Question

Suppose a natural gas distribution company has capital investments of $8 million and a capital cost r of 10%. The firm’s operating, billing, and maintenance costs are $200,000. The firm buys natural gas at the city gate price of $5/MCF to sell to its customers. The firm distributes gas to those customers through its existing pipeline network at close to zero marginal cost.

The firm faces the following (inverse) demand by customer type (recall that these are average demand per customer, so at any given price you have to multiply quantity by the number of customers to get total quantity demanded in that customer group):

Residential (10,000 customers): P = 50 − 5*q

Commercial (1,000 customers): P = 50 − q

Industrial (100 customers): P = 20 − 1/100 * q

Calculate the two-part tariff if the firm charges each customer the same two-part tariff and charges P = MC as the variable charge. What is the deadweight loss in this case?

Answer 1

1) The inverse demand functions for residential, commercial and industrial customers are given as follows,

P = 50-5q

P= 50-q & P = 20 -q/100

Therefore Total Revenue for all three consumer categories will be as follows,

TR₁ = Pq = 50q- 5q² ,

TR₂ = Pq = 50q -q² & TR₃ = Pq = 20q - q² /100

Therefore Marginal Revenue for residential customers will be d(TR₁)/dq = 50 - 10q

Similarly Marginal cost for commercial and industrial customers will be 50 - 2q and 20- q/50 respectively.

Under equilibrium condition,

           Marginal Revenue equals to Marginal Cost i.e., MR = MC

From the given condition, marginal cost is equal to 0 therefore,

50-10q = 0 or q = 5 for residential customers

For commercial customers, 50-2q = 0 or q = 25 and for industrial customers, 20-q/50 = 0 or q = 1000

Thus here average quantity demanded by each category of customer is different i.e., 5, 25 and 1000.

Therefore, two-part tariff will be, P= 50-q =50 -25 = 25 and P = 20-q/100 = 20- 1000/100 = 10

i.e., P = (25,10)

2) Total quantity demanded by residential customers = q * total number of customers = 5 * 10,000 = 50,000

Total quantity demanded by commercial customers = q * total number of customers = 25 * 1,000 = 25,000

and total quantity demanded by industrial customers = 1000 * 100 = 100,000

Deadweight loss = 1/2 * change in price * change in quantity demanded

At price P = 25, Average quantity demanded by industrial customers is

25 = 20- q/100 or. q = -500 i.e, q = 0 since quantity demanded cannot be negative.

Thus at, price P =25, total quanity demanded = 50,000 + 25,000 -0 = 75,000

when P = 10,

Average quantity demanded by residential customers is,

10 = 50- 5q or, q= 8 and Average quantity demanded by commercial customers is,

10= 50- q or q = 40

Thus at P= 10, total quantity demanded by customers is

    8* 10,000 + 40 * 1,000 + 100,000 = 220,000

Therefore, change in quantity demanded = 220,000 - 75,000 = 145,000

Therefore, deadweight loss in this case = 1/2 * (25-10) * 145,000 = $108,75,000