Surplus with Different Buyer and Seller Prices

Why Two Prices Matter

A per-unit tax or subsidy can cause the price paid by a buyer to differ from the amount received by a seller. The distinction matters for surplus calculations: buyers’ decisions depend on the price they pay, while sellers’ decisions depend on the price they receive. The gap between those prices changes the quantity traded and can prevent some mutually beneficial trades—or encourage trades whose cost exceeds their benefit.

The analysis below uses the standard competitive-market model. It separates the change in who pays or receives money from the change in the gains created by trade. A tax payment is a transfer to the government, and a subsidy is a transfer from it; neither is itself a resource gain or loss when the public funds are valued dollar for dollar. The loss in total surplus comes from the trades that no longer occur or the additional trades that should not occur.

Definition: Let $Q$ be the quantity traded, $P_b$ the price paid by buyers, and $P_s$ the price received by sellers. Define the per-unit wedge as $t=P_b-P_s$. A positive $t$ is a tax wedge; a negative $t$ means sellers receive more per unit than buyers pay, as with a subsidy. Wedge revenue is $tQ$, so it is positive for a tax and negative for a subsidy.

Equilibrium with a Constant Wedge

Let inverse demand $P_b(Q)$ give buyers’ willingness to pay for the marginal unit, and let inverse supply $P_s(Q)$ give sellers’ marginal cost of supplying it. In a competitive equilibrium with a fixed per-unit wedge, the market quantity satisfies $P_b(Q)-P_s(Q)=t$. Demand is evaluated at the price buyers pay, and supply at the price sellers receive. The equilibrium prices are found by substituting the equilibrium quantity into the respective curves.

For linear inverse demand $P_b(Q)=a-bQ$ and inverse supply $P_s(Q)=c+dQ$, where $b>0$ and $d>0$, an interior equilibrium has $a-bQ-(c+dQ)=t$. Solving gives the quantity with the wedge. Without the wedge, the efficient quantity is where marginal willingness to pay equals marginal cost, or $P_b(Q)=P_s(Q)$. This benchmark assumes there are no external costs or benefits, so private demand and supply capture all benefits and costs relevant to total surplus.

Q_t=\frac{a-c-t}{b+d}, \qquad Q^*=\frac{a-c}{b+d}, \qquad Q_t-Q^*=-\frac{t}{b+d}.

These expressions apply when the resulting quantities are feasible and the market has an interior solution. A positive wedge reduces quantity; a negative wedge increases it. If the formula gives a negative quantity, the interior calculation is not a valid market outcome: the relevant boundary is typically zero trade, and the specific market setup must be considered.

Surplus and the Wedge Transfer

Consumer surplus is the area between buyers’ willingness to pay and the price they pay, over the units purchased. Producer surplus is the area between the price sellers receive and their marginal costs, over the units supplied. For linear demand and supply, these areas can be calculated as triangles. Wedge revenue is $tQ$; it is a government receipt for a tax and a government expenditure with a minus sign for a subsidy.

CS=\int_0^Q \bigl(P_b(q)-P_b(Q)\bigr)\,dq, \qquad PS=\int_0^Q \bigl(P_s(Q)-P_s(q)\bigr)\,dq.

Total surplus includes consumer surplus, producer surplus, and the net wedge revenue. Including that transfer is essential: otherwise a tax payment would appear to vanish from the calculation even though buyers pay more than sellers receive. The transfer cancels between private parties and the government, leaving total surplus equal to the gains from trade.

TS=CS+PS+tQ =\int_0^Q \bigl(P_b(q)-P_s(q)\bigr)\,dq.

Why Deadweight Loss Is the Lost Gains from Trade

The marginal gain from trading unit $q$ is buyers’ willingness to pay minus sellers’ marginal cost, $P_b(q)-P_s(q)$. At the efficient quantity $Q^*$, this gap is zero. Units below $Q^*$ have positive gains from trade; units beyond it have negative gains when demand and supply slope in the usual directions. A wedge moves the equilibrium away from $Q^*$, so deadweight loss is the net gains from trade that the wedge removes.

The following justification also explains why wedge revenue is not the deadweight loss. At any quantity, consumer surplus subtracts the buyer price on every unit and producer surplus adds the seller price. Adding $tQ=(P_b-P_s)Q$ cancels those price terms. What remains is the area between willingness to pay and marginal cost, not the rectangle of tax receipts or subsidy spending.

Write the three components.
$CS=\int_0^Q P_b(q)\,dq-P_b(Q)Q$, $PS=P_s(Q)Q-\int_0^Q P_s(q)\,dq$, and wedge revenue is $(P_b(Q)-P_s(Q))Q$.

Add and cancel.
The terms $-P_b(Q)Q+P_s(Q)Q+(P_b(Q)-P_s(Q))Q$ sum to zero. Thus $TS=\int_0^Q(P_b(q)-P_s(q))\,dq$.

