Applied Cooperative Game Theory: Pareto Optimality in Microeconomics

Applied cooperative game theory: Pareto optimality in microeconomics

Harald Wiese

University of Leipzig

April 2010

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 1/30 Pareto optimality in microeconomics overview

1 Introduction: Pareto improvements 2 Identical marginal rates of substitution 3 Identical marginal rates of transformation 4 Equality between marginal rate of substitution and marginal rate of transformation

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 2/30 Pareto optimality in microeconomics overview

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 3/30 Introduction: Pareto improvements

Judgements of economic situations Ordinal utility ! comparison among di¤erent people Vilfredo Pareto, Italian sociologue, 1848-1923:

De…nition Situation 1 is called Pareto superior to situation 2 (a Pareto improvement over situation 2) if no individual is worse o¤ in the …rst than in the second while at least one individual is strictly better o¤. Situations are called Pareto e¢ cient, Pareto optimal or just e¢ cient if Pareto improvements are not possible.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 4/30 Pareto optimality in microeconomics overview

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 5/30 MRS = MRS The Edgeworth box for two consumers

Francis Ysidro Edgeworth (1845-1926): “Mathematical Psychics”

A x2 B w1 B B x1

indifference curve B

exchange lens indifference curve A A B w2 w2

A x1 A A w1 B x2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 6/30 MRS = MRS The Edgeworth box for two consumers

A x2 B w1 B B Problem x1 indifference Which bundles of goods curve B does individual A prefer exchange lens to his endowment? indifference curve A A B w2 w2

A x1 A A w1 B x2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 7/30 MRS = MRS The Edgeworth box for two consumers

A x2 B w1 B Problem x B 1 Which bundles of goods indifference curve B does individual A prefer exchange to his endowment? lens indifference curve A A B Solution w2 w2 All those bundels xA to

A the right and above the x1 A A w1 indi¤erence curve x B 2 crossing ωA.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 8/30 MRS = MRS The Edgeworth box for two consumers

Consider dxA dxB (3 =) 2 = MRSA < MRSB = 2 (= 5) dxA dxB 1 1

If A gives up a small amount of good 1, he needs MRSA units of good 2 in order to stay on his indi¤erence curve. If individual B obtains a small amount of good 1, she is prepared to give up MRSB units of good 2. MRS A +MRS B 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: April2010 9/30 MRS = MRS The Edgeworth box for two consumers

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

A x2 good 1 B B x1 contract curve

good 2 good 2

A x1 A good 1 B x2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 10/30 MRS = MRS The Edgeworth box for two consumers

Problem Two consumers meet on an exchange market with two goods. Both have the utility function U (x1, x2) = x1x2. Consumer A’sendowment is (10, 90), consumer B’sis (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¢ cient Pareto improvements!

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 11/30 MRS = MRS The Edgeworth box for two consumers

a) good 1 B Solution 100 80 60 40 20 0 0 A A 100 b) x1 = x2 , c) (70, 70) . 80 20 d) The exchange lens is dotted. The Pareto 60 40 e¢ cient Pareto good 2 good 2 improvements are 40 60 represented by the contract curve within 20 80 this lens.

100 0 0 20 40 60 80 100 A good 1

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 12/30 MR(T)S = MR(T)S The production Edgeworth box for two products

Analogous to exchange Edgeworth box MRTS = dC1 1 dL1

Pareto e¢ ciency

dC dC 1 = MRTS =! MRTS = 2 dL 1 2 dL 1 2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 13/30 MRS = MRS Two markets –one factory

A …rm that produces in one factory but supplies two markets 1 and 2. Marginal revenue MR = dR can be seen as the monetary marginal dxi willingness to pay for selling one extra unit of good i. Denominator good — > good 1 or 2, respectively Nominator good — > “money” (revenue). Pro…t maximization by a …rm selling on two markets 1 and 2 implies

dR dR = MR =! MR = dx 1 2 dx 1 2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 14/30 MRS = MRS Two …rms in a cartel

The monetary marginal willingness to pay for producing and selling one 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 ∂Π ∂Π 1,2 =! 0 =! 1,2 ∂x1 ∂x2 If ∂Π1,2 were higher than ∂Π1,2 ... ∂x2 ∂x1 How about the Cournot duopoly with FOCs

∂Π1 ! ! ∂Π2 = 0 = ? ∂x1 ∂x2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 15/30 Pareto optimality in microeconomics overview

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 16/30 MRT = MRT Two factories –one market

dC Marginal cost MC = dy is a monetary marginal opportunity cost of production dx transformation curve MRT = 2 dx 1

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 a speci…c group of agents!

