Consider a public utility providing water service as a natural monopoly to residents of a city. The market (as defined by the city's boundaries) contains 100,000 identical customers, each of which has a demand function q(p) = 0.2875-80,000 where q(p) is the number of megalitres (ML) of water demanded annually and p is the price per megalitre (1 ML = 1,000 cubic metres). Letting ? denote total output in the market in megalitres, the total annual cost to the utility of providing water is TC(w) = 81,900,000 + 5,000w + 0.1w². You are encouraged to utilize Excel to aid in answering this question.

a) Very briefly explain what characteristic about water service renders it a natural monopoly within the city and why? Your answer should refer to the MES and demand for water services.

b) Show that the inverse market demand function is given by p(w)= 23,000 -0.8w and that the efficient price, consumption level and output level are P* = $8,600, q*= 0.18 and w* = 18,000, respectively.

c) Calculate the deficit incurred by the utility if it charges p*. Illustrate the efficient outcome and the deficit in an appropriate diagram.

d) If the market is left unregulated, show that the equilibrium price, consumption level and output level are
p^e= $15,000, q^e= 0.1 and w^e= 10,000, respectively. Calculate the profit earned by the utility in equilibrium. Illustrate the equilibrium outcome and profit in an appropriate diagram.

e) Suppose the market is regulated by the city's waterworks commission and assume its information about the market limits it to imposing average cost pricing. Show that the price, consumption level and output level are
p^a= $12,600, q^a= 0.13 and  w^a= 13,000, respectively, under average cost pricing regulation. Calculate the profit earned by the utility in equilibrium. Illustrate the regulated outcome and profit in an appropriate diagram.