Eciency
(also Pareto called Pareto Optimality)
1
Denitions and notation
Recall some of our denitions and notation for preference orderings. Let X be a set (the set of
alternatives); we have the following denitions:
1. A relation R on X is a subset of X X . We often write xRy instead of (x, y ) R and we
say x is R-related to y .
2. If R is a relation on X , we denote the complement of R by R (instead of R, because
/
will be given another meaning). Thus, xR means that x is not R-related to y .
/y
3. A strict preordering of X is a transitive and irreexive relation P on X . We sometimes
write x
y for xP y , and also y
x. We say that x is preferred to y . A complete
preordering of X is a transitive and complete relation
on X .
4. If P is a strict preordering, we denote the corresponding indierence relation by I , dened
by xIy [xP & yP ]. We also write x y for xIy , and x
/y
/x
y (also y
x) for
[xP y or xIy ]. Note that is both reexive and symmetric, but it need not be transitive;
and that
is complete, but it need not be transitive. (Can you provide a counterexample
to show that transitivity may fail?) If is transitive , then
5. If
is a complete preordering.
is a complete preordering, then is transitive, and [x
y&y
z ] implies x
z for
any x, y, z X .
2
Aggregation of rankings into a single ranking
Let X be a set of alternatives, generically denoted by x; let N be a set of n individuals, generically denoted by i; and let P be a set of admissible preorderings (rankings, or preferences)
over X , generically denoted by P .
We want to have a rule we can use to aggregate a list P = (P1 , ..., Pn ) of individual rankings into
a single aggregate ranking, P. In other words, we want to have an aggregation function or rule
a : P n P ,
i.e.,
1
a
(P1 , ..., Pn ) P.
(1)
(Note the similarity with the notation for the sample mean of a list of n numbers: x =
1
n
i
xi .
The sample mean is a way of aggregating a list of numbers into a single representative number
i.e., its a function that maps a list of numbers into a single number).
Instead of framing the problem as one of aggregating a list of rankings into a single ranking, we
could alternatively frame the problem as one of aggregating a list of utility functions into a single
utility function. We will return to this idea in Section 7.
2.1
Examples
Here are several examples of sets X of alternatives for which we might wish to aggregate a list of
individual rankings into a representative ranking:
1. X is a set of allocations x = (x1 , ..., xn ) Rnl .
+
2. X is a set of candidates for a job, or for a political position.
3. X is a set of public policies.
4. X is a set of teams; for example, X = {A, B, C } , A : Arizona, B : Boston College, C :
California.
5. X is a set of tennis players; for example, X = {A, B, C } , A : Agassi, B : Becker, C : Chang.
In this last example the n individual rankings P1 , ..., Pn could be the rankings (i.e., the order of
nish) in each of n tournaments, and the problem is to aggregate these tournament results into a
single ranking of the players. This is exactly what the ATP Ranking is, an aggregation of the
players tournament nishes during the preceding year into a single ranking of the players. The
ATP ranking uses a specic rule (function, algorithm) a : (P1 , ..., Pn ) P to calculate P . The
ATP rule weights the various tournaments dierently, assigning more weight for example to the
so-called Grand Slam tournaments than to other tournaments. (ATP is the abbreviation used by
the Association of Tennis Professionals.)
3
The Pareto ranking
Denition: Let P = (P1 , ..., Pn ) be a list of preorderings of a set X of alternatives. We say that
x is a Pareto improvement upon x (which we write xPx), or that x Pareto dominates x, if
i : xP i x and i : xPi x. P is called the Pareto ranking or Pareto ordering associated with
/
the list P = (P1 , ..., Pn )
2
Remark: As above, the alternatives (the elements of X ) neednt be allocations they could be
political parties, candidates, athletic teams, etc. and the function (P1 , ..., Pn ) P is only
one of many possible ways to aggregate the list (P1 , ..., Pn ) of rankings into a single aggregate
ranking P.
Remark: If each Pi is transitive or irreexive, then so is P. But even if each Ii is transitive,
and each Pi is transitive and irreexive, I may fail to be transitive, as Examples 2 and 3 below
demonstrate, or I may be transitive but uninformative, as Example 1 demonstrates.
