Surplus from Demand and Supply Schedules

Reading Discrete Demand and Supply Schedules

A demand or supply schedule can list the value or cost of each successive unit rather than giving an equation or a graph. This format makes surplus calculations concrete: each unit bought creates a buyer’s gain, and each unit sold creates a seller’s gain. The key is to use the schedule values for the units that actually trade—not every unit listed.

Assume the demand schedule ranks buyers’ willingness to pay from highest to lowest, and the supply schedule ranks units by marginal cost from lowest to highest. Each row represents one unit. When units trade at a common price, a buyer’s surplus on a purchased unit is willingness to pay minus price; a seller’s surplus on a sold unit is price minus marginal cost. These calculations assume the units are allocated to the highest-value buyers and lowest-cost sellers. If allocation is different, the surplus amounts may differ.

Definition: Let $w_i$ be willingness to pay for the $i$th unit, $c_i$ its marginal cost, $P$ the common price, and $Q$ the number of units traded. With buyers ranked from highest to lowest willingness to pay and sellers from lowest to highest marginal cost, consumer surplus is $CS=\sum_{i=1}^{Q}(w_i-P)$, producer surplus is $PS=\sum_{i=1}^{Q}(P-c_i)$, and total surplus is $TS=CS+PS$. These formulas assume the first $Q$ buyer units and first $Q$ seller units are the ones traded.

The schedule values describe marginal units: the willingness to pay or cost associated with one additional unit. They are not totals for all units. In particular, do not multiply one row’s value by the number of units unless the problem explicitly says that the value applies to every unit.

A Reliable Calculation Method

When a problem gives both a price and a traded quantity, calculate surplus for exactly that many units. When it gives a price but not the traded quantity, use the schedules to count eligible units: buyers are willing to buy units with willingness to pay at least the price, while sellers are willing to sell units with marginal cost no greater than the price. In a simple setting where trades are limited by the smaller eligible side, that smaller count determines the number traded.

Read the ranking.
Check that willingness to pay is listed from high to low and marginal cost from low to high. If the data are not ranked, identify and rank the units before selecting those that trade.

Find the traded units.
Use the stated quantity, or count the units eligible at the stated price. Do not include units that are not traded.

Calculate each side’s surplus.
For each traded buyer unit subtract price from willingness to pay. For each traded seller unit subtract marginal cost from price.

Check the total.
Add consumer and producer surplus, then compare with total willingness to pay on traded units minus total marginal cost on traded units.

Worked Schedule Examples

Worked Example: Surplus When Price and Quantity Are Given

A market has the following schedules. The price is $18 per unit, and 3 units trade. Calculate consumer surplus, producer surplus, and total surplus.

Unit	Willingness to pay ($)	Marginal cost ($)
1	32	5
2	25	11
3	19	16
4	12	22

Method. The quantity is specified, so use the first 3 buyer units and the first 3 seller units. The price also confirms that the fourth unit does not qualify: its willingness to pay is $12, below $18, and its marginal cost is $22, above $18.

Consumer surplus. The three buyer gains are $32 − $18 = $14, $25 − $18 = $7, and $19 − $18 = $1. Therefore, $CS=14+7+1=\$22$.

Producer surplus. The three seller gains are $18 − $5 = $13, $18 − $11 = $7, and $18 − $16 = $2. Therefore, $PS=13+7+2=\$22$.

Total surplus. $TS=22+22=\$44$. Check directly: traded buyers’ total willingness to pay is $32+25+19=\$76$, and traded sellers’ total marginal cost is $5+11+16=\$32$. The difference is $76-32=\$44$, as required. A common mistake is to include unit 4 because it appears in the table; the stated quantity limits the calculation to 3 units.

Worked Example: Inferring Quantity from a Price

A different market has a price of $20 per unit. Its willingness-to-pay values are $40, $31, $24, and $17; its marginal costs are $4, $12, $18, and $26. Assume eligible buyers are matched with eligible sellers in the stated rankings. Find the quantity traded and all three surplus measures.

Find quantity. Buyers accept units whose willingness to pay is at least $20. The first three values, $40, $31, and $24, meet that condition; $17 does not. Thus, 3 units are demanded at this price. Sellers offer units whose marginal cost is no greater than $20. The first three costs, $4, $12, and $18, meet that condition; $26 does not. Thus, 3 units are supplied. The traded quantity is $Q=3$.

Consumer surplus. The three gains are $40-20=20$, $31-20=11$, and $24-20=4$, in dollars. Hence $CS=20+11+4=\$35$.

Producer surplus. The three gains are $20-4=16$, $20-12=8$, and $20-18=2$, in dollars. Hence $PS=16+8+2=\$26$.

