A manufacturer promises to give a prize to one of two dealers who wins a sales contest. The value of the prize is V=10 to each dealer. The dealers compete by spending sales effort xi. The probability of winning is of the Tullock form: p1(x1,x2)= x1/(x1+x2) and p2(x1,x2)= x2/(x1+x2). Dealers have the same constant marginal cost of effort, c=1.
1. Write down firm 1's expected payoff from participating in the sales contest.
2. Find the best response of firm 1.
3. Are effort levels strategic substitutes or complements? Explain.
4. By analogy, write the best response of firm 2.
5. The firms are symmetric in all respects. Using the symmetry, find the equilibrium sales efforts, x1 and x2.
6. What is the extent of rent dissipation (that is, what fraction of the prize value V is spent in rent-seeking efforts)?
7. Would dealers choose to participate in the sales contest? What are their expected equilibrium payoffs from participation in the contest?
8. Sketch the best-response functions or use Excel to draw them. Mark the Nash equilibrium.