In Examples 1 and 2, below, the set of alternatives is X = {A, B, C }. Here are two possible
interpretations of the examples:
A is Arizona, B is Boston College, C is California. Each Pi could be an individuals
ranking of these universities basketball teams, or their economics departments, or their
reputations as party schools, etc.
A is Agassi, B is Becker, C is Chang. Each Pi is their order of nish in a tournament.
Example 1: X = {A, B, C } ; A
1
B
1
C; B
2
C
2
A; C
3
A
3
B . Therefore AIB , B IC ,
and C IA. Thus, the aggregate indierence relation I is transitive, but not very useful.
Example 2: X = {A, B, C } ; A
1
C
1
B; B
2
A
2
C . Therefore A B, B C, and A
C.
Thus, the aggregate indierence relation I (or ) is not transitive.
Example 3: Figure 1 is an Edgeworth box with two consumers, each with a standard preference.
Nevertheless, the rankings of the allocations A, B, and C are just as in Example 2: A
and B
2
A
and A
C.
2
1
C
1
B
C , so that the Pareto ranking is also the same as in Example 2: A B, B C,
3
4
Pareto Eciency
Denition: Let X be a set of alternatives, and let (
n
i )1
be a list of preferences over X . An
alternative x is Pareto ecient if no alternative in X Pareto dominates x.
We often have a fundamental set X of alternatives for example, all the conceivable or well
dened alternatives but only a subset F
X of the alternatives are actually possible, or
feasible. Moreover, we generally want to allow the set F to vary and to see how the Pareto
ecient alternatives depend on the set F . Our standard allocation problem is a good example
of this: we take X = Rnl to be the set of all conceivable allocations, and this is the set over
+
which individuals preferences
i
are dened; but in order to say whether a given allocation (xi )n
1
is ecient we dont want to insist that it not be dominated by any conceivable allocation, only
that it not be dominated by any other feasible allocation i.e., by any other allocation that can
actually be achieved with existing resources.
Denition: Let (
n
i )1
be a list of preferences over a set X , and let F
X . An alternative x F
is Pareto ecient (with respect to F ) if it is not Pareto dominated by any other alternative
x F.
5
Characterizing Pareto ecient allocations
The denition of Pareto eciency is pretty awkward and clumsy to work with analytically. Wed
like to be able to characterize the ecient alternatives in some way thats more analytically
tractable or more economically intuitive for example, as the solution to an optimization problem,
or in terms of marginal rates of substitution. For our economic allocation problem we can actually
establish such a characterization.
So far, with the exception of Example 3 above, weve been dealing with alternatives in the abstract:
the alternatives could be just about anything. Dened at this level of generality, the idea of Pareto
eciency can be applied in many useful contexts. But for the economic allocation problem were
studying, the alternatives we want to compare are alternative allocations; moveover, when were
dealing with allocations, the individual preferences are typically representable by utility functions.
With the structure provided by Euclidean space (where the allocations live) and by using functions
instead of orderings, its pretty easy to characterize the Pareto ecient allocations as the solutions
to a constrained maximization problem. Well tackle that rst. And then, since we already know
how to characterize the solutions of a constrained maximization problem in terms of rst-order
conditions, well have solved the problem of characterizing the Pareto ecient allocations (or simply
4
the Pareto allocations) by rst-order conditions. Then well nd that its pretty straightforward
to translate the rst-order conditions into a set of economic marginal conditions thus giving us
a characterization of the Pareto allocations in terms of marginal conditions.
Lets begin by taking an allocation x = (xi )n thats Pareto ecient and well show that because x
1
is a Pareto allocation it must be a solution to a specic constrained maximization problem. The
constraints are of course the usual resource constraints, to which we add the requirement that any
alternative allocation x must make n 1 of the consumers no worse o than they would have
been at x. Then, since x is Pareto ecient, it must be providing the remaining consumer with the
greatest utility possible among all these alternatives x. Note that since x is given, each ui (xi ) in
the following proposition is just a real number. Throughout this section were assuming that the
preferences are representable by utility functions.
