Intermediate Microeconomics
Problem Set 1
to be handed in at the latest by 1PM on November 2nd 2021
Consumer Theory
Exercise 1
Suppose the demand and supply for beer in Springfield is given by
p = 120 − 0.7Qd
p = 3 + 0.2Qs
where the quantity is in million of liters and the price in cents per liter. Assume that
Springfield does not import or export beer.
a) Find the market equilibrium quantity Q∗ and the equilibrium price p∗ .
b) Find the consumer and producer surplus at the market equilibrium.
c) The Springfield breweries successfully lobby for a price floor of p̄ = 36 cents per
liter. What will be the new quantity sold in the market, Q̄?
d) Find the new consumer and producer surplus after the price floor.
e) What is the deadweight loss from the price floor?
f) Suppose the government doesn’t agree with a price floor, but proposes instead
to give a subsidy of 3 cents to the breweries for each liter sold. Find the new
equilibrium quantity Q∗∗ and the new equilibrium price p∗∗ . What is the total
subsidy amount spent by the government?
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Exercise 2
Barney spends all his monthly income of m = A
C500 on two goods, beers (x) and
burgers (y). Assume both goods are divisible. The price of a burger is px = A
C8.
Draw Barney’s budget constraint in the following three cases:
a) One beer costs px = A
C6 if he buys between 0 and 3 beers and px = A
C5 per
bottle if he buys more than 3 beers.
b) One beer costs px = A
C6 if he buys between 0 and 3 beers; if he buys more than
3 beers, each additional beer is half-price, i.e. px = A
C3
c) One day, Barney sees an advertisement stating: ”Buy one beer, get one free!
No limits per customer.” One beer costs px = A
C6.
Exercise 3
For each of the following utility functions describing Barney’s preferences for beers
(x) and burgers (y) calculate the marginal rate of substitution for positive levels of
both goods.
a) u(x, y) = (x + 5y)2
b) u(x, y) = 5 ln x + 2 ln y
√
c) u(x, y) = 12 x + 3 y
d) Assume now that Barney’s preferences for beers and burgers can be described by
the following utility function: u(x, y) = k(x − a)α (y − b)β , where k, a, b, α, β >
0. Show that the utility function
u(x, y) = δ ln(x − a) + (1 − δ) ln(y − b)
represents the same preferences let δ =
α
α+β
.
Exercise 4
• Barney invites his friends Homer and Moe for lunch. They can choose between
burgers (x), beers (y) and ice cream (z). Prove the following results.
a) If x y and y z, then x z.
b) If x ∼ y and y ∼ z, then x ∼ z.
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c) If x y and y ∼ z, then x z.
d) If x ∼ y and y z, then x z
In other words, if the ’parent’ binary relation % is transitive, what does it imply
about its ’offspring’, and ∼?
• Consider the above mentioned 3 alternatives x, y, z. How many comparisons
have to be specified with rationality? (You can ignore comparisons with the
alternative itself.) Consider now the general case with n alternatives. How
many alternatives have to be specified without rationality and how many have
to be specified with rationality.
Exercise 5
For each of the following utility functions, calculate Barney’s demand for burgers
and beers, hx (px , py , m) and hy (px , py , m), as functions of the prices and income. In
addition to that, verify that the demand functions satisfy the budget exhaustion
property and that they are homogeneous of degree zero in prices and income.
a) u(x, y) = 3 ln x + 5 ln y
b) u(x, y) = [min{x, 2y}]2
c) u(x, y) = 3x + 2 ln y
√
√
d) u(x, y) = x + y
Exercise 6
Find the demand functions for Barney if his preferences are given by the utility
function u(x, y) = x− y1 and calculate the price elasticities εxx , εyy , εxy , εyx and income
elasticities η1 and η2 as functions of px , py , and m.
Exercise 7
We know that Barney loves eating burgers and drinking beers. This year, due to
the mad cow disease, there was a sharp increase in the price of cattle meat - the main
ingredient of burgers. This, in turn resulted in an increase in the price of burgers.
Show how a change in the price of burgers effects Barney’s optimal consumption
choice.
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a) Illustrate the
• the total effect,
• the substitution effect,
• and the income effect
for an increase in the price of burgers (px ) graphically.
b) Explain the logic behind the total effect, the substitution effect and the income
effect.
c) Suppose Barney’s preferences are given by the utility function u(x, y) = x + y.
His monthly income is m = A
C500, the price of a burger is A
C8 and the price of a
beer is A
C5. Due to the mad cow disease, the price of a burger then increases,
such that px = A
C9. Decompose the price effects into the substitution and
income effects using either the Hicks-Allen or Slutsky decompositions.
Intertemporal choice
Exercise 8
Barney lives in two periods (period 1 and period 2).
Barney's utility is:
U (c1 , c2 ) = u(c1 ) + βu(c2 )
where c1 and c2 are the consumptions in period 1 and period 2 respectively, β < 1
is the discount factor, and u(c) = ln(c) with ln the natural logarithm. Barney has
incomes y1 in period 1 and y2 in period 2 with the possibility to borrow and lend at
the market interest rate r.
a) Write down the intertemporal budget constraint faced by Barney.
b) Find out the Euler equation.
c) Express the optimal saving as function of incomes and interest rate.
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