Consider a rent-seeking game with N ≥ 2 contestants. The effort for person i is denoted by xi for i = 1; ... ; N. The cost per unit of effort is C. All contestants are identical. They value the rent at V and each contestant can win the prize with a probability equal to their effort relative to the total effort of all contestants. Thus the payoff function of person i exerting effort xi is given by Exercise 11.10 Three firms have applied for the franchise to operate the cable TV system during the coming year. The annual cost of operating the system is $250 and the demand curve for its services is P = 500 - Q, where P is the price per subscriber per year and Q is the expected number of subscribers. The franchise is assigned for only one year, and it allows the firm with the franchise to charge whatever price it chooses. The government will choose the applicant that spends the most money lobbying the government members. If the applicants cannot collude, how much will each spend on lobbying? Note that the cost must be paid whether or not the prize is obtained. A Nash equilibrium is a lobbying effort for each contestant such that nobody would want to alter their expenditure given that of the other competitors. a. Find the derivative of player i’s expected utility and then set it equal to zero. Draw the resulting best-response function for player i when the N - 1 each of the others chooses x. b. In a symmetric equilibrium it must be the case that all contestants choose the same effort xi = x. Using this symmetry condition and the best-response function in part a, show that the equilibrium outcome is Show that the total cost and that the fraction of the rent that is dissipated (i.e., total cost relative to the value of the prize) is an increasing function of N. c. Suppose that there are four contenders (N = 4), the value of the prize is V = 20;000, and the cost of effort is C = 5; 000. What is the equilibrium level of rent-seeking activity x? What is the fraction of rent dissipation? d. Suppose that the cost of rent-seeking effort reduces from 5,000 to 2,500 with four competitors and a prize of 20,000. What is the impact on the common equilibrium level of rent-seeking activity? Does it affect the fraction of rent dissipation? Why or why not?
Consider a rent-seeking game with N ≥ 2 contestants. The effort for person i is denoted by xi for i = 1; ... ; N. The cost per unit of effort is C. All contestants are identical. They value the rent at V and each contestant can win the prize with a probability equal to their effort relative to the total effort of all contestants. Thus the payoff function of person i exerting effort xi is given by
Exercise 11.10
Three firms have applied for the franchise to operate the cable TV system during the coming year. The annual cost of operating the system is $250 and the demand curve for its services is P = 500 - Q, where P is the price per subscriber per year and Q is the expected number of subscribers. The franchise is assigned for only one year, and it allows the firm with the franchise to charge whatever price it chooses. The government will choose the applicant that spends the most money lobbying the government members. If the applicants cannot collude, how much will each spend on lobbying?
Note that the cost must be paid whether or not the prize is obtained. A Nash equilibrium is a lobbying effort for each contestant such that nobody would want to alter their expenditure given that of the other competitors.
a. Find the derivative of player i’s expected utility and then set it equal to zero. Draw the resulting best-response function for player i when the N - 1 each of the others chooses x.
b. In a symmetric equilibrium it must be the case that all contestants choose the same effort xi = x. Using this symmetry condition and the best-response function in part a, show that the equilibrium outcome is
Show that the total cost 
and that the fraction of the rent that is dissipated (i.e., total cost relative to the value of the prize) is an increasing function of N.
c. Suppose that there are four contenders (N = 4), the value of the prize is V = 20;000, and the cost of effort is C = 5; 000. What is the equilibrium level of rent-seeking activity x? What is the fraction of rent dissipation?
d. Suppose that the cost of rent-seeking effort reduces from 5,000 to 2,500 with four competitors and a prize of 20,000. What is the impact on the common equilibrium level of rent-seeking activity? Does it affect the fraction of rent dissipation? Why or why not?
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