Proposition: If the allocation x is Pareto ecient for the n consumers (ui , i )n , then x is a
x1
solution of the following maximization problem:
max u1 (x1 )
(xi )Rnl
+
k
xi
k
subject to
n
0,
xi
k
k ,
x
i=1
i
i
u (x )
i = 1, ..., n,
k = 1, ..., l
(P-Max)
k = 1, ..., l
ui (xi ),
i = 2, ..., n.
Proof. Suppose ( i )n is not a solution to (P-Max) i.e., there exist x1 , ..., xn Rl for which
x1
+
n
xi
k
i=1
i
i
u ( )
x
k ,
x
k = 1, ..., l
ui ( i ) i = 2, ..., n
x
u1 ( 1 ) > u1 ( 1 ).
x
x
Then ( i )n is clearly a Pareto improvement on ( i )n ; i.e., ( i )n is not Pareto ecient.
x1
x1
x1
A striking feature of this proposition is that it requires no assumptions about the consumers utility
functions. They neednt be convex, or continuous, or even increasing. And the same proof can
be used even if the utility functions arent selsh i.e., even if some of the consumers care about
others consumption levels.
In order to have a characterization of the Pareto ecient allocations, we have to establish the
converse of the proposition weve just established: we have to show that any solution of (P-Max)
is Pareto ecient. In fact, the converse isnt actually true under such general conditions. But if
5
the consumers utility functions are all continuous and strictly increasing, thats enough to ensure
that the converse is true.
Proposition: If every ui is continuous and strictly increasing, and if the allocation x is a solution
of the problem (P-Max), then x is Pareto ecient for (ui , i )n .
x1
Proof. Suppose ( i )n is not Pareto ecient; we will show that then ( i )n is not a solution of
x1
x1
(P-Max). Since ( i )n is not Pareto ecient, there exists a Pareto improvement upon ( i )n , lets
x1
x1
say ( i )n :
x1
n
xi
k
k ,
x
i=1
i
i
k = 1, ..., l
ui ( i ) i = 1, ..., n
x
u ( )
x
ui ( i ) > ui ( i ) for some i.
x
x
If u1 ( 1 ) > u1 ( 1 ), then ( i )n is not a solution of (P-Max), and the proof is complete. So assume
x
x
x1
that u1 ( 1 ) = u1 ( 1 ) and (wlog) u2 ( 2 ) > u2 ( 2 ).
x
x
x
x
Since u2 is continuous, we may choose
> 0 small enough that every x2 B (x2 ) Rl satises
+
u2 (x2 ) > u2 (2 ). And since u1 is strictly increasing, there is a bundle x 1 B (x1 ) Rl that
x
+
satises u1 (x 1 ) > u1 ( 1 ) = u1 ( 1 ).
x
x
For the
identied in the preceding paragraph, dene a new allocation (x i )n as follows:
1
x1
is as above:
x 1 B (x1 ) Rl and u1 (x 1 ) > u1 ( 1 )
x
+
x 2 = x2 + x1 x 1
x i = xi ,
i = 3, ..., n.
Thus, x 1 + x 2 = x 1 + x2 + x1 x 1 = x1 + x2 , so that
n
i=1
xki =
n
xi
k
k , k = 1, ..., l and
x
i=1
u2 (x 2 ) = u2 ( 2 + x1 x 1 ) > u2 ( 2 ), since x1 x 1 < . We also have ui (x i )
x
x
ui ( i ), i = 3, ..., n
x
and u1 (x 1 ) > u1 ( 1 ). In other words, (x i )n satises all the constraints in (P-max) and u1 (x 1 ) >
x
1
u1 ( 1 ); therefore ( i )n is not a solution of (P-max).
x
x1
Combining the two propositions weve just established gives us the following theorem.
Theorem: If every ui is continuous and strictly increasing, then an allocation x is Pareto ecient
for the economy (ui , i )n if and only if it is a solution of the problem (P-max).
x1
Its important to note that while the problem (P-max) as well as the theorem and the two propositions are all stated in terms of maximizing u1 , the theorem and the propositions are actually true
6
if we restate (P-max) using any one of the n utility functions ui as the maximand and of course use
the remaining 1 n utility functions in the constraints. We can see this in either of two ways: each
proof can obviously be altered in accordance with the change in the statement of the maximization
problem; or we could simply re-index the n individuals in the economy so that the utility function
to be maximized becomes u1 , and then the original maximization problem becomes the relevant
one.
