The Arrow-Debreu Model
Larry Blume
Cornell University & IHS Wien
April 28, 2020
PlanTitles are linked.
The Problem of Value
The Origins of Modern GE
Arrow-Debreu
Behavioral General Equilibrium
The Private Ownership Economy
Existence of Competitive Equilibrium
Pareto Optimality
The 1st Welfare Theorem
The 2nd Welfare Theorem
A Calculus Approach to the Welfare Theorems
Bibliography
The Problem of Value
The word VALUE, it is to be observed, has two different
meanings, and sometimes expresses the utility of some
particular object, and sometimes the power of purchasing
other goods which the possession of that object conveys.
The one may be called ‘value in use;’ the other, ‘value
in exchange.’ The things which have the greatest value
in use have frequently little or no value in exchange; and
on the contrary, those which have the greatest value in
exchange have frequently little or no value in use. Nothing
is more useful than water: but it will purchase scarce any
thing; scarce any thing can be had in exchange for it. A
diamond, on the contrary, has scarce any value in use;
but a very great quantity of other goods may frequently
be had in exchange for it.
Adam Smith, The Wealth of Nations
Bk. I ch. 4.
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Utility
What determines value in exchange?
I Carl Menger, 1871.I Value in use is determined by the lowest
value in which an object is being used.
(Diminishing marginal utility!)I When will someone trade an object?
When its value in exchange (price) is
higher than its value in use.
I Contrast this utility theory of value with Marx’s labor theory
of value.
I Pushing a little farther, we see that the value in use of an
object depends upon what other objects we are using, so the
exchange-use threshold must be determined simultaneously
for all goods.
2 / 48
Utility
Value in exchange expresses nothing
but a ratio, and the term should not
be used in any other sense. To speak
simply of the value of an ounce of
gold is as absurd as to speak of the
ratio of the number seventeen. What
is the ratio of the number seventeen?
The question admits no answer, for
theremust be another number named in order to make a ratio;
and the ratio will differ according to the number sug-
gested. What is the value of iron compared with that of
gold?—is an intelligible question. The answer consists in
stating the ratio of the quantities exchanged.
William S. Jevons, The Theory of
Political Economy, 1871, p.78.3 / 48
Utility and Demand
A few pages later, Jevons formulates the principle idea of
neoclassical demand theory:
MUxMUy
=pxpy.
4 / 48
Modern General Equilibrium
I Walras is pronounced “Valrasse”. He was
Alsatian.
I Introduces multi-market pure exchange
models.
I Existence proof is equality of equations and
unknowns.
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Two Views of General Equilibrium
To some people (including no doubt Walras himself) the system
of simultaneous equations determining a whole price-system
seems to have vast significance. They derive intense satisfac-
tion from the contemplation of such a system of subtly interre-
lated prices; and the further the analysis can be carried (in fact
it can be carried a good way)...the better they are pleased, and
the profounder the insight into the working of a competitive
economic system they feel they get.
John Hicks, Value and Capital, 1939, p.60.
The fundamental Anglo-Saxon quality is satisfaction with the
accumulation of facts. The need for clarity, for logical coher-
ence and for synthesis is, for an Anglo-Saxon, only a minor
need, if it is a need at all. For a Latin, and particularly a
Frenchman, it is exactly the opposite.
Maurice Allais, Traite d’Economie
Pure, 1952, p.58.
6 / 48
Two Systems
The GE models have consumers endowed with factors, production
processes that demand factors and produce consumer goods, and
equilibrium is a vector of prices that equilibrate supply and
demand in all markets. In the early models, production processes
are linear and prices are such that costs are just covered.
Walras-Cassel
I Demand functions for final
products
I Supply functions for factors
I Factor supply equals factor
demand
I Price equates revenues and
costs.
Edgeworth-Pareto
I Utility maximization
I Profit maximization
I Welfare economics
7 / 48
Edgeworth
Why price-taking? Edgeworth imagines a
recontract- ing process in which individuals are
never price- takers, but always looking for an edge.
In large markets (competitive fields) the outcome is
as if they were. Edgeworth’s view is justified by the
Debreu-Scarf limit theorem. His equilibrium is
Nash-like.
Equilibrium is attained when the existing contracts can neither
be varied without recontract with the consent of the existing
parties, nor by recontract within the field of competition. The
advantage of this general method is that it is applicable to the
particular cases of imperfect competition ; where the concep-
tions of demand and supply at a price are no longer appropriate.
