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B suppose all individuals live in community 2 then we

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(b) Suppose all individuals live in community 2, then we know that ¯ y 2 = 0 . 5 (4) Consider the poorest person with income equal to 0. His utility from staying in community 2 is U = (1 + 0) (1 + 0 . 5) - c 1 = 1 . 5 - c 1 - Δ (5) If he moves to community 1, mean income be equal to zero, i.e. ¯ y 1 = 0 and the utility in community 1 will be: U = (1 + 0) (1 + 0) - c 1 = 1 - c 1 (6) The pay-off in community 1 is higher if Δ > 0 . 5.
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(c) Consider two individuals with y l < y h . Assume y h lives in 1 and y l lives in 2 with ¯ y 1 < ¯ y 2 . We will show that this assumption will lead to a contradiction. Consider the individual with income y l . The utilities from living in community 1 and 2 are: U ( y l , ¯ y 2 ) = (1 + y l ) (1 + ¯ y 2 ) - c 2 = 1 + y l + ¯ y 2 + y l ¯ y 2 - c 1 - Δ (7) U ( y l , ¯ y 1 ) = 1 + y l + ¯ y 1 + y l ¯ y 1 - c 1 for this to be an equilibrium, we need U ( y l , ¯ y 2 ) > U ( y l , ¯ y 1 ) (8) which implies that ¯ y 2 + y l ¯ y 2 - Δ > ¯ y 1 + y l ¯ y 1 (9) and similarly for individual h , we need U ( y h , ¯ y 2 ) < U ( y h , ¯ y 1 ) (10) which implies ¯ y 2 + y h ¯ y 2 - Δ < ¯ y 1 + y h ¯ y 1 (11) how equation (9) and (11) imply that y l y 2 - ¯ y 1 ) > ¯ y 1 - ¯ y 2 + Δ > y h y 2 - ¯ y 1 ) (12) which implies that y l > y h which is a contradiction. (d) Let y * denote the household that is indifferent between living in community 1 and 2.
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