MICROECONOMICS II
PROBLEM SET 5: Pure Exchange General Equilibrium Model
Problem 1
The set of efficient allocations is gathered in the contract curve. The contract curve
shows all possible combinations of allocations that are Pareto optimal. That is, it gathers
all allocations such that it is not possible to make some individual better off without
lowering the utility of the other individual.
In an economy with 2 goods and 2 individuals, we can find the set of efficient
allocations equating the marginal rate of substitution of both individuals:
We know that the marginal rate of substitution of a generic individual is given by:
,
,
, .
Computing the marginal rate of substitution for each individual, we get that:
,
2
,
,
,
Adding up the endowment that individual A and individual B have of good 1, we get
that the total endowment of good 1 is 20 (=15+5). Similarly, the total endowment of
good 2 is 20 (=3+17). By feasibility, we know that
20
and
20
.
Equating both marginal rate of substitution for both individuals and introducing the
feasibility condition, we get that:
2
20
20
from this equation and obtain that the set of efficient allocations are
Finally, isolate
the allocations such that this equation is satisfied:
20
40
Similarly, we could get the contract curve as the set of efficient allocations that satisfy
the equation 40 / 20
. To do so, we should have introduced in the
equality of marginal rate of subtitution for both agents the feasibility conditions:
20
y
20
.
If we draw in an Edgeworth box the set of efficient allocations, we get that:
The red color represents individual A and the blue one represents B. The red numbers in
the Edgeworth box are the endowments of A and the blue numbers are the endowments
of B. The red curve represents preferences of A, while the blue one represents the
preferences of B. The numbers in black are the total endowment of good 1 and 2. The
purple curve is the contract curve, which shows the set of efficient allocations.
Problem 2
Firstly, plot the Edgeworth box:
The red preferences are the ones of agent A, while the blue ones correspond to B. The
red numbers are the endowment of A, while the blue ones correspond to B. The purple
line is the contract curve.
In this exercise, the goods are perfect complements for A and perfect substitutes for B.
Due to the shape of these preferences, it is not possible to use the tangency condition.
The efficient allocations are in the vertexes of the indifference curve of consumer A;
2 .
that is, the efficient allocations are given by:
The initial endowment is not in the contract curve, since all the exchanges in the lined
area are mutually beneficial for both consumers (they both reach better indifference
curves) and only the vertex is an efficient exchange.
Problem 3
a) The Walras’ law says that if preferences are monotone, the value of the excess
demand function is 0. Formally, monotonicity implies that
0.
=0, where
The proof is the following. Monotonicity implies that ∀ y ∀ , p
is a vector of goods (or components) that shows the net demand of
individual for each good. The net demand is the difference among the demand
of a good and its endowment, such that
, where
is a
vector of endowments of goods (or components). Adding p
=0 for
=0. Notice that the excess
agents in the economy, we get that p ∑
∑
demand function is
, such that
0.
Recall the in this exercise there are only 2 goods and 2 agents.
The intuition behind the Walras’ law consists in the fact that markets clear, since
the value of the excess demand function under monotonicity is 0. This law has 2
implications:
1) If a market has excess of supply, there exists another market with excess of
demand.
2) If
1 markets are in equilibrium, market so will be.
b) In order to compute the competitive equilibrium for this economy, firstly we
have to solve the maximization problem of agents A and B.
We start with individual A. His problem consists in:
15
3
The lagrangian of the problem is the following:
15
3
Take the first order condition of the Lagrangian with respect to
y , and get:
0⇔2
0⇔
Divide both conditions and obtain that:
2
Isolate
and get:
Introduce
2
in the budget constraint of A, and get:
Now, easily isolate
15
3
2
of this linear equation and get:
2
2
10
10
/ . With this, we are normalizing the price of good 2 to 1, and
Call
consider the price of good 1 relative to the price of good 2.
in the previous equation of
and get:
Now, introduce
2
10
5
1
2
Then, we have obtained the demand functions of individual A for both goods:
2
10
5
1
Now, we have to get the demand for individual B. Individual B solves the
following problem:
5
17
Since both problems are similar, the procedure will be omitted (is exactly the
same). Easily, you should get that:
8.5
2.5
2.5
8.5
Now analyze for which prices in the market of a good, the excess demand
function is 0. For instance, take the excess demand function of good 2 and
equate to 0:
=0
Then
5
1 2.5
8.5 3 17 0
Solving this equation, we get that P=1.4. Moreover, by the Walras’ law we know
that if the market of good 2 is in equilibrium, the market of good 1 is also in
equilibrium for these prices (you can check it clearing the market of good 1).
Finally, introduce the prices in the demand functions got previously and get that
the competitive equilibrium is given by:
2
10
11.43
5
2.5
2.5
Graphically:
1
8.5
8
8.57
8.5=12
Where the purple numbers represent the competitive equilibrium.
c) The allocation can be a competitive equilibrium for this economy. To see that,
introduce
the
proposed
equilibrium
in
the
contract
curve: , 40
/ 20
, and notice that the equalities hold. We see
that this allocation can be a competitive equilibrium if and only if agents have
different initial endowments than the ones given by the exercise. Then, keep
constant the endowments of good 2 and let’s determine under which distribution
of endowments for good 1 we would get the proposed equilibrium.
and
, we get that:
Firstly, notice that, for any generic endowment
2
1.5
8.5
2
Introduce in
the feasibility condition
20
. Moreover, we know
that in the proposed equilibrium,
10. Then, we have the following
system of equations with 2 unknowns:
2
10
1.5
20
8.5
10
2
12.75 and finally,
Solving the system, we get that
1.333 ,
20
20 12.75 7.25.
