Prof. Dr. Olivier Bochet
Room: A.314
Phone: 031 631 4176
E-mail: [email protected]
Webpage: http://staff.unibe.ch/bochet
Advanced Microeconomics
Fall 2010
Lecture Note 2
General Equilibrium I
General equilibrium is the study of the microfoundations of the aggregate.
All agents and markets interact with one another. So it views the economy
as an interrelated system in which equilibrium values of all variables are
determined simultaneously. It is a theory of the determination of prices
quantities in a system of perfectly competitive markets. The basic questions
we aim to address are whether markets work well, do market equilibrium
exist, what do we think about the outcome of a market process?
We will first introduce the notations and definitions that are useful for
this part of the course. At the end of the note, we will discuss in detail
several of the crucial assumptions under the theory of general equilibrium is
built. One of the two crucial assumptions are the existence of a complete
set of markets and price-taking behavior –i.e. the assumption that economic
agents have no influence whatsoever on the quoted prices.
1
1
1.1
Basic notations and definitions
A review of properties of preference relations
Let RL+ be the consumption set. A bundle x = (x1 , ..., xL ) ∈ RL+ is simply a
list of quantities of each good. A preference relation denoted is a binary
relation defined over the consumption set RL+ . A preference relation describes
an (ordinal) ranking of the possible bundles in RL+ . For each x, y ∈ RL+ , x y
reads as ”x is at least as good as y”; x y reads ”x is strictly preferred to
y”; and x ∼ y reads as ”x is indifferent to y”. For preference relation and
each x ∈ RL+ , let LC(, x) ≡ {y ∈ RL+ : x y} be the lower contour set at
x of ; U C(, x) = {y ∈ RL+ : y x} be the upper contour set at x of ;
SU C(, x) = {y ∈ RL+ : y x} be the strict upper contour set at x of ;
and IC(, x) = {y ∈ RL+ : x ∼ y} be the indifference set at x of .
We now introduce several properties of preference relations that will be
useful for this note.
1. Preference relation is complete if for each x, y ∈ RL+ , either
x y, or y x, or both.
2. Preference relation is transitive if for each x, y, z ∈ RL+ , x y
and y z implies that x z.
If property 1. and 2. are satisfied, we say that is rational. In the rest
of this note, we take for granted that preference relations are rational.
3. Preference relation is locally non-satiated if for each x ∈ RL+
and each > 0, there exists y ∈ RL+ such that ky − xk ≤ and y x.
The property of local non-satiation asserts that for any x, one can draw
a closed ball of radius centered at x, B,x ≡ {y ∈ RL+ : ky − xk ≤ }, and
find a bundle y within that ball such that y x. Notice that the definition
is silent regarding the direction in which one moves from x to find y. For
instance the definition does not rule out that y is located south-west of x –see
Figure 1 in the appendix. However, notice that although the definition does
not rule out that some goods are ”bad”, it is not possible that all goods are
”bad”. For suppose this is the case, then the best consumption point would
be the 0 bundle, in contradiction with local non-satiation. One important
2
implication of local non-satiation is that it rules out thick indifference sets.
This is shown in Figure 2.
4. Preference relation is monotonic if for each x, y ∈ RL+ such that
y x, we have y x.
Monotonicity of preferences require that for y to be strictly preferred to
x, y should contain more than x in every good. That is for each good `,
y` > x` . This is what means. Obviously, monotonicity of preferences
implies local non-satiation.
5. Preference relation is strictly monotonic if for each x, y ∈ RL+
such that y > x, we have y x.
Strict monotonicity requires that for each good `, y` ≥ x` with at least
one strict inequality for one good. Obviously, strict monotonicity implies
monotonicity.
We now turn to convexity assumptions.
6. Preference relation is convex if for each x, y, z ∈ RL+ such that
y x and z x, then
αy + (1 − αz) x for each α ∈ [0, 1]
A direct implication of the assumption of convexity is that upper contour
sets are convex. That is for each x ∈ RL+ , the set U C(, x) is a convex set.
Convexity is usually explained as diminishing marginal rates of substitutions
→ inclination of agents to diversify their consumptions. An important implication of the assumption of convexity is that it rules out ”wavy” indifference
curves.
7. Preference relation is strictly convex if for each x, y, z ∈ RL+
such that y x and z x, then
αy + (1 − αz) x for each α ∈ (0, 1)
An important implication of the assumption of strict convexity is that it
rules out indifference curves with flat segments. Obviously, strict convexity
implies convexity. See Figure 3.