(F. Edgeworth, 1881: p:31.)
. . . , we see how contract is more or less indeterminant accord-
ing as the field is less or more affected with. . . , limitation of
numbers. (ibid. p.42.)
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“The interesting fact deserves to be noticed that sogreat an influence could have been exerted by a manwho lived in resolute though hospitable seclusion in ashabby house full of cats (hence Villa Angora) that wasthen not convenient to visit.” Schumpeter, History ofEconomic Analysis
“I never found Cassel interested in slander. He neverslandered anybody himself, and he turned remarkblydeaf, long before his hearing was actually impaired, whenanyone else ventured slanderous remarks in his presence.On this point his personality was, of course, wonderfullyprotected by his almost complete lack of psychologicalinsight and interest.” Gunnar Myrdal (1945) [1963], p.7.
9 / 48
The Arrow and Debreu Model
Sometimes called the
neo-Walrasian approach,
A&D combine the insights
of Walras-Cassel and
Edgworth-Pareto. The
principle idea is this:
The problem is no longer conceived as that of proving
that a certain set of equations has a solution. It has been
reformulated as one of proving that a number of maxi-
mization of individual goals under independent restraints
can be simultaneously carried out.
T. C. Koopmans, 1957: p.60.
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Behavioral General Equibrium
Walras and Cassel posit demand and supply, and search for prices
to equilibrate the system. This is behavioral because individual
demands and firm supplies are simply behaviors.
A behavior is a rule that maps environments into actions. In GE
models, an environment for a consumer is a budget set. For a
firm it is a price vector and a set of production possibility set.
I In GE models, environments are budget sets, and behaviors
are (sets of) consumption bundles.
Consider this for exchange economies, where only demands are
present. What assumption on behaviors guarantees the existence
of equilibrium.
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The Exchange Model
I N consumers; L commodities. Prices are p > 0. An
allocation is an x ∈ RNL+ .
I Each consumer n has an endowment ωn > 0 of commodities.
ω =∑n ωn � 0 is the aggregate endowment, and the
endowment allocation is ω.
I Each consumer has a demand function dn(p, ωn). Her excess
demand is zn(p, ωn) = dn(p, ωn)− ωn, and aggregate excess
demand is Z (p,ω) =∑n dn(p, ωn)− ω.
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Assumptions
A.1. (Homogeneity). Z (p,ω) is homogeneous of degree 0 in
prices.
Is this consistent with utility maximization? What about other
behaviors? Its implication is that we can normalize prices to sum
to 1: p ∈ ∆L+.
A.2. (Walras’ Law). for all p ∈ ∆L+, p · Z (p,ω) = 0.
Is this consistent with utility maximization? What does it require
from other behaviors?
A.3. (Continuity). Z ( · ,ω) is continuous on ∆L+.
Does this assumption have observable implications?
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Competitive Equilibrium
A competitive equilibrium is a price vector p ∈ ∆L+ such that
Z (p,ω) ≤ 0 and p · Z (p,ω) = 0.
Complementary slackness: What is the equilibrium price of a good
in excess supply?
Big Math Tool (Brouwer). If C 6= ∅ is a compact, convex set and
f : C → C is a continuous function, there is a c ∈ C such that
f (c) = c .
Theorem. If A.1–3, then a competitive equilibrium exists.
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Proof
This proof is built on economic intuition: If a good is in excess
demand, its price should increase; if in excess supply, decrease.
Define f : ∆L+ → ∆L+ such that
fl(p) =pl + max{0,Zl(p,ω)}∑m pm + max{0,Zm(p,ω)} .
The denominator exceeds 0, for if not,
pm + max{0,Zm(p,ω)} = 0
for all m, and so∑m
pmZm(p,ω) + max{0,Zm(p,ω)}Zm(p,ω) = 0.
Walras’ Law implies that∑mmax{0,Zm(p,ω)}Zm(p,ω) = 0,
and so for all m, max{0,Zm(p,ω)} = 0, a contradiction.15 / 48
Proof
The set ∆L+ is convex and compact, so there is a p∗ such that
p∗ = f (p∗). That is,
λp∗m ≡
(∑m
(p∗m + max{0,Zm(p∗,ω)}
)− 1
)p∗m = max{0,Zm(p∗,ω)}.
Again, Walras’ Law implies that for all m,
max{0,Zm(p∗,ω)} = 0, and so each Zm(p∗,ω) ≤ 0.
