1. Economics 214 Lecture 35 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(x 1,x 2 ) is negative for any dx 1 and dx 2, 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(x 1,x 2 ) is positive for any dx 1 and dx 2, then that stationary point represents a local minimum of the function.

5. Deriving the second order conditions

7. Understanding the 2 nd Order condition

8. Understanding the 2 nd Order Conditions y 0 x x0x0 TyTy y0y0 TxTx

9. Maximum drawing A TyTy TxTx x y z dx>0,dy>0

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

11. 2 nd Order conditions for Bivariate Function

12. Example

13. Multiproduct Monopolist

14. 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/ http://www.compute.uwlax.edu/calc2D/ output/2112/