A restricted quantity does not identify who trades.
Under a binding control, surplus depends on which buyers receive the good and which sellers supply it. The standard area under demand assumes the highest-willingness-to-pay buyers obtain the units; the area above supply assumes the least-cost sellers produce them. State these assumptions. With fewer units traded, the relevant area may be a trapezoid, not the original full triangle.
- Buyers
- Adding the area between demand and the buyer price over the traded units assumes those units go to buyers with the highest willingness to pay. A queue may choose different buyers.
- Sellers
- Adding the area between the seller price and original supply assumes the least-cost sellers provide the units. Selecting higher-cost sellers can reduce producer surplus.
- Rationing
- Rationing decides who trades. The same controlled price and quantity can produce different surplus totals if different buyers or sellers are selected.
Restricted areas and licence payments
Why a trapezoid?
At the $4 ceiling, only 40 units trade. Over those units, demand falls from $10 to $8, so the gap above the price is $6 at one end and $4 at the other. This makes a trapezoid: average the two end heights, then multiply by 40. Do not include the 80 unfilled purchases in the surplus area.
Free licences
In the 40-unit quota example, sellers retain $320. The area under original supply represents $120 of variable costs, which depend on output. PS is $320-$120=$200; it is not the full receipts rectangle.
Auction comparison
If the same least-cost sellers instead pay total licence charges of $160 while the price, quantity and allocation stay unchanged, they retain PS of $40 after those charges and government receives $160. The free-licence $200 is not automatically all retained by producers.
Consumer allocation
The ceiling example gives goods to the highest-valuing buyers. If lower-valuing buyers receive some units instead, total willingness to pay and consumer surplus can be lower. Waiting costs would further change the account.
| Policy | CS | PS | Allocation assumption |
|---|---|---|---|
| None | 160 | 160 | Highest-valuing buyers and least-cost sellers |
| Tax 2 | 90 | 90 | Net seller price 5; buyer price 7 |
| Subsidy 2 | 250 | 250 | Seller receipt 7; buyer price 5 |
| Ceiling 4 | 200 | 40 | Highest-valuing buyers; no waiting cost |
| Floor 8 | 40 | 200 | Least-cost sellers; no procurement |
| Quota 40 | 40 | 200 | Free licences; least-cost sellers retain receipts |
Worked example: A ceiling, a floor and a quota
Demand is P=10-0.05 Q and supply is P=2+0.05 Q, with quantity in units per day. Give units to the highest-willingness-to-pay buyers and obtain them from the least-cost sellers. There is no government purchase or waiting cost. Quota licences are free and retained by sellers.
- At a $4 ceiling,40 units trade. Demand falls from $10 to $8 over those units, so CS=0.5x[(10-4)+(8-4)]x 40=$200. PS=0.5x(4-2)x 40=$40.
- At an $8 floor,40 units trade. CS=0.5x(10-8)x 40=$40. Original supply rises from $2 to $4, so PS=0.5x[(8-2)+(8-4)]x 40=$200.
- A quota of 40 also gives price $8. With the stated free licences and allocation, CS=$40 and PS=$200. Charging for licences would change who retains these receipts.
- Different rationing can allocate goods to lower-willingness-to-pay buyers; different seller selection can raise production costs. The quantities and legal price alone do not establish the same surplus totals.
Watch out for this
A price ceiling always raises total consumer surplus.
Successful buyers pay less, but fewer units trade and allocation matters. Even the direction of the total change needs analysis.
Check your understanding
Two ceilings produce the same price and quantity, but use different rationing rules. Must consumer surplus be identical?
- No: the buyers who receive the units may value them differently.
- Yes: price times quantity is consumer surplus.
- Yes: all buyers on a demand curve have the same willingness to pay.