Total and check. $TS=35+26=\$61$. Total willingness to pay on the traded units is $40+31+24=\$95$, and their total cost is $4+12+18=\$34$. The difference, $95-34=\$61$, checks the result. The common mistake is to count the fourth row as a trade merely because it is listed; both its buyer value and seller cost fail the price test.

Worked Example: A Shortage Limits the Number of Trades

At a price of $10, buyers’ willingness to pay for successive units is $27, $21, $16, and $11. Sellers’ marginal costs are $2, $7, $13, and $18. Suppose the highest-value buyers receive the available units and the lowest-cost sellers supply them. Calculate the traded quantity and surplus.

Determine eligible units. All 4 buyer values are at least $10, so buyers want 4 units at this price. Only the first 2 seller costs are no greater than $10, so sellers offer 2 units. With these assumptions, only 2 units trade: $Q=2$. The third buyer has willingness to pay above price, but there is no third eligible seller.

Consumer surplus. The two traded buyer gains are $27-10=17$ and $21-10=11$. Thus $CS=17+11=\$28$. The other buyers’ values do not count because they do not receive units.

Producer surplus. The two traded seller gains are $10-2=8$ and $10-7=3$. Thus $PS=8+3=\$11$.

Total and check. $TS=28+11=\$39$. The traded units have total willingness to pay $27+21=\$48$ and total marginal cost $2+7=\$9$, giving $48-9=\$39$. The trap is to calculate consumer surplus for all 4 willing buyers, which would count gains from units that were not actually traded.

Why the Total-Surplus Check Works

The individual calculations can be combined before evaluating any numbers. For each traded unit, the buyer’s gain is $w_i-P$, and the seller’s gain is $P-c_i$. Adding those gains cancels the payment:

$$ (w_i-P)+(P-c_i)=w_i-c_i $$

Across $Q$ traded units, this gives the following result.

CS+PS =\sum_{i=1}^{Q}(w_i-P)+\sum_{i=1}^{Q}(P-c_i) =\sum_{i=1}^{Q}w_i-\sum_{i=1}^{Q}c_i

Key result: For the same traded units, total surplus equals their total willingness to pay minus their total marginal cost. The price determines how the gains are divided between buyers and sellers, but it cancels when their surpluses are added.

This check depends on using the same units in both surplus calculations and on each unit trading at the same price. It also treats the schedule values as the relevant benefits and costs, with no additional transaction costs or external effects included. If a problem adds such effects, they must be accounted for separately rather than silently treated as part of the schedule.

Common Errors to Avoid

Including units that do not trade. Calculate gains only for the stated traded quantity or the units actually supplied and purchased under the problem’s assumptions.
Reversing the subtraction. Consumer surplus is willingness to pay minus price; producer surplus is price minus marginal cost. A negative result may signal that a unit was included despite not meeting the relevant condition.
Counting the price as a social benefit or cost. The buyer pays what the seller receives. For total surplus, these payments cancel; do not add the price to total willingness to pay or subtract it again from total cost.
Treating a marginal value as a total. A schedule’s entry applies to its listed unit. Sum the values for the units traded instead of multiplying one entry by quantity.
Ignoring allocation when units differ. If buyers are not selected in descending willingness-to-pay order, or sellers in ascending marginal-cost order, identify which specific units trade before summing their gains.

Check Your Understanding

For each problem, state the traded quantity if it is not given, then calculate the requested surplus. Assume the schedule rankings and allocation rules described in the question.

At a price of $10, willingness to pay is $21, $15, and $9, while marginal cost is $3, $8, and $14. The eligible buyer and seller units trade. Find quantity, consumer surplus, producer surplus, and total surplus. (Answer: $Q=2$, $CS=\$16$, $PS=\$9$, $TS=\$25$.)
A market price is $16. Willingness to pay is $35, $26, $18, and $12; marginal cost is $5, $12, $16, and $20. Find quantity and all three surplus measures. (Answer: $Q=3$, $CS=\$31$, $PS=\$15$, $TS=\$46$.)
At a price of $13, buyers have willingness-to-pay values of $28, $20, $14, and $8, and sellers have marginal costs of $2, $9, $15, and $21. The highest-value buyers receive available units. Find quantity and all three surplus measures. (Answer: $Q=2$, $CS=\$22$, $PS=\$15$, $TS=\$37$.)
A stated quantity of 3 units trades at $17. The three traded buyer values are $29, $23, and $19; the relevant seller costs are $4, $10, and $16. Calculate consumer surplus, producer surplus, and total surplus, and verify total surplus from values and costs. (Answer: $CS=\$20$, $PS=\$21$, $TS=\$41$.)
At a price of $12, willingness to pay is $30, $22, $15, and $11; marginal cost is $3, $8, $12, and $19. Eligible units are matched in schedule order. Find quantity and all three surplus measures. (Answer: $Q=3$, $CS=\$31$, $PS=\$13$, $TS=\$44$.)

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