Advanced Microeconomics P . v. Mouche Wageningen University 2016 - PowerPoint PPT Presentation

Microeconomics Game Theory Advanced Microeconomics P . v. Mouche Wageningen University 2016

Microeconomics Game Theory Outline Microeconomics 1 Motivation Reminder Core Walrasian equilibrium Welfare theorems Quiz Game Theory 2 Motivation Games in strategic form Games in extensive form

Microeconomics Game Theory Capitalism According to Adam Smith’s (the Wealth of nations 1776): a laissez-faire approach to economics is the essential way to ensure prosperity for a nation as a whole. Ultimately, when capitalism is allowed to run its course, the greed and self-interest of the capitalists would produce results in the economy that benefit not only the individual, but society as well. Scientific problem: prove such claims with models. This problem comes down to: general equilibrium exists, is efficient and stable.

Microeconomics Game Theory Overall notations Space of goods + = { x = ( x 1 , . . . , x n ) ∈ R n | x 1 ≥ 0 , . . . , x n ≥ 0 } . R n Utility function u : R n + → R . Prices p = ( p 1 , . . . , p n ) . All p i > 0 (if not stated otherwise). Note: prices do not appear in the utility function. Budget (income) m ≥ 0. (Marshallian) demand correspondences ˜ x i ( p ; m ) . Often these are functions. Producer theory will be less important for what follows.

Microeconomics Game Theory Concrete functions Cobb-douglas: u ( x ) = Ax α 1 1 · · · x α n n , α i m x i ( p ; m ) = ˜ . α 1 + · · · + α n p i Ces: u ( x ) = A ( α 1 x ρ 1 + · · · + α n x ρ n ) h /ρ .

Microeconomics Game Theory Concrete functions (cont.) Leontief: u ( x ) = min ( x 1 /α 1 , . . . , x n /α n ) . Example What are the marshallian demand functions for this utility function? Answer: α i m ˜ x i ( p ; m ) = . α 1 p 1 + · · · + α n p n

Microeconomics Game Theory Concrete functions (cont.) Solow (for n = 2): u ( x 1 , x 2 ) = α 1 x 1 + α 2 x 2 . Here there are marshallian demand correspondences (instead of functions). Maximum (for n = 2): u ( x 1 , x 2 ) = max ( α 1 x 1 , α 2 x 2 ) .

Microeconomics Game Theory Increasingness Remember the notations ≤ , <, ≪ . For instance: ( 3 , 4 ) ≤ ( 3 , 4 ) , ( 3 , 4 ) ≤ ( 3 , 5 ) . ( 3 , 4 ) < ( 3 , 5 ) , ( 3 , 4 ) < ( 4 , 5 ) . ( 3 , 4 ) ≪ ( 4 , 5 ) . Remember: u is increasing: x ≤ y ⇒ u ( x ) ≤ u ( y ) . u is strongly increasing: x < y ⇒ u ( x ) < u ( y ) . u is strictly increasing: u is increasing, and x ≪ y ⇒ u ( x ) < u ( y ) . Example Is the cobb-douglass utility function strongly increasing? Answer: no (but it is strictly increasing).

Microeconomics Game Theory Setting for core Pure exchange economy. Specified by: N consumers and n goods. for each consumer h a good bundle ω h = ( ω h 1 , . . . , ω h n ) > 0 (i.e. each consumer has something), to be called initial good bundle such that for each good k , N � ω h O k := k > 0 h = 1 (i.e. each good is present). for each consumer h a continuous utility function u h : R n + → R . Attention: there are (still) no prices!

Microeconomics Game Theory Allocations + ) N . Allocation: X := ( x 1 , . . . , x N ) ∈ ( R n Initial allocation: Ω := ( ω 1 , . . . , ω N ) . Feasible allocation: � N h = 1 x h k = O k ( 1 ≤ k ≤ n ) . Attention: not ≤ -sign! Feasible allocation is interior if 0 < x h k < O k for all h and k .

Microeconomics Game Theory Pareto efficiency Definition A feasible allocation X is called weakly pareto efficient if there is no feasible allocation Y with u h ( y h ) > u h ( x h ) ( 1 ≤ h ≤ N ) . (strongly) pareto efficient if there is no feasible allocation Y with u h ( y h ) ≥ u h ( x h ) ( 1 ≤ h ≤ N ) with at least one of these inequalities strict.

Microeconomics Game Theory Pareto efficient allocations Each strongly pareto efficient allocation is also weakly pareto efficient. So each weakly pareto inefficient allocation is strongly pareto inefficient. In fact weakly and strongly pareto efficiency make sense in other contexts. (See next Example.)

Microeconomics Game Theory Example Example Consider N agents for which there are a finite number of ’states of the world’. Denote by ( a 1 , a 2 , . . . , a N ) a state where agent i has ‘utility’ a i . Determine for the following situations which states are strongly pareto efficient and which are weakly pareto efficient. a. A = ( 5 , 10 ) , B = ( 6 , 9 ) , C = ( 6 , 11 ) , D = ( 4 , 12 ) . Answer: Weak: B , C , D . Strong: C , D . b. A = ( 6 , 6 ) , B = ( 6 , 7 ) , C = ( 3 , 2 ) , D = ( 7 , 6 ) , E = ( 5 , 6 ) , F = ( 11 , 1 ) . Answer: Weak: A , B , D , F . Strong: B , D , F . c. A = ( 5 , 4 ) , B = ( 9 , 1 ) , C = ( 3 , 8 ) . Answer: Weak: A , B , C . Strong: A , B , C .

