(20 marks) Consider a utility providing water service as a natural monopoly in a city. The market comprises ? identical households, each of which has an inverse demand function of ?(?) = 22,000 ? 70,000? where ? is the number of megalitres of water demanded annually and ? is the price per megalitre (1 megalitre = 1,000m3). Letting ? denote total annual output in megalitres, inverse market demand is ?(?) = 22,000 ? 0.7? and the utility's total cost is ??(?) = 60,000,000 + 4,000? + 0.05? 2 per annum. Thus, marginal revenue, average cost and marginal cost are, respectively, as follows: ??(?) = 22,000 ? 1.4?; ??(?) = 60,000,000/? + 4,000 + 0.05?; and ??(?) = 4,000 + 0.1?.

a. Very briefly explain why water service (treatment, transmission and distribution) is a natural monopoly within an urban area. Determine the value of n. b. Show that the efficient price, consumption level and output level are p* = $6,250, q' = 0.225 and w* = 22,500, respectively. Compute the deficit incurred by the utility if it charges p*. c. Show that the equilibrium price, consumption level and output level are pe = $13,600, q = 0.12 and we = 12,000, respectively, in the absence of regulation. Compute the profit earned by the utility in equilibrium. d. Show that the regulated price, consumption level and output level are p* = $8,000, q" = 0.2 and we = 20,000, respectively, if the utility is subject to a regulator that sets the price to maximize consumer surplus subject to enabling the utility to recover all its costs. Compute the profit earned by the utility under regulation. e. The total replacement value of the utility's infrastructure is $1,000,000,000. The infrastructure is financed by shareholders whom earn a return on investment (ROI), and it requires capital replacement at a rate of $10,000,000 per annum due to physical depreciation. Capital replacement and the required ROI are the only fixed costs to the utility; all other costs are variable. Determine the equilibrium rates of depreciation and ROI in percentage terms, and decompose that for ROI into its required (i.e. investment opportunity cost) and economic rent components