# 10 - Larry Karp Notes for Dynamics X Dynamic Games Types of...

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Larry Karp Notes for Dynamics X. Dynamic Games Types of problems 1) "Symmetric" players. Each moves at the same time during a "period" (i.e., at each instant). Ex. 1 Common property resource extraction. x = stock, h i = i ’s extraction policy x (0) given ˙x f ( x ) i h i Single "state variable", x. Do we have "natural boundary condition" as t →∞ ? Is f ( x ) 0? Is U i a function of h j , j i ? Ex. 2 Dynamic oligopoly with adjustment costs. K j is stock of firm i ’s capital (e.g. physical plant, or advertising capital). π i ( K i , K j )is i ’s restricted profit function, φ ( I i ) is adjustment costs. ˙ K i δ K i I i Here there are two state variables. (2) "Asymmetric" (hierarchical) players. (a) "Leader" moves before nonstrategic, but forward looking "follower". Ex. 3 Optimal tariff for nonrenewable resource, competitive seller with rational expectations. x 0 given (resource stock) h p 0 endogenous ˙p r p c ( x )

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Equation for p ˙ results from seller’s necessary condition for his optimization problem (Hotelling condition.) Ex. 4 Optimal protection of dying sector with costly adjustment of labour L ˙ = V ( q , τ ) L 0 given q ˙ = ??? q 0 endogenous τ is value of policy (wage subsidy, tariff) at a point in time. q is the value of being in the growing sector. (b) Leader moves before strategic (forward looking) player Ex. 5 Monopsonistic importer (leader) and monopolistic seller (follower) of nonrenewable resource. x ˙ =– hx 0 given λ ˙ = ??? λ 0 endogenous (c) Leader moves before 2 or more strategic agents who are playing a dynamic game. Ex. 6 Dynamic Brander and Spencer. Take example 2 and let government choose a subsidy to influence investment decisions. Types of Nash equilibria (strategies) 1) Open loop. Decisions are conditioned on time and initial value of state. Typically these are not subgame perfect. They may or may not be "dynamically consistent", or "time consistent" 2) Markov perfect (Feedback) Decisions are also conditioned on current state. These are subgame perfect. 2 ) Differentiable, or "Smooth" M.P. Strategies are differentiable functions of state. 3) History dependent (Nonstationary) subgame perfect strategies. "Folk theorem" type results. 10:2
4) Closed loop. Strategies are state dependent, but need not be subgame perfect. (Empty threats OK). Example that illustrates (1) and (2) Spencer’s capital accumulation game (follow Fudenberg and Tirole) illustrates difference between OLE and MPE, and nonuniqueness of MPE irreversible investment 0 a i a , a is finite ˙ K i a i is static Cournot level. K C i Firms maximize time average payoffs - only steady state matters.

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## This note was uploaded on 08/01/2008 for the course ARE 263 taught by Professor Karp during the Fall '06 term at Berkeley.

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10 - Larry Karp Notes for Dynamics X Dynamic Games Types of...

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