notes1 - The Kuhn-Tucker and Envelope Theorems Peter...

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The Kuhn-Tucker and Envelope Theorems Peter Ireland EC720.01 - Math for Economists Boston College, Department of Economics Fall 2010 The Kuhn-Tucker and envelope theorems can be used to characterize the solution to a wide range of constrained optimization problems: static or dynamic, under perfect foresight or featuring randomness and uncertainty. In addition, these same two results provide foun- dations for the work on the maximum principle and dynamic programming that we will do later on. For both of these reasons, the Kuhn-Tucker and envelope theorems provide the starting point for our analysis. Let’s consider each in turn, first in fairly general or abstract settings and then applied to some economic examples. 1 The Kuhn-Tucker Theorem References: Dixit, Chapters 2 and 3. Simon-Blume, Chapter 18. Acemoglu, Appendix A. Consider a simple constrained optimization problem: x R choice variable F : R R objective function, continuously di ff erentiable c G ( x ) constraint, with c R and G : R R , also continuously di ff erentiable. The problem can be stated as: max x F ( x ) subject to c G ( x ) Copyright c 2010 by Peter Ireland. Redistribution is permitted for educational and research purposes, so long as no changes are made. All copies must be provided free of charge and must include this copyright notice. 1
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This problem is “simple” because it is static and contains no random or stochastic elements that would force decisions to be made under uncertainty. This problem is also “simple” because it has a single choice variable and a single constraint. All these simplifications will make our statement and proof of the Kuhn-Tucker theorem as clean and intuitive as possible. But the results can be generalized along all of these dimensions and, later, we will work through examples that do so. Probably the easiest way to solve this problem is via the method of Lagrange multipliers. The mathematical foundations that allow for the application of this method are given to us by Lagrange’s Theorem or, in its most general form, the Kuhn-Tucker Theorem. To prove this theorem, begin by defining the Lagrangian: L ( x, λ ) = F ( x ) + λ [ c G ( x )] for any x R and λ R . Theorem (Kuhn-Tucker) Suppose that x maximizes F ( x ) subject to c G ( x ), where F and G are both continuously di ff erentiable, and suppose that G ( x ) = 0. Then there exists a value λ of λ such that x and λ satisfy the following four conditions: L 1 ( x , λ ) = F ( x ) λ G ( x ) = 0 , (1) L 2 ( x , λ ) = c G ( x ) 0 , (2) λ 0 , (3) and λ [ c G ( x )] = 0 . (4) Proof Consider two possible cases, depending on whether or not the constraint is binding at x . Case 1: Nonbinding Constraint. If c > G ( x ), then let λ = 0. Clearly, (2)-(4) are satisfied, so it only remains to show that (1) must hold. With λ = 0, (1) holds if and only if F ( x ) = 0 . (5) We can show that (5) must hold using a proof by contradiction. Suppose that instead of (5), it turns out that F ( x ) < 0 .
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