A small country considers the building of a national highway system. There are?n?citizens in this country. Every citizen has the same wealth?w. Let G = miles of paved roads in the highway system. Each citizen derives a payoff u_i(G, c_i) = A_i?log(G) + w ? c_i?where A_i?is i's valuation of his marginal utility of the good. Let c_i?be i?s contribution/tax for the highway system. (For the purposes of this problem, assume that log is the natural logarithm function.) Finally, assume G = b_i?c_i?where b > 0 is the marginal impact of an extra dollar collected to build the highway.

A citizen?i?can belong to one of three groups, High intensity users, Medium intensity users, and low intensity users of the highway. Specifically,

A_H?>A_M?>A_L.

There are n_H,n_M,n_L?members of High, Medium and Low groups, respectively. Obviously n_H?+ n_M?+ n_L?= n.

(a) Find the equilibrium contributions/taxes c_H* , c_M* , and c_L* that would be assigned by a utilitarian social planner to each group. Be sure to state the planner?s maximization problem. Show all steps.

(b) Find the contributions c_H* , c_M* , and c_L* that would emerge in an equilibrium in which each individual is free to choose his/her own contribution. Again, be sure to state each individual?s optimization problem.

(c) Compare your answers in (a) and (b). Are contributions higher in the planner?s problem or in the voluntary solution? Briefly explain.

(d) Suppose that the median voter belongs to group H. The median voter chooses a uniform tax rate ? so that each individual?s contribution is c_i?= ?w (so that ?w should stand in for ci in the citizen?s payoff). Find ?.

(e) Compare the contribution c_i?that citizen i pays in part (b) with the contribution ?w that each citizen pays in part (d). Describe the trade offs in payoffs between voluntarism and compulsory taxation from each group?s point of view. Which scheme is preferred by each group and why? How does your answer depend on the values of A_H?, A_M?, and A_L?