Compare with efficient trade.
Deadweight loss is efficient total surplus less total surplus at the wedged quantity. It is the area between demand and supply over the units by which quantity differs from $Q^*$.

With linear curves and a constant wedge, the marginal-benefit/marginal-cost gap at the wedged quantity has magnitude $|t|$, and it falls linearly to zero at $Q^*$. Thus the loss is a triangle, whether a tax causes undertrading or a subsidy causes overtrading. This triangle shortcut depends on the linear curves and constant wedge; for nonlinear curves, calculate the corresponding area by integration or another appropriate area method.

DWL=TS(Q^*)-TS(Q_t) =\frac{1}{2}|t|\,|Q_t-Q^*| =\frac{t^2}{2(b+d)}.

Worked Example: A Tax Wedge with Linear Curves

In a hypothetical market, inverse demand is $P_b=100-2Q$, inverse supply is $P_s=10+Q$, and a tax creates a per-unit wedge of $t=\$9$. Find the traded quantity, buyer and seller prices, consumer surplus, producer surplus, tax revenue, total surplus, and deadweight loss. Use the efficient market outcome as the benchmark.

Method. Set the difference between inverse demand and inverse supply equal to the wedge. Then calculate each surplus at its relevant price and check the total against the gains-from-trade benchmark.

Quantity and prices. Solve $100-2Q-(10+Q)=9$, so $90-3Q=9$, $3Q=81$, and $Q=27$. Buyers pay $P_b=100-2(27)=\$46$. Sellers receive $P_s=10+27=\$37$, and the price difference is $\$46-\$37=\$9$, as required.

Consumer surplus. Demand’s intercept is $\$100$, so the height above the buyer price is $100-46=\$54$. Therefore $CS=\tfrac12(27)(54)=\$729$.

Producer surplus. The supply intercept is $\$10$, so the height above marginal cost at zero quantity and below the seller price is $37-10=\$27$. Therefore $PS=\tfrac12(27)(27)=\$364.50$.

Tax revenue and total surplus. Tax revenue is $tQ=9(27)=\$243$. Hence $TS=729+364.50+243=\$1{,}336.50$.

Efficient benchmark and check. Set demand equal to supply: $100-2Q=10+Q$, so $Q^*=30$ and the efficient price is $\$40$. At that quantity, consumer surplus is $\tfrac12(30)(100-40)=\$900$, and producer surplus is $\tfrac12(30)(40-10)=\$450$. Efficient total surplus is $\$1{,}350$. The lost three units form a triangle with base $30-27=3$ and height $9$, giving $DWL=\tfrac12(3)(9)=\$13.50$. The check agrees: $\$1{,}350-\$1{,}336.50=\$13.50$.

Common mistake avoided. The tax revenue $\$243$ is not the deadweight loss. It is a transfer included in total surplus; the deadweight loss is only the $\$13.50$ of gains from the three foregone units.

Worked Example: A Subsidy That Encourages Extra Trade

In a hypothetical market, inverse demand is $P_b=80-Q$ and inverse supply is $P_s=8+Q$. A per-unit subsidy gives sellers $\$8$ more than buyers pay. Find the equilibrium prices and quantity, consumer and producer surplus, subsidy expenditure, total surplus, and deadweight loss.

Method. Because buyers pay less than sellers receive, the wedge as defined here is negative: $t=P_b-P_s=-8$. Use that signed wedge in the equilibrium condition, and subtract the subsidy expenditure when computing total surplus.

Quantity and prices. Solve $80-Q-(8+Q)=-8$. This gives $72-2Q=-8$, so $2Q=80$ and $Q=40$. Buyers pay $P_b=80-40=\$40$; sellers receive $P_s=8+40=\$48$. The difference $40-48=-\$8$ confirms the signed wedge.

Private surpluses. Consumer surplus is $\tfrac12(40)(80-40)=\$800$. Producer surplus is $\tfrac12(40)(48-8)=\$800$.

Subsidy expenditure and total surplus. The government pays $\$8$ per unit, so expenditure is $8(40)=\$320$, corresponding to wedge revenue $tQ=-\$320$. Total surplus is $800+800-320=\$1{,}280$.

Efficient benchmark and check. Set $80-Q=8+Q$, giving $Q^*=36$ and price $\$44$. Efficient total surplus is consumer surplus $\tfrac12(36)(80-44)=\$648$ plus producer surplus $\tfrac12(36)(44-8)=\$648$, or $\$1{,}296$. The subsidy induces four extra units. At $Q=40$, willingness to pay is $\$40$ and marginal cost is $\$48$, a gap of $\$8$; the gap was zero at $Q=36$. Therefore $DWL=\tfrac12(4)(8)=\$16$, which also equals $\$1{,}296-\$1{,}280$.

Common mistake avoided. Adding the subsidy payment to total surplus would count a transfer as a social gain. The expenditure is $\$320$, so it enters the total-surplus calculation as $-\$320$.

