The Cap — The Surplus, Plate xii

A ceiling promises a buyer that the money price will go no higher. If the good can be obtained at that price, the promise can be valuable: more of the benefit stays with its user and less passes to its owner. But a price also coordinates how much people offer and how much others seek. Hold it below the level at which those quantities would meet, and the promise of a lower payment is joined by a question about access. Who receives the available units? Who would have bought before but can no longer obtain one? What must people do to secure their place? A price control sets a condition on exchange; its consequences depend on the market and the allocation procedure to which that condition is attached. The surplus has to be followed through all of them.

Begin with a hypothetical competitive market, free of costs or benefits outside the exchange. The value of another unit to buyers is one hundred minus the quantity already supplied; the resource cost of another unit is twenty plus that quantity. At forty units the two meet at a price of sixty. Buyers keep eight hundred of surplus and sellers another eight hundred, giving sixteen hundred in all. Now set a ceiling of fifty, with the same demand and supply schedules. Sellers offer thirty units, while buyers seek fifty. The shortage is twenty, although the reduction in actual trade from its former level is ten. Suppose, for the moment, that the thirty units reach the buyers who value them most. Their surplus rises to one thousand and fifty; the sellers' falls to four hundred and fifty. The combined fifteen hundred is a hundred below the original total. Some buyers gain from the lower price, sellers lose, and the gains from ten previously worthwhile trades are no longer realized.¹

1. The schedules and figures are authored: demand price P = 100 − Q, supply price P = 20 + Q. The competitive solution is Q = 40 and P = 60. A ceiling of 50 gives supply 30 and demand 50. With the highest-valuing buyers served and the lowest-cost production used, buyer surplus is ½(100 + 70) × 30 − 50 × 30 = 1,050; seller surplus is 50 × 30 − ½(20 + 50) × 30 = 450. The lost triangle is ½ × 10 × 20 = 100. The money measure assumes that the stated demand represents the relevant willingness to pay, without additional income effects.

The example makes the distribution visible without assuming that all buyers share the gain. On the thirty units still sold, reducing the price from sixty to fifty transfers three hundred from sellers to buyers. But the ten units no longer traded would have provided fifty of buyer surplus and fifty of seller surplus. The buyers' net aggregate gain is therefore two hundred and fifty, while the sellers' loss is three hundred and fifty. Those totals conceal different positions within each group. A buyer who continues to receive a unit enjoys the price cut; a buyer excluded from a purchase he previously made loses its benefit. Nor does the ceiling itself guarantee the favorable allocation assumed in the calculation. Edward Glaeser and Erzo Luttmer distinguish the reduction in supply from the further loss that can arise when the available stock goes to people who value it less than other potential users. Even with the same number of units and the same permitted price, the allocation can change the amount of surplus created.²

2. Edward L. Glaeser and Erzo F. P. Luttmer, “The Misallocation of Housing Under Rent Control,” American Economic Review 93 (2003), introduction and §I, pp. 1027–1031, distinguish undersupply from allocation among consumers. The fixed-stock comparison here is an illustrative limiting case. The paragraph relies on that theoretical distinction; it imports no empirical estimate about a city or an existing policy. In the authored example, buyer change is 300 − 50 = 250 and seller change is −300 − 50 = −350.

⁂

A single unit is enough to show the allocation problem. Suppose supplying it costs twenty and one buyer values it at a hundred and another at sixty. At a controlled price of fifty, both want it. If the first receives it, buyer and seller together obtain eighty of surplus; if the second receives it, they obtain forty. The price and quantity are identical in the two outcomes. A lottery, a waiting list, a preference for existing customers or an assessment of need can select differently, and each has its own purpose and demands on information. People may also spend time or other resources competing for access. A queue can consume part of the saving in money, just as the previous plate's contest consumed part of its prize. A payment made to obtain access must still be distinguished from time consumed: one changes hands, while the other displaces an alternative use. The relevant inquiry follows the whole procedure, including its costs and intended distribution, rather than reading the effect from the posted price alone.³

3. The two-buyer example is authored: totals are 100 − 20 = 80 and 60 − 20 = 40. At price 50, the respective buyer surpluses are 50 and 10, while seller surplus is 30 in either case. Glaeser and Luttmer, §I, pp. 1028–1031, analyze misallocation and discuss search or queue costs alongside it. The example excludes resale; unrestricted costless resale could change the final allocation. Judgments about need require a welfare criterion beyond ranking monetary willingness to pay.

A floor holds the price in the other direction. Return to the competitive schedules and require a payment of at least seventy. Buyers now want thirty units while sellers would supply fifty; without a purchaser for the excess, thirty units are traded. If the lowest-cost sellers supply them, buyer surplus is four hundred and fifty and seller surplus is one thousand and fifty. The same hundred of potential surplus is lost, with the gain from the changed price now going toward the sellers who retain access to the market. Once again, the rule alone does not decide which sellers those will be. A guarantee that a public buyer will purchase the extra twenty units would constitute an additional policy, with additional production, a use or disposal decision, and financing to examine. The difference between offering to sell and making a sale is as important as the difference between wishing to buy and receiving a good. A floor cannot turn every desired sale into a completed transaction merely by naming a more favorable price.⁴

4. At a floor of 70, demand is 30 and supply is 50. With efficient selection of suppliers, buyer surplus is ½ × 30² = 450 and seller surplus is 70 × 30 − ½(20 + 50) × 30 = 1,050. The resulting 1,500 is 100 below the competitive total. No government purchases, disposal costs or enforcement costs are assumed. This is a competitive commodity illustration; the separate Floor examines the labor-market model and its empirical tests.

The competitive benchmark gives these calculations their force and marks their scope. A ceiling above the market-clearing price, or a floor below it, leaves the price free to reach that point in the stated model. A binding rule changes the result, but the size and direction of the response depend on the original market. When a seller or buyer has power to restrict trade, the starting quantity may already fall short of the efficient level. The Wage volume's Monopsony shows how a suitably placed floor can raise both a wage and employment under different assumptions. Time matters as well. A stock may be fixed for the period under study, removing the immediate supply reduction while leaving allocation among users consequential; investment or maintenance can introduce further margins in a longer comparison. Evidence about an actual control must establish which responses are available, whom the rule covers and how access is assigned. The diagram supplies a conditional account of a mechanism, while the policy question requires identifying the mechanism at work.⁵

5. Glaeser and Luttmer, pp. 1027–1031, explicitly separate their allocation question from supply and maintenance responses discussed in the literature; note 8 observes that misallocation can occur even with the right equilibrium quantity of each housing type. The fixed-stock case in this plate holds the supply response at zero as an illustrative limiting case. No inference about the magnitude of a long-run response is made here. The monopsony comparison changes market structure, so it does not contradict the competitive calculation. Outside effects, quality changes and the costs of administering a rule likewise require an expanded comparison rather than being silently included in its triangle.

The cap closes the volume by bringing its questions into one place. There is the surplus made possible by a good reaching someone who values it more than the resources needed to supply it. There is the price that divides that gain, the power that can alter the price, the rule that can restrain it and the procedure that decides who participates. Some of the resulting changes are transfers, some are lost opportunities, and some consume resources in the struggle for access. The accounts clarify what has happened without assigning every person's gain the same human importance. A household's inability to pay can express poverty even when its need is urgent; a money measure of benefit cannot settle that question by itself. The next volume, The Utility, turns to the thing these sums have been trying to represent. Behind the price and the surplus stands the satisfaction of a want—and the difficulty of saying how much that satisfaction is worth.