There are two routes for driving from A to B. One is a freeway, and the other consists of local roads. The benefit of using the freeway is constant and equal to 1.8, irrespective of the number of people using it. Local roads get congested when too many people use this alternative, but if not enough people use it, the few isolated drivers run the risk of becoming victims of crimes. Suppose that when a fraction x of the population is using the local roads, the benefit of this mode to each driver is given by
1 + 9x * 10x^2
(a) Draw a graph showing the benefits of the two driving routes as functions of x, regarding x as a continuous variable that can range from 0 to 1.
(b) Identify all possible equilibrium traffic patterns from your graph in part (a). Which equilibria are stable? Which ones are unstable? Why?
(c) What value of x maximizes the total benefit to the whole population?