For interior allocations, we can weaken the requirement that utility functions be strictly increasing,
requiring only that they be locally nonsatiated.
Denition: A preference
on a set X
Rl is locally nonsatiated if for any x X and any
neighborhood N of x, there is an x N that satises x
x.
Note: We would therefore say that a utility function u on a set X
Rl is locally nonsatiated if
for any x X and any neighborhood N of x, there is an x N that satises u( ) > u(x).
x
Theorem: If every ui is continuous and locally nonsatiated, then an interior allocation x is Pareto
ecient for the economy (ui , i )n if and only if it is a solution of the problem (P-max).
x1
Proof. In the proof given above, for strictly increasing utility functions, we can now choose small
enough that B (x1 ) Rl , because x is an interior allocation. The remainder of the proof is
+
identical.
Exercise: Provide a counterexample to show why, for interior allocations, the theorem requires
that utility functions be locally nonsatiated, and a counterexample to show why, at a boundary
allocation, local nonsatiation is not enough.
6
Calculus characterization of Pareto eciency: marginal conditions
Now that weve characterized Pareto ecient allocations as solutions to a constrained maximization
problem, it should be straightforward to use that maximization problem to characterize the Pareto
allocations in terms of rst-order conditions, and then to re-cast the rst-order conditions as
economic marginal conditions. First-order conditions are calculus conditions, and they require
some convexity i.e., second-order conditions so throughout this section we assume that each
consumers utility function ui is continuously dierentiable and quasiconcave. To simplify notation
we write ui for the partial derivative
k
ui
.
xi
k
We also assume that each ui is strictly increasing:
ui (xi ) > 0 for all i and k . Thus, only those allocations that fully allocate all the goods those
k
that satisfy
n
1
xi =
ni
x
1
could be Pareto allocations. You should be able to verify that under
these assumptions the Kuhn-Tucker Theorems second-order conditions and constraint qualication
7
are satised, so that the KT rst-order conditions are necessary and sucient for an allocation
( i )n to be a solution of (P-max).
x1
6.1
Interior Allocations
In the previous section we established that an allocation is Pareto ecient if and only if it is
a solution of the constrained maximization problem (P-max). Lets assign Lagrange multipliers
1 , ..., l to the l resource constraints in problem (P-max) and multipliers 2 , ..., n to the n 1
utility-level constraints ui (xi )
x1
ui ( i ), i = 2, ..., n. If all the xi s are strictly positive i.e., if ( i )n
x
k
is an interior allocation then the rst-order marginal conditions for ( i )n to be a solution of
x1
(P-max) are all equations:
2 , ..., n
0 and 1 , ..., l
0 such that for each k = 1, ..., l:
u1 = k
k
and 0 = k i ui ,
k
i = 2, ..., n
(FOMC)
We can rewrite the last n 1 equations as i ui = k (i = 2, ..., n; k = 1, ..., l). We also have
k
k > 0 for each k and i > 0 for each i = 2, ..., n (you should be able to show why this is so;
recall that the value of a constraints Lagrange multiplier is the constraints shadow value the
marginal increase in the objective value achievable by a one-unit relaxation of the constraints
right-hand-side). Therefore, for every consumer i and every pair of goods k and k , we have
ui
k
k
=
,
i
uk
k
i
i.e. M RSkk =
k
.
k
That last equation says that each consumers M RSkk between any two goods k and k is equal
to the relative shadow values of those two goods in the maximization problem (P-max). Clearly
then, for any pair of goods every consumer must have the same M RS :
1
i
n
M RSkk = ... = M RSkk = ... = M RSkk .
(EqualMRS)
Weve derived the equality of M RS s in (EqualMRS) from the Kuhn-Tucker rst-order conditions
for ( i )n to be a solution of (P-max). Therefore (EqualMRS) is a necessary condition for ( i )n
x1
x1
to be a Pareto allocation.