3
8. Preference relation is continuous if for each sequence of pairs
of bundles {(xm , y m }∞
lim xm = x, lim y m = y, and xm y m for
m=1 with m→∞
m→∞
all m, then x y.
Thus, the preference relation is continuous if it is preserved under
limits. An important implication of the assumption of continuity is that
upper contour sets and lower contour sets are closed sets. A second important
implication is that if the preference relation is continuous, there exists a
utility function u : RL+ → R such that for each x, y ∈ RL+ , x y ⇐⇒ u(x) ≥
u(y). That is, the function u represents the preference relation .
A classical example of a preference relation that is not continuous is the
lexicographic preference relation.
Example 1 Lexicographic preferences are not continuous
Consider the case L = 2 goods. Let be the lexicographic preference
relation over good 1. Then for each x, y ∈ R2+
x y if either x1 > y1 , or if x1 = y1 and x2 ≥ y2
Notice in particular that by definition if x1 > y1 , we necessarily have
x y. You can check that lexicographic preferences are rational, strictly
monotonic and strictly convex. However they are not continuous as shown
below.
Consider the two sequence xm = ( m1 , 0) and y m = (0, 2). Note that y m
is a constant sequence since it will not vary with m. For every m ≥ 1,
we have xm y m since m1 > 0. But observe that lim xm = (0, 0) ≡ x and
m→∞
lim y m = (0, 2) ≡ y. Hence in the limit we obtain that y x, a contradiction
m→∞
with continuity. The lexicographic preference relation puts infinite weight to
infinitesimal changes. An important feature of the lexicographic preference
relation is that indifference sets are singletons, i.e. no two bundles x and y
are indifferent. Obviously, its lack of continuity implies that upper contour
sets are not closed.
4
1.2
Exchange Economies
There is a set of agents N = {1, .., n} and L infinitely divisible goods. The
consumption set of each agent i ∈ N is RL+ . For each i ∈ N , a bundle
xi = (x1 , ..., xL ) ∈ RL+ is simply a list of quantities of each good. A preference
relation for agent i, denoted i. Each i ∈ N has some initial endowment –a
L
The aggregate endowment is
stock of resources–
P ωi = (ω1i , ..., ωLi ) ∈ R+ \{0}.
L
denoted ω̄ = i∈N ωi . We assume that ω̄ ∈ R++ , i.e. each good is available
in some quantities. The endowment point ω = (ω1 , ..., ωn ) ∈ RLn
+ is the list
of agents’ initial endowments.
An exchange economy is E = h(i )i∈N , (ωi )i∈N i. An allocation x =
(x1 , ..., xn ) ∈ RLn
+ is aPlist of bundles, one for each agent. An allocation
Ln
x ∈ R+ is feasible if i∈N xi ≤ ω̄. This definition of feasibility implicitly
implies that there is free disposal of goods, i.e. there is a technology available
to destroy goods if necessary.
The set of feasible allocations for economy E
P
x
≤
ω̄}. The set of non-wasteful allocations is
:
is FE ≡ {x ∈ RLn
+P
i∈N i
Ln
F̄E ≡ {x ∈ R+ : i∈N xi = ω̄}. Notice that F̄E is nothing more than the
Edgeworth box for E.
We now want to investigate among all allocations that are feasible, the
ones that are economically meaningful. For this, we introduce the central
notion of economic efficiency, namely Pareto efficiency.
Pareto efficiency: Allocation x ∈ FE is Pareto efficient if there does
not exist another allocation y ∈ FE such that
yi i xi for each i ∈ N
yj j xj for at least one j ∈ N
(1)
(2)
By looking at Figure 4, we quickly notice that an allocation can be Pareto
efficient only when the intersection of the strict upper contour sets of agents
is empty. That is, allocation x is efficient if ∩ SU C(i , xi ) = ∅.
i∈N
At an interior allocation1 , indifference curves must be tangent at x if x is
efficient. This is illustrated in Figure 5. On the other hand, this claim is not
true for allocations that lie on one of the boundary of the Edgeworth box
as shown in Figure 6.2 There, the tangency condition is not necessary. In
1
An allocation is interior if for each i ∈ N , xi 0.
The boundary of the Edgeworth box is ∂ F̄E = {x ∈ F̄E : there exists i ∈ N for whom
x`i = ω̄` for at least one `}.
2
5
Figure 6, we clearly see that the slopes of the two indifference curves at x are
not equal. However, it remains true that SU C(1 , x1 ) ∩ SU C(2 , x2 ) = ∅.
Let us now look at a computational example of the set of Pareto efficient
allocations.
