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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #24: TUE, NOV 20, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT1) Contents 7. Foundations of Comparative Statics 1 7.1. The envelope theorem for unconstrained maximization 1 7.2. The envelope theorem for constrained maximization 4 7.3. Application of the envelope theorem for constrained maximization 10 7. Foundations of Comparative Statics 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 . 1 2 LECTURE #24: TUE, NOV 20, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT1) 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.) 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 . Mathematical proof is trivial d f ( b,x * ( b )) d b = f b ( b,x * ( b )) + f x ( b,x * ( b )) d x * ( b ) d b Necessary condition for x to maximize f ( b, ) is that f x ( b,x ) = 0; this is how x * ( b ) is defined; hence f x ( b,x * ( b )) = 0 by definition of x * ( b ). The picture is much more important; note that in the picture, the golden rule is broken, for display purposes: the first component of the function is pictured on the horizontal axis....
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.
- Fall '07