One and Two are competing for a gold medal in the 100-meter dash. Both can take steroids to enhance their speed. If neither takes steroids, then the probability of winning is 50% for both. Let x and y be quantities of steroids used by One and Two. The probability that One wins the race is x/(x+y), and the probability that Two wins the race is y/(x+y). The value of winning the race is 100,000, and the cost, inclusive of adverse effects on the runner's own health, is 100 per unit of steroids consumed.
a. Given y, find the value of x that maximizes One's net value: [100000x/(x+y)-100x].
b. Find the equilibrium values of x and y.
c. Explain the nature of the externality problem and show that the equilibrium is not Pareto optimal.
d. Would these runners embrace an institution that limited their ability to use steroids?
e. What happens if steroids get banned, their black market price rises, and the cost to the athletes is now 1000 per unit?
f. Show that the incentive for an institution grows with a higher number of athletes (n=10 for example). Will the consumption of steroids by each runner increase?