Person 1 and person 2 are the only two residents of an economy. Person i (where i is either 1 or 2) has the utility function Here, Yi is person i’s income and β is parameter between 0 and 1. Assume that the social welfare function is where α is a parameter between 0 and 1. Initially, person 1’s income is 1 and person 2’s income is  2. a) Express W in terms of Y1 and Y2. If a social indifference curve shows all the pairs (Y1, Y2) that yield the same value of W, what woul

Person 1 and person 2 are the only two residents of an economy. Person i (where i is either 1 or 2) has the utility function

Here, Y_i is person i’s income and β is parameter between 0 and 1. Assume that the social welfare function is

where α is a parameter between 0 and 1. Initially, person 1’s income is ₁ and person 2’s income is  ₂.

a) Express W in terms of Y₁ and Y₂. If a social indifference curve shows all the pairs (Y₁, Y₂) that yield the same value of W, what would a social indifference curve look like if it were drawn in the (Y₁, Y₂) quadrant? Find an algebraic expression for the slope of a social indifference curve.

b) Imagine that income redistribution is costless, in the sense that the economy can reach any pair (Y₁, Y₂) that satisfies the condition

Draw a graph of the attainable pairs in the (Y₁, Y₂) quadrant. This set is called the “utility possibility frontier.” Using this frontier and the social indifference curves, find the best attainable income distribution. Show that this distribution is the same for every initial distribution satisfying the condition

Where  is a constant. Show that the best attainable income distribution is income equality

c) Now imagine that income redistribution is costly, in the sense that taking $1 away from one person allows k dollars to be given to the other, where 0

i) If k ≤ ₂/₁ ≤ 1/k, the optimal income redistribution policy is to do nothing.

ii) If ₂/_{1 2} is equal to kY₁; and if ₂/₁ > 1/k, the optimal policy is to redistribute income from person 2 to person 1 until Y₁ is equal to kY₂

d) Finally, imagine that income redistribution is costly, in the sense that taking z dollars away from one person allows ƒ(z) dollars to be given to the other person, where the function  has these properties:

That is, taking the first dollar from one person allows $1 to be given to the other person, but each successive dollar taken from the first person allows successively smaller amounts to be given to the second person. Show that income equality is optimal only if the initial income distribution is completely equal. Show that if the initial income distribution is not completely equal, some redistribution of income from rich to poor is always optimal.