# Consider a representative consumer with CobbDouglas preferences

## Intermediate Microeconomics Homework Set 6 (2015)

Intermediate Microeconomics Homework Set 6 (2015)

Question
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)

2

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
.

3

Intermediate Microeconomics Homework Set 6 (2015)

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