The Arrow-Debreu Model - Cornell University

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.

1 / 48

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.

5 / 48

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.)

8 / 48

“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.

10 / 48

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.

11 / 48

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.

12 / 48

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?

13 / 48

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.

14 / 48

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?

16 / 48

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


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

).

18 / 48


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.

19 / 48

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.

20 / 48

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

−.

21 / 48

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



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.

22 / 48

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.

23 / 48

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.

24 / 48

Pareto Optimality

��

��

A

Y

Y

2

1

An open set of feasible

consumptions.

��

��

��

��

��

��

2

A

1Y

Y

An unbounded feasible set.

25 / 48

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.

26 / 48

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.

27 / 48

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 .

28 / 48

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.

29 / 48


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



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.

30 / 48


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 .

31 / 48

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}.

32 / 48

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.

33 / 48

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}.

34 / 48


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.

35 / 48


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.

36 / 48


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.

37 / 48

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.

38 / 48

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 .

39 / 48

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.

40 / 48

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.

41 / 48

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.

42 / 48

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?

43 / 48


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.

44 / 48


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.

45 / 48

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.

46 / 48

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

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