lecture18_slides

# Denote the optimal solution and their respective

• Notes
• 30

This preview shows page 15 - 22 out of 30 pages.

denote the optimal solution and their respective Lagrange multipliers as a function of the parameters c D .c 1 ; : : : ; c m / . Thus, we have V.c/ D f .x .c// C m X i D 1 i .c/.g i .x .c// c i //: and @f @x j .x .c// C m X i D 1 i .c/ @g i @x j .x .c// D 0; by the first-order conditions of Lagrange’s method.

Subscribe to view the full document.

Digression: the envelope theorem By the chain rule of calculus, @V.c/ @c 1 D X j @f @x j @x j @c 1 C m X i D 1 @ i @c 1 .g i .x .c// c i // C m X i D 1 i .c/ X j @g i @x j @x j @c 1 1 D X j @x j @c 1 h @f @x j C X i i @g i @x j ƒ‚ D 0 i C m X i D 1 @ i @c 1 .g i .x .c// c i / ƒ‚ D 0 / 1 : and hence, @V.c/ @c 1 D 1 :
Digression: the envelope theorem Envelope Theorem. Under suitable technical conditions (omitted), @V @c i .c 1 ; : : : ; c m / D i .c 1 ; : : : ; c m /; where i is the Lagrande multiplier of constraint i corresponding to the optimal solution.

Subscribe to view the full document.

Characterizing the MRT Recall that . N y 1 ; N y 2 / 2 PPF if and only if N y 1 D max J X j D 1 f j 1 .I j;1 1 ; : : : ; I j;1 N / subject to J X j D 1 f j 2 .I j;2 1 ; : : : ; I j;2 N / D N y 2 . w/ Lagrange mult. / J X j D 1 I j;1 n C I j;2 n D N I n ; . w/ Lagrange mult. n /
Characterizing the MRT Hence, by the envelope theorem, MRT D d N y 2 d N y 1 D 1 d N y 1 d N y 2 D 1 D MP j;2 n MP j;1 n :

Subscribe to view the full document.

Pareto efficiency Proposition. In a Pareto efficient allocation, production must be in the PPF. Why? If production were not in the PPF, then it is possible to produce weakly more of both goods and strictly more of at least one good, without violating the technological constraints and the resource constraints. Then, we can give the extra units produced to a single consumer to obtain a Pareto improvement.
Characterization of Pareto efficiency An allocation is Pareto efficient if and only if it solves, for each u B in B’s utility possibility set, max .x A 1 ;x A 2 ;x B 1 ;x B 2 ;y 1 ;y 2 / u A .x A 1 ; x A 2 / subject to u B .x B 1 ; x B 2 / D u B . w/ Lagrange mult. / x A 1 C x B 1 D ! 1 C y 1 . w/ Lagrange mult.

Subscribe to view the full document.

You've reached the end of this preview.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern