Why Compare Allocations by Total Surplus?
Two market arrangements can trade different quantities, involve different buyers and sellers, or use different prices. To compare the gains created by those arrangements, calculate total surplus: the gains to buyers and sellers combined, including any transfer recipient when buyers and sellers pay and receive different prices. Comparing only the buyer’s or seller’s gain can make an arrangement look better simply because it shifts gains from one side to another.
This comparison uses a private-value model: each buyer’s willingness to pay represents the benefit of the purchase, and each seller’s cost represents the opportunity cost of supplying it. Assume these benefits and costs are measured in the same monetary units, and that there are no external costs or benefits or administrative resource costs unless stated. Under those assumptions, total surplus measures the net gains from the trades in the model. If important costs or benefits are outside the model, market surplus alone is not a complete measure of overall social gains.
Adding Gains from Trade
For a single trade, the gain from trade is the buyer’s willingness to pay minus the seller’s cost. With multiple units, add that difference for every unit traded. The order of units matters when their values or costs differ: the gain from each additional unit is its marginal willingness to pay minus its marginal cost.
For continuous inverse demand and supply curves, let \(P_D(Q)\) be buyers’ marginal willingness to pay at quantity \(Q\), and let \(P_S(Q)\) be sellers’ marginal opportunity cost. If the first \(q\) units are traded and the curves represent private benefits and costs, total surplus is the area between the two curves from zero to \(q\):
For a discrete set of units, the corresponding calculation is a sum rather than an integral. Both approaches compare the benefits and opportunity costs of the units actually traded; neither requires that every unit be traded at the same price.
where \(W\) is the total willingness to pay for the units traded, \(C\) is their total opportunity cost, and \(R\) is the transfer recipient’s revenue. The price terms cancel because a payment by one party is a receipt by another. This cancellation does not apply to a real resource cost: for example, if administering the arrangement uses resources worth \(K\), that cost must be subtracted from the gains.
Worked Examples: Comparing Allocations
Worked Example: A Discrete Market with Two Quantities
A market has three buyers whose willingness to pay for successive units is $50, $42, and $34. Sellers’ costs for those units, in the same order, are $18, $24, and $30. Allocation A trades three units at a uniform price of $32. Allocation B trades two units at a uniform price of $35. Which allocation has more total surplus, and how are the gains divided?
Method. First check that each stated price allows the specified buyers and sellers to participate. Then add the buyer and seller gains, or calculate the gain from trade for each unit.
For Allocation A, the lowest willingness to pay among the three buyers is $34, and the highest cost among the three sellers is $30. The price of $32 is between those amounts, so all three trades are consistent with voluntary participation. Buyer surplus is:
Seller surplus is:
Thus \(TS_A=30+24=\$54\). Directly adding gains from trade gives \((50-18)+(42-24)+(34-30)=32+18+4=\$54\), which checks the result.
For Allocation B, the second buyer is willing to pay $42 and the second seller’s cost is $24. The price of $35 lies between them, so both trades are consistent with voluntary participation. Buyer surplus is \((50-35)+(42-35)=15+7=\$22\), and seller surplus is \((35-18)+(35-24)=17+11=\$28\). Therefore, \(TS_B=22+28=\$50\). The direct check is \((50-18)+(42-24)=32+18=\$50\).
Allocation A creates $4 more total surplus. Allocation B gives sellers a larger share than buyers receive, but that distribution does not make its total larger. A common error would be to compare only buyer surplus, $30 versus $22, and treat that difference as the difference in total gains. The correct comparison includes seller surplus: the totals are $54 and $50.
Worked Example: Comparing Quantities on Linear Curves
Suppose inverse demand is \(P_D(Q)=100-Q\), and inverse supply is \(P_S(Q)=20+Q\), with price measured in dollars per unit and quantity in units. Compare allocations of 30 and 40 units. For both allocations, the transaction price is $60. Calculate consumer surplus, producer surplus, and total surplus.
Method. Integrate willingness to pay and opportunity cost over the traded quantity, then subtract the relevant payment or cost. The stated price is consistent with each allocation: at 30 units, demand price is $70 and supply price is $50; at 40 units, both equal $60.
At \(q=30\), total willingness to pay is:
Buyers pay \(60(30)=\$1{,}800\), so \(CS=2{,}550-1{,}800=\$750\). Total seller cost is:
Sellers receive $1,800, so \(PS=1{,}800-1{,}050=\$750\). Thus \(TS(30)=750+750=\$1{,}500\).
At \(q=40\), total willingness to pay is \(100(40)-40^2/2=4{,}000-800=\$3{,}200\). Buyers pay \(60(40)=\$2{,}400\), so \(CS=3{,}200-2{,}400=\$800\). Total seller cost is \(20(40)+40^2/2=800+800=\$1{,}600\). Sellers receive $2,400, so \(PS=2{,}400-1{,}600=\$800\). Hence \(TS(40)=800+800=\$1{,}600\).
The 40-unit allocation has $100 more total surplus. The direct check for either quantity is the area between the curves: \(P_D(Q)-P_S(Q)=80-2Q\). Integrating gives \(TS(q)=80q-q^2\), so \(TS(30)=2{,}400-900=\$1{,}500\) and \(TS(40)=3{,}200-1{,}600=\$1{,}600\). The common mistake is to compare the two allocations using consumer surplus alone: consumer surplus rises by only $50, while producer surplus also rises by $50. The total increase is $100.
