Exercise 1 A monopolist has an inverse demand curve given ... - IDEA

12/ As long as the welfare function is increasing in each individual’s utility,
a welfare maximum will be Pareto efficient. Furthermore, every Pareto
efficient allocation can be thought of as maximizing some welfare function.
13/ If externalities are present, the outcome of a competitive equilibrium
market is unlikely to be Pareto efficient.
Exercise 6
Consider a pure exchange economy with two goods (1 and 2) and two
consumers (A and B) whose utility functions and initial endowments are
given by:
U A (x 1 , x 2 ) = √ x 1 + x 2 , w A = (3, 7)
U B (x 1 , x 2 ) = x 1 + 2x 2 , w B = (2, 4)
a) Find all the Pareto-efficient allocations (Contract curve).
b) Consider good 1 as the numéraire (p 1 = 1), what is the equilibrium
price of good 2 Find the equilibrium allocation.
c) Is the equilibrium a fair allocation
d) Find the fair allocations.
Answers
a) Any Pareto efficient allocation must satisfy: MRS A = MRS B and the
following feasibility conditions:
x A 1 + x B 1 = w1 A + w1 B = 3 + 2 = 5
x A 2 + x B 2 = w2 A + w2 B = 7 + 4 = 11
MRS A =
√
1/(2
x A 1 )
[
∂U A /∂x A 1
∂U A /∂x A 2
= 1/(2 √
]
x A 1 )
1
=
[
]
1
= ∂U B /∂x B 1
= MRS
2 ∂U B /∂x B B
2
= 1 ⇒ 1 2 xA 1 = 1
The contract curve (the set of Pareto efficient allocations) is given by
x 1 = 1 (This is a straight vertical line in the Edgeworth box). The Pareto
set is P = { (x A 1 , x A 2 ); x A 1 = 1 }
We know that MRS A = MRS B = p 1
p 2
Since p 1
p 2
= 1 and p 2 1 = 1, then p 2 must be equal to 2. There is only one
couple of prices (p 1 , p 2 ) that leads to Pareto efficiency.
b) Consumer A’s problem √
max U A (x A 1 , x A 2 ) = x A 1 + x A 2
subject to p 1 x A 1 + p 2 x A 2 = m A (with p 1 = 1 and m A = 3 + 7p 2 )
6