Unformatted text preview: Econ 121.
Intermediate Microeconomics. Eduardo Faingold
Yale University Lecture 12 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 Primitives of the model
Two consumers, labeled A and B ; two goods, labeled 1 and 2
A
A
Consumer A’s bundle: .x1 ; x2 /
B
B
Consumer B ’s bundle: .x1 ; x2 /
B
B
A
A
Utility functions: uA.x1 ; x2 / and uB .x1 ; x2 /, for A and B , respectively
A
A
B
B
Endowments: ! A D .!1 ; !2 /, ! B D .!1 ; !2 / Net supplies:
B
A
!1 D !1 C !1 ;
A
B
!2 D !2 C !2 : Feasible allocations
B
B
A
A
An allocation is a vector x D ..x1 ; x2 /; .x1 ; x2 //, that is, a consumption
bundle for each consumer. An allocation x is feasible if
B
A
x` C x` D !`; for ` D 1; 2: Feasible allocations can be viewed as points in the Edgeworth box. Edgeworth box xB1 xB2
w2 xA2 xA1
w1 x2A Indiﬀerence curves in the Edgeworth box x1B B uB = const uA = const x1A A x2B Goal of analysis
We address the following questions:
1. Which allocations can emerge as outcomes of some efﬁcient trading
process?
2. Which allocations would a social planner like to implement?
3. Which allocations can emerge as outcomes of some decentralized,
competitive market?
4. Which policies can a social planner introduce to induce a desirable
market outcome? Pareto efﬁciency
An allocation x is Pareto dominated by an allocation x 0 if for each consumer
i D A; B , ui .x 0i ; x 0i /
2
1 i
i
ui .x1 ; x2 /; with strict inequality for some consumer i .
A feasible allocation is Pareto efﬁcient if there does not exist a feasible
allocation x 0 that Pareto dominates x . Pareto efﬁciency
Two equivalent interpretations of Pareto inefﬁciency:
Gains from trade: if an allocation x is Pareto inefﬁcient, there are
mutually beneﬁcial gains from trade. Thus, we “expect” A and B to
negotiate some mutually beneﬁcial terms of trade that would make them
both better off at some allocation other than x .
Social planner: if an allocation x is Pareto inefﬁcient, a benevolent social
planner would not like to implement that allocation, as he can ﬁnd some
other allocation that makes all agents weakly better off, and at least
some agent strictly better off. A Pareto improvement xB1 xB2
w2 xA2 xA1
w1 Pareto efﬁciency
In the Edgeworth, can plot the set of Pareto efﬁcient allocations. This set is
called the contract curve.
If preferences are monotonic, convex and smooth then:
A feasible allocation x is Pareto efﬁcient if and only if the indifference
curves of A and B are tangent at x in the Edgeworth box. The Contract Curve B uA
uB
A Utility Possibility Set
The utility possibility set, U , is the set of all pairs of utility levels
B
B
A
A
.uA.x1 ; x2 /; uB .x1 ; x2 //; where x ranges over all feasible allocations.
The utility possibility set of consumer i , U i , is the set of all utility levels
i
i
ui .x1; x2/ of consumer i , where x ranges over all feasible allocations. Utility Possibility Set uB U UA uA Characterization of Pareto efﬁciency
For each utility level uB in consumer B ’s utility possibility set U B consider
the maximization problem P.uB / below: max AABB
.x1 ;x2 ;x1 ;x2 / A
A
uA.x1 ; x2 / subject to
B
B
uB .x1 ; x2 / uB A
B
x1 C x1 D !1 A
B
x2 C x2 D !2 Characterization of Pareto efﬁciency
B
B
A
A
Theorem. Let x D ..x1 ; x2 /; .x1 ; x2 // be an allocation. (a) If x is Pareto efﬁcient then x solves the maximization problem P.uB / for
some uB 2 U B .
