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Unformatted text preview: Econ 121.
Intermediate Microeconomics. Eduardo Faingold
Yale University Lecture 13 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 Competitive Equilibrium
So far we have talked about allocations in an exchange economy without
worrying about how these allocations emerge.
Now we will consider trade in this economy.
i
i
Let .!1; !2/ designate consumer i ’s endowment and let
A
B
!1 D !1 C !1
B
A
!2 D !2 C !2 Write p1 for the price of good 1 and p2 for the price of good 2.
We will now deﬁne a competitive equilibrium in this economy. A competitive (or Walrasian) equilibrium is an allocation
B
B
A
A
..x1 ; x2 /; .x1 ; x2 // and prices .p1; p2/ that satisfy the following conditions:
A
A
I. Optimization: the bundle .x1 ; x2 / solves max .x1 ;x2 / uA.x1; x2/ subject to p1x1 C p2x2 and likewise for consumer B .
II. Market clearing:
B
A
x1 C x1 D !1 B
A
x2 C x2 D !2 A
A
p 1 !1 C p 2 !2 ; The First Welfare Theorem
Throughout we shall maintain the assumption that utility functions are
strictly increasing.
Theorem. Let x D .x A; x B / be an allocation and let p D .p1; p2/ ¤ 0. If
.x; p/ is a competitive equilibrium then x is Pareto efﬁcient.
In words, the allocation of every competitive equilibrium is Pareto efﬁcient. The First Welfare Theorem
The FWR is one of the most important results of Economics.
Benchmark for understanding role of institution design / policy making.
The theorem is so important that we will see two alternative proofs of it.
One simple proof is for the case in which utility functions are smooth,
strictly increasing and quasiconcave (i.e. preferences are convex).
Another, much more general, proof is for the case in which utility functions
satisfy only a very weak nonsatiation condition. Proof when preferences are smooth and convex
In a Walrasian equilibrium .x; p/ each consumer maximizes his utility
subject to his budget constraint, taking the prices p D .p1; p2/ as given.
Under the assumption that the utility functions are smooth and that x is
strictly positive, this implies
A
A
MRS A.x1 ; x2 / D and MRS B B
B
.x1 ; x2 / D p1
p2
p1
:
p2 Therefore,
A
A
B
B
MRS A.x1 ; x2 / D MRS B .x1 ; x2 /: Proof when preferences are smooth and convex
Moreover, by the market clearing condition, x is a feasible allocation.
Hence, the allocation x satisﬁes all the necessary ﬁrstorder conditions of
the maximization problem P.u/ (with u D uB .x B /) that characterizes
Pareto efﬁciency (cf. lecture 12).
Since we are assuming that the consumers’ preferences are convex, the
ﬁrstorder conditions are also sufﬁcient conditions for an optimum.
Therefore, x must be a solution of P.u/, and hence Pareto efﬁcient, as was
to be shown.
Q.E.D. General proof using revealed preference
We will prove the contrapositive, i.e. that if x is not Pareto efﬁcient then
.x; p/ cannot be a competitive equilibrium.
Suppose x is not Pareto efﬁcient. Then there is a feasible allocation x 0 such
that either
uA.x 0A/ > uA.x A/ and uB .x 0B / uB .x B /;
or uA.x 0A/ uA.x A/ and uB .x 0B / > uB .x B /: General proof using revealed preference
For now, let us “cheat” a little bit and assume strict inequality for both A and
B:
uA.x 0A/ > uA.x A/ and uB .x 0B / > uB .x B /:
Suppose .x; p/ were a competitive equilibrium, for some p D .p1; p2/ ¤ 0.
By revealed preference, x 0A cannot be affordable at prices p D .p1; p2/,
because x A maximizes A’s utility subject to her budget constraint, taking
the equilibrium prices p as given. Hence,
A
A
p 1 x 0 A C p 2 x 0 A > p 1 !1 C p 2 !2 :
1
2 Likewise, x 0B cannot be affordable at prices p :
B
B
p 1 x 0 B C p 2 x 0 B > p 1 !1 C p 2 !2 :
2
1 General proof using revealed preference
Adding the above inequalities yields
! p1 x 0 A
1 C x 0B
1 C p2 x 0 A
2 C x 0B
2 > p1 ! 2
1
‚ …„ ƒ
‚ …„ ƒ
A
A
B
B
!1 C !1 C p 2 !2 C !2 But since x 0 is a feasible allocation, x 0A C x 0B D !1 and x 0A C x 0B D !2
2
2
1
1
and therefore, p 1 !1 C p 2 !2 > p 1 !1 C p 2 !2 ;
which is impossible.
Thus, .x; p/ cannot be a competitive equilibrium and this concludes the
proof (except for the “cheating” above).
Q.E.D. The missing step in the proof
Let us go back to the beginning of the proof, where we have: uA.x 0A/ uA.x A/ and uB .x 0B / > uB .x B /;
without loss of generality.
Now we will not assume a strict inequality for both. Instead, let us assume uA.x 0A/ D uA.x A/ and uB .x 0B / > uB .x B /:
0
0
Consider the bundle x 00A D .x1A C "; x2A C "/, where " > 0 is arbitrary.
Since uA is strictly increasing, then uA.x 00A / > uA.x 0A/ D uA.x A/: The missing step in the proof
Then, assuming that .x; p/ is a competitive equilibrium,
A
A
p1x 00A C p2x 00A > p1!1 C p2!2 :
2
1 and
B
B
p 1 x 0 B C p 2 x 0 B > p 1 !1 C p 2 !2 :
2
1 But, the ﬁrst inequality can be rewritten as
A
A
p 1 x 0 A C p 2 x 0 A > p 1 !1 C p 2 !2
2
1 .p1 C p2/": The missing step in the proof
Since this must hold for every " > 0, we have
A
A
p 1 x 0 A C p 2 x 0 A p 1 !1 C p 2 !2 :
2
1 Summing the weak inequality for A and the strict inequality for B yields a
strict inequality:
! p1 x 0 A
1 C x 0B
1 C p2 x 0 A
2 C x 0B
2 > p1 Then, the proof proceeds as before... ! 1
2
‚ …„ ƒ
‚ …„ ƒ
A
A
B
B
!1 C !1 C p 2 !2 C !2 : Implicit assumptions behind the FWT
(a) No market power
(b) Private goods
(c) No externalities
(d) No information asymmetries
(e) Complete markets
When either of these assumptions is violated, the FWT breaks down.
Signiﬁcance of the FWT is that it provides a benchmark to study market
failures and role of institution design / government policy. ...
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This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.
 Spring '09
 SAMUELSON
 Microeconomics

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