Problem. Two goods, N consumers each with Cobb-Douglas
preferences. Find the equilibrium prices. Is this case covered by
the theorem?
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An Extension
Clearly continuity must be relaxed.
A.3’ Z (p,ω) <∞ on ∆L++, and is continuous on its domain. For
a p ∈ ∂∆L++ for which Z (p,ω) is not defined, and any sequence
of prices {pk} such that limk→∞ pk = p,
limk→∞
∑l
Zl(pk ,ω) = +∞.
What does this say about behavior?
A.4. (Bound). For all ω there is a number z < 0 such that for all
p ∈ ∆L+ and goods l , Zl(p,ω) ≥ z.
How general is this assumption?
Theorem, If A.1–2,3’, and 4, then a competitive equilibrium
exists.17 / 48
The Private Ownership Economy
I N consumers, M firms, L goods.
I Consumer n has a preference order �n defined on a
consumption set Xn ⊂ RL, an endowment bundle ωn, and a
vector θn = (θnm)Mm=1 representing the share of firm M
consumer n owns.
I Each firm is characterized by a production set Ym ⊂ RL.
Negative terms represent inputs and positive terms represent
outputs.
A private ownership economy is a tuple((Xn,�n, θn, ωn)Nn=1, (Ym)Mm=1
).
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The Private Ownership Economy
An allocation (x , y) is a specification of a consumption allocation
for each consumer n, a vector xn ∈ Xn, and a production
allocation for each firm m, a vector ym ∈ Ym. An allocation is
feasible iff∑n xn = ω +
∑m ym. A ⊂ RL(N+M) is the set of
feasible allocations.
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Let E =((Xn,�n, θn, ωn)Nn=1, (Ym)Mm=1
)denote a private
ownership economy. A competitive equilibrium for the economy Eis an allocation (x∗, y∗) and a price vector p∗ such that
I For every firm m, y∗m maximizes profits among all feasible
production plans in Ym:
p∗y∗m ≥ p∗ym for all ym ∈ Ym.
I For every consumer n, x∗n is preference-maximal among all
affordable consumption plans. That is, x∗n �n xn for all xn in
the set
{xn : xn ∈ Xn and p∗xn ≤ p∗ωn +∑m
θnmp∗y∗m}.
I (x∗, y∗) ∈ A.
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Existence of Competitive Equilibrium
Theorem. A competitive equilibrium for the private ownership
economy E exists if for every consumer n,
1. Xn is closed, convex and bounded from below,
2. �n is non-satiated in Xn,
3. �n is continuous,
4. If x ′n �n xn, then for all 0 < t < 1, tx ′n + (1− t)xn �n xn,
5. there is an x0n in Xn such that ωn � x0
n ;
and for every firm m,
1. 0 ∈ Ym,
2. the aggregate production set Y =∑m Ym is closed and
convex,
3. Y ∩ (−Y ) = {0},4. Y ⊃ RL
−.
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Competitive Equilibrium with Transfers
A competitive equilibrium with transfers for the economy E is an
allocation (x∗, y∗), a price vector p∗ and an assignment of wealths
(w∗1 , . . . ,w∗I ) to consumers such that
1. For every firm m, y∗m maximizes profits among all feasible
production plans in Ym:
p∗y∗m ≥ p∗ym for all ym ∈ Ym.
2. For every consumer n, x∗n is preference-maximal among all
affordable consumption plans. That is, x∗n �n xn for all xn in
the set
{xn : xn ∈ Xn and p∗xn ≤ w∗n}.
3. (x∗, y∗) ∈ A.
4.∑n w∗n =
∑n p∗ω +
∑m p∗y∗m.
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Pareto Optimality
The economist’s notion of social desirability is the Pareto order:
A consumption plan x is Pareto-better than consumption plan x ′,
written x �P x ′, iff for all n, xn �n x ′n, and for some consumer k ,
xk � x ′k . An allocation z = (x , y) is Pareto optimal iff it is
feasible, and if for no other feasible consumption plan z ′ = (x ′, y ′)is it true that x ′ �P x .
How do we know an optimum exists? In exchange economies this
is not hard. The set of feasible allocations is obviously compact,
so suitable continuity assumptions on preferences should do the
trick. When production is possible, compactness of the set of
feasible allocations is not so obvious. Debreu (Theory of Value,
Ch. 6.2.) gives us an answer.
