CHAPTER 17
EXCHANGE
In Chapter 13 we discussed the economic theory of a single market. We
saw that when there were many economic agents each might reasonably
be assumed to take market prices as outside of their control. Given these
exogenous prices, each agent could then determine his or her demands and
supplies for the good in question. The price adjusted to clear the market,
and at such an equilibrium price, no agent would desire to change his or
her actions.
The single-market story described above is a part ial equilibrium model
in that all prices other than the price of the good being studied are assumed
to remain fixed. In the general equilibrium model all prices are variable,
and equilibrium requires that all markets clear. Thus, general equilibrium
theory takes account of all of the interactions between markets, as well as
the functioning of the individual markets.
In the interests of exposition, we will examine first the special case of
the general equilibrium model where all of the economic agents are con-
sumers. This situation, known as the case of pu re exchange, contains
many of the phenomena present in the more extensive case involving firms
and production.
314 EXCHANGE (Ch. 17)
In a pure exchange economy we have several consumers, each described
by their preferences and the goods that, they possess. The agents trade the
goods among themselves according to certain rules and attempt to make
themselves better off.
What will be the outcome of such a process? What are desirable out-
comes of such a process? What allocative mechanisms are appropriate for
achieving desirable outcomes? These questions involve a mixture of both
positive and normative issues. It is precisely the interplay between the
two types of questions that provides much of the interest in the theory of
resource allocation.
17.1 Agents and goods
The concept of good considered here is very broad. Goods can be dis-
tinguished by time, location, and state of world. Services, such as labor
services, are taken to be just another kind of good. There is assumed to
be a market for each good, in which the price of that good is determined.
In the pure exchange model the only kind of economic agent is the con-
sumer. Each consumer i is described completely by his preference, ki (or
his utility function, ui), and his initial endowment of the k commodities,
wi. Each consumer is assumed to behave competitively-that is, to take
prices as given, independent of his or her actions. We assume that each
consumer attempts to choose the most preferred bundle that he or she can
afford.
The basic concern of the theory of general equilibrium is how goods are
allocated among the economic agents. The amount of good j that agent i
holds will be denoted by 4. Agent i's consumption bundle will be
denoted by xi = (xi, . . . , xf); it is a k-vector describing how much of each
good agent i consumes. An allocation x = (xl , . . . , x,) is a collection
of n consumption bundles describing what each of the n agents holds. A
feasible allocation is one that is physically possible; in the pure exchange
case, this is simply an allocation that uses up all the goods, i.e., one in
which C;=, xi = xr=, wi. (In some cases it is convenient to consider an
allocation feasible if xr=l Xi I xy=L=l ~ i . )
When there are two goods and two agents, we can use a convenient way of
representing allocations, preferences, and endowments in a two-dimensional
form, known as the Edgeworth box. We've depicted an example of an
Edgeworth box in Figure 17.1.
Suppose that the total amount of good 1 is w1 = w: + wi and that
the total amount of good 2 is w2 = wf + wi. The Edgeworth box has a
width of w1 and a height of w2. A point in the box, (xi, x:), indicates how
much agent 1 holds of the two goods. At the same time, it indicates the
amount that agent 2 holds of the two goods: (xi, xg) = (wl - x i , w2 - x:).
Geometrically, we measure agent 1's bundle from the lower left-hand corner
WALRASIAN EQUILIBRIUM 315 - CONSUMER 2
CONSUMER 1 -
GOOD 1
Edgeworth box. The length of the horizontal axis measures Figure
the total amount of good 1, and the height of the vertical axis 17.1
measures the total amount of good 2. Each point in this box is
a feasible allocation.
of the box. Agent 2's holdings are measured from the upper right-hand
corner of the box. In this way, every feasible allocation of the two goods
between the two agents can be represented by a point in this box.
We can also illustrate the agents' indifference curves in the box. There
will be two sets of indifference curves, one set for each of the agents. All of
the information contained in a two-person, two-good pure exchange econ-
omy can in this way be represented in a convenient graphical form.
17.2 Walrasian equilibrium
We have argued that, when there are many agents, it is reasonable to
suppose that each agent takes the market prices as independent of his or her
actions. Consider the particular case of pure exchange being described here.
We imagine that there is some vector of market prices p = (pl, . . . , pk), one
price for each good. Each consumer takes these prices as given and chooses
the most preferred bundle from his or her consumption set; that is, each
consumer i acts as if he or she were solving the following problem:
max 'LL,(x,) x,
such that px, = pw,.
The answer to this problem, x,(p, pw,), is the consumer's demand func-
t ion, which we have studied in Chapter 9. In that chapter the consumer's
income or wealth, m, was exogenous. Here we take the consumer's wealth
ahh
-
-
market
value of
given puns
and we adorned
'
'
316 EXCHANGE (Ch. 17)
to be the market value of his or her initial endowment, so that mi = pwi.
We saw in Chapter 9 that under an assumption of strict convexity of pref-
erences, the demand functions will be well-behaved continuous functions.
Of course, for an arbitrary price vector p , it may not be possible actually
to make the desired transactions for the simple reason that the aggregate
demand, xi xi(p , pw,), may not be equal to the aggregate supply, xi wi.
