33. Welfare - Solutions

# 33. Welfare - Solutions - Chapter 33 NAME Welfare...

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Chapter 33 NAME Welfare Introduction. Here you will look at various ways of determining social preferences. You will check to see which of the Arrow axioms for ag- gregating individual preferences are satisFed by these welfare relations. You will also try to Fnd optimal allocations for some given social welfare functions. The method for solving these last problems is analogous to solving for a consumer’s optimal bundle given preferences and a budget constraint. Two hints. Remember that for a Pareto optimal allocation inside the Edgeworth box, the consumers’ marginal rates of substitution will be equal. Also, in a “fair allocation,” neither consumer prefers the other consumer’s bundle to his own. Example: A social planner has decided that she wants to allocate income between 2 people so as to maximize Y 1 + Y 2 where Y i is the amount of income that person i gets. Suppose that the planner has a Fxed amount of money to allocate and that she can enforce any income distribution such that Y 1 + Y 2 = W ,whe re W is some Fxed amount. This planner would have ordinary convex indi±erence curves between Y 1 and Y 2 and a “budget constraint” where the “price” of income for each person is 1. Therefore the planner would set her marginal rate of substitution between income for the two people equal to the relative price which is 1. When you solve this, you will Fnd that she sets Y 1 = Y 2 = W/ 2. Suppose instead that it is “more expensive” for the planner to give money to person 1 than to person 2. (Perhaps person 1 is forgetful and loses money, or perhaps person 1 is frequently robbed.) ²or example, suppose that the planner’s budget is 2 Y 1 + Y 2 = W . Then the planner maximizes Y 1 + Y 2 subject to 2 Y 1 + Y 2 = W . Setting her MRS equal to the price ratio, we Fnd that Y 2 Y 1 =2 . So Y 2 =4 Y 1 . Therefore the planner makes Y 1 = W/ 5and Y 2 =4 W/ 5. 33.1 (2) One possible method of determining a social preference relation is the Borda count , also known as rank-order voting. Each voter is asked to rank all of the alternatives. If there are 10 alternatives, you give your Frst choice a 1, your second choice a 2, and so on. The voters’ scores for each alternative are then added over all voters. The total score for an alternative is called its Borda count. ²or any two alternatives, x and y , if the Borda count of x is smaller than or the same as the Borda count for y ,then x is “socially at least as good as” y . Suppose that there are a Fnite number of alternatives to choose from and that every individual has complete, reﬂexive, and transitive preferences. ²or the time being,

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410 WELFARE (Ch. 33) (a) Is the social preference ordering deFned in this way complete? Yes. Reﬂexive?
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## This note was uploaded on 07/16/2010 for the course ECON 21 taught by Professor Johng.sessions during the Summer '09 term at Dartmouth.

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33. Welfare - Solutions - Chapter 33 NAME Welfare...

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