Course: ECO 3553

Consider the following game. We have a group consisting of N rich people and one poor person. The rich have an income of y_R and the poor have an income of y_P . The rich in this economy are altruistic to the poor. Each richi{1,2,...,N} can give a gift of d_i to the poor. In such a setting, the rich man's utility depends on his net income, x_R = y_R - d_i and the poor man's net incomexp=yp+j=1ndj . Suppose that the utility of each rich person can be represented by U(xR;xP)=ln(xR)+ln(xP)

(a) How much will a rich person donate to the Nash equilibrium of this game? b) Is the individual donation increasing or decreasing in the number of players? c) Is the total donation received by the poor player increasing or decreasing in the number of players? players? d) Is there a minimum income difference between the rich and poor for the rich to make a non-zero donation? rich to make a non-zero donation? Now assume that there is a central utilitarian planner who coordinates the donation decisions of the rich. of the rich. Suppose that this planner places no weight on the utility of the poor but has a choice function that depends on the utility of the N rich. This function is:sw(U1,U2...,UN)=i=1NUi

where Ui is the utility of rich i. e) How much of a gift will the planner impose on each rich person? f) Is there a minimum income difference between the rich and poor for the central planner to impose a non-zero donation? Compare this condition to the one found in (b). g) Is the donation amount greater when a central planner makes the decision? h) Is the decentralized charity in this example Pareto efficient?