Structure of the Correspondence Principle at an Extremum Point

Abstract

One of the most powerful principles of microeconomics is that comparative static properties of a given behavioural equation can often be derived from the postulate that it is a solution function of some extremum problem. Applying this principle, economists have individually analyzed a wide variety of specific economic models-usually through complicated derivations involving bordered Hessians. The main purpose of the present article is to derive general formulae of comparative statics for the extremum problem with many constraint functions, where the same shift parameters appear in both target and constraint functions and all of the constraint and target functions are non-linear. This problem contains most of the models whose comparative statics have been studied so far. Our method of comparative statics is related to those proposed by Houthakker (1951-52), Samuelson (1960, 1965), Diamond and McFadden (1974), Silberberg (1974) and Gorman (1976). Threefeatures distinguish the present approach from those ofthis literature. First, our method yields satisfactory comparative static conditions when constraints contain parameters. In particular, they contain the theorems of recent applied economic models whose shift parameters appear in both target and constraint functions.' Second, only unconstrained extremum conditions are used in our derivation of comparative statics for constrained extremum problems; the Lagrangean multiplier method is dispensed with as an analytical tool. Third, the present framework yields heuristic proofs of both the Lagrangean method and the Kuhn-Tucker conditions. The method may be outlined as follows. We transform the original constrained extremum problem into an unconstrained extremum problem of the associated gain function. This function contains as the variables vectors of both decision variables and parameters of the original problem. The unconstrained extremum conditions of the gain function with respect to the decision variables establish the Lagrangean method, while those with respect to the parameter vector yield the comnparative static formulae. Our method of derivation may be called the gain function method. In the present paper we will nowhere assume convexity, nor will we employ theories on convex sets and functions. The next section illustrates the gain function approach in the simplest possible situation by proving the Lagrangean method. Comparative static results are derived in Section 2. The Le Chatelier-Samuelson Principle is generalized in Section 3. The gain function method is applied to maximization problems with inequality constraints in Section 4. The final section presents economic applications of the results.