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Pareto Optimality
The private ownership economy E has an optimum if
1. for all n, Xn is closed and bounded from below and ωn ∈ Xn,
2. for every x ′n ∈ Xn, the set {xn ∈ Xn : xn � x ′n} is closed,
3.∑m Ym is closed, convex, has free disposal, and
Y ∩ −Y = {0}, and
4. ω ∈∑n Xn −
∑m Ym.
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Pareto Optimality
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The 1st Welfare Theorem
The First Welfare Theorem gives conditions guaranteeing that a
competitive equilibrium allocation is Pareto optimal.
Recall that a preference order �n is locally non-satiated at x∗n if in
every open neighborhood of x∗n there is an x ′n �n x∗n .
First Welfare Theorem. Let E be a private ownership economy
with an equilibrium (p∗, x∗, y∗). Suppose for all n, �n is
everywhere locally non-satiated. Then (x∗, y∗) is a
Pareto-optimal allocation.
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Failure of the 1st Welfare Theorem
Proving the First Welfare Theorem requires that in any
equilibrium, any consumption bundle which is better for consumer
n costs more. This is just what preference maximization on the
budget set means. The proof requires more; specifically, than any
bundle which is at least as good costs at least as much. This is
exactly what fails in the figure below — a thick indifference curve.
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Proof
Lemma. If �n is locally non-satiated at bundle x∗n which is
preference-maximal on the set {xn ∈ Xn : pxn ≤ px∗n}, and if
x ′n �n x∗n , then px ′n ≥ px∗n .
Proof Since �n is locally non-satiated at x ′n, there is a sequence
of consumption bundles xkn with limit x ′n such that xkn �n x ′n.
Transitivity implies that xkn �n x∗n . Preference maximality implies
that pxkn > px∗n . Taking limits, px ′n ≥ px∗n .
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Proof
Suppose that (x ′, y ′) is Pareto-superior to (x∗, y∗). Then for all
n, x ′n �n x∗n , and for some individual this ranking is strict. This
means that p∗x ′n ≥ p∗x∗n for all i , with strict inequality for some i .
Furthermore, for each j , p∗y ′m ≤ p∗y∗m since each firm profit
maximizes in equilibrium. Thus
p∗ω = p∗∑n
x∗n − p∗∑m
y∗m < p∗∑n
x ′n − p∗∑m
y ′m.
The equality is a consequence of feasibility of the equilibrium
allocation, and the inequality follows from the relations just
established. Consequently, ω 6=∑n x ′n −
∑m y ′m. That is, the
allocation (x ′, y ′) is not feasible.
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The 2nd Welfare Theorem
Let �(xn) = {x ′n ∈ Xn : x ′n �n xn} and
�(xn) = {x ′n ∈ Xn : x ′n �n xn}.
A quasi-equilibrium for the economy E is an allocation (x∗, y∗)and a price vector p∗ such that
1. For every firm m, y∗m maximizes profits among all feasible
production plans in Ym:
p∗y∗m ≥ p∗ym for all ym ∈ Ym.
2. For every consumer n, x∗n is expenditure-minimal on the ’no
worse than’ set. That is, p∗x∗n ≤ p∗xn for all xn in the set
�n(x∗n ).
3. (x∗, y∗) ∈ A.
A quasi-equilibrium is sometimes called a compensated
equilibrium.
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The 2nd Welfare Theorem
Second Welfare Theorem. Let (x∗, y∗) be a Pareto Optimal
allocation for a private ownership economy E with the properties
that
1. for all n, Xn is convex,
2. the sets �(x∗n ) are convex,
3. for some consumer k , �(x∗k ) is convex and �k is locally
non-satiated at x∗k ,
4. Y is convex.
Then there is a p∗ such that (x∗, y∗, p∗) is a quasi-equilibrium
for E .
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Proof
Define the set G =∑n 6=k �n(x∗n )+ �k(x∗k )− Y . This set is
convex and ω is not in G because the allocation is Pareto
optimal. Thus there is a vector p∗ such that p∗ω ≤ p∗g for all
g ∈ G . Since consumer k is locally non-satiated, there is a
sequence of consumption plans x ik with limit x∗k , each element of
which is better for k than x∗k . Then for all n the vector
gi =∑n 6=k
x∗n + x ik −∑m
y∗m
is in G , and the sequence gi converges to∑n 6=k
x∗n + x∗k −∑m
y∗m = ω.
Thus p∗ω = inf{p∗g : g ∈ G}.
