Monopoly Graph: Marginal Revenue, Profit and Deadweight Loss

One family owns the only hot spring on a snowy mountain. Every skier wants a soak, and nobody else can sell one. So why not charge a million dollars a soak? This is Module 4.2A, Monopoly: the graph, profit and deadweight loss. You'll see the price they pick, and who loses.

The family is a monopoly: the only seller, with no close substitutes. A barrier to entry keeps rivals out: here, owning a key resource, the spring. Patents and licenses, which are legal barriers, and huge start-up costs work too. That was Module 4.1.

As the only seller, the spring faces the whole market demand curve, which slopes down. To sell more soaks, it must cut its price. Its demand schedule: at $140, nobody soaks. At $130, skiers buy 1 soak a day; at $120, 2; down to $60 for 8.

Quick refresher: total revenue is price times quantity, and marginal revenue, MR, is the change in total revenue from one more unit. 1 soak brings in $130; 2 bring in $120 times 2, $240. So the 2nd soak's MR is $110.

But the 2nd soak sells for $120. Why only $110? Two effects. Quantity effect: the new soak brings in its price, plus $120. Price effect: to sell it, the first soak's price drops from $130 to $120, minus $10. Net: $110. By the 4th soak, the price effect is bigger: it sells for $100, but 3 earlier soaks each drop $10, minus $30. MR is $70. Only the first soak's MR equals its price. After that, MR is below the price, because selling one more means cutting the price on every unit.

So what just happened? One: selling one more soak means cutting the price on all of them. Two: the new soak adds its price; the cut takes some back. Three: so at the 4th soak, MR is $70, below the $100 price.

MR keeps falling: $50, $30, $10, then negative $10 for the 8th soak. Total revenue climbs while MR is positive, peaks at $490 with 7 soaks, then falls. The 7th adds $10, the 8th loses $10, so on the graph MR hits zero at 7: QR, the revenue-maximizing quantity.

Quick refresher from Unit 2, the total revenue test: if a price cut raises total revenue, demand is elastic; if it lowers it, inelastic; if it doesn't change, unit elastic.

Cut the price from $80 to $70: revenue rises, $480 to $490, so demand is elastic. From $70 to $60: revenue falls to $480, inelastic. At QR, where MR is zero: unit elastic.

So what just happened? One: revenue rose while MR was positive. Two: it peaked at $490 at 7 soaks, where MR hits zero: QR. Three: elastic to the left, unit elastic at QR, inelastic to the right.

Now the graph, in the order it's scored. Axes: Price, Cost up the side; Quantity along the bottom. Draw demand, D, down from $140. Draw MR from the same point, below D, falling faster and reaching the quantity axis at 7. Add marginal cost, MC: it dips, then rises.

Quick refresher from Module 3.5: make each unit whose MR is above its MC; stop where MR = MC. The 4th soak: MR $70, MC $55, so make it. The 5th: MR $50, MC $65, so skip it.

On the graph, each soak's MR and MC sit halfway between counts, so the lines cross at 4 soaks, at $60, between those numbers. That's QM, the profit-maximizing quantity; drop a dashed line down. Then go straight up from QM to demand, and across: PM is $100, the most skiers will pay for 4 soaks. That answers the million dollars: demand slopes down, so a sky-high price sells nothing. The spring picks QM, then the highest price that sells it. And $100 is above the $60 marginal cost: a monopoly's price sits above MC.

So what just happened? One: MR meets MC at 4 soaks: that's QM. Two: go up to demand: PM is $100, not a million. Three: the $100 price is above the $60 MC.

Exam trap: reading the price at the MR and MC crossing, $60. On recent AP exams, about one student in three lost the price point. Always go up to demand. Exam trap: MR drawn on top of demand, or stopping short of the axis. MR starts where D starts, stays below it, and reaches the quantity axis.

Compare a competitive market with the same costs: it would produce where demand meets MC, 6 soaks at $80. The monopoly makes less and charges more.

