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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 #25: TUE, NOV 27, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT2) Contents 7. Foundations of Comparative Statics (cont) 1 7.3. Implicit function Theorem. 1 7. Foundations of Comparative Statics (cont) 7.3. Implicit function Theorem. Hardest concept in the course. It is also the basis for all of comparative statics. If you know how to do KTC and use the implicit function theorem you can read 90% of the journals. A notation problem : Dealing with the implicit function theorem creates an inescapable notation problem. It is all about endogenous variables and exogenous variables. • Economists always , always , always write the endogenous variables first, i.e., f ( x ; α ) where α is the vector of exogenous variables. • The implicit function theorem is all about writing x as a function of α . • Mathematicians always , always , always write x n +1 as a function of the first n variables. Translating to economic language, they always , always , always put the exogenous variables first. That is, if they used the α and x notation, they would write f ( α ; x ) • So we have a problem either way. • I’ll go with the mathematicians, since you will be reading math textbooks to understand the implicit function theorem, i.e., I’ll write f ( α ; x ). 1 2 LECTURE #25: TUE, NOV 27, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT2) The implicit function theorem and comparative statics: The solution to any economic system can be represented as the level set of some function. Here’s a simple economic model: S = S ( p, t ), D = D ( p, y ), S = D , where p denotes market price, t denotes a tax rate paid by the producer and y denotes consumer income level. The solution to this model can be represented as the level set f ( α , x ) ≡ 0, where f = S- D , α = t, y and x = p . When we do comparative statics, we change α and ask how x has to change in order to stay on the same level set....
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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 Berkeley.
- Fall '07