Handout 11th Lecture, ECON 5200,
General equilibrium & Welfare
Kjell Arne Brekke
November 8, 2010
1
General outline
First we will discuss some of the welfare properties of general equilibrium. In this context
it is also worth going somewhat deeper into the notion of social welfare.
2
General equilibrium and Pareto Optimality
An economic allocation (x1 ; ::; xI ; y1 :::; yJ ) is feasible if
X
X
xli ! l +
ylj
I
J
An allocation (x; y) is Pareto e¢ cient if there is no alternative feasible allocation
(x ; y 0 ) such that
0
ui (x0i )
ui (xi ) for all i with ui (x0i ) > ui (xi ) for some i
A Private ownership equilibrium is an allocation (x ; y ) and a vector of prices
p such that
(i) For all …rms j
yj solves max p yj
yj 2Yj
(ii) For all consumers i
xi solves max ui (xi )
X
p !i +
ij (p yj )
s.t.p xi
J
(iii) Market clearing: For each good l we have
X
X
xli = ! l +
ylj
i
J
1
Note here that ij is consumer i0 s ownership of …rm j. Clearly, any …rm j is totally
owned by consumers
X
ij = 1
i
We also want to consider the quilibria that are possible through redistribution:
A price equilibrium with transfers is an allocation (x ; y ) and a vector of prices
p such that
(i) For all …rms j
yj solves max p yj
yj 2Yj
(ii) For all consumers i
xi solves max ui (xi )
wi
s.t.p xi
(iii) Market clearing: For each good l we have
X
X
xli = ! l +
ylj
i
J
Note that no transfer corresponds to
wi = p ! i +
X
ij (p
yj )
J
We assume that preferences are locally nonsatiated.
2.1
First Fundamental Theorem of Welfare Economics
Theorem 1 If preferences are locally nonsatiated and (x ; y ), p is a price equilibrium
with transfer, then the allocation is Pareto Optimal.
Proof in class.
Intuitively: To make someone better o¤ we need to give them an allocation at higher
value under current prices, and none can get anything with a lower value. But the …rms
have maximized pro…t and cannot provide higher values at these prices.
2.2
Second welfare theorem
The problem with going the other way is that not all Pareto Optimal allocation can be
supported by an equilibrium. There are two main reasons: Non-convexities, and more
subtle ones. The way the textbook handle these is to introduce a quasi-equilibrium
A price quasi-equilibrium with transfers is like a price equilibrium with transfers except
(ii) For all consumers i if
xi
then p xi
2
xi
wi
Note that if xi xi and p xi = wi then i would not choose xi ; so the choice xi may
not be optimal. Note however, that the indi¤erence curve cannot cross the budget line
since then we could also …nd an xi xi with p xi < wi .
Thus quasi-equilibriums are not real equilibrium, but comes pretty close.
Theorem 2 Suppose that the technology is convex and preferences are convex and locally
nonsatiated, then for every Pareto optimal allocation there is a price quasi equilibrium
(with transfers).
Proof in class
2.2.1
Edgeworth box illustrations of the problems:
Non-convexities may yield jumps in the o¤er curve.
The subtler problems like Figure 15.B.10. in the textbook.
2.3
Social welfare
The welfare theorem only talk about Pareto e¢ ciency, can we get more, in terms of social
welfare?
