Monopoly, Marginal Revenue, and Price Discrimination

A monopolist is the only seller, but that does not make demand disappear. Buyers still decide how many units they will purchase at each price. The firm may choose a point on market demand, not a price and quantity independently. Here is a market demand curve. Quantity runs across the bottom, price runs up the side, and the downward slope says that reaching more buyers requires a lower price. At quantity twenty, demand allows a price of sixty. The point twenty, sixty is therefore one feasible price-quantity choice, and the rectangle beneath it is the firm's total revenue. Read that choice algebraically. Demand gives sixty dollars per unit, and twenty units at sixty dollars produce total revenue of twelve hundred dollars. Now ask the firm to sell one additional unit. Demand says quantity twenty-one can be sold only if the market price falls from sixty to fifty-eight. The extra unit brings in fifty-eight dollars. That is the narrow green strip. But the firm must also cut the price by two dollars on each of the twenty units it was already selling. The red band is that lost revenue: twenty earlier units times a two-dollar reduction, or forty dollars lost. Marginal revenue is the green gain minus the red loss, not simply the new unit's price. Fifty-eight gained minus forty lost leaves only eighteen dollars of additional revenue. The new unit sells for fifty-eight, yet its marginal revenue is eighteen. That gap between price and marginal revenue is the central monopoly constraint. Expanding sales earns revenue on the new unit, but lowering one market price sacrifices revenue on every earlier unit. The same logic can be written for every quantity on a linear demand curve. Let inverse demand be price equals a minus b Q. The intercept a is the price of the first infinitesimal unit. The coefficient b records how quickly the market price must fall as total output rises. Total revenue is price times quantity. Substitute demand into that product, then multiply out. Revenue is a Q minus b Q squared. Differentiate revenue with respect to quantity. The derivative of a Q is a, while the derivative of minus b Q squared is minus two b Q. So marginal revenue has the same vertical intercept a as demand, but its slope is minus two b instead of minus b. It falls twice as fast because the price reduction applies to the old units as well as the new one. For our numerical demand, price is one hundred minus two Q. Marginal revenue is one hundred minus four Q. Draw the red marginal-revenue curve. Both curves begin at one hundred. Demand reaches the quantity axis at fifty, while marginal revenue reaches it halfway across, at twenty-five. The algebra and the graph tell the same story. Linear demand falls by two dollars per additional unit; marginal revenue falls by four because it includes both the new sale and the price cut imposed on existing sales.

Now turn marginal revenue into a quantity decision. We will keep one posted price, constant marginal cost of twenty, and no fixed cost. Those assumptions make the allocation and the welfare accounting transparent. The blue curve is demand, the red curve is marginal revenue, and the green line is marginal cost. Demand describes buyers. Marginal revenue and marginal cost describe the consequence of producing one more unit. At quantity ten, marginal revenue exceeds marginal cost. Another unit adds more to revenue than to cost, so stopping there would leave profitable units unproduced. Push output to twenty-five. Marginal revenue has fallen below marginal cost, so the last units destroy profit. Move back until the red and green readings meet, at quantity twenty. The interior profit maximum therefore satisfies marginal revenue equal to marginal cost. Substitute the numerical curves: one hundred minus four Q equals twenty. Solving gives the monopoly quantity, twenty. Notice what has been chosen so far: quantity, not price. To find the price, move vertically from quantity twenty to the demand curve. Demand says buyers will pay sixty for each of those twenty units. This order matters. Marginal revenue equal to marginal cost selects quantity. Demand then supplies the single market price. Reading price from marginal revenue would confuse an incremental revenue with what buyers actually pay. For the competitive benchmark, price equals marginal cost. Demand reaches twenty dollars at quantity forty, so competition produces forty units at a price of twenty. Place the two outcomes beside the same market. Monopoly stops at twenty and charges sixty. Competition continues to forty and charges twenty. Under monopoly, the yellow triangle is consumer surplus. Buyers receive the difference between willingness to pay and the sixty-dollar price on the twenty units sold. The magenta rectangle is producer surplus. With constant marginal cost twenty and price sixty, the firm receives forty dollars of surplus on each of twenty units, for eight hundred. The red triangle contains quantities twenty through forty. For every unit there, willingness to pay exceeds marginal cost, yet the monopolist does not sell it because expansion would force down the price on earlier units. Those are mutually beneficial trades that never occur. Their lost total surplus is deadweight loss, four hundred. Monopoly total surplus is twelve hundred. Switch to competition. Output reaches forty, the point where willingness to pay just equals marginal cost. The formerly missing trades now occur. With price equal to constant marginal cost and no fixed cost, consumer surplus is sixteen hundred and producer surplus is zero. Total surplus is sixteen hundred, and deadweight loss is zero. The competitive benchmark maximizes total surplus because every unit worth at least its cost is produced. The monopoly restriction lowers quantity from forty to twenty and destroys four hundred of that possible value. Two questions must remain separate. Efficiency asks how much total value is created. Distribution asks whether buyers or the firm capture that value. Monopoly changes both, but the missing red triangle is a loss to everyone, not a transfer between them.

