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Economics 2301

Economics 2301. Lecture 34 Multivariate Optimization. Second Total Differential of Bivariate Function. Example of Second Total Differential. Second Order Conditions.

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Economics 2301

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  1. Economics 2301 Lecture 34 Multivariate Optimization

  2. Second Total Differential of Bivariate Function

  3. Example of Second Total Differential

  4. Second Order Conditions • If the second total differential evaluated at a stationary point of a function f(x1,x2) is negative for any dx1 and dx2, then that stationary point represents a local maximum of the function. • If the second total differential evaluated at a stationary point of a function f(x1,x2) is positive for any dx1 and dx2, then that stationary point represents a local minimum of the function.

  5. Deriving the second order conditions

  6. Deriving the second order conditions

  7. Understanding the 2nd Order condition

  8. Understanding the 2nd Order Conditions Ty y y0 Tx x 0 x0

  9. Maximum drawing z Ty A Tx x dx>0,dy>0 y

  10. Our Slice of the cone A dx>0,dy>0 dx<0,dy<0

  11. 2nd Order conditions for Bivariate Function

  12. Example

  13. Multiproduct Monopolist

  14. Multiproduct Monopolist Cont.

  15. Multiproduct Monopolist Cont.

  16. Saddle Point

  17. Figure 10.4 A Saddle Point

  18. Example of Saddle Point

  19. Example 2 from last Lecture

  20. Example 2 Continued

  21. Graph of Previous Slide • http://www.compute.uwlax.edu/calc2D/output/2112/

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