NE applications

# NE applications - Econ 171 Spring 2010 Notes on...

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Econ 171 Spring 2010 Notes on Applications of Nash Equilibrium 1 April 16 Overview These notes explore some applications of Nash equilibrium and are intended to supplement Chapter 10 of Watson. The ﬁrst section presents an extension of the Cournot and Bertrand duopoly models to more general oligopolistic settings. It largely consists of a guided tour through Watson problems 10.1 and 10.2. The second section looks at strategic considerations in Political Economy. It looks at the way politicians choose their platforms through the lens of a simple location model and analyzes voter turnout. Finally, the last section looks at strategic social situations, such as public good provision and reporting crime. Oligopoly One of the primary concerns of microeconomic theory is how market structure aﬀects market outcomes and consumer and producer welfare. The standard market models that we use to study the two extremes of perfect competition and monopoly provide help us understand how prices and quantities depend upon market structure and the conditions under which goods will be provided eﬃciently by the market. In reality, the structure of many industries lies somewhere in between in these two extremes: non-monopolistic ﬁrms competing with each other, but not so many that one ﬁrm can disregard the eﬀect of its own choices on others, or the impact of other ﬁrms’ behavior on its own outcomes. Standard market models do not capture this interdependence, so we turn to the tools of game theory to study oligopoly. In previous microeconomic theory courses, you may have studied simple Cournot and Bertrand duopoly models similar to those presented in Chapter 10. In this section we will make the leap from duopoly to more general oligopoly, extending the Courtnot and Bertrand models to allow an arbitrary number of ﬁrms. What we accomplish will be very close to solving Watson Problems 10.1 and 10.2. Cournot Suppose there are n ﬁrms that simultaneously and independently choose quantities, with q i 0 being ﬁrm i ’s quantity. The price p is determined by (inverse) market demand, which is given by p = a - bQ , where Q = n i =1 q i , and each ﬁrm faces a constant marginal cost c . Assume that a > c > 0 and b > 0 and that each ﬁrm seeks to maximize proﬁts. First, let us formalize this game by writing the strategy spaces and payoﬀ functions. For each i , S i = [0 , ). The payoﬀ functions are written as u i ( Q ) = p ( Q ) q i - cq i or U i ( q i ,Q - i ) = ( a - bq i - bQ - i ) q i - cq i , where Q - i = Q - q i . 1 These notes rely heavily on Markus Moebius’ course materials and on Martin Osborne’s Introduction to Game Theory , Oxford, 2004. 1

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Let’s analyze the model by applying each of the two main solution concepts that we’ve studied so far. Rationalizability with
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## This note was uploaded on 12/26/2011 for the course ECON 171 taught by Professor Charness,g during the Fall '08 term at UCSB.

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NE applications - Econ 171 Spring 2010 Notes on...

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