A.Welfare
Assume there exists a certain amount of money, X, that can be allocated across three individuals, George, John and Paul. A typical allocation would be (xG, xJ, xP) with xG + xJ + xP ≤ X. The utility functions are,
uJ(xJ ) = xJ

uG(xG) = xG

uP(xP ) = xP

1. Explain which allocation would maximize the classical utilitarian welfare function?
2. Explain which allocation would maximize the Rawlins welfare function?
3. Discuss which allocation would maximize the welfare function, W (x) = uG(xG) · uJ(xJ) · uP(xP) (the product of the utilities)?
4. Discuss how would your answers to parts a), b) and c) change (if it does at all) if the utility functions were,

uG(xG) = 4xG
uP(xP) = 3xP
uJ(xJ ) = xJ

Q2 Externalities

Suppose A and B are runners competing for a gold medal in the 100 meter dash. Both can take steroids to improve their speed. Let x and y indicate the quantities of steroids used by the A and B. If neither party takes any steroids each wins the race with probability 1/2. If at least one of x and y is non zero, the probability that A wins the race is (x/x+y) and the probability that B wins the race is(y/x+y).  The value of winning the race is 100 and the cost, inclusive of the adverse effects on the runner's own health, is 1 per unit of steroids consumed. Thus A's objective function is (100x/(x+y)) - x and B's objective function is (100y/(x+y)) - y.

1. Determine the Nash Equilibrium values of x and y for this game if A and B make their choice regarding steroids simultaneously.
2. Is the equilibrium efficient? If not, state an outcome that Pareto dominates it.
3. Discuss the nature of the externality problem in this scenario.