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Proof
Now show that (x∗, y∗, p∗) is quasi-equilibrium. Anything at least
as good costs at least as much, and profit maximization.
For n 6= k , and for any x ′n ∈�(x∗n ), let
gi =∑j 6=k,n
x∗n + x ′n + x ik −∑m
y∗m.
ω =∑j 6=k,n
x∗n + x∗n + x∗k −∑m
y∗m.
Each gi ∈ G , so p∗gi ≥ p∗ω. Taking limits and subtracting,
p∗x ′n ≥ p∗x∗n . Apply the same argument to y∗m to see that
−p∗y∗m ≥ −p∗y ′m for all ym ∈ Ym; y∗m is profit-maximizing. For
consumer k , we can see directly by subtraction that p∗x ′k ≥ p∗x∗kfor all x ′k �k x∗k , and the conclusion for all x ′k � x∗k follows from
local non-satiation.
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From Quasi- to Competitive Equilibrium
Cheaper Point Lemma. Suppose at a price p, x ′n minimizes
expenditure on �n(x ′n). Suppose that �n(x ′n) is open and that
there is an x0n ∈ Xn such that px0
n < px ′n. Then x ′n is
preference-maximal on the set {x ′′n ∈ Xn : px ′′n ≤ px ′n}.
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From Quasi- to Competitive Equilibrium
Proof. If x ′n is expenditure minimizing on �n(x ′n), then x ′′n �n x ′n
implies px ′′n ≥ px ′n. We must show that the inequality is strict.
Suppose to the contrary that p′′xn = px ′n. Since px0n < px ′n,
x ′′n �n x ′n �n x0n . For all 0 < t < 1, p(tx ′′n + (1− t)x0
n ) < px ′n, and
for t near enough to 1, (tx ′′n + (1− t)x0n ) �n x ′n, contradicting
expenditure minimization.
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From Quasi- to Competitive Equilibrium
The existence of x ′i is referred to as the cheaper point
assumption. Figure 2 demonstrates what can go wrong with the
duality between expenditure minimization and utility maximization
when the cheaper point assumption does not hold.
No cheaper point.
The consumption is R2+ in which
the open triangle with vertices
(0, 0), (1, 0) and (0, 1) has been
removed. Prices and wealth are
such that the budget set is the
lower 45 degree line. The
indicated consumption bundle is
expenditure minimizing on its ’no
worse than’ set, but it is not
preference maximal on the
budget set.
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From Quasi- to Competitive Equilibrium
Theorem. Suppose that (x∗, y∗, p∗) is a quasi-equilibrium for the
private ownership economy E . Suppose that for all consumers n
and for all xn ∈ Xn, the set �n (xn) is open. Then (x∗, y∗, p∗) is a
competitive equilibrium with transfers.
Proof.Take w∗n = p∗x∗n . The theorem is then a consequence of
the definition of a quasi-equilibrium and the cheaper point
Lemma.
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A Calculus Approach to the Welfare Theorems
Suppose that each of N consumers has preferences which are
represented by strictly concave, C 2 and strictly increasing utility
functions u1, . . . , uN defined on Xn which is convex and has
non-empty interior in RL That is, D2un is negative definite and
Dun � 0 on Xi . Suppose too each consumer has a strictly
positive endowment.
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A Calculus Approach to the Welfare TheoremsOptimality
If x∗ is a Pareto optimal allocation, then there is no reallocation
that can increase the utility of any consumer without decreasing
the utility of anyone else. let un(x∗n ) = u∗n. Then x∗ solves the
optimization problem on∏n Xn:
PO : max u1(x1)
s.t. un(xn) ≥ u∗n for i = 2, . . . , I ,∑n
xn =∑n
x∗n .
Since the un are strictly increasing, the weak inequalities can be
assumed to be equalities. Let us, for simplicity, consider an
allocation in which each x∗n is interior to X ∗n .
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A Calculus Approach to the Welfare TheoremsOptimality
The first order conditions are
Du1(x∗1 ) = λ
νnDun(x∗n ) = λ for n 6= 1.
From these conditions the usual equality conditions for marginal
rates of substitution follow. These conditions, along with the
constraints, are sufficient for an allocation to be Pareto optimal.
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A Calculus Approach to the Welfare TheoremsEquilibrium
Now suppose an allocation x ′1, . . . , x ′I is a competitive equilibrium
at price vector p. Then∑n x ′n =
∑n ωn, and for each i the
bundle x ′n solves the optimization problem
CEn : max un(xn)
s.t. pxn ≤ pωn.
