PARETO EFFICIENCY of
GENERAL EQUILIBRIUM
with COMPLETE MARKETS
Our original hypothesis that utility is money measurable, and independent of the
amount of money, is highly suspect. Who believes he can measure another man’s
happiness, much less equate it with a quantity of money? But if we give up the
money measurable utility hypothesis, what is the argument for laissez faire?
The answer was provided by Wilfredo Pareto, an Italian economist from the early
1900s, who discovered the concept of Pareto e
ﬃ
ciency. One of the earliest proofs
of the Pareto e
ﬃ
ciency of competitve equilibrium was given by the Polish socialist
Oskar Lange. (Lange was Polish ambassador to the United States in the 1940s). A
pictorial proof in two dimensions due to the Oxford economist Edgeworth is also very
instructive. The best, shortest, and most general proof was given independently by
Kenneth Arrow and Gerard Debreu in 1951.
Another important property of equilibrium for
f
nance is arbitrage e
ﬃ
ciency. Two
goods that are perfect substitutes will trade at the same price. Arbitrage e
ﬃ
ciency
is much weaker than Pareto e
ﬃ
ciency.
Arrow emphasized that if some goods are not traded, then equilibrium will prob
ably not be Pareto e
ﬃ
cient. But it will still be arbitrage e
ﬃ
cient.
Geanakoplos and Polemarchakis proved that if some goods are not traded in a
dynamic economy, then typically the goods that are traded will not be traded Pareto
e
ﬃ
ciently, even taking into account the constraint on reallocating the untraded goods.
1 Preferences and Monotonic Transformations
Edgeworth’s family of indi
f
erence curves completely describes the
preferences
of
agent
i
, namely it provides enough information to determine which of any two bun
dles gives agent
i
more welfare (or whether the two bundles are indi
f
erent). Some
years later economists saw the importance of an obvious mathematical fact: there can
be many di
f
erent welfare functions that give rise to the same family of indi
f
erence
curves. Indeed, consider the welfare function
U
(
x, y
)=2
W
(
x, y
)
obtained by doubling the welfare function
W
.I
ti
sev
iden
ttha
t
U
and
W
describe the
same preferences between commodity bundles. In fact if
f
is any increasing function
(called a monotonic function), then the welfare function
U
(
x, y
)=
f
(
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View Full Documentalso represents the same preferences as
W
does. In particular, letting
f
(
w
)=
e
w
,
the
behavior of
W
(
x, y
)=
a
log
x
+
b
log
y
U
(
x, y
)=
f
(
W
(
x, y
)) =
x
a
y
b
are identical.
Incidentally, it will be useful later to observe that the welfare functions
W
(
x, y
)=
u
(
x
)+
dv
(
y
)
U
(
x, y
)=
qu
(
x
)+(1
−
q
)
v
(
y
)
represent the same preferences if
q
is chosen so that
q
=1
/
(1 +
d
)
.Inpa
r
t
icu
la
r
,i
f
W
(
x, y
)=
a
log
x
+
b
log
y
we will get the same behavior by replacing
a
with
α
=
a/
(
a
+
b
)
and
b
with
β
=
b/
(
a
+
b
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 '09
 GEANAKOPLOS,JOHN
 Economics, Utility, competitive equilibrium, edgeworth box

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