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 17/30 MRT = MRT International trade

David Ricardo (1772–1823) “comparative cost advantage”

dW P dW E 4 = MRT P = > = MRT E = 2 dCl dCl

Lemma

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

df (x ) MC MRT (x ) = 1 = 1 . 1 dx MC 1 2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 18/30 MRT = MRT International 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 are constant along the transformation curve:

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

Forming the derivative yields

∂C ∂C df (x ) + 1 = 0. ∂x1 ∂x2 dx1 Solving for the marginal rate of transformation yields

df (x ) MC MRT = 1 = 1 . dx1 MC2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 19/30 MRT = MRT International trade

Before Ricardo: England exports cloth and imports wine if

E P MCCl < MCCl and E P MCW > MCW

hold. Ricardo: E P MCCl MCCl E < P MCW MCW su¢ ces for pro…table international trade.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 20/30 Pareto optimality in microeconomics overview

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 21/30 MRS = MRT Base case

Assume

dx indi¤erence curve dx transformation curve MRS = 2 < 2 = MRT dx dx 1 1

If 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: April2010 22/30 MRS = MRT Perfect competition - output space

FOC output space p =! MC Let good 2 be money with price 1 MRS is consumer’smonetary marginal willingness to pay for one additional unit of good 1 equal to p for marginal consumer MRT is the amount of money one has to forgo for producing one additional 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: April2010 23/30 MRS = MRT Perfect competition - input space

FOC output space dy MVP = p =! w dx where the marginal value product MVP is the monetary marginal willingness to pay for the factor use and w, the factor price, is the monetary marginal opportunity cost of employing the factor.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 24/30 MRS = MRT Cournot 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 the dR marginal revenue MR = dy – and the monetary marginal opportunity cost of production, the marginal dC cost MC = dy

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 25/30 MRS = MRT Household optimum

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

m p1 x2 = f (x1) = x1. p2 p2 Hence, p MRS =! MRT = MOC = 1 p2

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 26/30 Sum of MRS = MRT Public goods

De…nition: non-rivalry in consumption Setup: A and B consume a private good x (xA and xB ) and a public good G Optimality condition

MRSA + MRSB indi¤erence curve indi¤erence curve dxA dxB = + dG dG

transformation curve d xA + xB = ! = MRT dG

A B Assume MRS + MRS < MRT . Produce one additional unit of the public good ...

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 27/30 Sum of MRS = MRT Public goods

Good x as the numéraire good (money with price 1) Then, the optimality condition simpli…es: sum of the marginal willingness’to pay equals the marginal cost of the good.

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 28/30 Sum of MRS = MRT Public goods

Problem In a small town, there live 200 people i = 1, ..., 200 with identical preferences. Person i’sutility 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-optimal quantity of the public good.

Solution

200 d (∑ xi ) MRT = i=1 equals pG = 10 = 10. dG px 1

1 i indi¤erence curve MRS for inhabitant i is dx = MUG = 2pG = 1 . dG MUxi 1 2pG

Hence: 200 1 =! 10 and G = 100. 2pG

Harald Wiese (Chair of Microeconomics) Applied cooperative game theory: April2010 29/30 Further exercises: Problem 1

Agent A has preferences on (x1, x2), that can be represented by A A A A u (x1 , x2 ) = x1 . Agent B has preferences, which are represented by the B B B B A A utility function u (x1 , x2 ) = x2 . Agent A starts with ω1 = ω2 = 5, and B B B has the initial endowment ω1 = 4, ω2 = 6. (a) Draw the Edgeworth box, including ω, an indi¤erence curve for each agent through ω! A A B B (b) Is (x1 , x2 , x1 , x2 ) = (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: April2010 30/30