It is natural to think of an equilibrium price vector as being one that
clears all markets; that is, a set of prices for which demand equals supply
in every market. However, this is a bit too strong for our purposes. For
example, consider the case where some of the goods are undesirable. In
this case, they may well be in excess supply in equilibrium.
For this reason, we typically define a Walrasian equilibrium to be a
pair (p*,x*), such that
That is, p* is a Walrasian equilibrium if there is no good for which there is
positive excess demand. We show later, in Chapter 17, page 318, that if all
goods are desirable---in a sense to be made prec ise then in fact demand
will equal supply in all markets.
17.3 Graphical analysis
Walrasian equilibria can be examined geometrically by use of the Edge-
worth box. Given any price vector, we can determine the budget line of
each agent and use the indifference curves to find the demanded bundles
of each agent. We then search for a price vector such that the demanded
points of the two agents are compatible.
In Figure 17.2 we have drawn such an equilibrium allocation. Each agent
is maximizing his utility on his budget line and these demands are compat-
ible with the total supplies available. Note that the Walrasian equilibrium
occurs at a point where the two indifference curves are tangent. This is
clear, since utility maximization requires that each agent's marginal rate
of substitution be equal to the common price ratio.
Another way to describe equilibrium is through the use of offer curves.
Recall that a consumer's offer curve describes the locus of tangencies be-
tween the indifference curves and the budget line as the relative prices
vary-i.e., the set of demanded bundles. Thus, at an equilibrium in the
Edgeworth box the offer curves of the two agents intersect. At such an
intersection the demanded bundles of each agent are compatible with the
available supplies.
MRS
-
- pnumtv
.
•=
EXISTENCE OF WALRASIAN EQUILIBRIA 317
+ CONSUMER 2
GOOD 2
CONSUMER 1 *
GOOD 1
A
Walrasian equilibrium in the Edgeworth box. Each agent Figure
is maximizing utility on his budget line. 17.2
offer curve
17.4 Existence of Walrasian equilibria
Will there always exist a price vector where all markets clear? We will
analyze this question of the existence of Walrasian equilibria in this
section.
Let us recall a few facts about this existence problem. First of all, the
budget set of a consumer remains unchanged if we multiply all prices by any
positive constant; thus, each consumer's demand function has the property
that xi (p, pwi) = xi (kp, kpwi) for all k > 0; i.e., the demand function is
homogeneous of degree zero in prices. As the sum of homogeneous functions
is homogeneous, the aggregate excess demand function,
is also homogeneous of degree zero in prices. Note that we ignore the fact
that z depends on the vector of initial endowments, (wi) , since the initial
endowments remain constant in the course of our analysis.
If all of the individual demand functions are continuous, then z will be a
continuous function, since the sum of continuous functions is a continuous
function. Furthermore, the aggregate excess demand function must satisfy
a condition known as Walras' law.
Walras' law. For any price vector p, we have pz(p) 3 0; i.e., the value
of the excess demand is identically zero.
318 EXCHANCE (Ch. 17)
Proof. We simply write the definition of aggregate excess demand and
multiply by p:
since x i (p , pi) must satisfy the budget constraint pxi = pwi for each
agent i = 1, ..., n. I
Walras' law says something quite obvious: if each individual satisfies his
budget constraint, so that the value of his excess demand is zero, then the
value of the sum of the excess demands must be zero. It is important to re-
alize that Walras' law asserts that the value of excess demand is identically
zero--the value of excess demand is zero for all prices.
Combining Walras' law and the definition of equilibrium, we have two
useful proposit ions.
Market clearing. If demand equals supply i n k - 1 markets, and pk > 0,
then demand must equal supply in the kth market.
Proof. If not, Walras' law would be violated. I
Free goods. If p* is a Walrasian equilibrium and z j ( p * ) < 0 , then
p; = 0. That is, if some good is in excess supply at a Walrasian equilibrium
it must be a free good.
Proof. Since p* is a Walrasian equilibrium, z ( p * ) 5 0. Since prices are
k nonnegative, p*z(p*) = C,=l p:zi(p*) 5 0. If z, ( p * ) < 0 and p; > 0, we
would have p*z(p*) < 0, contradicting Walras' law. I
This proposition shows us what conditions are required for all markets to
clear in equilibrium. Suppose that all goods are desirable in the following
sense:
Desirability. If pi = 0, then z , (p ) > 0 for i = 1,. . . , k .
The desirability assumption says that if some price is zero, the aggregate
excess demand for that good is strictly positive. Then we have the following
proposition:
#
O
EXISTENCE OF AN EQUILIBRIUM 319
Price simplices. The first panel depicts the one-dimensional Figure
price simplex S1; the second panel depicts S2. 17.3
Equality of demand and supply. If all goods are desirable and p* is
a Walrasian equilibrium, then z(p*) = 0.
Proof. Assume zi(p*) < 0. Then by the free goods proposition, pf = 0.
But then by the desirability assumption, zi(p*) > 0, a contradiction. I
To summarize: in general all we require for equilibrium is that there is
no excess demand for any good. But the above propositions indicate that
if some good is actually in excess supply in equilibrium, then its price must
be zero. Thus, if each good is desirable in the sense that a zero price implies
it will be in excess demand, then equilibrium will in fact be characterized
by the equality of demand and supply in every market.