Now suppose the seller can separate customers into two observable groups. Group A has stronger demand than group B. The product and marginal cost are the same, but willingness to pay differs. This possibility requires strong assumptions. The firm must identify each group, keep customers from pretending to belong to the other group, and prevent low-price buyers from reselling to high-price buyers. The blue curve is demand in group A: price equals one hundred minus Q A. The yellow curve is demand in group B: price equals forty minus Q B. The green marginal-cost line is twenty in both markets. Before allowing separate prices, force the firm to post one common price. That benchmark will show exactly which trades discrimination restores. At prices of forty or more, group B buys nothing. Only group A remains, with quantity one hundred minus price. Profit is price minus marginal cost, times group A quantity. Differentiate with respect to price. The high-price region has its maximum at a common price of sixty. At sixty, group A buys forty units, group B buys none, and profit is sixteen hundred. The gray line and point mark this one-price outcome. Could a lower common price that serves both groups do better? Below forty, total quantity is one hundred forty minus two P. Its unconstrained optimum would lie above the permitted range, so the best feasible point in that region is the boundary price forty. At that boundary profit is only twelve hundred, below sixteen hundred. Therefore the best uniform price is sixty, even though it excludes every buyer in group B. Now allow a separate price in each group. The firm treats each market's marginal revenue independently, while comparing both with the same marginal cost. In group A, marginal revenue equals twenty at quantity forty. Demand then gives price sixty, exactly the outcome group A already had. In group B, marginal revenue equals twenty at quantity ten. Demand gives a group-B price of thirty, low enough to create ten sales that the uniform price had excluded. Total output rises from forty to fifty, and profit rises from sixteen hundred to seventeen hundred. In this example, segmentation restores trades rather than merely reallocating a fixed total. Now separate efficiency from distribution. Under one price, forty units are sold, consumer surplus is eight hundred, producer surplus is sixteen hundred, and total surplus is twenty-four hundred. The one-price deadweight loss is one thousand. Eight hundred comes from missing group-A trades beyond quantity forty, and two hundred comes from excluding the entire efficient range in group B. With two prices, group A remains at forty units while group B gains ten. Total quantity becomes fifty. The restored group-B trades create one hundred fifty dollars of surplus. Fifty goes to group-B consumers as the yellow triangle, and one hundred goes to the firm as the magenta rectangle. Consumer surplus across both groups rises to eight hundred fifty, producer surplus rises to seventeen hundred, and total surplus rises to twenty-five hundred fifty. Deadweight loss falls from one thousand to eight hundred fifty. It does not disappear: group A still stops at forty, and group B still stops at ten instead of the efficient quantity twenty. This welfare improvement is an example, not a universal theorem. Third-degree discrimination can raise or lower total output, and it can redirect units toward groups with higher or lower willingness to pay. What is reliable is the firm's incentive: if separation is voluntary and feasible, its profit rises. What happens to total surplus depends on which trades appear, which disappear, and how production is reallocated. Who captures the gains is a further question, distinct from whether gains exist at all.

Perfect, or first-degree, price discrimination is a much stronger benchmark. The firm knows each buyer's exact willingness to pay and can charge that amount unit by unit. It must also prevent resale. Otherwise a buyer offered a low price could resell to someone facing a high price, and the entire pricing scheme would unravel. The blue demand curve now has a second interpretation. Its height at each quantity is the willingness to pay for that marginal unit. The green line remains marginal cost twenty. Begin near the left. The fifth unit is worth ninety dollars to its buyer. Under perfect discrimination, the firm can charge ninety for that unit without lowering the prices paid for the units before it. That last clause removes the price-cut loss that pushed marginal revenue below demand under one-price monopoly. Each unit contributes its own willingness to pay, less its own production cost. Move to unit twenty. Its willingness to pay is sixty. Earlier buyers may still pay more than sixty because their individual prices do not have to match this buyer's price. Move farther, to unit thirty-five. That buyer is willing to pay thirty, still ten dollars above marginal cost, so producing the unit creates ten dollars of total surplus. Approach unit forty. At unit forty, willingness to pay is twenty, exactly marginal cost. Beyond forty, willingness to pay would be below cost, so those units should not be produced. The rule is now different from one-price monopoly. Revenue from the marginal unit equals that unit's own demand price because selling it does not reduce the prices charged on earlier units. Produce while willingness to pay is at least marginal cost. For our market, set one hundred minus two Q equal to twenty. The result is quantity forty, the same efficient quantity produced under competition. Perfect discrimination restores every trade whose value covers its cost. Efficiency has returned, but competitive pricing has not. There is no one price here. Early units carry high prices, later units carry lower prices, and the final unit is priced at marginal cost. Return briefly to the one-price monopoly. It sold twenty units at sixty. The yellow triangle was consumer surplus, the magenta rectangle was producer surplus, and the red triangle was deadweight loss. Under perfect discrimination, output extends from twenty to forty. The missing red trades return, so deadweight loss falls from four hundred to zero. But each buyer is charged exactly what the unit is worth to that buyer. The difference between willingness to pay and marginal cost is therefore captured by the firm, not left with consumers. Producer surplus becomes the entire magenta triangle, sixteen hundred. Consumer surplus is zero because every buyer pays their full willingness to pay. Total surplus is also sixteen hundred, the same efficient total as under competition. The firm has not created extra value beyond the efficient allocation. It has changed who receives that value. That completes the distinction. A one-price monopolist restricts quantity because expanding sales lowers the price on earlier units. Third-degree discrimination may restore some trades by separating groups. Perfect discrimination restores every efficient trade, eliminates deadweight loss, and transfers the available surplus to the monopolist.