# Dynamic Economics: Optimization by the Lagrange Method

By Gregory C. Chow | Go to book overview

CHAPTER SIX Dynamic Games

6.1 A Formulation of Models of Dynamic Games

In section 5.6, dynamic game theory was applied to study oligopoly pricing. This section can serve as an introduction to the subject of dynamic games. From the viewpoint of dynamic optimization, which is a main theme of this book, it is convenient to introduce a model of dynamic games by specifying two decision makers (players) each trying to solve the standard dynamic optimization problem of section 2.3 with the vector state variable xt+1 depending on xt and the vector control variables u1t and u2t of both players; thus,

xt+1 = f(xt, u1t, u2t ) + εt+1 (6.1)

By using the notation of section 2.3, the Lagrangean expression of the ith player's optimization problem is

(i = 1, 2) (6.2)

in which the discount factor βi and the return function ri may be player specific, and in which ut is a vector of control variables consisting of u1t and u2t. Each player is assumed to maximize the first part of his or her Lagrangean expression ℒi, subject to the constraint given in the second part.

Three solution concepts have found important applications, depending on the empirical problems at hand. By the Nash solution, each player solves his or her optimization problem by taking the other player's strategy or decision function as given. This is the competitive solution in a market economy when each firm, in setting its price, assumes that the prices of all other firms to be given. By the Stakleberg solution, one player is specified as the dominant player. Imagine the dominant player to play first. Given the dominant player's strategy, the other player or the follower tries to optimize by choosing his or her strategy. This being the case, the dominant player or leader will take the follower's reaction into account in choosing his or her strategy. To find the

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