100 commuters need to use a bridge. They all have big SUVs because driving big cars gives them a sense of prestige. But heavy cars also cause damage to the bridge. If'w' is the weight of the car, the benefit from driving the car is given by 5wand the cost is given by (1/2)w^2. The damage caused to bridge by driving a car of weight wis (2/3)w^3.

a.What weight of cars will be chosen in competitive equilibrium?

b.What is the social optimum?

c.If you have to design a toll for using the bridge, how much would you impose if you want the drivers to choose the social optimum?

Suppose now tastes have changed and the drivers no longer care about the size (weight) of the car they drive. They are more concerned about the cost of driving which in turn depends upon the number of axles present in the car. The petrol costs per 100 km driven is (1/6)A^2, where 'A'is the number of axles. The government re-estimates the damage to the bridge from driving and finds that the damage per 100km driven is (20/A). The pollution caused for a 100km trip is (1/3)A^2.

a.What is the socially optimal number of axles?

b.How can the government enforce this social optimum?