A gym charges a membership fee plus a price per class; an amusement park charges admission plus a price per ride. With identical customers, Monopoly with Calculus and Consumer Surplus as an Integral show how to set both parts.
Key idea
With identical customers, the profit-maximizing two-part tariff sets the per-unit price equal to marginal cost and the entry fee equal to each customer's consumer surplus at that price. The firm captures the whole surplus, and output is efficient.
Why it works
Profit from one customer at unit price $p$ is the per-unit margin plus the largest fee she will pay, her consumer surplus: $$\pi(p) = (p - c)\,q(p) + CS(p).$$ Since $CS'(p) = -q(p)$ (raising the price removes a strip of surplus as wide as the quantity), $$\pi'(p) = q(p) + (p - c)\,q'(p) - q(p) = (p - c)\,q'(p),$$ which is zero at $p = c$. Any price above cost shrinks the surplus that the fee collects by more than it adds in margin; the difference is the deadweight-loss triangle.
Worked example: Setting the tariff
Each customer has $p = 20 - q$; MC is \$4.
- Per-unit price $= 4$, so each customer buys $q = 16$.
- Fee $= CS = \tfrac{1}{2}(16)(20 - 4) = 128$.
- Profit per customer: \$128.
Answer: \$4 per unit plus a \$128 fee.
Worked example: Compared with a single price
Same customers, but only one price per unit and no fee.
- $MR = 20 - 2q = 4$ gives $q = 8$ and $p = 12$.
- Profit per customer: $(12 - 4)(8) = 64$.
- The two-part tariff earns twice as much: it takes the whole triangle of surplus, where the single price gets a rectangle half its size and leaves a deadweight loss.
Answer: \$64, half of \$128.
The two-part tariff is efficient: every unit worth more than its cost to a customer gets made, so there is no deadweight loss. What changes is who keeps the surplus. Under competition the customers would keep the \$128 per head; here the firm takes it all through the fee. With 500 customers, profit is $500 \times 128 = \$64{,}000$.
Common mistake
Keeping the monopoly per-unit price and adding a fee. At $p = 12$ the customer's surplus is only $\tfrac{1}{2}(8)(8) = 32$, so the most profit is $64 + 32 = 96$, less than 128. Check: compare total profit (margin plus fee) at your price with profit at $p = MC$.
With customers who differ, the answer changes: a fee high enough to take a big user's surplus drives out small users, so the best tariff usually sets the per-unit price somewhat above marginal cost and the fee at the smallest customers' surplus. The identical-customer case shows the core logic.