Applied cooperative game theory:Pareto optimality in microeconomics

Harald Wiese

University of Leipzig

April 2010

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 1 / 30

Pareto optimality in microeconomicsoverview

1 Introduction: Pareto improvements2 Identical marginal rates of substitution3 Identical marginal rates of transformation4 Equality between marginal rate of substitution and marginal rate oftransformation

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 2 / 30

Pareto optimality in microeconomicsoverview

1 Introduction: Pareto improvements2 Identical marginal rates of substitution3 Identical marginal rates of transformation4 Equality between marginal rate of substitution and marginal rate oftransformation

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 3 / 30

Introduction: Pareto improvements

Judgements of economic situations

Ordinal utility! comparison among di¤erent people

Vilfredo Pareto, Italian sociologue, 1848-1923:

De�nitionSituation 1 is called Pareto superior to situation 2 (a Paretoimprovement over situation 2) if no individual is worse o¤ in the �rstthan in the second while at least one individual is strictly better o¤.

Situations are called Pareto e¢ cient, Pareto optimal or just e¢ cient ifPareto improvements are not possible.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 4 / 30

Pareto optimality in microeconomicsoverview

1 Introduction: Pareto improvements2 Identical marginal rates of substitution3 Identical marginal rates of transformation4 Equality between marginal rate of substitution and marginal rate oftransformation

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 5 / 30

MRS = MRSThe Edgeworth box for two consumers

Francis Ysidro Edgeworth (1845-1926): �Mathematical Psychics�

A

B

Ax1

Ax2

Bx2

indifferencecurve B

indifferencecurve A

Bx1

exchangelens

A1ω

B1ω

B2ωA

2ω

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 6 / 30

MRS = MRSThe Edgeworth box for two consumers

A

B

Ax1

Ax2

Bx2

indifferencecurve B

indifferencecurve A

Bx1

exchangelens

A1ω

B1ω

B2ωA

2ω

Problem

Which bundles of goodsdoes individual A preferto his endowment?

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 7 / 30

MRS = MRSThe Edgeworth box for two consumers

A

B

Ax1

Ax2

Bx2

indifferencecurve B

indifferencecurve A

Bx1

exchangelens

A1ω

B1ω

B2ωA

2ω

ProblemWhich bundles of goodsdoes individual A preferto his endowment?

SolutionAll those bundels xA tothe right and above theindi¤erence curvecrossing ωA.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 8 / 30

MRS = MRSThe Edgeworth box for two consumers

Consider

(3 =)

����dxA2dxA1���� = MRSA < MRSB = ����dxB2dxB1

���� (= 5)If A gives up a small amount of good 1,he needs MRSA units of good 2 in order to stay on his indi¤erencecurve.

If individual B obtains a small amount of good 1,she is prepared to give up MRSB units of good 2.MRSA+MRSB

2 units of good 2 given to A by B leave both better o¤

Ergo: Pareto optimality requires MRSA = MRSB

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 9 / 30

MRS = MRSThe Edgeworth box for two consumers

Pareto optima in the Edgeworth box�contract curve or exchange curve

A

B

Ax1

Ax2

Bx2

Bx1

contract curve

good 1

good 1

good 2 good 2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 10 / 30

MRS = MRSThe Edgeworth box for two consumers

ProblemTwo consumers meet on an exchange market with two goods. Both havethe utility function U (x1, x2) = x1x2. Consumer A�s endowment is(10, 90), consumer B�s is (90, 10).a) Depict the endowments in the Edgeworth box!b) Find the contract curve and draw it!c) Find the best bundle that consumer B can achieve through exchange!d) Draw the Pareto improvement (exchange lens) and the Pareto-e¢ cientPareto improvements!

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 11 / 30

MRS = MRSThe Edgeworth box for two consumers

a)

good 110080604020

0100

80

60

40

20

0204060

60

80

80

100

100

20

40

00

good 1

good 2 good 2

A

B Solution

b) xA1 = xA2 ,

c) (70, 70) .d) The exchange lens isdotted. The Paretoe¢ cient Paretoimprovements arerepresented by thecontract curve withinthis lens.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 12 / 30

MR(T)S = MR(T)SThe production Edgeworth box for two products

Analogous to exchange Edgeworth box

MRTS1 =��� dC1dL1

���Pareto e¢ ciency����dC1dL1

���� = MRTS1 != MRTS2 =

����dC2dL2

����

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 13 / 30

MRS = MRSTwo markets � one factory

A �rm that produces in one factory but supplies two markets 1 and 2.

Marginal revenue MR = dRdxican be seen as the monetary marginal

willingness to pay for selling one extra unit of good i .

Denominator good � > good 1 or 2, respectivelyNominator good � > �money� (revenue).

Pro�t maximization by a �rm selling on two markets 1 and 2 implies���� dRdx1���� = MR1 !

= MR2 =

���� dRdx2����

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 14 / 30

MRS = MRSTwo �rms in a cartel

The monetary marginal willingness to pay for producing and sellingone extra unit of good y is a marginal rate of substitution.

Two �rms in a cartel maximize

Π1,2 (x1, x2) = Π1 (x1, x2) +Π2 (x1, x2)

with FOCs∂Π1,2

∂x1!= 0

!=

∂Π1,2

∂x2

If ∂Π1,2∂x2

were higher than ∂Π1,2∂x1

...

How about the Cournot duopoly with FOCs

∂Π1

∂x1!= 0

!=

∂Π2

∂x2?

