(Optional—requires calculus) Recall the political campaign advertising example from Section 1.C concerning parties L and R. In that example, when L spends $x million on advertising and R spends $y million, L gets a share x/(x 1 y) of the votes and R gets a share y/ (x 1 y). We also mentioned that two types of asymmetries can arise between the parties in that model. One party—say, R—may be able to advertise at a lower cost or R’s advertising dollars may be more effective in generating votes than L’s . To allow for both possibilities, we can write the payoff functions of the two parties as
These payoff functions show that R has an advantage in the relative effectiveness of its ads when k is high and that R has an advantage in the cost of its ads when c is low.
(a) Use the payoff functions to derive the best-response functions for R (which chooses y) and L (which chooses x).
(b) Use your calculator or your computer to graph these best-response functions when k = 1 and c = 1. Compare the graph with the one for the case in which k = 1 and c 5 0.8. What is the effect of having an advantage in the cost of advertising?
(c) Compare the graph from part (b), when k = 1 and c = 1 with the one for the case in which k = 2 and
c = 1. What is the effect of having an advantage in the effectiveness of advertising dollars?
(d) Solve the best-response functions that you found in part (a), jointly for x and y, to show that the campaign advertising expenditures in Nash equilibrium are
(e) Let k = 1 in the equilibrium spending-level equations and show how the two equilibrium spending levels vary with changes in c (that is, interpret the signs of dx dc and dy dc). Then let c = 1 and show how the two equilibrium spending levels vary with changes in k (that is, interpret the signs of dx dk and dy dk). Do your answers support the effects that you observed in parts (b) and (c) of this exercise?