Example 2 Computing the Pareto set
Let n = L = 2, ω̄ = (1, 1), and preferences of agents be represented by
utility functions as follows
u1 (x11 , x21 ) = x11 x21
1
2
u2 (x12 , x22 ) = (x12 ) 3 (x22 ) 3
Let’s start by asking ourselves the following question. Can an allocation
at which one agents gets 0 of one of the good be Pareto efficient? Notice that
this agent would get a 0 utility level. Actually the indifference curves of these
two utility functions never touch the axis –i.e. both axis are asymptotes.
Therefore, each agent is indifferent between getting the 0 bundle and any
bundle that gives 0 unit of one of the good. We conclude that Pareto efficient
allocations are interior exception made of both origins –preferences being
strictly monotonic, an agent getting all of the two goods while the other gets
nothing is obviously efficient.
Hence at an interior efficient allocations, we have
−2
2
1
(x12 ) 3 (x22 ) 3
x21
= 32
M RS1 = M RS2 ⇐⇒
−1
1
x11
(x12 ) 3 (x22 ) 3
3
Rearranging,
x21
x22
=
(3)
x11
2x12
We now want to compute an equation of x21 as a function of x11 , that
describes the set of efficient allocations (from the point of view of agent 1).
Because preferences are monotonic, we know that efficient allocations
must be non-wasteful, i.e. any efficient allocation x is in F̄E . Hence
x11 + x12 = ω̄1 = 1
x21 + x22 = ω̄2 = 1
6
Substituting into equation (1) above, we have
x21
1 − x21
=
x11
2(1 − x11 )
We obtain that the set of efficient allocations is described as
x21 (x11 ) =
x11
2 − x11
What is the shape of this curve? We already know that it connects both
origins.
2
dx21 (x11 )
=
>0
dx11
(2 − x11 )2
And
4
d2 x21 (x11 )
=
> 0 since x11 ≤ 1
dx211
(2 − x11 )3
The Pareto set is an increasing and convex curve –and it lies below the
45 line. This is shown in Figure 7.
◦
We now turn our attention to the outcomes delivered by market processes. Notice that in determining the Pareto set, the initial distribution of
resources in the economy played no role whatsoever. Only the size of the
Edgeworth box mattered. This will not be true of market processes since
initial endowments will play a central role: along with prevailing prices, they
determine the initial wealth of each agent.
For each i ∈ N , let Bi (p) = {xi ∈ RL+ : p · xi ≤ p · ωi } be agent i’s budget
set at prices p.
Walrasian equilibrium: Given an economy E, A price-allocation pair
(p , x∗ ) ∈ RL \ {0} × RLn
+ is a Walrasian equilibrium if the following two
conditions hold
X
1)
x∗i = ω̄
∗
i∈N
|
2) For each i ∈
{z
}
Supply=Demand
N , x∗i i x0i for
|
all x0 ∈ Bi (p)
{z i
}
x∗i is maximal for i over Bi (p)
7
There are two components in the definition of a Walrasian equilibrium.
The first item states that supply is equal to demand. The second one states
that each agent i is utility maximizing at x∗i . Any bundle strictly preferred
to x∗i must be unaffordable. Notice that the definition does not rule out
that there may be more than one such maximizer –e.g. indifference curves
that have flat segment. Also nothing guarantees that an economy has a single
Walrasian equilibrium. First, existence is not guaranteed (we will study later
on the conditions that guarantee existence). Second, in general an economy
may have several Walrasian equilibria.
The following five figures illustrate the definition and the possibility for
multiplicity of equilibria. Figure 8 shows an example in which the maximization of preferences, given prices p and endowments ω, violates the
supply = demand requirement contained in the definition of a Walrasian
equilibrium. Figure 9 shows an example of a Walrasian equilibrium. Notice
that the maximizers of each agent’s preferences, given p and ω, is a singleton.
This is so because preferences are strictly convex. On the other hand, Figure
10 shows an example of a Walrasian equilibrium where preferences of agent
1 are convex but not strictly convex. We see that the set of best bundles
that agent 1 can obtain, given p and ω1 , indeed contains x1 but this is not
the only bundle that maximizes preferences with respect to agent 1’s budget
constraint.
In Figure 11, x is a Walrasian allocation that is on the boundary of
the Edgeworth box. In Figure 12, x is a boundary allocation that is not
Walrasian. If we restrict ourselves to looking only inside the Edgeworth box,
then x seems Walrasian. But there are bundles that are affordable and better
than x1 for agent 1 and that lie outside of the Edgeworth box. Obviously
these bundles violate feasibility but observe that the preference maximization
part in the definition of a Walrasian equilibrium does not require that the
preference maximizing bundle be feasible. This is a very important detail to
understand. What happens outside of the box is crucial to understand what
is going on inside!
