The Smith and Jones families live next to each other. Each is planning to buy a car. We'll use S to stand for the amount the Smiths spend, and J for the amount the Joneses spend (both in thousands of dollars). The utility benefit the Smiths receive from their car is
40S + 600 S / J
And the utility benefit the Joneses receive from theirs is
40J + 600 (J / S)
Notice that both families care not only about the quality of the car they purchase, but also about how it compares to the other family's car. For these functions, the marginal benefit of spending more on a car for the Smiths is
40 + 600 / J
And for the Joneses
Spending more money on a car entails a utility cost, because it reduces the amount of other goods consumed. That cost is S2 for the Smiths and J2 for the Joneses. For these functions, the marginal cost of spending more on a car is 2S for the Smiths and 2J for the Joneses. Suppose they choose S and J simultaneously. Find a Nash equilibrium in which both families spend the same amount on their cars. What is the amount? Can they do better? Would they be happier if both spent more than in the Nash equilibrium, or less?