Consider a representative consumer with CobbDouglas preferences

Intermediate Microeconomics
Homework Set 6: Intertemporal Choice and Competitive Equilibrium
(Due 11/06)
Exercise 1 (20 pts)
There are two goods in this economy, Cigarettes and “all other goods,” sold at a
per unit price of pc and po = $1. Consider a representative consumer with CobbDouglas
preferences u(xc, xo) = x
1/3
c x
2/3
o , where xc is the quantity of cigarettes and xo
is the quantity of “all other goods” (assume both amounts can be any positive real
number). This consumer has a $9 income. Suppose congress enacts a $0.75 quantity
tax on cigarettes (a quantity tax is a per unit tax). Show the original and new budget
constraint for a representative consumer.
(1) Compute the pre- and post-tax optimal bundle. Then decompose the change in
cigarette consumption into income and substitution effects.
(2) What is the maximum amount the smoker would have been willing to pay the
government not to impose the quantity tax? What is the name of this concept?
(3) What amount of income would have to be given to the smoker after the tax to
restore his or her utility to the pre-tax level? What is the name of this concept?
(4) What is amount of taxes the government collects from the consumer? Show
that the government could levy a lump sum tax on the consumer without changing
the price of cigarettes, collect the same amount of revenue, and leave the consumer
better off.

Exercise 4 (20 pts)
Consider an exchange economy with two consumers and two goods. The consumption
sets of the consumers are R
2
+. Each consumer has the same utility function u(x1, x2) =
min{x1, x2}. Provide a complete characterization of all the allocations which are in the
core of this economy. (Recall that an allocation is in the core if it is i) Pareto optimal
and ii) Individually rational (weakly preferred to the endowment))

Exercise 6 (20 pts)
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Consider an exchange economy with two consumers, A and B, and two goods, 1 and
2. Consumer A’s initial endowment is (1, 2), and B’s initial endowment is (2, 1). Their
preferences are represented by the utility functions uA(x
A
1
, xA
2
) = (x
A
1
)
0.25(x
A
2
)
0.75 and
uB(x
B
1
, xB
2
) = 2(x
B
1
)
0.1
(x
B
2
)
0.9
.
(1) Represent the initial situation in an Edgeworth box (draw the indifference curves
through the endowment).
(2) What is the equation of the contract curve? Draw it on the diagram.
(3) Compute the Walrasian equilibrium of this economy.
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