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Download Chapter Twenty-Six - Uniwersytet Warszawski
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Summary (almost) everything you need to know about micro theory in 30 minutes Production functions Q=f(K,L) Short run: at least one factor fixed Long run: anything can change Average productivity: APL=q/L Marginal productivity: MPL=dq/dL Ave prod. falls when MPL<APL MPL falls, eventually (the „law” of diminishing marginal productivity) Isoquants All combinations of factors that allow same production Stolen from: prenhall.com Substitution MRTSKL=-MPK/MPL (how many units of labor are necessary to replace one unit of capital) MRTS is the inverse of the slope of the isoquant Economies of scale f(zK,zL)><=zf(K,L), z>1 Shows whether large of small production scale more efficient Example: Cobb-Douglas: (zK)α(zL)β=z(α+β)KαLβ Thus economies of scale are constant (increasing, decreasing) if α+β equal to (greater than, smaller than) 1. Costs Economist’s and accountant’s view Opportunity costs Sunk costs („bygones are bygones”) TC(q)=VC(q)+FC ATC(q)=TC(q)/q MC(q)=dTC(q)/dq MC assumed to go up, eventually AVC(q) and ATC(q) minimum when equal to MC Cost minimization Cost minimization with fixed production Dual problem to maximizing production with fixed costs Perfect competition Assumptions – Many (small) firms – New firms can enter in the long run – Homegenous product – Prices known – No transaction or search costs – Prices of factors (perceived as) constant – Market price perceived as constant (firm is a „pricetaker”) – Profit maximisation – Decreasing economies of scale Main feature: perfectly elastic demand for a single firm Perfect competition-analysis Magical formula: MC(q)=P Defines inverse supply f. for a single firm Aggregate supply: S(P)=ΣSi(P) In the long run: – Profit=0 – P=min(AC) – S=D Efficiency: – Lowest possible production cost – Production level appropriate given preference Monopoly Sources of monopolistic power – Administrative regulations (e.g. Poczta Polska) – Natural monopoly (railroad networks) – Patents – Cartels (the OPEC) – Economies of scale The magic formula: MR(q)=MC(q) Monopoly-cont’d By increasing production, monopoly negatively affects prices Thus MR lower than AR(=p) E.g. with P=a+bq: TR=Pq=(a+bq)q=aq+bq2 MR=a+2bq Another useful formula: link with demand elasticity: MR=P(q)(1+1/ε) Thus always chooses such q that demand is elastic Inefficiency: production lower than in PC, price higher – deadweight loss Plus, losses due to rent-seeking Monopoly: price discrimination Trying to make every consumer pay as much as (s)he agrees to pay 1st degree (perfect price disc. – every unit sold at reservation price), – production as in the case of a perfectly competitive market – (thus no inefficiency) – No consumer surplus either Price discrimination-cont’d 2nd degree: different units at different prices but everyone pays the same for same quantity Examples: mineral water, telecom. 3rd degree: different people pay different prices – (because different elasticities) – E.g.: discounts for students Two-part tarifs Access fee + per-use price Examples: Disneyland, mobile phones, vacuum cleaners Homogenous consumers: – Fix per-use price at marginal cost – Capture all the surplus with the access fee Different consumer groups – Capture all the surplus of the „weaker” group – Price>MC – OR: forget about the „weaker” group Game theory Used to model strategic interaction Players choose strategies that affect everybody’s payoffs Important notion: (strictly) Dominant strategy – always better than other strategy(ies) Example left middl e 4,1 right Strategy „left” is dominated by „right” up 2,2 1,3 Will not be played 2,5 2,2 up, down, middle and dow 6,1 right are rationalizable n Nash equilibrium: two strategies that are mutually best-responses (no profitable unilateral deviation) No NE in pure strategies here NE in mixed strategies to be found by equating expected payoffs from strategies Repeated games Same („stage”) game played multiple times If only one equilibrium, backward induction argument for finite repetition What if repeated infinitly with some discount factor β? Repeated games-cont’d „prisoner’s dillema” Low price High price Low price 1,1 High price 3,0 0,3 2,2 Consider „trigger” stragegy: I play high but if you play low once, I will always play low. If you play high, you will get 2+2β+2β2+… If you play low, you will get 3+β+β2+… Collusion (high-high) can be sustained if our βs are .5 or higher (though low-low also an equilibrium in a repeated game) Sequential games A tree (directed graph with no cycles) with nodes and edges Information sets Subgame: a game starting at one of the nodes that does not cut through info sets SPNE: truncation to subgames also in equilibrium Backward induction: start „near” the final nodes Example: battle of the sexes Oligopoly: Cournot Low number of firms Firms not assumed to be price-takers Restricted entry Nash equilibrium Cournot: competition in quantities Example: duopoly with linear demand Cournot duopoly with linear demand P=a-bQ=a-b(q1+q2) Cost functions: g(q1), g(q2) Π1=q1(a-b(q1+q2))-g(q1) Optimization yields q1=(a-bq2MC1)/2b (reaction curve of firm 1) Cournot eq. where reaction curves cross Useful formula: if symmetric costs: q1 =q2 =(a-MC)/3b Oligopoly: Stackelberg First player (Leader) decides on quantity Follower react to it SPNE found using backward induction: Π2=q2(a-b(q1+q2))-TC2 Reaction curve as in Cournot: q2= (a-bq1-MC2)/2b For constant MC we get: q1 =2q2 =(a-MC)/2b Comparing Cournot and Stackelberg Firm 2 reacts optimally to q1 in either But firm 1 only in Cournot Firm 1 will produce and earn more in vS Firm 2 will produce and earn less Production higher, price lower in Stackelberg if cost and demand are linear Oligopoly: plain vanilla Bertrand Both firms set prices Basic assumption: homogenous goods (firm with lower price captures the whole market) Undercutting all the way to P=MC If firms not identical, the more efficient one will produce and sell at the other’s cost More realistic: heterog. goods Competitor’s price affects my sales negatively (but not drives them to 0 when just slightly lower than mine) Example: q1=12-P1+P2 TC1=9q1, TC2=9q2 q1=12-P2+P1 P1=P2=10>MC Before the exam Look up www.miq.woee.pl (password: miq) for questions, tests and more