Finally, Figure 13 displays an example of an economy with multiple Walrasian equilibria. Notice that the number if equilibria is odd. This is a
general observation –we will come back to it in the next lecture note.
Example 3 The algebra of equilibrium
8
Let us go back to our previous example and add now the initial distribution of resources in the economy with ω1 = (1, 0) and ω2 = (0, 1). In order
to find the (unique) Walrasian equilibrium of this economy, we need to first
compute the demand functions of both agents.
Agent 1 : M ax u1 subject to p · x ≤ p · ω1 = p1
x11 ,x21
Since u1 is strictly increasing, the constraint must hold with equality.
Also since u1 = 0 whenever the bundle contains 0 unit of one of the two
goods, we know that the solution to the maximization problem is interior.
Hence, at the optimum,
p1
x21
=
M RS1 =
x11
p2
We know then that p2 x21 = p1 x11 . We substitute this information into
the budget constraint and solve for the demand functions. We obtain that
1
2
p1
x21 (p, p · ω1 ) =
2p2
x11 (p, p · ω1 ) =
and
p2
3p1
2
x22 (p, p · ω2 ) =
3
x12 (p, p · ω2 ) =
This takes care of the utility maximization part in the definition of a
Walrasian equilibrium. Now at equilibrium, we must also have supply equal
to demand. Let us normalize p1 = 1 –only relative prices matter. Let us look
at the market for good 1 –we only need to solve for one of the two markets:
whenever L − 1 markets are in equilibrium, then so is the nth market; can
you say why?
1 p2
3
+
= 1 ⇐⇒ p2 =
2
3
2
Hence the equilibrium price vector is p = (1, 32 ). We now compute the
demands at these prices and check that supply is indeed equal to demand.
9
We obtain that
1
2
1
=
3
1
=
2
2
=
3
x11 =
x21
x12
x22
The Walrasian equilibrium (p∗ , x∗ ) is
1, 23 , ( 12 , 13 ); ( 12 , 23 ) .
Given our investigation of markets processes, we may want to know more
regarding this equilibrium allocation. In particular, we would like to know
whether markets deliver an allocation that is efficient. The equation of the
Pareto set for this economy was
x21 =
x11
2 − x11
1/2
At x∗ , x11 = 12 and hence x21 = 2−1/2
= 13 . We conclude that x∗ is
efficient. This turns out to be a general observation under some very weak
assumptions regarding the primitives of the economy.
Theorem 4 (First welfare theorem) Let E be an economy. Suppose preferences are locally non-satiated. Then any Walrasian equilibrium is Pareto
efficient.
Proof. Pick an economy E and pick (p∗ , x∗ ) a Walrasian equilibrium. Assume by contradiction that x∗ is not efficient. Then there exists y ∈ F̄E such
that yi i x∗i for each i ∈ N , and yj j x∗j for at least one j ∈ N . The
preference maximization part in the definition of a Walrasian equilibrium
implies that yj j x∗j =⇒ p∗ · yj > p∗ · ωj . Local non-satiation implies also
an additional property: if yi i x∗i , then p · yi ≥ p · ωi . Suppose this is not
true. That is there exists yi i x∗i and p · yi < p · ωi . By local non-satiation,
there exists yi0 and > 0, arbitrarily small, such that kyi0 − yi k ≤ , yi0 i yi ,
and p · yi0 ≤ p · ωi . By transitivity yi0 i x∗i . But this is in contradiction with
10
x∗i being a maximal element in agent i’s budget set. Hence the claim is true.
This gives us that
X
X
X
p · yi >
p · ωi = p ·
ωi = p · ω̄
i∈N
i∈N
i∈N
Thus,
p·
X
yi > p · ω̄
i∈N
This inequality can be true if and only if
X
yi > ω̄
i∈N
But then y ∈
/ F̄E , a contradiction with our initial assumption. We con∗
clude that x is Pareto efficient.
Q.E.D.
Figure 14 shows what is called the contract curve: the intersection between the Pareto set and the set of allocations that are individually rational.3
Indeed, no one would accept to trade to an allocation that makes one worse
than keeping one’s endowments. Contracts (trades) can only occur within
the individually rational region. By the first welfare theorem, trades must
lead to an efficient allocation of resources.