17.5 Existence of an equilibrium
Since the aggregate excess demand function is homogeneous of degree zero,
we can normalize prices and express demands in terms of relative prices.
There are several ways to do this, but a convenient normalization for our
purposes is to replace each absolute price pi by a normalized price
This has the consequence that the normalized prices pi must always sum
up to 1. Hence, we can restrict our attention to price vectors belonging to
the k - 1-dimensional unit simplex:
- lfpieo
-74-4170
KO
XP V -Ruuoutfrunecate -
-
→ Etim
-
320 EXCHANGE (Ch. 17)
For a picture of S1 and S2 see Figure 17.3.
We return now to the question of the existence of Walrasian equilibrium:
is there a p* that clears all markets? Our proof of existence makes use of
the Brouwer fixed-point theorem.
Brouwer fixed-point theorem. Iff : Skpl -+ Sk-l is a continuous
function from the unit simplex to itself, there is some x in Sk-I such that
x = f(x).
Proof. The proof for the general case is beyond the scope of this book;
a good proof is in Scarf (1973). However, we will prove the theorem for
k = 2.
In this case, we can identify the unit 1-dimensional simplex S1 with the
unit interval. According to the setup of the theorem we have a continuous
function f: [O,1] -+ [O, 11 and we want to establish that there is some x in
[O, 11 such that x = f (x).
Consider the function g defined by g(x) = f (x) -x. Geometrically, g just
measures the difference between f (x) and the diagonal in the box depicted
in Figure 17.4. A fixed point of the mapping f is an x* where g(x*) = 0.
Figure Proof of Brouwer's theorem in two dimensions. In the
17.4 case depicted, there are three points where x = f (x).
Now g(0) = f (0) - 0 > 0 since f (0) is in [O,l], and g(1) = f (1) - 1 < 0
for the same reason. Since f is continuous, we can apply the intermediate
value theorem and conclude that there is some x in [O,1] such that g(x) =
f (x) - x = 0, which proves the theorem. I
We are now in a position to prove the main existence theorem.
EXISTENCE OF AN EQUILIBRIUM 321
Existence of Walrasian equilibria. If z : Skpl + R~ is a continuous
function that satisfies Walrus' law, pz(p) = 0, then there exists some p*
i n Sk--' such that z(p*) 5 0.
Proof. Define a map g : Sk-I -+ Sk-' by
gi(p) = pi + max(0, z i ( ~ ) ) for i = 1, . . . , k.
1 + c;=1 max(0, z,(P))
Notice that this map is continuous since z and the max function are con-
tinuous functions. Furthermore, g(p) is a point in the simplex Sk-' since xi gi(p) = 1. This map also has a reasonable economic interpretation: if
there is excess demand in some market, so that zi(p) > 0, then the relative
price of that good is increased.
By Brouwer's fixed-point theorem there is a p* such that p* = g(p*);
i.e.,
p; = P,' + max(0, zi(p*)) for i = 1, . . . , k. (17.1)
1 + C, mado, zj (P* ))
We will show that p* is a Walrasian equilibrium. Cross-multiply equation
(17.1) and rearrange to get
Now multiply each of these k equations by zi(p*):
.i (p*)p: x max(0, r j (p*)) = zi(p*) max(0, zi (p*)) i = 1,. . . , k. [,a ]
Sum these k equations to get
k Now p f z i ( ~ * ) = 0 by Walras' law so we have
Each term of this sum is greater than or equal to zero since each term is
either 0 or ( ~ ~ ( p * ) ) ~ . But if any term were strictly greater than zero, the
2¥40
⇒
322 EXCHANGE (Ch. 17)
equality wouldn't hold. Hence, every term must be equal to zero, which
says
zi(p*) 5 0 for i = 1,. . . ,k.
I
It is worth emphasizing the very general nature of the above theorem.
All that is needed is that the excess demand function be continuous and
satisfy Walras7 law. Walras' law arises directly from the hypothesis that
the consumer has to meet some kind of budget constraint; such behavior
would seem to be necessary in any type of economic model. The hypothe-
sis of continuity is more restrictive but not unreasonably so. We have seen
earlier that if consumers all have strictly convex preferences then their de-
mand functions will be well defined and continuous. The aggregate demand
function will therefore be continuous. But even if the individual demand
functions display discontinuities it may still turn out the aggregate demand
function is continuous if there are a large number of consumers. Thus, con-
tinuity of aggregate demand seems like a relatively weak requirement.
However, there is one slight problem with the above argument for exis-
tence. It is true that aggregate demand is likely to be continuous for positive
prices, but it is rather unreasonable to assume it is continuous even when
some price goes to zero. If, for example, preferences were monotonic and
the price of some good is zero, we would expect that the demand for such a
good might be infinite. Thus, the excess demand function might not even
be well defined on the boundary of the price simplex-i.e., on that set of
price vectors where some prices are zero. However, this sort of disconti-
nuity can be handled by using a slightly more complicated mathematical
argument.
