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mathCompStat1-07-draft

mathCompStat1-07-draft - ARE211 Fall 2007 LECTURE#24 TUE...

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Preliminary draft only: please check for final version 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, we’re often interested in a function which has two arguments; the second is a function of the first. As we’ve 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, we’ll 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
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2 LECTURE #24: TUE, NOV 20, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT1) Example π ( 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., you’ve learnt to tell the difference between df and ∂f ; now you find that in this case, there isn’t 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. The line in the domain x * ( b ) has the property that vertically above points on this line, the function f ( b, · ) is maximized in the x direction. Now as you move out along the line x * ( b ) there are in general two contributors to the change in f ; f changes because b changes AND because x changes. In this particular case, f doesn’t change when x changes, but it does change when b changes.
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ARE211, Fall 2007 3 db b f ( b 1 , x * ( b 1 )) ( b 2 , x * ( b 2 )) ( b 3 , x * ( b 3 )) x * ( b ) df dx x When you evaluate df , the dx term has no effect because it is multiplied by zero df = ∂f ∂b ( b, x * ( b )) db + ∂f ∂x ( b, x * ( b )) dx bracehtipupleft
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