代写 ECON 301 Microeconomic Theory

Assignment 5

Assignment 5

ECON 301 Microeconomic Theory 2

Jean Guillaume Forand ∗

Winter 2016, Waterloo

1. Consider a two-consumer economy in which ω A = (1,1), ω B = (2,3), u A (x A

1 ,x A 2 ) = x A 1

+x A

2

and u B (x B

1 ,x B 2 ) = 2x B 1

+ x B

2 .

(a) Derive fair allocations for this economy.

(b) Derive a competitive equilibrium for this economy that generates the allocations you

found in (a).

2. Consider a two-consumer economy in which ω A = (2,2), ω B = (2,2), u A (x A

1 ,x A 2 ) = x

A 1

2

1

x A

1

2

1

and u B (x B

1 ,x B 2 ) = x B 1

+ x B

2 .

(a) Derive fair allocations for this economy.

(b) Derive a competitive equilibrium for this economy that generates the allocations you

found in (a).

3. Consider a two-consumer economy in which ω A = (1,1), ω B = (1,1), u A (x A

1 ,x A 2 ) = x A 1

+x A

2

and u B (x B

1 ,x A 2 ) = x B 1

− 3x A

2 .

(a) Illustrate this economy in an Edgeworth box. Label your axes carefully.

(b) Suppose that there are only two markets: one for good 1 and one for good 2. Show

that there exists a competitive equilibrium with p ∗

1

= p ∗

2

= 1.

(c) Show that the equilibrium allocations from (a) are not Pareto-efficient.

(d) Complete the missing market from the equilibrium in (a) by introducing a market for

the externality generated by the consumption of good 2 by consumer A. Suppose that

the allocation of rights is such that consumer A has all the rights over the consumption

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1

of good 2, i.e., such that (ω A

R ,ω

B

R ) = (2,0).

Show that there exists a competitive

equilibrium with p ∗

2

= 0 and p ∗

1

= p ∗

R

= 1.

(e) Show that the equilibrium allocations from (d) are Pareto-efficient.

(f) Find a competitive equilibrium for this economy if the allocation of rights is such

that consumer B has all the rights over the consumption of good 2, i.e., such that

(ω A

R ,ω

B

R ) = (0,2). Show that the allocations in this equilibrium are Pareto-efficient.

(g) Now suppose that the government expropriates all units of good 2 in the economy,

and establishes a permit system: consumption of one unit of good 2 requires a permit

which costs c > 0. All government revenue from permits is returned in equal shares

to the two consumers, and there is a competitive market that determines the price of

good 1. Define a competitive equilibrium for this economy.

(h) Derive a competitive equilibrium for the economy in (g).

(i) For what values of the permit price c are the equilibrium allocations from (h) Pareto-

efficient?

4. Consider a two-consumer, two-period economy, in which the consumption of consumer J in

period i = 1,2 is c J

i

≥ 0. In each period, there is 1 unit of the consumption good available,

which is entirely owned by consumer A in period 1 and entirely owned by consumer B in

period 2. The utility of consumer A from consumption bundles (c A

1 ,c A 2 ) and (c B 1 ,c B 2 ) is

q

c A

1

+ β

q

c A

2 , while that of consumer B is

代写 ECON 301 Microeconomic Theory

代写 ECON 301 Microeconomic Theory

q

c B

1

+ β

q

1 − c A

2 .

(a) Illustrate this economy in an Edgeworth box. Label your axes carefully.

(b) Suppose that there are only two markets: one for the consumption good in period 1

and one for the consumption good in period 2. Derive a competitive equilibrium.

(c) Show that the equilibrium allocations from (a) are not Pareto-efficient.

(d) Complete the missing market from the equilibrium in (a) by introducing a market

for the externality generated by consumption in period 2 by consumer A. Suppose

that the allocation of rights is such that both consumer have rights over

1 / 2 units of

consumption in the second period, i.e., such that (ω A

R ,ω

B

R ) = ( 1 / 2 ,

1 / 2 ). Show that

there exists a competitive equilibrium with p ∗

2

= 0.

(e) Show that the equilibrium allocations from (d) are Pareto-efficient.

(f) Now suppose that the government expropriates all units of the consumption good in

period 2, and establishes a permit system: consumption of one unit of the good in

period 2 requires a permit which costs c > 0. All government revenue from permits is

returned in equal shares to the two consumers, and there is a competitive market that

determines the price of consumption in period 1. Define a competitive equilibrium for

this economy.

2

(g) Derive a competitive equilibrium for the economy in (f).

(h) For what values of the permit price c are the equilibrium allocations from (g) Pareto-

efficient?

3

代写 ECON 301 Microeconomic Theory

代写 ECON 301 Microeconomic Theory