EXAMPLE: The Cobb-Douglas Economy
Let agent 1 have utility function ul(x$, xi) = ( X : ) ~ ( X ? ) ~ - ~ and endowment
wl = (1 ,O) . Let agent 2 have utility function u2(x;, x;) = (x;)~(x;)'-~ and
endowment ~2 = (0,l). Then agent 1's demand function for good 1 is
am1 5 : ( ~ 1 , ~ 2 , m l ) = -* Pl
At prices (pl, p2), income is ml = pl x 1 + pz x 0 = pl. Substituting, we
have
Similarly, agent 2's demand function for good 1 is
THE FIRST THEOREM OF WELFARE ECONOMICS 323
The equilibrium price is where total demand for each good equals total
supply. By Walras' law, we only need find the price where total demand
for good 1 equals total supply of good 1:
Note that, as usual, only relative prices are determined in equilibrium.
17.6 The first theorem of welfare economics
The existence of Walrasian equilibria is interesting as a positive result inso-
far as we believe the behavioral assumptions on which the model is based.
However, even if this does not seem to be an especially plausible assumption
in many circumstances, we may still be interested in Walrasian equilibria
for their normative content. Let us consider the following definitions.
Definitions of Pareto efficiency. A feasible allocation x is a weakly
Pareto efficient allocation if there is no feasible allocation x' such that all
agents strictly prefer x' to x. A feasible allocation x is a strongly Pareto
efficient allocation if there is no feasible allocation x' such that all agents
weakly prefer x' to x, and some agent strictly prefers x' to x.
It is easy to see that an allocation that is strongly Pareto efficient is also
weakly Pareto efficient. In general, the reverse is not true. However, under
some additional weak assumptions about preferences the reverse implica-
tion is true, so the concepts can be used interchangeably.
Equivalence of weak and strong Pareto efficiency. Suppose that
preferences are continuous and monotonic. Then an allocation is weakly
Pareto efficient if and only if i t is strongly Pareto eficient.
Proof. If an allocation is strongly Pareto efficient, then it is certainly
weakly Pareto efficient: if you can't make one person better off without
hurting someone else, you certainly can't make everyone better off.
We need to show that if an allocation is weakly Pareto efficient, then it
is strongly Pareto efficient. We prove the logically equivalent claim that if
an allocation is not strongly efficient, then it is not weakly efficient.
Suppose, then, that it is possible to make some particular agent i better
off without hurting any other agents. We must demonstrate a way to make
everyone better off. To do this, simply scale back i's consumption bundle
324 EXCHANGE (Ch. 17)
by a small amount and redistribute the goods taken from i equally to the
other agents. More precisely, replace i's consumption bundle xi by Oxi and
replace each other agent j 's consumption bundle by x j + (1 - O)xi/(n - 1).
By continuity of preferences, it is possible to choose 6 close enough to 1 so
that agent i is still better off. By monotonicity, all the other agents are
made strictly better off by receiving the redistributed bundle.
It turns out that the concept of weak Pareto efficiency is slightly more
convenient mathematically, so we will ge~~erally use this definition: when we
say "Pareto efficient" we generally mean "weakly Pareto efficient." How-
ever, we will henceforth always assume preferences are continuous and
monotonic so that either definition is applicable.
Note that the concept of Pareto efficiency is quite weak as a normative
concept; an allocation where one agent gets everything there is in the econ-
omy and all other agents get nothing will be Pareto efficient, assuming the
agent who has everything is not satiated.
Pareto efficient allocations can easily be depicted in the Edgeworth box
diagram introduced earlier. We only need note that, in the two-person
case, Pareto efficient allocations can be found by fixing one agent's utility
function at a given level and maximizing the other agent's utility func-
tion subject to this constraint. Formally, we only need solve the following
maximization problem:
such that u2(x2) 2 &
X l + x2 = W l + w2.
This problem can be solved by inspection in the Edgeworth box case. Sim-
ply find the point on one agent's indifference curve where the other agent
reaches the highest utility. By now it should be clear that the resulting
Pareto efficient point will be characterized by a tangency condition: the
marginal rates of substitution must be the same for each agent.
For each fixed value of agent 2's utility, we can find an allocation where
agent 1's utility is maximized and thus the tangency condition will be
satisfied. The set of Pareto efficient points-the Pa re to set-will thus be
the locus of tangencies drawn in the Edgeworth box depicted in Figure 17.5.
The Pa re to set is also known as the contract curve, since it gives the
set of efficient "contracts" or allocations.
The comparison of Figure 17.5 with Figure 17.2 reveals a striking fact:
there seems to be a one-to-one correspondence between the set of Wal-
rasian equilibria and the set of Pareto efficient allocations. Each Walrasian
equilibrium satisfies the first-order condition for utility maximization that
the marginal rate of substitution between the two goods for each agent be
equal to the price ratio between the two goods. Since all agents face the
THE FIRST THEOREM OF WELFARE ECONOMICS 325
C CONSUMER 2
CONSUMER 1 I +
GOOD 1
Pareto efficiency in the Edgeworth box. The Pareto set, Figure
or the contract curve, is the set of all Pareto efficient allocations. 17.5
same price ratio at a Walrasian equilibrium, all agents must have the same
marginal rates of substitution.
Furthermore, if we pick an arbitrary Pareto efficient allocation, we know
that the marginal rates of substitution must be equal across the two agents,
and we can thus pick a price ratio equal to this common value. Graphically,
given a Pareto efficient point we simply draw the common tangency line
separating the two indifference curves. We then pick any point on this
tangent line to serve as an initial endowment. If the agents try to maximize
preferences on their budget sets, they will end up precisely at the Pareto
efficient allocation.
