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Unformatted text preview: Aggregation (Part 2) Aggregation Why we care Properties of Aggregate Demand Social Welfare Aggregation (Part 2) September 29, 2011 Aggregation (Part 2) Aggregation Why we care Properties of Aggregate Demand Social Welfare What is it? I consumers with preferences & i Choice set X R L + Price vector p 2 R L + Can we generate a "representative consumer" as a concise description of markets for ... Predicting market outcomes (Positive Representative Agent) Evaluating Welfare (Normative Representative Agent) Aggregation (Part 2) Aggregation Why we care Properties of Aggregate Demand Social Welfare Basic Ideas Let us just aggregate together demand by x ( p , w 1 , ..., w I ) = x ( p , w ) = I i = 1 x i ( p , w i ) w = I i = 1 w i tracks wealth distribution for us Aggregation (Part 2) Aggregation Why we care Properties of Aggregate Demand Social Welfare Basic Ideas De&nition x ( p , w ) satis&es the Weak Axiom if p & x ( p , w ) w and x ( p , w ) 6 = x ( p , w ) implies p & x ( p , w ) > w for any ( p , w ) and ( p , w ) So for any ( w 1 , ..., w I ) and ( w 1 , ..., w I ) such that I i = 1 w i = w and I i = 1 w i = w we have p & x ( p , w ) = I i = 1 p & x i ( p , w i ) I i = 1 w i = w p & x ( p , w ) = I i = 1 p & x i ( p , w i ) > I i = 1 w i = w Aggregation (Part 2) Aggregation Why we care Properties of Aggregate Demand Social Welfare Basic Ideas De&nition x ( p , w ) satis&es the Weak Axiom if p & x ( p , w ) w and x ( p , w ) 6 = x ( p , w ) implies p & x ( p , w ) > w for any ( p , w ) and ( p , w ) The problem is that some agents might have p & x ( p , w i ) w i and others might have p & x ( p , w i ) w i and so naively adding up doesnt work Another way of putting it: As we go from ( p , w ) to ( p , w ) , we know WA , Compensated law of demand. Consumers "disagree" about this ... [Picture 4.C.1 p. 110] Aggregation (Part 2) Aggregation Why we care Properties of Aggregate Demand Social Welfare Uncompensated Law of Demand De&nition x i ( p , w i ) satis&es the Uncompensated Law of Demand (ULD) if ( p & p ) ( x i ( p , w i ) & x i ( p , w i )) for any p , p and &xed w i Basically, wealth e/ects are not too extreme We think of "most preferences" as satisfying ULD Alternate: D p x i is negative semide&nite, then x i satis&es ULD Aggregation (Part 2) Aggregation Why we care Properties of Aggregate Demand Social Welfare...
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