In order to show that (EqualMRS) is also a sucient condition for Pareto eciency we need to
determine values of the Lagrange multipliers k and i for which the equations (FOMC) all hold
when the derivatives ui are evaluated at x. Thus,
k
for each k, let k = u1 ( 1 ),
kx
and for each i, let i =
8
l
.
i
ul ( i )
x
For each k and each i we have k > 0 and i > 0 and therefore, since the equations (EqualMRS)
are satised at ( i )n , we have
x1
ui
u1
k
k
= k= ,
i
1
ul
ul
l
which yields
k =
l i
u = i ui ,
k
ui k
l
which are exactly the rst-order marginal conditions (FOMC) for ( i )n to be a solution of (P-max).
x1
We have succeeded in characterizing the interior solutions of (P-max) as the allocations that
satisfy the condition (EqualMRS). In the preceding section we characterized the interior Pareto
allocations as the solutions to (P-max). Therefore we have the following characterization of the
Pareto allocations in terms of marginal conditions:
Theorem: If every ui is strictly increasing, quasiconcave, and dierentiable, then an interior
allocation x is Pareto ecient for the economy (ui , i )n if and only if it satises (EqualMRS) and
x1
n
1
6.2
xi =
ni
x
1.
Boundary Allocations
Typically many of the Pareto allocations are boundary allocations: some consumers bundles
dont include positive amounts of all the goods. We want our marginal conditions to tell us which
boundary allocations are Pareto ecient and which arent, in the same way as the conditions
weve just developed do for interior allocations. Since were dealing with continuous and strictly
increasing utility functions, we know that a boundary allocation, just like an interior allocation, is
Pareto ecient if and only if its a solution of (P-max). So all we need to do is adapt the rst-order
conditions (FOMC) to cover boundary allocations: we have to allow for the equations in (FOMC)
to be inequalities when theyre associated with variables that have the value zero. Thus, we have
1 , ..., n
0 and 1 , ..., l
i ui
k
0 such that for each k = 1, ..., l and each i = 1, ..., n:
k ,
and i ui = k if xi > 0
k
k
(FOMC)
Of course these inequalities dont yield the nice equality of all consumers M RS s for any pair
of goods that we obtained in (EqualMRS) for interior allocations. Lets see how these rst-order
inequalities translate into marginal conditions, for any pair of goods and for any pair of consumers.
Without loss of generality, we consider the two goods k = 1, 2. For each consumer (and omitting
superscripts for the moment), (FOMC) yields
9
u1
u2
u1
If x2 > 0, then
u2
If x1 > 0, then
1
;
2
1
;
2
i.e., M RS
i.e., M RS
1
.
2
1
.
2
(2)
(3)
Combining (2) and (3) for any two consumers (wlog, lets say theyre i = 1, 2), we have the
following two M RS conditions that must be satised at a Pareto ecient allocation:
(A)
If x1 > 0 and x2 > 0, then M RS 1
2
1
M RS 2 .
(B)
If x1 > 0 and x2 > 0, then M RS 1
2
1
M RS 2 .
Together, these two conditions cover every combination of positive and zero values of these two
goods in the bundles assigned to consumers i = 1, 2, as the following table describes. Note that all
interior allocations are Case (1) in the table i.e., the case in which both (A) and (B) above apply,
so that we have M RS 1 = M RS 2 . All the other eight cases in the table are boundary allocations.
And its always useful to remember that a consumers M RS at a bundle is the personal value
one of the goods has to him, measured in terms of another good, i.e., it tells us how much of the
other good the consumer would be willing to give up to get a marginal increase in the good in
question. This is extremely useful in trying to nd Pareto improvements, and in seeing when no
Pareto improvements are possible.
