Problem Set 8. General equilibrium analysis
EconS 526
1. We have two agents with indirect utility functions: 𝑣𝑣1 = ln𝑚𝑚1 − 𝑎𝑎ln𝑝𝑝𝑥𝑥 − (1 − 𝑎𝑎)ln𝑝𝑝𝑦𝑦 and 𝑣𝑣2 =
ln𝑚𝑚2 − 𝑏𝑏ln𝑝𝑝𝑥𝑥 − (1 − 𝑏𝑏)ln𝑝𝑝𝑦𝑦 where 𝑚𝑚1 and 𝑚𝑚2 are income levels for person 1 and person 2
respectively, while 𝑝𝑝𝑥𝑥 and 𝑝𝑝𝑦𝑦 are the prices of goods 𝑥𝑥 and 𝑦𝑦, respectively. The initial
endowments for person 1 and person 2 are 𝜔𝜔1 = (𝑥𝑥 = 1, 𝑦𝑦 = 1) and 𝜔𝜔2 = (𝑥𝑥 = 1, 𝑦𝑦 = 1),
respectively. Calculate the market clearing prices.
We only need to do this for one good since only relative prices matter. First, derive demand for good 1
using Roy’s identity,
Person 1: 𝑥𝑥 1 =
𝑎𝑎𝑚𝑚1
𝑝𝑝𝑥𝑥
𝑏𝑏𝑚𝑚2
.
𝑝𝑝𝑥𝑥
and Person 2: 𝑥𝑥 2 =
person 2 it is 𝑚𝑚2 = 1𝑝𝑝𝑥𝑥 + 1𝑝𝑝𝑦𝑦 = 𝑝𝑝𝑥𝑥 + 𝑝𝑝𝑦𝑦 .
Wealth of person 1 is 𝑚𝑚1 = 1𝑝𝑝𝑥𝑥 + 1𝑝𝑝𝑦𝑦 = 𝑝𝑝𝑥𝑥 + 𝑝𝑝𝑦𝑦 and for
So aggregate demand is,
𝑥𝑥 1 + 𝑥𝑥 2 =
𝑎𝑎𝑚𝑚1
𝑝𝑝𝑥𝑥
+
𝑏𝑏𝑚𝑚2
𝑝𝑝𝑥𝑥
=
𝑎𝑎(𝑝𝑝𝑥𝑥 +𝑝𝑝𝑦𝑦 )
𝑝𝑝𝑥𝑥
+
𝑏𝑏(𝑝𝑝𝑥𝑥 +𝑝𝑝𝑦𝑦 )
𝑝𝑝𝑥𝑥
Since aggregate supply is 2, we will have,
= 𝑎𝑎 + 𝑏𝑏 + (𝑎𝑎 + 𝑏𝑏)
𝑎𝑎 + 𝑏𝑏 + (𝑎𝑎 + 𝑏𝑏)
Which means,
𝑝𝑝𝑦𝑦
𝑝𝑝𝑥𝑥
𝑝𝑝𝑦𝑦
=2
𝑝𝑝𝑥𝑥
𝑝𝑝𝑦𝑦
2
=
−1
𝑝𝑝𝑥𝑥 𝑎𝑎 + 𝑏𝑏
2. Consider an economy with 15 consumers and 2 goods. Consumer 3 has a Cobb- Douglas utility
function 𝑢𝑢 = ln𝑥𝑥 + ln𝑦𝑦. At a certain Pareto efficient allocation, x*, consumer 3 holds (x=10,
y=5). What are the competitive prices that support the allocation x*?
In equilibrium the MRS for all consumers will be equal to each other and the relative price. In this case,
3
𝑀𝑀𝑀𝑀𝑀𝑀𝑥𝑥𝑥𝑥
Since x=10, y=5 we have,
1
𝑀𝑀𝑀𝑀𝑀𝑀
𝑦𝑦
𝑝𝑝𝑥𝑥
=−
= − 𝑥𝑥 = − = −
1
𝑝𝑝𝑦𝑦
𝑀𝑀𝑀𝑀𝑀𝑀
𝑥𝑥
𝑦𝑦
5
𝑝𝑝𝑥𝑥
=
= 0.5
𝑝𝑝𝑦𝑦 10
3. Person A has a utility function 𝑢𝑢𝐴𝐴 = x + y and person B has a utility function 𝑢𝑢𝐵𝐵 = max(𝑥𝑥, 𝑦𝑦).
a. Illustrate the situation in an Edgeworth box diagram.
b. What is the equilibrium relationship between px and py?
c. What is the equilibrium allocation?
a
B
A
b.
Here, since there is going to be a corner solution and preferences for one good over the other are
equally weighted in the utility function, we expect Px=Py.
c. We expect at least one of the goods to all go to one person. If they end up in the corner of the box, all
will go to one person, while the other gets the rest.