Q1 Welfare
Suppose 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 functionsare,
uG(xG) = xG
uJ(xJ ) = xJ
uP(xP ) = xP
a) Which allocation(s) would maximize the classical utilitarian welfare function?
b) Which allocation(s) would maximize the Rawlsian welfare function?
c) Which allocation(s) would maximize the welfare function, W (x) = uG(xG) · uJ(xJ) · uP(xP) (theproduct of the utilities)?
d) How would your answers to parts a), b) and c) change (if it does at all) if the utility functions were,
uG(xG) = 4xG
uJ(xJ ) = xJ
uP(xP) = 3xP
Q2 Externalities
A and B are runners competing for a gold medal in the 100 meter dash. Both can take steroids to enhance their speed. Let x and y denote quantities of steroids used by A and B. 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). If neither party takes any steroids each wins the race with probability 1/2. 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.
a) If A and B make their choice regarding steroids simultaneously, what is the Nash Equilibrium values of x and y for this game?
b) Is the equilibrium efficient? If not, state an outcome that Pareto dominates it.
c) Explain the nature of the externality problem in this scenario.
d) Would these runners embrace an institution that limited their ability to use steroids?