(Contributing to a public good) Consider the model in this section when ui(c1, c2) is the sum of three parts: the amount c1 + c2 of the public good provided, the amount wi − ci person i spends on private goods, and a term (wi − ci)(c1 + c2) that reflects an interaction between the amount of the public good and her private consumption—the greater the amount of the public good, the more she values her private consumption. In summary, suppose that person i’s payoff is c1 + c2 + wi − ci + (wi − ci)(c1 + c2), or
Where j is the other person. Assume that w1 = w2 = w, and that each player i’s contribution ci may be any number (positive or negative, possibly larger than w). Find the Nash equilibrium of the game that models this situation. (You can calculate the best responses explicitly. Imposing the sensible restriction that ci lie between 0 and w complicates the analysis, but does not change the answer.) Show that in the Nash equilibrium both players are worse off than they are when they both contribute one half of their wealth to the public good. If you can, extend the analysis to the case of n people. As the number of people increases, how does the total amount contributed in a Nash equilibrium change? Compare the players’ equilibrium payoffs with their payoffs when each contributes half her wealth to the public good, as n increases without bound.