Microeconomics Game Theory Example (cont.) Example d. A = ( − 4 , 8 ) , B = ( − 4 , 3 ) , C = ( − 5 , − 3 ) , D = ( 6 , 0 ) . Answer: Weak: A , B , D . Strong: A , D . e. A = ( 1 , 2 , 6 , 4 ) , B = ( 4 , 8 , 3 , 2 ) , C = ( 1 , 8 , 1 , 2 ) , D = ( 0 , 0 , 0 , 0 ) . Answer: Weak: A , B , C . Strong: A , B . f. A = ( 1 , 3 , 5 ) , B = ( 1 , 3 , 5 ) , C = ( 2 , 4 , 3 ) . Answer: Weak: A , B , C . Strong: A , B , C . g. A = ( 1 , 3 , 5 ) , B = ( 1 , 3 , 5 ) , C = ( 2 , 4 , 3 ) , D = ( 1 , 3 , 6 ) . Answer: Weak: A , B , C , D . Strong: C , D . h. A = ( 1 ) , B = ( − 8 ) , C = ( 137 ) . Answer: Weak: C . Strong: C .

Microeconomics Game Theory Strong versus weak pareto efficiency Theorem If each utility function is continuous and strongly increasing, then the set of weak and strong pareto efficient allocations is the same. Proof. This is a technical result. We omit here its proof. (If wished, see exercise 5.44 in the text book).

Microeconomics Game Theory Pareto

Microeconomics Game Theory Pareto (ctd.) Vilfredo Pareto (1848-1923): Italian engineer, economist and sociologist. Very good knowledge of mathematics. For 20 years director of two Italian railway companies. Later, motivated by Walras to switch to economic research. After disenchantment in economics, switched to sociology. His articles are difficult to read.

Microeconomics Game Theory Barter equilibrium What would be a reasonable feasible allocation X = ( x 1 , . . . , x N ) when the N consumers exchange goods? 1. X is individually rational. This is defined as follows: X is individually rational for consumer h , if u h ( x h ) ≥ u h ( ω h ) and X is individually rational if X is for each consumer individually rational. Is that all? If an allocation is strongly pareto inefficient, then there is another feasible allocation making someone better off and no one worse off: then a trade can be arranged to which no consumer will object. So: 2 X is strongly pareto efficient. Is that all?

Microeconomics Game Theory Core NO: 3 X should belong to the core.

Microeconomics Game Theory Definition Let X = ( x 1 , . . . , x N ) be a feasible allocation. We say that a coalition S can improve upon x if there are good bundles y h ( h ∈ S ) such that h ∈ S y h = � h ∈ S ω h ; � 1 u h ( y h ) ≥ u h ( x h ) for all h ∈ S with at least one inequality 2 strict. Definition A feasible allocation belongs to the (strong) core if there is no coalition that can improve upon it. Attention: the set of pareto efficient allocations only depends on O 1 , . . . , O n , but the core depends on the whole initial allocation!

Microeconomics Game Theory Properties of core Theorem Each element of the core is individually rational and pareto efficient. Proof. Suppose X is in the core. S = { i } cannot improve upon X . In particular for y i = ω i it does not hold that u h ( y h ) ≥ u h ( x h ) for all h ∈ S with at least one inequality strict. So u i ( y i ) ≤ u i ( x i ) , i.e. u i ( ω i ) ≤ u i ( x i ) follows. Thus X is individual rational for consumer player i . As this holds for every i , X is individual rational. Also S = { 1 , . . . , N } cannot improve upon X . So for any feasible allocation Y it does not hold that u h ( y h ) ≥ u h ( x h ) for all h ∈ S with at least one inequality strict, i.e. that Y is a pareto improvement of X . Thus X is pareto-efficient.

Microeconomics Game Theory Core for N = 2 Theorem For N = 2 the core is the set of pareto efficient individually rational allocations. Proof. Because S = { 1 } , S = { 2 } or S = { 1 , 2 } .

Microeconomics Game Theory Non-empty core? Very fundamental question: is the core non-empty? Intuitively one may think that this is always the case. However, this is not true. (Not so easy to find good counter-examples.) We shall see: Theorem Each pure exchange economy where each utility function is continuous, strictly quasi-concave and strongly increasing has a non-empty core. Remark: result also holds for cd-function (see Exercise 5.14 in the text book).

Microeconomics Game Theory Box of Edgeworth Box of Edgeworth: D := { x ∈ R 2 | 0 ≤ x k ≤ O k ( 1 ≤ k ≤ 2 ) } . One can identify a feasible allocation with the corresponding point in D . The set of pareto efficient allocations in the box is the contract curve.

Microeconomics Game Theory Pareto efficient allocations: necessary condition Theorem If X = ( x 1 , . . . , x N ) is an interior pareto efficient allocation, then under mild differentiability conditions, equality for each consumer of each specific marginal rate of substitution holds. Proof. We omit the proof which can be given with the method of Lagrange and only illustrate with a figure the idea.