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# 1 x a 2 c x b 2 d 2 c y 2 w lagrange mult 2 t y 1 y 2

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1 / x A 2 C x B 2 D ! 2 C y 2 . w/ Lagrange mult. 2 / T .y 1 ; y 2 / D 0 . w/ Lagrange mult. /

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Characterization of Pareto efficiency Thus, the Lagrangean is L D u A .x A 1 ; x A 2 / C u B .x B 1 ; x B 2 / u B C C T x A 1 C x B 1 ! 1 ; x A 2 C x B 2 ! 2 Taking FOC: MU A 1 D @T @y 1 and MU A 2 D @T @y 2 and likewise for B . Hence, Pareto efficiency is equivalent to: MRS A D 1 2 D MRT D MRS B :
Competitive Equilibrium: Definition Allocation: .x A ; x B ; I 1;1 ; : : : ; I J;1 ; I 1;2 ; : : : ; I J;2 ; y 1 ; : : : ; y J /; where x i D .x i 1 ; x i 2 /; y j D .y j 1 ; y j 2 / and I j;` D .I j;` 1 ; : : : ; I j;` n /: for i D A; B , j D 1; : : : ; J and ` D 1; 2 . Prices for the consumption goods: p D .p 1 ; p 2 / Prices for the inputs: w D .w 1 ; : : : ; w n / such that...

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Competitive Equilibrium: Definition .... such that: consumers take prices p and w as given and maximize their utility subject to their budget constraint; firms take prices p and w as given and maximize their profit subject to their technological constraint; the markets for input goods and consumption goods clear.
Competitive Equilibrium: Definition Consumer maximization: x A D .x A 1 ; x A 2 / solves: max u A .x 1 ; x 2 / subject to p 1 x 1 C p 2 x 2 D p 1 ! A 1 C p 2 ! A 2 C N X n D 1 w n N I A n C J X j D 1 j;A j and likewise for B . Here, j designates the profit of firm j in equilibrium; ! A 1 , ! A 2 and N I A n are A ’s endowments; and the j;A ’s are his shares of profits.

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Competitive Equilibrium: Definition Firm maximization: y j D .y j 1 ; y j 2 / , I j;1 D .I j;1 1 ; : : : ; I j;1 N / and I j;2 D .I j;2 1 ; : : : ; I j;2 N / solves j D max p 1 y 1 C p 2 y 2 N X n D 1 w n .I 1 n C I 2 n / subject to y 1 D f 1 .I 1 1 ; : : : ; I 1 N / y 2 D f 2 .I 2 1 ; : : : ; I 2 N /:
Competitive Equilibrium: Definition Market clearing conditions: x A 1 C x B 1 D ! A 1 C ! B 1 C X j y j 1 x A 2 C x B 2 D ! A 2 C ! B 2 C X j y j 2 X j I j;1 n C I j;2 n D N I A n C N I B n ; for n D 1; : : : ; N:

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First Welfare Theorem with Production Theorem. The allocation of every competitive equilibrium is Pareto efficient.
First Welfare Theorem with Production Proof using FOC.

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