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601-compstats - E 601 1 Comparative statics Suppose an...

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E frqrplfv 601 1 Comparative statics Suppose an agent is maximizing her objective function f ( x, a ) , where x is the choice variable and a is a parameter. Assuming the maximization problem is well- behaved, and all we are interested in is the sign of dx/da then the method of com- parative statics (the implicit function theorem) tells us that Sign dx da = Sign 2 f ( x, a ) a x . The fi rst-order condition for this problem is f ( x, a ) x = 0 which implicitly de fi nes the optimal level of x as a function of a . Taking a total di ff erential of this equation obtain 2 f ( x, a ) x 2 dx + 2 f ( x, a ) a x da = 0 or dx da = 2 f ( x, a ) / a x 2 f ( x, a ) / x 2 The result follows immediate from this equation and that the second-order con- ditions for the maximization require 2 f ( x, a ) / x 2 < 0 . EXAMPLE Consider a two-period life-cycle model of a single individual whose endowment is e 1 in period 1 and half the time e 2 + δ in period 2 and half the time e 2 δ in period 2 , where δ is a positive number. The individual must use an asset that pays a known real rate of return, r, to carry purchasing power across the periods. With no uncertainty, her lifetime utility function is U = v
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