The next two theorems give this correspondence precisely. First, we
restate the definition of a Walrasian equilibrium in a more convenient form:
Definition of Walrasian equilibrium. A n allocation-price pair (x, p)
is a Walrasian equilibrium i f (1) the allocation is feasible, and (2) each
agent is making an optimal choice from his budget set. I n equations:
n n
(1) Exz = xu,.
i=l i=l
(2) If x: is preferred by agent i to x,, then px: > p i .
This definition is equivalent to the original definition of Walrasian equi-
librium, as long as the desirability assumption is satisfied. This definition
allows us to neglect the possibility of free goods, which are a bit of a nui-
sance for the arguments that follow.
326 EXCHANGE (Ch. 17)
First Theorem of Welfare Economics. If ( x , p) is a Walrasian equi-
librium, then x is Pareto eficient.
Proof. Suppose not, and let x' be a feasible allocation that all agents prefer
to x. Then by property 2 of the definition of Walrasian equilibrium, we
have
pxi > pwi for i = 1,. . . , n.
Summing over i = 1,. . . , n, and using the fact that x' is feasible, we have
which is a contradiction. I
This theorem says that if the behavioral assumptions of our model are
satisfied then the market equilibrium is efficient. A market equilibrium is
not necessarily 'Loptimal" in any ethical sense, since the market equilibrium
may be very "unfair." The outcome depends entirely on the original dis-
tribution of endowments. What is needed is some further ethical criterion
to choose among the efficient allocations. Such a concept, the concept of a
welfare function, will be discussed later in this chapter.
17.7 The second welfare theorem
We have shown that every Walrasian equilibrium is Pareto efficient. Here
we show that every Pareto efficient allocation is a Walrasian equilibrium.
Second Theorem of Welfare Economics. Suppose x* is a Pareto
eficient allocation in which each agent holds a positive amount of each
good. Suppose that preferences are convex, continuous, and monotonic.
Then x* is a Walrasian equilibrium for the initial endowments wi = xi* for
i = 1, ..., n.
Proof. Let
Pi = { x i in R ~ : xi + i xi) .
This is the set of all consumption bundles that agent i prefers to xf . Then
define
n n
z : z = x x i w i t h x i i n p i
i=l i=l
P is the set of all bundles of the k: goods that can be distributed among
the n agents so as to make each agent better off. Since each Pi is a convex
THE SECOND WELFARE THEOREM 327
set by hypothesis and the sum of convex sets is convex, it follows that P
is a convex set.
Let w = xy=l xxf be the current aggregate bundle. Since x* is Pareto
efficient, there is no redistribution of x* that makes everyone better off.
This means that w is not an element of the set P.
Hence, by the separating hyperplane theorem (Chapter 26, page 483)
there exists a p # 0 such that
n
p z > p ~ x ~ for a l l z i n P.
i=l
Rearranging this equation gives us
We want to show that p is in fact an equilibrium price vector. The proof
proceeds in three steps.
(1) p is nonnegative; that is, p 2 0.
To see this, let ei = (0,. . . , I , . . . ,0) with a 1 in the ith component. Since
preferences are monotonic, w + ei must lie in P; since if we have one more
unit of any good, it is possible to redistribute it to make everyone better
off. Inequality (17.2) then implies
p(w +ei - W) 2 0 for i = 1,. . . , k.
Canceling terms,
pei 2 0 for i = 1,. . . , k.
This equation implies pi 2 0 for i = 1,. . . , k.
(2) If y, > j x,?, then pyj > px,?, for each agent j = 1 , . . . , n.
We already know that, if every agent i prefers yi to xf, then
Now suppose only that some particular agent j prefers some bundle y j
to xj . Construct an allocation z by taking some of each good away from
agent j and distributing it to the other agents. Formally, let 8 be a small
number, and define the allocations z by
328 EXCHANGE (Ch. 17)
For small enough 8, strong monotonicity implies the allocation z is Pareto
preferred to x', and thus Cy=, zi lies in P. Applying inequality (17.2), we
have
n n
This argument demonstrates that if agent j prefers yj to xj+, then yj can
cost no less than x> It remains to show that we can make this inequality
strict.
(3) If yj >j xj*, we must have pyj > pxj*.
We already know that pyj 2 px;; we want to rule out the possibility
that the equality case holds. Accordingly, we will assume that pyJ = px;
and try to derive a contradiction.
From the assumption of continuity of preferences, we can find some 8
with 0 < 0 < 1 such that 8yj is strictly preferred to xj+ By the argument
of part (2), we know that 8yJ must cost at least as much as xj':
One of the hypotheses of the theorem is that xj* has every component
strictly positive; from this it follows that pxj' > 0.
Therefore, if pyJ - pxj = 0, it follows that 8pyJ < pxj'. But this
contradicts (17.3), and concludes the proof of the theorem. I
It is worth considering the hypotheses of this proposition. Convexity and
continuity of preferences are crucial, of course, but strong monotonicity can
be relaxed considerably. One can also relax the assumption that xf >> 0.
A revealed preference argument
There is a very simple but somewhat indirect proof of the Second Welfare
Theorem that is based on a revealed preference argument and the existence
theorem described earlier in this chapter.
