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/