Is it profitable? Add average total cost, ATC: U-shaped, with MC rising through its lowest point.

Quick refresher: ATC is total cost divided by quantity; profit is price minus ATC, times quantity. Total cost for 4 soaks is $280, so ATC is $280 divided by 4: $70.

Profit: $100 minus $70 is $30 a soak, times 4 soaks, $120 a day. On the graph, it's the rectangle from $70 up to $100, out to QM.

So what just happened? One: ATC at QM is $70. Two: each soak earns $100 minus $70, $30. Three: times 4 soaks, that's $120 of profit, the rectangle.

The question tells you which case to draw. Positive profit: ATC below the price at QM. Negative profit: ATC above demand, so above the price at QM. Exam trap: on the standard monopoly graph, this point is missed most; on recent AP exams, only about half earned it. The misses: ATC at or above the price with positive profit, or MC missing ATC's lowest point. Barriers keep rivals out, so this profit can last into the long run.

Who gains, who loses? Consumer surplus is the triangle below demand and above PM, out to QM.

Quick refresher from Unit 2: a triangle's area is one half times base times height. Here: one half times 4 soaks times $40, from $140 down to $100. That makes $80.

Producer surplus is the area below PM and above MC, out to QM. Soaks 5 and 6 are worth more to skiers than they cost to provide: demand sits above MC there. But at $100, nobody buys them. That lost value is deadweight loss: the triangle from QM, between demand above and MC below, out to where MC meets demand. Its corners: $100 and $60 at QM, and $80 where MC meets demand at 6. Height: $100 minus $60, $40. Base: 6 minus 4, 2 soaks. One half times $40 times 2: $40 a day.

Some consumer surplus went to the firm, and $40 of total surplus is gone: the price stops the market short of the efficient amount.

So what just happened? One: competition makes 6 soaks; the monopoly, 4. Two: soaks 5 and 6 are worth more than they cost, but never sold. Three: that lost triangle, $40, is deadweight loss.

Exam trap: deadweight loss ends where MC meets demand, not where ATC crosses demand, and ATC is never its bottom edge.

Exam moment: two questions the way the AP exam asks them, then one free-response part.

The graph shows a profit-maximizing monopolist's D, MR, MC and ATC curves. What price will the monopolist charge? A) $12 B) $16 C) $22 D) $27 E) $37.

The answer is D. MR meets MC at Q2. Go straight up to demand, then across: $27. A: the MR and MC crossing itself. B: ATC's lowest point. C: where MC meets demand, the competitive price. E: where ATC crosses demand.

A monopolist's MR equals MC at 10 units, MR is zero at 15 units, and MC meets demand at 18 units. Which quantity maximizes total revenue, and how elastic is demand there? A) 10 units, elastic B) 15 units, elastic C) 15 units, unit elastic D) 15 units, inelastic E) 18 units, inelastic.

The answer is C. Total revenue peaks where MR is zero, at 15 units, and demand is unit elastic there. A: 10 units maximizes profit, not revenue. B and D: right quantity, but demand there is unit elastic. E: 18 units is where MC meets demand.

Free response. A firm holds the only license to run boat tours on a lake; the graph shows its D, MR, MC and ATC with QM and PM. On the graph, completely shade the area of deadweight loss.

To earn the point, shade one triangle: left edge, the vertical line at QM; top edge, demand; bottom edge, MC; and it ends where MC meets demand. Shade all of it. The costly mistake: shading out to where ATC crosses demand, or down to ATC. On a recent AP exam, only about half of students earned this point.

So what earned the point? One: start at QM. Two: demand on top, MC on the bottom. Three: stop where MC meets demand, and shade the whole triangle.

Recap: MR sits below demand; produce where MR = MC and read the price up on demand, above MC. Profit is P minus ATC, times Q. Deadweight loss runs from QM to where MC meets demand. Why does a town get one water company, and who sets its price? Next: Module 4.2B, Natural Monopoly, Regulation and Policy.