PARETO EFFICIENCY AND CALCULUS 329
Second Theorem of Welfare Economics. Suppose that x* is a Pareto
eficient allocation and that preferences are nonsatiated. Suppose further
that a competitive equilibrium exists from the initial endowments oi = x;
and let it be given by (p', x'). Then, i n fact, (p', x*) is a competitive
equilibrium.
Proof. Since xf is in consumer i's budget set by construction, we must
have xl ki x f . Since x* is Pareto efficient, this implies that x; --i xl.
Thus if xi is optimal, so is xf. Hence, (p', x*) is a Walrasian equilibrium.
I
This argument shows that if a competitive equilibrium exists from a
Pareto efficient allocation, then that Pareto efficient allocation is itself a
competitive equilibrium. The remarks following the existence theorem in
this chapter indicate that the only essential requirement for existence is
continuity of the aggregate demand function. Continuity follows from ei-
ther the convexity of individual preferences or the assumption of a "large"
economy. Thus, the Second Welfare Theorem holds under the same cir-
cumst ances.
17.8 Pareto efficiency and calculus
We have seen in the last section that every competitive equilibrium is
Pareto efficient and essentially every Pareto efficient allocation is a compet-
itive equilibrium for some distribution of endowments. In this section we
will investigate this relationship more closely through the use of differential
calculus. Essentially, we will derive first-order conditions that characterize
market equilibria and Pareto efficiency and then compare these two sets of
conditions.
The conditions characterizing the market equilibrium are very simple.
Calculus characterization of equilibrium. If (x*, p*) is a market
equilibrium with each consumer holding a positive amount of every good,
then there exists a set of numbers ( X I , . . . , A,) such that:
Proof. If we have a market equilibrium, then each agent is maximized on
his budget set, and these are just the first-order conditions for such utility
maximization. The Xi's are the agents' marginal utilities of income. I
The first-order conditions for Pareto efficiency are a bit harder to formu-
late. However, the following trick is very useful.
330 EXCHANGE (Ch. 17)
Calculus characterization of Pareto efficiency. A feasible allocation
x* is Pareto efficient if and only if x* solves the following n maximization
problems for i = 1,. . . , n:
max ui(xi)
(x: 9.;)
n
such that Ex: < wg g = 1, ..., k
Proof. Suppose x* solves all maximization problems but x* is not Pareto
efficient. This means that there is some allocation x' where everyone is
better off. But then x* couldn't solve any of the problems, a contradiction.
Conversely, suppose x* is Pareto efficient, but it doesn't solve one of the
problems. Instead, let x' solve that particular problem. Then x' makes
one of the agents better off without hurting any of the other agents, which
contradicts the assumption that x* is Pareto efficient. 1
Before examining the Lagrange formulation for one of these maximization
problems, let's do a little counting. There are k + n - 1 constraints for
each of the n maximization problems. The first k constraints are resource
constraints, and the second n - 1 constraints are the utility constraints. In
each maximization problem there are kn choice variables: how much each
of the n agents has of each of the k goods.
Let qg, for g = 1,. . . , k, be the Kuhn-Tucker multipliers for the resource
constraints, and let aj , for j # i, be the multipliers for the utility con-
straints. Write the Lagrangian for one of the maximization problems.
Now differentiate L with respect to x: where g = 1,. . . , k and j =
1,. . . , n. We get first-order conditions of the form
au, (x; ) -- q g = o g = 1, ..., k ax:
auj (x;)
a~ - q
g= O j # i ; g = l ,... ,k.
At first these conditions seem somewhat strange since they seem to be
asymmetric. For each choice of i , we get different values for the multipliers
(qg) and (aj). However, the paradox is resolved when we note that the
PARETO EFFICIENCY A N D CALCULUS 331
relative values of the qs are independent of the choice of i. This is clear
since the above conditions imply
ax:
Since x* is given, q9/qh must be independent of which maximization prob-
lem we solve. The same reasoning shows that ai/aj is independent of which
maximization problem we solve. The solution to the asymmetry problem
now becomes clear: if we maximize agent i's utility and use the other
agent's utilities as constraints, then it is just as if we are arbitrarily setting
agent i's Kuhn-Tucker multiplier to be ai = 1.
Using the First Welfare Theorem, we can derive nice interpretations of
the weights (ai) and (qg): if x* is a market equilibrium, then
However, all market equilibria are Pareto efficient and thus must satisfy
From this it is clear that we can choose p* = q and ai = l /Xi . In words,
the Kuhn-Tucker multipliers on the resource constraints are just the com-
petitive prices, and the Kuhn-Tucker multipliers on the agent's utilities are
just the reciprocals of their marginal utilities of income.
If we eliminate the Kuhn-Tucker multipliers in the first-order conditions,
we get the following conditions characterizing efficient allocations:
This says that each Pareto efficient allocation must satisfy the condition
that the marginal rate of substitution between each pair of goods is the
same for every agent. This marginal rate of substitution is simply the ratio
of the competitive prices.
The intuition behind this condition is fairly clear: if two agents had dif-
ferent marginal rates of substitution between some pair of goods, they could
arrange a small trade that would make them both better off, contradicting
the assumption of Pareto efficiency.