Again one can take the inequality to be an equality. The first
order conditions are
Dun(x∗n ) = ηnp
These too are sufficient because of the concavity assumptions.
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A Calculus Approach to the Welfare TheoremsThe Welfare Theorems
The proof of the welfare theorems amounts to showing:
First Welfare Theorem If for all i , x∗n , ηn solves the first order
conditions for CEn with prices p, then x∗ and
λ = η1p, νn = η1/ηn solves the PO first order
conditions.
Second Welfare Theorem If x∗, ν and λ solve PO, then taking
ν1 = 1, x∗, ηn = 1/νn and p = λ solve all the CEnfirst order conditions.
This is simple algebra.
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The Meaning of Pareto Optimality
What things are good for humans? The answer to this question
begins with individual welfare or, distinctly, well-being.
Economists started with hedonism — the good is a mental state,
e.g. pleasure, and most now hold that welfare concerns the
satisfaction of preferences.
I There is a distinction between welfare and well-being.
1. Mistaken beliefs.
2. One may prefer to sacrifice well-being for an alternative goal.
3. How stable are preferences?
4. Preferences are not fundamental, they may be shaped by
circumstances or manipulation.
I Should we account for all possible preference orders?
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The Meaning of Pareto Optimality
I The “old welfare economics”
Problem.—To find (α) the distribution of means and (β)of labor, the (γ) quality and (δ) number of population,
so that there may be the greatest possible happiness.
. . . Greatest possible happiness is the greatest possible in-
tegral of the differential ‘Number of enjoyers × duration
of enjoyment× degree thereof’. . . . 2
2 The greatest possible value of∫ ∫ ∫
dp dn dt (where dp corresponds to a just perceivable incre-
ment of pleasure, dn to a sentient individual, dt to an instant of time).
Edgeworth (1881), pp. 56–57.
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The Meaning of Pareto Optimality
I The “new welfare economics”
We have taken this thing called pleasure, value in use,
economic utility, ophelimity, to be a quantity; but a
demonstration of this has not been given. Assuming this
demonstration accomplished, how would this quantity be
measured? It is an error to believe we could in general
deduce the value of ophelimity from the law of supply and
demand. . . .
Hereafter, when we speak of ophelimity it must always be
understood that we simply mean oone of the systems of
indices of ophelimity.
Pareto (1927) [1971] p.112.
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The “new welfare economics”
Although the economic system can be regarded as a
mechanism for adjusting means to ends, the ends in ques-
tion are ordinarily not a single system of ends, but as
many independent systems as there are ” individuals ” in
the community. This appears to introduce a hopeless ar-
bitrariness into the testing of efficiency. You cannot take
a temperature when you have to use, not one thermome-
ter, but an immense number of different thermometers,
working on different principles, and with no necessary cor-
relation between their registrations. How is this difficulty
to be overcome?
Hicks (1939), p.699.
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The “new welfare economics”
The “new welfare economics” of Hicks, Kaldor and Scitovsky
abandoned the “old welfare economics” of Marshall, Pigou and
Lerner by giving up on interpersonal utility comparisons. This left
only the Pareto principle, which is silent on tradeoffs across
people. Since few policy choices are Pareto improvements, Kaldor
(1939) and Hicks (1939) proposed an extension of the Pareto
principle through compensation tests:
I Kaldor: Can the gainors compensate the losers ex post
I Hicks: Can the losers compensate the gainers ex ante for
keeping the status quo?
These don’t work. See Chipman and Moore (1978).
47 / 48
Bibliography
Chipman, J. and J. Moore. (1978), “The New Welfare Economics
1938–1974.” International Economic Review 19:3 pp. 547-84.
Edgeworth, F. 1881 (1932) Mathematical Psychics. London:
London School of Economics and Political Economy.
Hicks, J. 1939. “The Foundations of Welfare Economics”,
Economic Journal 48 pp.696–712.
Lange. O. (1942) “The Foundations of Welfare Economics”,
Econometrica 10:3 p.215–28.
Myrdal, G. (1945) [1963], “Gustav Cassel in Memoriam
(1866-1945), Bulletin of the Oxford University Institute of
Economics & Statistics, 25:1 p.1–10.
Pareto, V. 1907 (1971) Manual of Political Economy. New York:
Augustus M. Kelly.48 / 48