slides_03_SWFs.pdf - Econ 473 Class 3 Theory and...

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Econ 473 Class 3 Theory and Measurement
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Next Economic theory and the measurement of inequality: I Social Welfare Functions I The leaky bucket I Inequality aversion and risk aversion I The Atkinson index: an SWF-based measure of inequality I A discussion of utilitarianism I But first: An important aside.
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A very important consideration: The size of the cake I (Almost) all our measures will ignore the size of the cake. I Obviously, the size of the cake matters for welfare. I Therefore, comparisons can only be made assuming the size of the cake is constant. I Some measures do lend themselves to comparisons across varying cake sizes. They need to be explicitly designed for this. Hence the importance of theory in measurement.
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Social Welfare Functions
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Social Welfare Function Often, we would like to I compare the desirability of different distributions or I determine an optimal distribution or an optimal policy, when policies have distributional implications. Example: optimal income taxes on top earners.
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Social Welfare Function Often, we would like to I compare the desirability of different distributions or I determine an optimal distribution or an optimal policy, when policies have distributional implications. Example: optimal income taxes on top earners. For this, we can use a Social Welfare Function . I A mathematical object that assigns values to society’s evaluation of all possible “states”. I What are relevant states? For example an income distribution. I No need to worry about where the state comes from: just apply the measure to it. I Allows comparisons even when there is no Lorenz dominance. I Can also use it to obtain a theory-based inequality measure.
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A Social Welfare Function Let y iA be person i ’s “economic position” (e.g. income, adjusted income, consumption) in state A . Features of a SWF: 1. individualistic and non-decreasing: W A = W ( y 1 A , ... y iA , ... y NA ) and y iB y iA implies (ceteris paribus) W B W A . If any income increases, W cannot decline.
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A Social Welfare Function Features of a SWF: 1. individualistic and non-decreasing: W A = W ( y 1 A , ... y iA , ... y NA ) and y iB y iA implies (ceteris paribus) W B W A . If any income increases, W cannot decline. 2. symmetric: W ( y 1 A , y 2 A , ... y NA ) = W ( y 2 A , y 1 A , ... y NA ) = W ( y NA , y 2 A , ... y 1 A )
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A Social Welfare Function Features of a SWF: 1. individualistic and non-decreasing: W A = W ( y 1 A , ... y iA , ... y NA ) and y iB y iA implies (ceteris paribus) W B W A . If any income increases, W cannot decline. 2. symmetric: W ( y 1 A , y 2 A , ... y NA ) = W ( y 2 A , y 1 A , ... y NA ) = W ( y NA , y 2 A , ... y 1 A ) 3. additive: W ( y 1 A , y 2 A , ... y NA ) = N X i =1 U i ( y iA ) This strengthens the first assumption, and rules out spillovers. It implies that W /∂ y 1 A is independent of all other y iA .
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A Social Welfare Function Features of a SWF: 1. individualistic and non-decreasing 2. symmetric 3. additive If all three hold, W ( y 1 A , y 2 A , ... y NA ) = N X i =1 U ( y iA ) , where U ( y i ) is the same for all i and increases in y i .
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