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Unformatted text preview: Econ 121.
Intermediate Microeconomics. Eduardo Faingold
Yale University Lecture 14 Outline of the course
I. Introduction
II. Individual choice
III. Competitive markets
IV. Market failure Outline of the course
I. Introduction
II. Individual choice
III. Competitive markets Exchange economies (Ch. 31)
IV. Market failure Failure of the FWT: monopolist
Suppose that consumer A behaves as a monopolist.
He goes to consumer B and makes him a takeitorleaveit offer:
“Either we trade at prices .p1; p2/ or we do not trade at all.” Failure of the FWT: monopolist
If consumer B accepts he will choose the bundle that maximizes his utility
subject to the budget constraint associated with the prices that were offered
by A.
If consumer B rejects, he will have to consume his endowment.
So, he always accepts. Failure of the FWT: monopolist
A knows that whatever price offer he makes, consumer B will accept.
Thus, A knows that consumer B will choose a bundle on his offer curve.
Hence, the monopolist,A, must maximize his utility over all feasible
allocations where B ’s consumption bundle belongs to B ’s offer curve.
By the ﬁrstorder conditions (tangency), the allocation that arises under a
monopolist is such that A’s indifference curve is tangent to B ’s offer curve,
not to B ’s in difference curve (as required by Pareto efﬁciency). x2
The
Oﬀer
Curve
Endowment
x1
x2
Oﬀer
curve
Indiﬀerence
curve
Endowment
x1
Second Welfare Theorem
The First Welfare Theorem states that given any competitive equilibrium
.x; p/, the allocation x must be Pareto efﬁcient.
Thus, competitive markets always allocate goods efﬁciently.
How about the converse?
Is it true that for any Pareto efﬁcient allocation x there is a vector of prices
p D .p1; p2/ such that .x; p/ is a competitive equilibrium? Second Welfare Theorem
The answer is no.
First, in a competitive equilibrium, each consumer must be getting at least
as much utility as he would get form consuming his endowment. endowment
Second Welfare Theorem
So, let us change the question a bit:
Is it true that for any Pareto efﬁcient allocation x that gives each consumer
at least as much utility as he gets from his endowment there is a vector of
prices p D .p1; p2/ such that .x; p/ is a competitive equilibrium? Second Welfare Theorem
The answer is still no, as shown in the following picture. B
endowment
A
Second Welfare Theorem
The Second Welfare Theorem states that the answer to the former question
becomes yes provided we allow a social planner (the government) to
design an appropriate tax system.
A lumpsum tax/subsidy is a tax/subsidy that enters additively in the
consumer’s budget constraint:
i
i
p1x1 C p2x2 i
i
p1!1 C p2!2 C : A system of lumpsum taxes/subsidies i (i D A; B ), is called purely
redistributive if A C B D 0. Second Welfare Theorem
Theorem. Assume preferences are convex. If x is a Pareto efﬁcient
allocation then there exist prices p D .p1; p2/ and purely redistributive
lumpsum transfers D .i /i DA;B such that .x; p/ is a competitive
equilibrium under .
A partial converse to the First Welfare Theorem.
Pareto efﬁciency is a socially desirable feature for an allocation, but does
not pin down a single allocation.
A social planner might like some allocations in the contract curve better
than others, due to considerations of fairness or some other issues (outside
the scope of Economics).
The SWT highlights the role of taxes/redistribution as a policy instrument to
implement a target allocation in the contract curve—the favorite allocation
of a social planner, say. Proof assuming preferences are smooth
If x is a Pareto efﬁcient allocation then MRS A.x A/ D MRS B .x B /:
Set p2 D 1 and p1 D MRS A.x A/. Hence, MRS equals minus price ratio for both consumers and x clears the
market.
Now for each i D A; B deﬁne i such that:
i
i
i
i
p1x1 C p2x2 D p1!1 C p2!2 C i Summing across i yields
A
B
A
B
p1.x1 C x1 / C p2.x2 C x2 / D p1!1 C p2!2 C A C B and hence A C B D 0: Q.E.D. General proof
While we will not go over the general proof without assuming smoothness,
it is important to note that the only important step in the proof was to ﬁnd
prices that deﬁne a budget line that “separate” the upper contour sets of the
two consumers in the Edgeworth box.
Such “separation argument” can be made with much greater generality,
using more powerful tool from mathematics.
The main idea is the Separating Hyperplane Theorem, illustrated in the
following picture. Separa&ng
Hyperplane
Theorem
...
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 Spring '09
 SAMUELSON
 Microeconomics

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