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# notes2008 - LECTURE NOTES ON ECONOMIC DYNAMICS Peter N...

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Copyright (c) 2008 by Peter N. Ireland. Redistribution is permitted for educational and research purposes, so long as no changes are made. All copies much be provided free of charge and must include this copyright notice. LECTURE NOTES ON ECONOMIC DYNAMICS Peter N. Ireland Department of Economics Boston College [email protected] http://www2.bc.edu/~irelandp/ec720.html

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Two Useful Theorems Two theorems will prove quite useful in all of our discussions of dynamic optimization: the Kuhn-Tucker Theorem and the Envelope Theorem. Let’s consider each of these in turn. 1 The Kuhn-Tucker Theorem References: Dixit, Chapters 2 and 3. Simon-Blume, Chapter 18. Consider a simple constrained optimization problem: x R choice variable F : R R objective function, continuously di f erentiable c G ( x ) constraint, with c R and G : R R , also continuously di f erentiable. The problem can be stated as: max x F ( x ) subject to c G ( x ) 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 de f ning 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 f erentiable, and suppose that G 0 ( x ) 6 =0 .T h e n there exists a value λ of λ such that x and λ satisfy the following four conditions: L 1 ( x F 0 ( x ) λ G 0 ( x )=0 , (1) 1
L 2 ( x )= 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 ) ,thenlet λ =0 . Clearly, (2)-(4) are satis f ed, so it only remains to show that(1)mustho ld .W ith λ , (1) holds if and only if F 0 ( 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 0 ( x ) < 0 . Then, by the continuity of F and G , there must exist an ε> 0 such that F ( x ε ) >F ( x ) and ( x ε ) . But this result contradicts the assumption that x maximizes F ( x ) subject to c G ( x ) . Similarly, if it turns out that F 0 ( x ) > 0 , then by the continuity of F and G there must exist an 0 such that F ( x + ε ) ( x ) and ( x + ε ) . But, again, this result contradicts the assumption that x maximizes F ( x ) subject to c G ( x ) . This establishes that (5) must hold, completing the proof for case 1. Case 2: Binding Constraint. If c = G ( x ) ,thenle t λ = F 0 ( x ) /G 0 ( x ) . This is possible, given the assumption that G 0 ( x ) 6 .C lear ly ,(1) ,(2) ,and(4)aresat is f ed, so it only remains to show that(3)mustho λ = F 0 ( x ) /G 0 ( x ) , (3) holds if and only if F 0 ( x ) /G 0 ( x ) 0 . (6) We can show that (6) must hold using a proof by contradiction. Suppose that instead of (6), it turns out that F 0 ( x ) /G 0 ( x ) < 0 .

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notes2008 - LECTURE NOTES ON ECONOMIC DYNAMICS Peter N...

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