The first welfare theorem delivers a strong message. Under very weak assumptions, the outcome of the interactions of agents through markets gives
an efficient allocation of resources. This is nothing less than the ”invisible
hand” described by Adam Smith. However, observe that we have no information regarding distribution of resources. All we know is that this is done in
an efficient manner. But could markets be biased toward some distributions,
thereby favoring some agents (or group of agents) over others? The answer
is negative and this will be the content of the second welfare theorem –the
most powerful of the two. Before going to its definition, we first conclude
on the first welfare theorem. The central assumption on the primitive of the
economy is that preferences satisfy local non-satiation. Unfortunately, this
assumption cannot be dispensed with, as show in Figure 15. There agent
3
An allocation x is individually rational if for each i ∈ N , xi Ri ωi .
11
1 has a thick indifference curve at x. Although x is a Walrasian, it is not
efficient because it is located inside the shaded area determined by agent 1’s
thick indifference curve.
We now look at the second welfare theorem and its basic message. We will
leave it unproved at this stage. Its proof will be the object of the next lecture
note in which we introduce economies with production –private ownership
economies. Before stating formally the second welfare theorem, we define a
more general notion of price equilibrium of which Walrasian equilibrium is a
special case.
Price equilibrium with transfers: Given an economy E, allocation x∗
is supportable as a price equilibrium with transfers
if there exists p 6= 0 and
P
a system of transfers T = (T1 , ..., Tn ) with i∈N Ti = 0 and such that
1)
X
x∗i = ω̄
i∈N
2) For each i ∈ N , x∗i i yi for all yi such that p · yi ≤ p · ωi + Ti
This notion of equilibrium is indeed more general than the Walrasian
equilibrium notion. The initial distribution of resources is not necessarily
the one that prevails since the planner may perform wealth transfers across
agents. However, if Ti = 0 for each i ∈ N , then the definition of a price
equilibrium with transfers coincides with the one of a Walrasian equilibrium.
Transfers Ti can be positive, negative or simply 0, but notice that transfers
are balanced in the sense that they sum to 0. No wealth goes out of the
system, and no additional wealth comes in.
Theorem 5 (Second welfare theorem) Let E be an economy. Suppose preferences are continuous, convex and strongly monotonic. Then any Pareto
efficient allocation can be supported as a price equilibrium with transfers.
Figure 16 illustrates the mechanic of the second welfare theorem. Its
message is very powerful since it assesses that markets are unbiased. Any
efficient distribution of resources can be reached through the operation of
markets. Simply redistribute wealth across agents in an appropriate manner,
and let the invisible hand do its work.
12
Observe that the second welfare theorem relies on much stronger assumptions that the first welfare theorem. What happens if we drop convexity or
strong monotonicity?
Figure 17 and 18 exemplify the possible failures of the second welfare
theorem if one of these two assumptions is not met. In Figure 17, agent 1
has preferences that are not convex –”wavy” indifference curves. Allocation
x is efficient since there is tangency between both agents’ indifference curves
at x, but it cannot be supported as a Walrasian equilibrium. In Figure 18, we
see another kind of failure. Because agent 2 likes only good 1 and holds all
of good 1, the only possible Walrasian equilibrium is one in which there is no
trade, i.e. ω is the Walrasian allocation. Obviously this allocation is Pareto
efficient. But it cannot be a Walrasian equilibrium. Suppose that p 0.
Then agent 1 can always afford a better bundle than x1 . On the other hand,
having one of the price being equal to 0 generates an infinite demand for at
least one agent. This shows that not only ω cannot be supported but there
exist no transfers that would make this possible. Notice also how the second
welfare theorem is linked to the existence of Walrasian equilibria.
13
2
appendix
14
Figure 1: Local non-satiation I
15
Figure 2: Violation of local non-satiation
16
Figure 3: Convex and non-convex indifference curves
17
Figure 4: An allocation that is not Pareto efficient
18
Figure 5: Tangency condition
19
Figure 6: No tangency on the boundary
20
Figure 7: Pareto set
21
Figure 8: Supply is not equal to demand
22
Figure 9: An interior Walrasian equilibrium I
23
Figure 10: An interior Walrasian equilibrium II
24
Figure 11: A boundary Walrasian equilibrium
25
Figure 12: A boundary allocation that is not Walrasian
26
Figure 13: Multiplicity of Walrasian equilibria
27
Figure 14: The contract curve
28
Figure 15: Failure of the 1st welfare theorem
29
Figure 16: The second welfare theorem
30
Figure 17: Non-convex preferences and the second welfare theorem
31
Figure 18: Monotonic preferences and the second welfare theorem
32