10
Table 1
x1
1
x1
2
x2
1
x2
2
Required Relation between MRSs
Cases
(1)
+
+
+
+
M RS 1 = M RS 2
(A) & (B)
(2)
0
+
+
+
M RS 1
M RS 2
(B)
(3)
+
+
0
+
M RS 1
M RS 2
(A)
(4)
+
0
+
+
M RS 1
M RS 2
(A)
(5)
+
+
+
0
M RS 1
M RS 2
(B)
(6)
0
+
+
0
M RS 1
M RS 2
(B)
(7)
+
0
0
+
M RS 1
M RS 2
(A)
(8)
0
0
+
+
-
(9)
+
+
0
0
-
11
7
Maximizing a Social Welfare Function
An alternative approach to making welfare comparisons of alternative allocations is to evaluate
the allocations according to a social welfare function. We could in principle use any real-valued
function W dened on the space Rnl of allocations (xi )n . Of course, we would want to use a
+
1
function that somehow reects the preferences of the n consumers, so well dene a social welfare
function as any weighted sum of the consumers utilities.
Denition: A social welfare function for the economy (ui , i )n is a function of the form
x1
W (x) =
n
i=1
i ui (xi ) for some numbers (weights) 1 , ..., n > 0.
This may seem to be an ill-advised approach, because the social welfare function W adds up
individual utilities that arent really comparable: the consumers utility functions have no cardinal
meaning, because the underlying preferences can be represented by any monotone transforms of
the given utility functions. But lets nevertheless see what the implications of using a social
welfare function would be. Note that the map taking proles of utility functions to a social welfare
function, (u1 , ..., un ) W (), is a particular way of aggregating proles of utility functions into an
aggregate utility function, as promised in Section 2. In keeping with our notation for aggregating
preference relations, it would be natural to denote the social welfare function as u(); we use W ()
instead, because thats the conventional notation for a social welfare function.
The rst thing we see is that any allocation that maximizes a social welfare function is Pareto
ecient:
Theorem: If an allocation x Rnl is a solution of the problem
+
max W (x) =
(xi )Rnl
+
k
n
i=1
i ui (xi )
xi
k
subject to
n
0,
xi
k
k ,
x
i = 1, ..., n,
k = 1, ..., l
(W-Max)
k = 1, ..., l
i=1
for some numbers 1 , ..., n > 0, then x is a Pareto allocation for the economy (ui , i )n .
x1
In fact, this result is much more general. It holds not just for our economic allocation problem,
but for any situation in which we want to aggregate individual preferences into a single aggregate
preference and in which the individual preferences can each be represented by a utility function. As
the proof below makes clear, the result follows immediately from the denition of Pareto eciency.
As in the denition, the set X of alternatives here can be any set whatsoever.
12
Theorem: If the alternative x is a solution of the problem
maxW (x) =
xX
n
i=1
i ui (x),
(4)
for some numbers 1 , ..., n > 0, then x is Pareto ecient in X .
Proof. Suppose x is not Pareto ecient in X : let x satisfy
i N : ui () ui () and j N : uj () > uj ().
x
x
x
x
(5)
Then for any 1 , ..., n > 0 we have
i ui () >
x
i N
i ui ()
x
(6)
i N
i.e., there are no values of the i for which x maximizes W () on X , contrary to assumption.
What about the converse? For any Pareto allocation x, can we always nd weights 1 , ..., n for
which x maximizes the social welfare function max W (x) =
(xi )Rnl
+
k
n
i=1
i ui (xi ) ? The answer is no;
the following exercise asks you to construct a counterexample.
Exercise: In a two-person, two-good exchange economy, assume that uA (xA , yA ) = xA yA and that
uB (xB , yB ) = xB yB and that the total resources are and . Depict the set of Pareto allocations
x
y
in the Edgeworth box. Then show that if = there are exactly two allocations that maximize
the social welfare function W (xA , yA , xB , yB ) = uA (xA , yA ) + uB (xB , yB ). Use this result, along
with the corresponding result for = , to establish that this example is a valid counterexample.
Suggestion: Write r for the ratio / and show that Pareto eciency and maximization of W
yx
each require that yi = rxi for i = A, B . This allows you to express uA , uB , and W in terms of just
xA and xB , and now you can draw the constraint and the contours of W in the two-dimensional
xA xB -space and easily establish the conclusion both geometrically and algebraically. Do it rst for
the case = , where there are two (and only two) allocations that maximize W .
13