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micro-fall2007-17

micro-fall2007-17 - y B ± ˆ u B B derives a utility of at...

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General Equilibrium Pure exchange economy Economy: Two goods: x , y Two consumers: A , B Preferences: u A ( x A ; y A ) ; u B ( x B ; y B ) Initial endowments: ( ¯ x A ; ¯ y A ; ¯ x B ; ¯ y B ) Allocation: ( x A ; y A ; x B ; y B ) Ani Guerdjikova Fall 2007 172 / 181
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General Equilibrium Edgeworth-box 6 6 - 6 ° 6 ? 0 A 0 B x A y A x B y B r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IC A ¯ x A ¯ x B ¯ y A ¯ y B IC B ............................................................. Ani Guerdjikova Fall 2007 173 / 181
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General Equilibrium Individual Rationality De°nition An allocation ( x A ; y A ; x B ; y B ) is called individually rational if u A ( x A ; y A ) ° u A ( ¯ x A ; ¯ y A ) and u B ( x B ; y B ) ° u B ( ¯ x B ; ¯ y B ) Ani Guerdjikova Fall 2007 174 / 181
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General Equilibrium Pareto-Optimality De°nition An allocation ( x A ; y A ; x B ; y B ) is Pareto-optimal if it is feasible there is no other feasible allocation which would increase the utility of (at least) one consumer without decreasing the utility of any other consumer. Ani Guerdjikova Fall 2007 175 / 181
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General Equilibrium Finding Pareto-optimal allocations The solution of the following optimization problem is a Pareto-optimal allocation: max x A ; y A ; x B ; y B u A ( x A ; y A ) , s.t. u B ( x B ; y B ) ° u B ( ˆ x B ; ˆ y B ) ± ˆ
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Unformatted text preview: y B ) ± ˆ u B ( B derives a utility of at least ˆ u B ) and x A + x B = ¯ x A + ¯ x B y A + y B = ¯ y A + ¯ y B ( feasibility of the allocation ( x A ; x B ; y A ; y B ) ) Solution: The contract curve is the geometric place of all Pareto-optimal allocations. Ani Guerdjikova ECON 313 Fall 2007 176 / 181 General Equilibrium Example u A ( x A ; y A ) = x A y A u B ( x B ; y B ) = 4 x B + y B 6 6-6 & 6 ? A B x A y A x B y B IC A q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¯ x A ¯ x B ¯ y A ¯ y B @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ l l l l l l l l l l l l & & & & & & & & & & & IC B CC Ani Guerdjikova ECON 313 Fall 2007 177 / 181...
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