Denote the optimal solution and their respective

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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.
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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 :
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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.
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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 /
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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 :
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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.
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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.
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