# 代写 ECON 301 Microeconomic Theory

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• 代写 ECON 301 Microeconomic Theory

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
∗ Room 131, Department of Economics, University of Waterloo, Hagey Hall of Humanities, Waterloo, Ontario,
Canada N2L 3G1. Office phone: 519-888-4567 x. 33635. Email: jgforand@uwaterloo.ca. Website: http://arts.
uwaterloo.ca/ ~ jgforand
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

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