Worked Example: Same Quantity, Different Buyers and Sellers
Two alternative arrangements each make two trades, but they select different participants. Three buyers have willingness to pay of $90, $70, and $50. Three sellers have costs of $20, $40, and $60. Arrangement A trades with the $90 and $70 buyers and the $20 and $40 sellers, at a price of $55 for each trade. Arrangement B trades with the $90 and $50 buyers and the $20 and $40 sellers; its two prices are $50 and $45, respectively. Compare the arrangements.
Method. Verify participation, then calculate buyer and seller gains for each trade. Prices can differ across trades; total surplus still equals the sum of willingness to pay minus costs.
In Arrangement A, both buyers accept $55 because their willingness to pay is at least $70, and both sellers supply because their costs are no more than $40. Buyer surplus is \((90-55)+(70-55)=35+15=\$50\). Seller surplus is \((55-20)+(55-40)=35+15=\$50\). Total surplus is \(50+50=\$100\), also equal to \((90-20)+(70-40)=70+30=\$100\).
In Arrangement B, the first trade has price $50, between the $90 buyer’s value and the $20 seller’s cost. The second has price $45, between the $50 buyer’s value and the $40 seller’s cost. Buyer surplus is \((90-50)+(50-45)=40+5=\$45\). Seller surplus is \((50-20)+(45-40)=30+5=\$35\). Total surplus is \(45+35=\$80\). Directly, the gains are \((90-20)+(50-40)=70+10=\$80\).
Arrangement A creates $20 more surplus, even though both arrangements trade the same number of units. The difference comes from who participates: A pairs the second trade with a buyer who values the unit $20 more than the buyer in B. Counting trades without checking the values and costs attached to them would miss this difference.
How to Make a Reliable Comparison
The comparison should keep the accounting boundary consistent across alternatives. If an arrangement has a buyer price different from a seller price, include the recipient of the difference. If a cost consumes real resources, include it as a cost rather than treating it as a transfer. Then compare the totals using the same definition of benefits and costs for every option.
Record the quantity traded and, when relevant, which buyers and sellers participate. Do not assume that equal quantities imply equal gains.
Use willingness to pay for the units buyers receive and the costs of the resources sellers use. State any additional resource costs or benefits that are included.
Add consumer and producer surplus and any transfer recipient’s revenue, or add the gains from trade unit by unit.
The larger total indicates more gains within the stated model. The separate surplus amounts show who receives those gains.
A useful check is to calculate total surplus in two ways when the data allow it: add the parties’ surpluses, then independently sum willingness to pay minus opportunity cost. A disagreement often signals that a payment, cost, or transfer recipient has been omitted or counted twice.
Common Mistake: Treating a Transfer as a Lost Resource
Suppose buyers pay $80 per unit, sellers receive $70 per unit, and 10 units are traded. The $10 gap per unit produces a $100 receipt for the intermediary. It is incorrect to subtract the full $800 paid by buyers and also ignore the $700 received by sellers and the $100 received by the intermediary. Those amounts are transfers between parties, not resource costs.
For example, if total willingness to pay is $1,200 and total seller cost is $500, consumer surplus is \(1{,}200-800=\$400\), producer surplus is \(700-500=\$200\), and the intermediary receives $100. Total surplus is \(400+200+100=\$700\), exactly \(1{,}200-500=\$700\). If the intermediary also uses resources worth $25 to administer the arrangement, that is a genuine cost: net surplus is then $675.
Check Your Understanding
For each problem, compare the stated allocations using the values and costs given. Treat prices as transfers unless a real resource cost is specified.
- Three units have successive willingness to pay of $48, $39, and $31, and corresponding seller costs of $12, $25, and $35. Allocation A trades two units; Allocation B trades three. Find total surplus under each and identify the larger total. (Answer: A: $50; B: $46; Allocation A is larger by $4.)
- Inverse demand is \(P_D(Q)=80-Q\) and inverse supply is \(P_S(Q)=20+Q\). Compare total surplus at 20 and 30 units by integrating the difference between the curves. (Answer: \(TS(20)=\$800\); \(TS(30)=\$900\), so 30 units has $100 more surplus.)
- For 10 trades, total willingness to pay is $1,200 and total seller cost is $500. Buyers pay $90 per unit and sellers receive $60 per unit; the recipient keeps the difference. Calculate consumer surplus, producer surplus, the recipient’s revenue, and total surplus. (Answer: $300, $100, $300, and $700, respectively.)
- Arrangement A trades with buyers valued at $85 and $65 and sellers costing $15 and $35. Arrangement B trades with buyers valued at $85 and $45 and the same two sellers. Calculate total surplus for each and compare them. (Answer: A: $100; B: $80; Arrangement A is larger by $20.)
- Inverse demand is \(P_D(Q)=120-2Q\), and inverse supply is \(P_S(Q)=20+Q\). Compare total surplus at quantities 20, 30, and 40, and identify which has the highest total. (Answer: \(TS(20)=\$1{,}400\), \(TS(30)=\$1{,}650\), and \(TS(40)=\$1{,}600\); 30 units has the highest total.)