It is often useful to note that the first-order conditions for a Pareto
efficient allocation are the same as the first-order conditions for maximizing
332 EXCHANGE (Ch. 17)
a weighted sum of utilities. To see this, consider the problem
n
max c aiui (xi)
i=l
n
suchthat czpi=l
The first-order conditions for a solution to this problem are
which are precisely the same as the necessary conditions for Pareto effi-
ciency.
As the set of "welfare weights" (al,. . . , an) varies, we trace out the set of
Pareto efficient allocations. If we are interested in conditions that charac-
terize all Pareto efficient allocations, we need to manipulate the equations
so that the welfare weights disappear. Generally, this boils down to ex-
pressing the conditions in terms of marginal rates of substitution.
Another way to see this is to think of incorporating the welfare weights
into the definition of the utility function. If the original utility function for
agent i is ui(xi), take a monotonic transformation so that the new utility
function is vi(xi) = aiui(xi). The resulting first-order conditions char-
acterize a partzcular Pareto efficient allocation-the one that maximizes
the sum of utilities for a particular representation of utility. But if we
manipulate the first-order conditions so that they are expressed in terms
of marginal rates of substitution, we will typically find a condition that
characterizes all efficient allocations.
For now we note that this calculus characterization of Pareto efficiency
gives us a simple proof of the Second Welfare Theorem. Let us assume that
all consumers have concave utility functions, although this is not really
required. Then if x* is a Pareto efficient allocation, we know from the
first-order conditions that
1 Dui(x*) = -q for i = 1,. . . , n. ai
Thus, the gradient of each consumer's utility function is proportional
to some fked vector q. Let us choose q to be the vector of competitive
prices. We need to check that each consumer is maximized on his budget
set {xi : qx, 5 qxy). But this follows quickly from concavity; according
to the mathematical properties of concave functions:
Thus, if xi is in the consumer's budget set, u(x) < u(xf).
WELFARE MAXIMIZATION 333
17.9 Welfare maximization
One problem with the concept of Pareto efficiency as a normative criterion
is that it is not very specific. Pareto efficiency is only concerned with
efficiency and has nothing to say about distribution of welfare. Even if we
agree that we should be at a Pareto efficient allocation, we still don't know
which one we should be at.
One way to resolve these problems is to hypothesize the existence of
some social welfare function. This is supposed to be a function that
aggregates the individual utility functions to come up with a LLsocial utility."
The most reasonable interpretation of such a function is that it represents
a social decision maker's preferences about how to trade off the utilities of
different individuals. We will refrain from making philosophical comments
here and just postulate that some such function exists; that is, we will
suppose that we have
W : Rn + R,
so that W(ul,. . . , u,) gives us the "social utility" resulting from any distri-
bution (ul, . . . , un) of private utilities. To make sense of this construction
we have to pick a particular representation of each agent's utility which
will be held fixed during the course of the discussion.
We will suppose that W is increasing in each of its arguments-if you
increase any agent's utility without decreasing anybody else's welfare, social
welfare should increase. We suppose that society should operate at a point
that maximizes social welfare; that is, we should choose an allocation x*
such that x* solves
max W(ui (xi), . . . , un(xn))
n
such that Ex: 5 wg g = l , . . , k .
i i l
How do the allocations that maximize this welfare function compare to
Pareto efficient allocations? The following is a trivial consequence of the
monotonicity hypothesis:
Welfare maximization and Pareto efficiency. If x* maximizes a
social welfare function, then x* is Pareto efficient.
Proof. If x* were not Pareto efficient, then there would be some feasi-
ble allocation x' such that ui(x:) > u , ( x ~ ) for i = 1, . . . , n. But then
W ( ~ ~ ( x ~ ) ~ . . . + n ( x ~ ) ) > W ( ~ i ( x ; ) , . . . ,un(x;)). 1
Since welfare maxima are Pareto efficient, they must satisfy the same
first-order conditions as Pareto efficient allocations; furthermore, under
334 EXCHANGE (Ch. 17)
convexity assumptions, every Pareto efficient allocation is a competitive
equilibrium, so the same goes for welfare maxima: every welfare maximum
is a competitive equilibrium for some distribution of endowments.
This last observation gives us one further interpretation of the com-
petitive prices: they are also the Kuhn-Tucker multipliers for the welfare
maximization problem. Applying the envelope theorem, we see that the
competitive prices measure the (marginal) social value of a good: how
much welfare would increase if we had a small additional amount of the
good. However, this is true only for the choice of welfare function that is
maximized at the allocation in question.
We have seen above that every welfare maximum is Pareto efficient, but is
the converse necessarily true? We saw in the last section that every Pareto
efficient allocation satisfied the same first-order conditions as the problem
of maximizing a weighted sum of utilities, so it might seem plausible that
under convexity and concavity assumptions things might work out nicely.
Indeed they do.
Pareto efficiency and welfare maximization. Let x* be a Pareto
eficient allocation with xf >> 0 for i = 1, . . . , n. Let the utility functions
ui be concave, continuous, and monotonic functions. Then there is some
choice of weights a f such that x* mw5mizes C afu i (x i ) subject to the re-
source constraints. f irthemore, the weights are such that af = 1/Xf where
X f is the ith agent's marginal utility of income; that is, i f mi is the value
of agent i 's endowment at the equilibrium prices p*, then
A* = dvi (P* mi) dmi
Proof. Since x* is Pareto efficient, it is a Walrasian equilibrium. There
therefore exist prices p such that each agent is maximized on his or her
budget set; this in turn implies
Dui(xz ) = Xip* for i = 1,. . . , n.