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 15 / 30

Pareto optimality in microeconomicsoverview

1 Introduction: Pareto improvements2 Identical marginal rates of substitution3 Identical marginal rates of transformation4 Equality between marginal rate of substitution and marginal rate oftransformation

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 16 / 30

MRT = MRTTwo factories � one market

Marginal cost MC = dCdy is a monetary marginal opportunity cost of

production

MRT =

����dx2dx1����transformation curve

One �rm with two factories or a cartel in case of homogeneous goods:

MC1!= MC2.

Pareto improvements (optimality) have to be de�ned relative to aspeci�c group of agents!

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 17 / 30

MRT = MRTInternational trade

David Ricardo (1772�1823)

�comparative cost advantage�

4 = MRTP =

����dWdCl����P > ����dWdCl

����E = MRT E = 2Lemma

Assume that f is a di¤erentiable transformation function x1 7! x2. Assumealso that the cost function C (x1, x2) is di¤erentiable. Then, the marginalrate of transformation between good 1 and good 2 can be obtained by

MRT (x1) =

����df (x1)dx1

���� = MC1MC2

.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 18 / 30

MRT = MRTInternational trade

Proof.Assume a given volume of factor endowments and given factor prices.Then, the overall cost for the production of goods 1 and 2 areconstant along the transformation curve:

C (x1, x2) = C (x1, f (x1)) = constant.

Forming the derivative yields

∂C∂x1

+∂C∂x2

df (x1)dx1

= 0.

Solving for the marginal rate of transformation yields

MRT = �df (x1)dx1

=MC1MC2

.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 19 / 30

MRT = MRTInternational trade

Before Ricardo:England exports cloth and imports wine if

MCECl < MCPCl and

MCEW > MCPW

hold.

Ricardo:MCEClMCEW

<MCPClMCPW

su¢ ces for pro�table international trade.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 20 / 30

Pareto optimality in microeconomicsoverview

1 Introduction: Pareto improvements2 Identical marginal rates of substitution3 Identical marginal rates of transformation4 Equality between marginal rate of substitution and marginalrate of transformation

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 21 / 30

MRS = MRTBase case

Assume

MRS =

����dx2dx1����indi¤erence curve < ����dx2dx1

����transformation curve = MRTIf the producer reduces the production of good 1 by one unit ...

Inequality points to a Pareto-ine¢ cient situation

Pareto-e¢ ciency requires

MRS!= MRT

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 22 / 30

MRS = MRTPerfect competition - output space

FOC output space

p!= MC

Let good 2 be money with price 1

MRS is

consumer�s monetary marginal willingness to pay for one additional unitof good 1equal to p for marginal consumer

MRT is the amount of money one has to forgo for producing oneadditional unit of good 1, i.e., the marginal cost

Thus,

price = marginal willingness to pay!= marginal cost

which is also ful�lled by �rst-degree price discrimination.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 23 / 30

MRS = MRTPerfect competition - input space

FOC output space

MVP = pdydx

!= w

where

the marginal value product MVP is the monetary marginal willingnessto pay for the factor use and

w , the factor price, is the monetary marginal opportunity cost ofemploying the factor.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 24 / 30

MRS = MRTCournot monopoly

For the Cournot monopolist, the MRS!= MRT can be rephrased as the

equality between

the monetary marginal willingness to pay for selling � this is themarginal revenue MR = dR

dy �and

the monetary marginal opportunity cost of production, the marginalcost MC = dC

dy

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 25 / 30

MRS = MRTHousehold optimum

Consuming household �produces�goods by using his income to buy them,m = p1x1 + p2x2, which can be expressed with the transformation function

x2 = f (x1) =mp2� p1p2x1.

Hence,MRS

!= MRT = MOC =

p1p2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 26 / 30

Sum of MRS = MRTPublic goods

De�nition: non-rivalry in consumptionSetup:

A and B consume a private good x (xA and xB )and a public good G

Optimality condition

MRSA +MRSB

=

����dxAdG����indi¤erence curve + ����dxBdG

����indi¤erence curve!=

�����d�xA + xB

�dG

�����transformation curve

= MRT

Assume MRSA +MRSB < MRT . Produce one additional unit of thepublic good ...

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 27 / 30

Sum of MRS = MRTPublic goods

Good x as the numéraire good (money with price 1)

Then, the optimality condition simpli�es: sum of the marginalwillingness�to pay equals the marginal cost of the good.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 28 / 30

Sum of MRS = MRTPublic goods

ProblemIn a small town, there live 200 people i = 1, ..., 200 with identicalpreferences. Person i�s utility function is Ui (xi ,G ) = xi +

pG, where xi is

the quantity of the private good and G the quantity of the public good.The prices are px = 1 and pG = 10, respectively. Find the Pareto-optimalquantity of the public good.

Solution

MRT =

���� d(∑200i=1 xi)dG

���� equals pGpx = 101 = 10.

MRS for inhabitant i is��� dx idG

���indi¤erence curve = MUGMUx i

=1

2pG1 = 1

2pG.

Hence: 200 � 12pG

!= 10 and G = 100.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 29 / 30

Further exercises: Problem 1

Agent A has preferences on (x1, x2), that can be represented byuA(xA1 , x

A2 ) = x

A1 . Agent B has preferences, which are represented by the

utility function uB (xB1 , xB2 ) = x

B2 . Agent A starts with ωA

1 = ωA2 = 5, and

B has the initial endowment ωB1 = 4,ω

B2 = 6.

(a) Draw the Edgeworth box, including

ω,an indi¤erence curve for each agent through ω!

(b) Is (xA1 , xA2 , x

B1 , x

B2 ) = (6, 0, 3, 11) a Pareto-improvement compared to

the initial allocation?

(c) Find the contract curve!

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April 2010 30 / 30