(b) Conversely, if for some uB the allocation x solves the maximization
problem P.uB / and the utility functions are continuous and strictly
increasing, then x is Pareto efﬁcient. Characterization of Pareto efﬁciency
Proof of (a).
Let x be a Pareto efﬁcient allocation and let us show that it solves the
maximization problem corresponding to uB D uB .x B /.
We will prove the contrapositive, i.e., we will show that if x is not a solution
of P.uB / with uB D uB .x B /, then x is not Pareto efﬁcient.
Indeed, if x does not solve the maximization problem corresponding to
uB D uB .x B /, then there exists a feasible allocation x 0 such that uA.x 0A/ > u.x A/
and uB .x 0B /
and hence x is not Pareto efﬁcient. uB D uB .x B /; Characterization of Pareto efﬁciency
Proof of (b).
Again, we will prove the contrapositive. If x is not Pareto efﬁcient, then there
exists a feasible allocation x 0 that Pareto dominates x . This means x 0 is
feasible and satisﬁes uA.x 0A/ uA.x A/ uB .x 0B / uB .x B /; and
with at least one strict inequality. Characterization of Pareto efﬁciency
Proof of (b).
First case: uA.x 0A/ > uA.x A/ and uB .x 0B / uB .x B /. Fix any uB in B ’s utility possibility set with uB Ä uB .x B /. Then, we must
also have uB .x 0B / uB . Therefore, allocation x 0 satisﬁes all the
constraints of the maximization problem. Since uA.x 0A/ > uA.x A/ we
conclude that x cannot solve the maximization problem corresponding to
uB . Characterization of Pareto efﬁciency
Proof of (b).
Second case: uA.x 0A/ uA.x A/ and uB .x 0B / > uB .x B /. By the strict monotonicity of uA (and the continuity of uB ) we can ﬁnd a
feasible allocation x 00 such that uA.x 00A/ > uA.x 0A/ uA.x A/ and uB .x 00B / > uB .x B /;
and so we are back to the ﬁrst case. Q.E.D. Characterization of Pareto efﬁciency
Back to the maximization problem that characterizes the contract curve: max AABB
.x1 ;x2 ;x1 ;x2 / A
A
uA.x1 ; x2 / subject to uB B
B
uB .x1 ; x2 /
B
A
x1 C x1 D !1 A
B
x2 C x2 D !2 If we could replace the inequality by an equality, then we could apply the
method of Lagrange. Digression: Lagrange’s method
Consider the maximization problem in n variables and m constraints. max .x1 ;:::;xn / f .x1; : : : ; xn/ subject to g1 .x1; : : : ; xn/ D c1 g2 .x1; : : : ; xn/ D c2 gm .x1; : : : ; xn/ D cm Digression: Lagrange’s method
The Lagrange method is to introduce m new variables 1; : : : ; m , called
Lagrange multipliers, and solve the ﬁrst order conditions of the Lagrangean
function L, deﬁned as L.x1 ; : : : ; xn;
C 1; : : : ; m/ D f .x1; : : : ; xn/ C 1 .g1 .x1 ; : : : ; xn / c1/ C : : : C m .gm .x1 ; : : : ; xm / cm/: Digression: Lagrange’s method
Thus, the Lagrange method is to solve: @L
.x1; : : : ; xn;
@x1 1; : : : ; m/ D 0 @L
.x1; : : : ; xn;
@xn 1; : : : ; m/ D 0 @L
.x1; : : : ; xn;
@1 1; : : : ; m/ D 0 @L
.x1; : : : ; xn;
@m 1; : : : ; m/ D 0 and Note that the last m conditions are equivalent to
g1.x1 ; : : : ; xn/ D c1; : : : ; gm.x1; : : : ; xn/ D cm. Digression: Lagrange’s method
If f; g1; : : : ; gm are differentiable and .x1 ; : : : ; xn/ is a solution of the
maximization problem such that the m vectors @g1
.x1; : : : ; xn/; : : : ;
@x1
:::
@gm
.
