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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 ARE 211, FALL 2007 LECTURE #28: THU, DEC 6, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT5) Contents 7. Implicit Function Theorem and the Envelope Theorem (cont) 1 7.4. Genericity and Transversality 1 7. Implicit Function Theorem and the Envelope Theorem (cont) 7.4. Genericity and Transversality Mas-Colell Whinston and Green have a difficult section on genericity, local uniqueness and the transversality theorem (pages 593-595). Fig. 1 and the following discussion is intended to shed some light on those pages. Definition: (Definition 17.D.3. in MWG) a system of M equations in N unknowns f ( v ) is regular if rank Df ( v ) = M whenever f ( v ) = 0. Fig. 1 graphs two functions τ and μ of α and x . The right panel is a 3-D version of MWG’s figure 17.D.4. Note that if you fix α and consider μ as a function of x only (that is M = N = 1), then μ is not regular when α = ¯ α , since there is a solution to μ (¯ α, · ) = 0 where the rank of the determinant is zero (i.e., at the point of tangency): in this case, v = ( α,x ) and Dμ ( v ) is just μ x = 0. One way= 0....
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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