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Unformatted text preview: Econ 511 Professor: John Nachbar November 30, 2008 Monotone Comparative Statics 1 Overview Given an optimization problem indexed by some parameter , comparative statics seeks a qualitative understanding of how the solution changes with . If, for example, wages decrease, does a firm hire more labor? One way to obtain results of this type is to assume (or use the Implicit Function theorem to establish the existence of) a differentiable solution function, and then, having substituted the solution function into the first order condition, apply the Chain Rule to the first order condition to try to determine the sign of the derivatives of the solution function. These notes briefly survey an alternative approach. Relative to the calculus- based approach, the approach described here has certain advantages. It makes assumptions that are weaker, either necessary or close to necessary. In particular, the results below can be applied in settings where differentiability, or even continuity of the solution function cannot be assumed. The approach below yields arguments that are often remarkably concise. And those arguments are often also relatively transparent. Work in this area was pioneered by Topkis in the 1970s. Over the subsequent two decades, economists gradually became persuaded that Topkiss approach was the right one for handling certain types of problems in economics. Key papers on comparative statics include Topkis (1978), Milgrom and Roberts (1990a), Milgrom and Shannon (1994), Athey (2002), and Quah and Strulovici (2008). Topkis (1998) is the standard general reference. There is a related literature analyzing equilibria, especially equilibria in games. Key papers there include Topkis (1979), Vives (1990), Milgrom and Roberts (1990b), and Milgrom and Roberts (1994). Finally, the lattice machinery that is characteristic of these literatures has proved important in some applications in competitive general equilibrium theory; see Aliprantis and Brown (1982) and Mas-Colell (1986). 2 The Basic Montonicity Result The material in this section and the next owes to Topkis. Topkis (1998) provides a comprehensive overview of the state of the art at the time of its publication. To simplify notation, I take function domains to be all of some R K . But the definitions and theorems below extend immediately to functions whose domains are nice subsets of R K . R K ++ is an example of a nice subset. 1 Given a function f : R 2 R , say that f is supermodular if, whenever x x * and * , f ( x * , * )- f ( x , * ) f ( x * , )- f ( x , ) . That is, fixing x x * , the function g ( ) = f ( x * , )- f ( x , ) , which measures the benefit (which could be negative) of switching from x to x * , is weakly increasing in . f is submodular iff- f is supermodular. In these notes, I focus primarily on supermodular functions....
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This note was uploaded on 05/20/2010 for the course ECON 511 taught by Professor Johnnachbar during the Fall '08 term at Washington University in St. Louis.
- Fall '08