We will below consider the notion of a social welfare function
U (x; y) = W (u1 (x1 ); :::; un (xn ))
Note though that these are de…ned in terms of utility functions. Can we return to the
ordinal approach - using preferences. This is the social choice problem of Arrow:
2.4
Social Choice theory
Suppose we want to rank a set of possible allocation X, at least 3 di¤erent. We ask
all individuals to rank the alternatives. This may re‡ect the utility of their private consumption, but they may have more non-sel…sh preferences as well. This gives a list of
preferences
( 1 ; :::; n )
and we want to aggregate these into one social preference
= F(
1 ; :::;
n ):
Now we want this mapping F to satisfy certain criteria:
Universal domain: Any individual ranking i is admissible
Pareto: if for some x; y 2 X it is the case that x i y for all i, then x
y
Pairvise independence (Independence of irrelevant alternatives). The ranking of
x versus y only depend on individual ranking of x and y. Formally
If
and if
x
= F(
i and
y i¤ x
1 ; :::; n )
0
i rank x
0
y
3
and 0 = F ( 01 ; :::; 0n )
and y the same way then
No dictator: No individual h such that
y if x
x
h
y
Theorem 3 There is no social choice rule satisfying these conditions
We will not prove this theorem, but suppose that you enter an ordinal store. You
are given a choice between a …xed set of commodity baskets. But you are not told how
much there is of each commodity in each basket, you are only given ordinal information
about the content of the baskets. You get information like: In x there is more juice than
in y. In x there is less co¤e than in y. Etc... How could you make a choice?
What is the point of IIA? This rules out Borda Count. Suppose that there are 5
individual and two kind of preferences over three alternatives
2 individual hold x
3 individual holds y
z
x
y count 3-2-1
z count 3-2-1
Now Borda counting would yield
x = 2 3 + 3 2 = 12
y = 2 1 + 3 3 = 11
so x win. But this is due ot the ranking of z and thus violates IIA and also essentially
introduces cardinality! The IIA insist on only ordinal information.
2.5
Harsanyis Utilitarianism
Suppose that both individual preferences and social preferences satis…es the axioms of
expected utility. Now, there is an expected utility function for each individual ui and an
expected utility function for society U , both can only be transformed by linear transforms.
Adding the Pareto criterion Harsanyi shows that
X
U= +
i ui
or by deleting
and transforming individual utility to u~i =
X
U=
ui
i ui
:
Harsanyi also produces a separate arguement for utilitariansim by arguing that to be
uninhibited, the observaer should assign equal probability to ending up in each possible
possition in society. The expected utility in this original position is
U=
n
X
1
ui
n
i=1
which is just a scaling away from utilitarianism again.
This argument has led to much discussion and critizism. Do the uninhibited person
with the possibilit of becoming anyone have preferences of his own? Or if we take the
…rst argument, how do we scale the utilities?
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2.6
Equality of what?
The problems on interpresonal comparison has led many to discard utilitarianism, and
rather argue for an equal distribution of something that is measurable. John Rawls argues
for equal distribution of what he call primary goods. Other have other suggestions. Equal
distribution of income would be unfair if the work load is not equally distributed, and
incentives to work are of course at stake.
One illustration of some of the problems that may arise is with the idea to share all
individuals time equally. Just like owning a share of each …rm diversi…es risk. Now if j
owns n1 of i0 ’s time. Unequal distribution of talent would not matter. Even if we disregard
incentive problems this is no good solution: The result is the slavery of the talented. Bill
Gates will have to pay more than 1000 $ just to take the time o¤ to pick up something
from the street. But he is not richer than anyone else.
2.7
Pragmatic measures of utils?
What about assuming the marginal utility is the same for all. This is probably not true
but maybe a good approximation? Consider then the alternative: The marginal utility
of an environmental good (say climate change) is the same for all. The two assumption
makes dramatically di¤erent conclusions about total willingness to pay for environmental
quality: factors of 10 000 and more. Now, money are di¤erent from environmental uinits
in at least two respects: They can be transferred and they are very universal. Moreover,
we can make adjustment for marginal utility of income as this is observable. Still, the
results are not comforting.
2.8
Welfare functions and Pareto Optimality
There is however an manner in which welfare functions serves a purely ordinal concept.
We denote the utility possibility set
U = f(u1 ; :::; un ) : There is a feasible allocation (x; y) with ui
ui (xi )g
Next the the Pareto-frontier is
U P = f(u1 ; :::; un ) 2 U : There is no u~ 2 U with u~i > ui for any ig
If U is convex, then
u 2 U P i¤ there exist
0: u
u0 for all u0 2 U
Thus there is a link between welfare maximization and the …rst and second welfare
theorem.
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