With these endowments and prices, we would get the proposed competitive
equilibrium.
Problem 4
a) Firstly, we solve the problem graphically. The problem has the same data that
exercise 2, so that the Edgeworth box is the following:
We need to find a vector of prices so that the constraint of both agents is tangent
to the marginal rate of substitution of the preferences of both. Due to the nature
of the preferences, the tangency will not arise. In any case, this constraint should
pass through the vertex of the preferences of A and stay over the preferences of
B. As the preferences of B are indifferent in a 1 to 1 relation among both goods,
the prices of both goods should be the same so that the restriction overlaps the
linear preferences.
Then, taking the price of good 2 as numeraire, P=1 clears the market, so that the
graphical equilibrium would be:
Now we solve the problem analytically. The problem of A is given by:
min 2 ,
2
8
Due to the fact that preferences are perfect complements, we obtain that
(it corresponds to the contract curve, due to the fact that the
2
preferences of B are linear). Introducing this equality in the budget constraint
and isolating, we obtain that:
2
8
2
4
16
2
By the previous reasoning, P=1, so that:
10/3
20/3
While what A does not consume is consumed by B:
5/3
10/3
Notice that with exchange, the utility of A has risen from 2 to 10/3, while the
utility of B keeps constant with the exchange (since we move along the
indifferent curve).
b) The new proposed allocation can be part of a competitive equilibrium. To see
that, notice that ((x1A,x2A), (x1B,x2B)) = ((3.5,7),(1.5,3)) is part of the contract
curve, given by 2
(simply introduce the allocation in the contract curve
and check that the equality holds).
Since preferences haven’t changed for any agent, and since the constraint must
overlap preferences of B and pass through the vertex of A, the prices must be the
same than in the previous exercise: P=1.
To get this equilibrium, then, the endowment must be such that they overlap the
indifference curve of B that pass through the proposed equilibrium. In particular,
) are such that
4.5,
this equilibrium will be reached if ( ,
5,
10.
satisfying
Graphically:
Problem 5
Briefly, we explain analytically why there does not exist a vector of prices such that the
market clears. Thereafter, we do it graphically.
Firstly, solve the problem of agent A, which is given by:
,
min
2
.
Due to the fact that preferences are perfect complements, we get that
Introducing this equality in the budget constraint and isolating, we get:
2
Normalize with
/ . Obtain that:
2
1
Now, we solve for agent B. His problem is the following:
2
Solving the problem exactly as we did in exercise 3, we get that:
2
1
2
1
1
Clearing the markets, for instance, for good 2, as we did in exercise 3, we get that:
4
4
3 0
If we tried to solve this 3rd degree equation, we would see there is no positive solution.
Then, there does not exist a vector of prices that clears the market.
The reason why it happens is that the function of agent B is concave. To see this, take
the function of agent B:
Isolate
and get:
Taking the first and second derivative with respect to , you will see they are negative.
This implies that the utility function of agent B is concave.
Then, if we represent both preferences in the Edgeworth box, we get that:
The red preferences are the ones of agent A, while the blue ones are the ones of B. The
lined area represents the exchange zone, and the purple line is the contract curve. From
the endowment point, we should be able to find a hyperplane tangent to both
indifference curves that represents the vector of prices. We see that it is impossible to
plot such a hyperplane. Then, there is no vector of prices that clears the market.
This is due to the nature of the preferences of individual B. In particular, these
preferences are concave, violating the necessary conditions of the welfare theorems.
Problem 6
a) False. The partial equilibrium analyzes for a unique good the functions of supply
and demand exclusively as a function of its price, without taking into account
the price of other goods. However, the general equilibrium analyzes the way in
which the conditions of demand and supply of different markets determine
jointly the prices of several goods.
b) True. In a pure exchange economy, the individuals have endowments of goods
and they exchange among them, but there is no production.
c) False. An allocation is feasible if the decisions of Exchange are compatible with
the endowments:
d)
e)
f)
g)
h)
That is, for all good l, the allocation is feasible if the quantity that A and B get of
each good is lower or equal than the total endowment of that good.
True. The Walras’ Law says that the value of the excess demand function is 0 in
case of monotony. It has 2 implications. One is that if there is a market with
excess of supply, there exists another market with excess of demand. Therefore,
it follows that if n-1 markets are in equilibrium, the nth market must also be.
True. The first welfare theorem says that in case the utility functions are
monotone, the competitive equilibrium is Pareto optimal. By definition, the
Pareto Optimum is a situation in which the utility of a consumer cannot improve
without making the other agent worse off.
False. It depends on the preferences of agents. For instance, think in the case in
which an individual considers one of the goods as a “bad”, while the other one
considers it as a good. In such a case, the individual will give all its bad to the
other agent, and the origin will never be part of the contract curve.
False. If an allocation is part of a competitive equilibrium, by the first welfare
theorem, is pareto optimal, so that it is impossible to improve the utility of and
agent without making another agent worse off. In other words, there does not
exist an allocation such that makes both agents better off.
False. It depends on the preferences of agents. It would be the case if both agents
have the same usual Cobb-Douglas preferences. But it may be not the case, for
instance, if preferences for both agents are linear with different valuation of each
good (the contract curve would be some of the borders of the Edgeworh box).