.x1; : : : ; xn/; : : : ;
@x1
. @g1
.x1 ; : : : ; xn//
@xn
@gm
.x1; : : : ; xn//
@xn are linearly independent, then there exist 1 ; : : : ; m such that
.x1; : : : ; xn; 1 ; : : : ; m/ solves the ﬁrstorder conditions of the Lagrangean.
Hence, the Lagrange method provides necessary conditions for
constrained optima.
Under suitable assumptions of f; g1 ; : : : ; gm, these conditions are also
sufﬁcient. Characterization of Pareto efﬁciency
Back to the maximization problem that characterizes the contract curve: max AABB
.x1 ;x2 ;x1 ;x2 / A
A
uA.x1 ; x2 / subject to
B
B
uB .x1 ; x2 / D uB .w/ Lagrange mult. / B
A
x1 C x1 D !1 .w/ Lagrange mult. 1/ A
B
x2 C x2 D !2 .w/ Lagrange mult. 2/ Thus, the Lagrangean is
A
A
B
B
L.x1 ; x2 ; x1 ; x2 ; ; 1; C 2/ D 1 B
B
uB .x1 ; x2 / uB A
A
uA.x1 ; x2 / C B
A
x1 C x1 !1 C 2 B
A
x2 C x2 !2 C Characterization of Pareto efﬁciency
Firstorder conditions: @L
D0
A
@x1 H) M U1A C 1 D0 H) M U1A D 1 @L
D0
A
@x2 H) M U2A C 2 D0 H) M U2A D 2 @L
D0
B
@x1 H) M U1B C 1 D0 H) M U1B D 1 @L
D0
B
@x2 H) M U2B C 2 D0 H) M U2B D
2 @L
D0
@
@L
D0
@1
@L
D0
@2 H) B
B
uB .x1 ; x2 / D uB H) A
B
x1 C x1 D !1 H) B
A
x2 C x2 D !2 Characterization of Pareto efﬁciency
Therefore, MRS A D M U1A
D
A
M U2 1
2 D M U1B
D MRS B ;
M U2B that is,
a feasible allocation is Pareto efﬁcient if and only if the marginal rate of
substitution is equalized across all consumers.
Recalling that the MRS is the slope of the indifference curve, the above
condition is just the tangency condition we saw in the Edgeworth box. 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
A
B
B
A competitive equilibrium is an allocation ..x1 ; x2 /; .x1 ; x2 // and prices
.p1; p2/ that satisﬁes the following two conditions:
A
A
I. Optimization: the bundle .x1 ; x2 / solves max .x1 ;x2 / uA.x1; x2/ subject to p1x1 C p2x2 Ä 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/ be a
strictly positive vector of prices. If .x; p/ is a competitive equilibrium then x
is Pareto efﬁcient.
The most important result of neoclassical Economics.
Benchmark for understanding role of policy making. The First Welfare Theorem
This is so important that we will see two proofs of this result!
Today we will present a simple proof for the case in which utility functions
are smooth, strictly increasing and quasiconcave (i.e. preferences are
convex). We will also assume that x is strictly positive (all coordinates are
positive). Proof when preferences are smooth and convex
In a competitive 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 MRS A A
A
.x1 ; x2 / D p1
p2 D p1
:
p2 and MRS B B
B
.x1 ; x2 / Therefore,
B
B
A
A
MRS A.x1 ; x2 / D MRS B .x1 ; x2 /: Also, by the market clearing condition, x is a feasible allocation. Hence, the
allocation x satisﬁes the necessary ﬁrstorder conditions of the
maximization problem that characterizes Pareto efﬁciency. Since we are
assuming that the preferences are convex the ﬁrstorder conditions are also
sufﬁcient conditions, and hence x is Pareto efﬁcient, as required.
Q.E.D. ...
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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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