Two construction companies are lobbying to obtain a share of work on repaving ci

Question

Two construction companies are lobbying to obtain a share of work on repaving city streets. The share of the project going to each firm depends on money contributions to the mayor's reelection fund. The project has value V in total. The mayor is somewhat biased in favor of firm 1, which is run by his niece. If firm 1 contributes x1 and firm 2 contributes x2, then the shares of the project going to firms 1 and 2 are s1 and s2, respectively, where
s1(x1,x2) = (2x1)/(2x1 + x2) and s2(x1,x2) = (x2)/(2x1 + x2);
The payoff to firm i is
pi(x1,x2) = si(x1, x2)V - xi, i = 1, 2.
Calculate the Nash equilibrium contributions for each firm if they make contributions simultaneously.

Explanation / Answer

We have: s1(x1,x2) = (2x1)/(2x1 + x2) s2(x1,x2) = (x2)/(2x1 + x2) pi(x1,x2) = si(x1, x2)V - xi, i = 1, 2 Let's find the payoff function form firm 1. p1(x1,x2) = s1(x1, x2)V - x1 p1(x1,x2) = V*(2x1)/(2x1 + x2) - x1 We can set the partial derivative with respect to x1 to zero. Remember the quotient rule. V*[(2x1 + x2)*2 - (2x1)*2] - 1 = 0 V*[4x1 + 2x2 - 4x1] - 1 = 0 V*2x2 - 1 = 0 x2 = 1/(2V) Now, let's find the payoff function for firm 2 p2(x1,x2) = s2(x1, x2)V - x2 p2(x1,x2) = V*(x2)/(2x1 + x2) - x2 Again, set the partial derivative of payoff with respect to x2 equal to zero. V*[(2x1 + x2)-(x2)] - 1 = 0 V*2x1 - 1=0 x1 = 1/(2V) So, both firms make the same contribution even though they have different payoff functions.

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