Question

Consider a utility providing water service as a natural monopoly to residents of a city. The market comprises ? identical households, each of which has an inverse demand function of ?(?) = 27,500 − 80,000? where ? is the number of megalitres (ML) of water demanded annually and ? is the price per megalitre (1 ML = 1,000m3). Letting ? denote total output in megalitres, inverse market demand is ?(?) = 27,500 − 0.8? and the annual total cost to the utility of providing water is ??(?) = 90,625,000 + 5,000? + 0.1? 2 . Thus, marginal revenue, average cost and marginal cost are, respectively, as follows: ??(?) = 27,500 − 1.6?; ??(?) = 90,625,000/? + 5,000 + 0.1?; and ??(?) = 5,000 + 0.2?.

a. Briefly explain why water service (treatment, transmission and distribution) is a natural monopoly within the city. Determine the value of ?. b. Show that the efficient price, consumption level and output level are ? ∗ = $9,500, ? ∗ = 0.225 and ? ∗ = 22,500, respectively. Calculate the deficit incurred by the utility if it charges ? ∗ . c. Show that the equilibrium price, consumption level and output level are ? ? = $17,500, ? ? = 0.125 and ? ? = 12,500, respectively, in the absence of regulation. Calculate the profit earned by the utility in equilibrium. d. 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 $10,625,000 per annum due to physical depreciation. Depreciation 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 ROI into its required (i.e. investment opportunity cost) and economic rent components.

Answer 1

a. The water service (treatment, transmission and distribution) is a natural monopoly within the city because it helps keeping the cost in check, and the monopolist can provide the whole demand of the city at without competitive prices.

b. In a Monopolist firm, equilibrium is reached when MR=MC, (P(w)= MR)

27500-0.8w= 5000+0.2w => w=22500

Now, p(w)= 27500-0.8*22500 => p=9500

q=w/100000= 22500/100000= 0.225

Now with p=$9500

TC(w)= 253,750,000 (putting value of w in TC(w))

TR(w)= w*p = 22500*9500= 213,750,000

Utility's deficit= TR-TC= -40,000,000

c. Under No regulation,

MR=MC=> 27500-1.6w= 5000+0.2w => w=12500

p= p(w)=27500-0.8*12500=> 17500

q= 0.125

Profit earned

TR(w)= w*p => 12500*17500 =>218750000

TC(W)= 168,750,000

Profit= TR-TC => 50,000,000.