Externalities. Suppose we are trying to figure out the optimal congestion tax.

Suppose have the following:

PMB : Cost = 50050n

PMC : Cost = 100 + 50n

SMC : Cost = 100 + 100n

where PMB = private marginal benefit (or demand), PMC = private marginal cost and SMC = social marginal cost. Furthermore,

Cost is the trip cost and n is the number of vehicles on the road.

(a) Compute the private equilibrium number of cars (n^*) and cost (Cost^*)

(b) Compute the socially optimal equilibrium number of cars (n^*s) and cost (Cost^*s)

(c) Compute the socially optimal (pigouvian) tax.

(d) Suppose there is actually two demand curves that reflect different commuting patterns throughout the day, Cost l= 1000100n

and Cost h= 2000100n, where l is for low and h is for high. In this example, does the socially optimal tax vary with demand? Explain your answer. Draw a graph, if you would find it helpful.