Two-Part Tariff Problem
Suppose that each of a firm’s customers has the following demand curve: P = 20 – 2Q.
Suppose also that the firm’s total cost function is TC = 8Q.
The firm is considering three pricing strategies.
Strategy 1: A single per unit fee. (No entrance fee)
Strategy 2: An entrance fee with no per unit charge.
Strategy 3: An entrance fee and a per unit fee equal to marginal cost.
I. Strategy 1: A single per unit fee. (No entrance fee)
a. Determine the firm’s total revenue function. (Note that this function is actually the total
revenue from each customer, since the demand curve was for each customer.)
b. Determine the marginal revenue function.
c. Determine the marginal cost function.
d. Set marginal revenue equal to marginal cost to determine this strategy’s profit-maximizing
output level.
e. Determine the profit-maximizing price using this strategy.
f. Determine the total revenue at the price and quantity found in parts d and e above.
g. Determine the total cost per customer at the output level in part d above.
h. Determine the profit that the firm will make from each customer.
II. Strategy 2: An entrance fee with no per unit charge.
a. Use the demand curve equation to determine how much each customer will purchase if there
is no per unit charge (that is, P = 0).
b. Determine the consumer surplus that the firm can charge each customer as an entrance fee.
c. What is the total cost per customer at the quantity consumed by each customer in part a.
d. Determine the profit that the firm will make from each customer. (Remember that the total
revenue per customer will be the entrance fee since there is no per unit charge.)
III. Strategy 3: An entrance fee and a per unit fee equal to marginal cost.
a. What is the marginal cost and therefore the price charged per unit based on this strategy?
b. At the price found in part a, what quantity will each customer purchase?
c. Determine the firm’s revenue from the per unit charge, using the answers to parts a and b.
d. Determine the consumer surplus that the firm can charge each customer as an entrance fee.
e. Determine the total revenue from each customer using the answers to parts c and d.
f. Determine the total cost at the quantity found in part b.
g. Determine the profit that the firm will make from each customer.
IV. Which of the three strategies will provide this firm with the greatest profit per customer?
I. Strategy 1: A single per unit fee (No entrance fee)
a. To determine the firm's total revenue function, we need to multiply the price (P) by the quantity demanded (Q). The demand curve given is P = 20 - 2Q. We substitute this into the revenue function: R = P * Q = (20 - 2Q) * Q = 20Q - 2Q^2.
b. The marginal revenue (MR) function is the derivative of the total revenue function with respect to quantity (Q). We differentiate the total revenue function: MR = dR/dQ = 20 - 4Q.
c. The marginal cost (MC) function is given as TC = 8Q. The marginal cost is the derivative of the total cost function with respect to quantity: MC = dTC/dQ = 8.
d. To determine the profit-maximizing output level, we set MR equal to MC: 20 - 4Q = 8. Solving for Q, we find Q = (20 - 8) / 4 = 3.
e. To determine the profit-maximizing price, we substitute the quantity back into the demand curve: P = 20 - 2(3) = 20 - 6 = 14.
f. The total revenue at the price and quantity found in parts d and e is obtained by substituting the values into the revenue function: R = (20 - 2Q) * Q = (20 - 2(3)) * 3 = 54.
g. The total cost per customer at the output level in part d is given as TC = 8Q. Substituting the quantity, we find TC = 8(3) = 24.
h. To determine the profit per customer, we subtract the total cost from the total revenue: Profit = R - TC = 54 - 24 = 30.
II. Strategy 2: An entrance fee with no per unit charge.
a. When there is no per unit charge (P = 0), we can substitute this into the demand curve P = 20 - 2Q: 0 = 20 - 2Q. Solving for Q, we find Q = 10.
b. The consumer surplus that the firm can charge each customer as an entrance fee is obtained by substituting the quantity into the demand curve: CS = 20 - 2(10) = 20 - 20 = 0.
c. The total cost per customer at the quantity consumed by each customer in part a is given as TC = 8Q. Substituting the quantity, we find TC = 8(10) = 80.
d. Since there is no per unit charge, the total revenue per customer will be equal to the entrance fee: TR = 0.
The firm does not make any profit per customer under this strategy.
III. Strategy 3: An entrance fee and a per unit fee equal to marginal cost.
a. The marginal cost is given as MC = 8. Therefore, the price charged per unit based on this strategy is equal to the marginal cost: P = MC = 8.
b. To determine the quantity each customer will purchase, we substitute the price into the demand curve: 8 = 20 - 2Q. Solving for Q, we find Q = (20 - 8) / 2 = 6.
c. The firm's revenue from the per unit charge is obtained by multiplying the price (P) by the quantity (Q): R = P * Q = 8 * 6 = 48.
d. The consumer surplus that the firm can charge each customer as an entrance fee is obtained by substituting the quantity into the demand curve: CS = 20 - 2(6) = 20 - 12 = 8.
e. The total revenue from each customer is the sum of the revenue from the per unit charge and the entrance fee: Total Revenue = R + CS = 48 + 8 = 56.
f. The total cost at the quantity found in part b is given as TC = 8Q. Substituting the quantity, we find TC = 8(6) = 48.
g. To determine the profit per customer, we subtract the total cost from the total revenue: Profit = Total Revenue - TC = 56 - 48 = 8.
IV. To determine which of the three strategies will provide the greatest profit per customer, we compare the profit per customer obtained in each strategy:
- Strategy 1: $30
- Strategy 2: $0
- Strategy 3: $8
Therefore, Strategy 1 provides the firm with the greatest profit per customer.