Monopoly pricing: MR = MC
Learning goals
After this page, you can:
- Explain why a single-price seller’s marginal revenue is below its price: to win one more customer, it must cut the price for every customer it already has.
- Find the best plan in two steps — the quantity where MR = MC, then the price read off the demand line — and compute profit, knowing that fixed cost changes profit but not the plan.
- Describe who a monopoly does not serve, and show that a competitive market would serve twice as many customers on a straight demand line.
The idea
In many rural counties, one cable company is the only wired internet a household can buy. It still does not charge $300 a month. Push the price too high and families switch to a phone hotspot, a satellite dish, or the library’s Wi-Fi. Having no rival does not remove the demand curve. It means the company gets to choose its point on it.
A monopoly is the only seller of a product that has no close substitute. More generally, a firm has market power when it can set its price above its marginal cost without losing all of its customers. In Week 4, a price taker could sell as much as it liked at the market price. A monopoly is the opposite case. It faces the whole downward-sloping demand line. If it wants more customers, it must lower its price.
Why marginal revenue is below price. Marginal revenue (MR) is the extra revenue from selling one more unit. For a price taker, MR equals the price. For a single-price seller — a firm that charges every customer the same price — it does not. Take a fictional cable company, Ridgeline Cable. It has 2,000 subscribers at $75 a month. To sign 100 more, it must cut the price to $72.50 — for everyone. The 100 new subscribers bring in $7,250. But the 2,000 existing subscribers now pay $2.50 less each, which is $5,000. Revenue rises by only $2,250. That is $22.50 per new subscriber, even though each one pays $72.50. The gap is the price cut on the customers you already had. (Week 6 shows how sellers try to avoid that cut by charging different prices to different customers.)
The MR line. On a straight demand line P = a − b·Q, marginal revenue is MR = a − 2b·Q. It starts at the same point as demand but falls twice as fast. Take this as a rule; no calculus is needed here. MR reaches zero halfway to where demand reaches zero. Past that point, selling more lowers revenue.
The two-step rule.
- Step 1 — the quantity. Keep adding customers while one more brings in more than it costs (MR above MC). Stop where MR = MC.
- Step 2 — the price. Go straight up from that quantity to the demand line. That is the highest price at which customers will buy that many. Do not read the price off the MR line or the MC line — that is the most common mistake.
On a straight demand line, the best price is always halfway between a (the price where nobody buys) and c (marginal cost).
Profit, and what fixed cost does. Profit = (P − MC) × Q − F, where fixed cost (F) is the cost that does not change with the number of customers, like keeping the network running. Fixed cost does not change MR or MC, so it cannot move the best plan. A higher fixed cost lowers profit and nothing else. Whether to stay in business at all is the Week 4 question.
Not the revenue-maximizing price. Revenue is highest where MR = 0. But the customers you add past MR = MC bring in less than they cost to serve. A profit-maximizer stops earlier, at a higher price.
Who is not served. At its best plan Ridgeline serves 2,000 homes at $75. If the price were equal to marginal cost ($25), as competition would push it, 4,000 homes would buy. The 2,000 homes in between would happily pay more than $25 — more than it costs to serve them — but not $75. They go without. The value of those missing deals is the deadweight loss: trades that would help both sides but never happen. On a straight demand line, the competitive quantity is always twice the monopoly quantity.
The markup rectangle. On the chart, the shaded rectangle is (P − MC) × Q: the markup on each unit times the number of units. It is what the markup earns before fixed cost. Playbook P5 asks you to draw your own company’s version.
Demand: P = a − b·Q · Marginal revenue: MR = a − 2b·Q
Step 1: MR = MC → Q* = (a − c) ÷ (2b) · Step 2: P* = a − b·Q* = (a + c) ÷ 2
Profit = (P* − c) × Q* − F · Competitive benchmark: P = c → Qc = (a − c) ÷ b = 2 × Q*
Try it
Part 1: the cable company’s chart. The blue dot is a point on the demand line; the purple dot below it is MR at that quantity.
- Set the subscriber slider to 2,000 and press Sell 100 more subscribers. Read the arithmetic. Why do 100 new subscribers add only $22.50 each?
- Move the slider from 1,000 to 3,000. Watch the purple MR dot cross the orange MC line. Where does the verdict say “this is the best plan”?
- Raise fixed cost from $35,000 to $60,000. Does the best plan move? What happens to profit?
- Turn on Show the unserved households. Who are they, and how many?
Part 2: mark your own price. Press Use my numbers to bring in your company’s current price and marginal-cost estimate. (You get the marginal-cost estimate from the Lerner calculator on the markup and elasticity page.) Export the picture for Playbook P5.
Worked example
Problem. Ridgeline Cable (fictional) faces demand P = 125 − 0.025·Q, where Q is subscribers. Each subscriber costs $25 a month to serve. Keeping the network running costs $35,000 a month. What price earns the most, and how much does it earn?
Step 1. Why MR is below price. At 2,000 subscribers the price is 125 − 0.025 × 2,000 = $75, and revenue is $150,000. At 2,100 subscribers the price is $72.50, and revenue is $152,250. Revenue rises $2,250: the new subscribers bring $7,250, but existing ones pay $5,000 less. That is $22.50 per new subscriber, below the $25 they cost → profit would fall $250. Going the other way, from 1,900 to 2,000 subscribers adds $2,750 of revenue ($27.50 each, above $25) → worth it.
Step 2. The quantity. MR = 125 − 0.05·Q. Set MR = MC: 125 − 0.05·Q = 25, so Q* = 100 ÷ 0.05 = 2,000 subscribers.
Step 3. The price. Go up to the demand line: P* = 125 − 0.025 × 2,000 = $75. Check: halfway between 125 and 25.
Step 4. Profit. (75 − 25) × 2,000 − 35,000 = 100,000 − 35,000 = $65,000 a month. If fixed cost rose to $60,000, the plan would not change (2,000 at $75). Profit would fall to $40,000.
Step 5. Not the revenue peak. Revenue peaks where MR = 0: 125 − 0.05·Q = 0 gives 2,500 subscribers at $62.50. But each of those last 500 subscribers costs $25 and adds less than $25 of revenue. Profit is lower there.
Step 6. Who is not served. At P = MC = $25, demand gives Qc = 100 ÷ 0.025 = 4,000 subscribers — twice Q*. The 2,000 households in between would pay more than $25 but less than $75. They value the service above its cost and still go without. That is the deadweight loss, in words.
Check yourself
A cable provider has 2,000 subscribers at $75 a month. To sign 100 more, it must cut the price to $72.50 for everyone. How much extra monthly revenue do the 100 new subscribers bring in, net of the price cut?
With one price for everyone, the cut on existing customers is why marginal revenue is below price.
Those 100 extra subscribers would add $2,250 of monthly revenue and cost $25 each per month to serve. What should the provider do?
Sell one more unit only while marginal revenue is at least marginal cost.
Demand is P = 125 − 0.025·Q, so marginal revenue is MR = 125 − 0.05·Q. Marginal cost is $25. What is the profit-maximizing price?
Find the quantity where MR = MC, then go up to the demand line for the price.
The same provider (best plan: 2,000 subscribers at $75, marginal cost $25) sees its fixed network cost rise from $35,000 to $60,000 a month. What happens to its best plan?
Fixed cost changes profit, not the profit-maximizing price or quantity.
On the monopoly chart, the shaded rectangle runs from marginal cost up to the price, across the best number of subscribers. What does its area measure?
The markup rectangle is (P − MC) × Q, the money the markup earns before fixed costs.
Video
Video coming soon.