Recall the political campaign advertising example from Section 1.C concerning parties L and R. In that example,

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Question:

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 + y) of the votes and R gets a share y/(x + 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

VL=x/x+ky and Vn=(ky/x+ky)-cy, where k>0 and c<0

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 = 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 advan- tage 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

x=ck/(c+k)^2 and y=k/(c+k)^2

(e) Let k = 1 in the equilibrium equations in part (d) 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 equilib- rium 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?