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Unformatted text preview: ARE211, Fall 2007 COMPSTAT1: TUE, NOV 20, 2007 PRINTED: FEBRUARY 7, 2008 (LEC# 24) Contents 7. Foundations of Comparative Statics 1 7.1. The envelope theorem for unconstrained maximization 2 7.2. The envelope theorem for constrained maximization 4 7.3. Applications of the envelope theorem: Hotellings and Shephards lemmas. 12 7.3.1. Hotellings Lemma 12 7.3.2. Shephards Lemma 14 7.4. Another Application of the envelope theorem for constrained maximization 15 7. Foundations of Comparative Statics General: (1) Implicit function theorem: used to get relationship between endog variables and exog vari ables. (2) Envelope theorem: shorthand way to get relationship between Value function and exog variables (3) Envelope theorem: if the endog variables are determined by exog variables by solving a constrained optimization problem, then sometimes you can shortcircuit the long and tedious implicit function theorem computation and use this theorem to get a relationship 1 2 COMPSTAT1: TUE, NOV 20, 2007 PRINTED: FEBRUARY 7, 2008 (LEC# 24) between endog variables and exog variables. The sometimes depends on the functional form of the constrained optimization program. 7.1. The envelope theorem for unconstrained maximization In economics, were often interested in a function which has two arguments; the second is a function of the first. As weve discussed in a previous lecture (CALCULUS2), economists typically deal with this by invoking the (unfortunately named) concept of the total derivative : if f ( b,x ) = f ( b,x ( b )), then total derivative of f x.r.t b is df/db = f b ( b,x ( b )) + f x ( b,x ( b )) x prime ( b ), where b and x are here scalars. When b changes in this case, there is a change in f due to two factors: first b changes, also, x changes as b changes. In this lecture, well consider the case in which the second argument is a special kind of function of b ; x * ( ) is the value of x that maximises (or minimizes) f for each value of b . The function f ( b,x * ( b )) is then called the value function . Example ( p,q * ( p )); for each p , pick the q that maximizes profits for that p ; call this function q * ( p ). Now ask how profit adjust as price changes and the producer adjusts quantity. Other examples of value functions in economics are the expenditure function and the indirect utility function. Answer is given by the envelope theorem which says that in this case, df/db = f/b . (i.e., youve learnt to tell the difference between df and f ; now you find that in this case, there isnt any difference.) ARE211, Fall 2007 3 The envelope theorem: Varian : Given f : R 2 R 1 (differentiable) and a function x * : R 1 R 1 (differentiable) defined by the condition that for each b , x * ( b ) maximizes f ( b, ). Then the total derivative of the function f ( b,x * ( b )) with respect to b is df ( ,x * ( )) /db = f ( ,x * ( )) /b ....
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