Consider now the welfare maximization problem
max C a i q (xi)
such that x xf 5 xf *
Notes 335
According to the sufficiency theorem for concave constrained maximiza-
tion problems (Chapter 27, page 504), x* solves this problem if there exist
nonnegative numbers (ql, . . . , qk) = q such that
If we choose ai = l / X i , then the prices p serve as the appropriate nonneg-
ative numbers and the proof is done. I
The interpretation of the weights as reciprocals of the marginal utilities
of income makes good economic sense. If some agent has a large income at
some Pareto efficient allocation, then his marginal utility of income will be
small and his weight in the implicit social welfare function will be large.
The above two propositions complete the set of relationships between
market equilibria, Pareto efficient allocations, and welfare maxima. To
recapitulate briefly:
(1) competitive equilibria are always Pareto efficient;
(2) Pareto efficient allocations are competitive equilibria under convexity
assumptions and endowment redistribution;
(3) welfare maxima are always Pareto efficient;
(4) Pareto efficient allocations are welfare maxima under concavity as-
sumptions for some choice of welfare weights.
Inspecting the above relationships we can see the basic moral: a com-
petitive market system will give efficient allocations but this says nothing
about distribution. The choice of distribution of income is the same as the
choice of a reallocation of endowments, and this in turn is equivalent to
choosing a particular welfare function.
Notes
The general equilibrium model was first formulated by Walras (1954).
The first proof of existence was due to Wald (1951); more general treat-
ments of existence were provided by McKenzie (1954) and Arrow & De-
breu (1954). The definitive modern treatments are Debreu (1959) and Ar-
row & Hahn (1971). The latter work contains numerous historical notes.
The basic welfare results have a long history. The proof of the first welfare
theorem used here follows Koopmans (1957). The importance of convexity
in the Second Theorem was recognized by Arrow (1951) and Debreu (1953).
The differentiable treatment of efficiency was first developed rigorously by
336 EXCHANGE (Ch. 17)
Samuelson (1947). The relationship between welfare maxima and Pareto
efficiency follows Negisihi (1960).
The revealed preference proof of the Second Welfare Theorem is due to
Maskin & Roberts (1980).
Exercises
17.1. Consider the revealed preference argument for the Second Welfare
Theorem. Show that if preferences are strictly convex, then x', = x: for all
i = 1, ..., n.
17.2. Draw an Edgeworth box example with an infinite number of prices
that are Walrasian equilibria.
17.3. Consider Figure 17.6. Here x* is a.Pareto efficient allocation, but
x* cannot be supported by competitive prices. Which assumption of the
Second Welfare Theorem is violated?
CONSUMER 1 GOOD 1
Figure Arrow's exceptional case. The allocation x* is Pareto effi-
17.6 cient but there are no prices at which x* is a Walrasian equilib-
rium.
17.4. There are two consumers A and B with the following utility functions
and endowments:
2 2 uA(x;,xA) = a l n x ; + ( l - a ) l n x A UA = (0 , l )
1 2 uB(xB, xB) = min(xh, xg) UB = (1,O).
Calculate the market clearing prices and the equilibrium allocation.
Exercises 337
17.5. We have n agents with identical strictly concave utility functions.
There is some initial bundle of goods w. Show that equal division is a
Pareto efficient allocation.
17.6. We have two agents with indirect utility functions:
and initial endowments
Calculate the market clearing prices.
17.7. Suppose that all consumers have quasilinear utility functions, so that
vi (p, mi) = vi (p) + mi. Let p* be a Walrasian equilibrium. Show that the
aggregate demand curve for each good must be downward sloping at p*.
More generally, show that the gross substitutes matrix must be negative
semidefinite.
17.8. Suppose we have two consumers A and B with identical utility func-
tions U A ( X ~ , X ~ ) = uB(xl, x2) = max(xl, 22). There are 1 unit of good 1
and 2 units of good 2. Draw an Edgeworth box that illustrates the strongly
Pareto efficient and the (weakly) Pareto efficient sets.
17.9. Consider an economy with 15 consumers and 2 goods. Consumer
3 has a CobbDouglas utility function us(x!j,xi) = Inxi + Inxi. At a
certain Pareto efficient allocation x*, consumer 3 holds (10,5). What are
the competitive prices that support the allocation x*?
17.10. If we allow for the possibility of satiation, the consumer's budget
constraint takes the form pxi 5 pwi. Walras' law then becomes pz(p) 5 0
for all p 2 0. Show that the proof of existence of a Walrasian equilibrium
given in the text still applies for this generalized form of Walras' law.
17.11. Person A has a utility function of UA (XI, 52) = XI + 2 2 and person
B has a utility function u ~ ( x ~ , x ~ ) = max(xl,x2). Agent A and agent B
have identical endowments of (1/2,1/2).
(a) Illustrate this situation in an Edgeworth box diagram.
(b) What is the equilibrium relationship between pl and p2?
(c) What is the equilibrium allocation?

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