(Optional—requires calculus) Recall the political campaign advertising example from Section 1.C…

(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

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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?

 

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