Worked Example: A Different Supply Slope

For a hypothetical market, inverse demand is $P_b=70-Q$, inverse supply is $P_s=10+2Q$, and the per-unit tax wedge is $\$6$. Calculate both market prices, the three surplus components, total surplus, and deadweight loss.

Method. Equate the demand–supply price gap to the wedge. Since the supply curve rises by $\$2$ per unit and demand falls by $\$1$ per unit, the vertical gap between them closes by $\$3$ for each additional unit.

Quantity and prices. Solve $70-Q-(10+2Q)=6$, or $60-3Q=6$. Thus $3Q=54$ and $Q=18$. Buyers pay $70-18=\$52$, and sellers receive $10+2(18)=\$46$. The difference is $\$6$.

Surplus components. Consumer surplus is $\tfrac12(18)(70-52)=\tfrac12(18)(18)=\$162$. Producer surplus is $\tfrac12(18)(46-10)=\tfrac12(18)(36)=\$324$. Tax revenue is $6(18)=\$108$. Total surplus is $162+324+108=\$594$.

Efficient benchmark and check. Efficient trade satisfies $70-Q=10+2Q$, so $Q^*=20$ and the price is $\$50$. Efficient consumer surplus is $\tfrac12(20)(70-50)=\$200$; efficient producer surplus is $\tfrac12(20)(50-10)=\$400$. Thus efficient total surplus is $\$600$. The wedge reduces quantity by two units; the initial marginal-benefit/marginal-cost gap at $Q=18$ is $\$52-\$46=\$6$. Therefore $DWL=\tfrac12(2)(6)=\$6$, which matches $\$600-\$594$.

Common mistake avoided. Do not assume the supply curve has slope one when finding the quantity change. Here the demand and supply slopes together close the gap by $1+2=3$ dollars per unit, so the quantity falls by $6/3=2$, not by six units.

Common Calculation Traps

Using one price for both sides. Calculate consumer surplus using $P_b$ and producer surplus using $P_s$. A wedge means these prices are not interchangeable.
Giving subsidy expenditure the wrong sign. With $t=P_b-P_s<0$, wedge revenue $tQ$ is negative. Add this signed amount to $CS+PS$, or equivalently subtract the positive subsidy expenditure.
Calling the transfer the deadweight loss. A tax receipt or subsidy payment is a transfer under the assumptions used here. Deadweight loss is the surplus from trades lost or wrongly induced.
Using the wedge as the quantity change. The quantity response depends on the combined slopes of demand and supply. For linear curves, $|Q_t-Q^*|=|t|/(b+d)$, not simply $|t|$.
Taking a triangle from the wrong units. For a tax, the missing trades lie between $Q_t$ and $Q^*$. For a subsidy, the excess trades lie between $Q^*$ and $Q_t$. The triangle’s height is the demand–supply gap at the wedged quantity.

Key takeaway: Use the buyer price for demand and the seller price for supply. Add signed wedge revenue to consumer and producer surplus to obtain total surplus. Deadweight loss is the gains from trade omitted by the wedge-distorted quantity, not the tax revenue or subsidy expenditure.

Check Your Understanding

For each hypothetical market, use $t=P_b-P_s$. Check that your surplus calculation and deadweight loss agree with the efficient benchmark.

Inverse demand is $P_b=50-Q$, inverse supply is $P_s=10+Q$, and the per-unit tax is $\$4$. Find quantity, both prices, tax revenue, and deadweight loss. (Answer: $Q=18$, $P_b=\$32$, $P_s=\$28$, tax revenue $=\$72$, $DWL=\$4$.)
Inverse demand is $P_b=72-Q$, inverse supply is $P_s=12+Q$, and sellers receive a subsidy of $\$6$ per unit. Find quantity, both prices, subsidy expenditure, and deadweight loss. (Answer: $Q=33$, $P_b=\$39$, $P_s=\$45$, expenditure $=\$198$, $DWL=\$9$.)
Inverse demand is $P_b=84-2Q$, inverse supply is $P_s=12+Q$, and the tax wedge is $\$9$. Calculate equilibrium quantity and prices, total surplus including tax revenue, and deadweight loss. (Answer: $Q=21$, $P_b=\$42$, $P_s=\$33$, total surplus $=\$850.50$, $DWL=\$13.50$.)
Inverse demand is $P_b=90-2Q$, inverse supply is $P_s=18+Q$, and a subsidy gives sellers $\$6$ more than buyers pay. Calculate the equilibrium quantity, prices, consumer surplus, producer surplus, net wedge revenue, total surplus, and deadweight loss. (Answer: $Q=26$, $P_b=\$38$, $P_s=\$44$, $CS=\$676$, $PS=\$338$, net wedge revenue $=-\$156$, total surplus $=\$858$, $DWL=\$6$.)

What You'll Learn

Why Two Prices Matter

Equilibrium with a Constant Wedge

Surplus and the Wedge Transfer

Why Deadweight Loss Is the Lost Gains from Trade

Worked Example: A Tax Wedge with Linear Curves

Worked Example: A Tax Wedge with Linear Curves

Worked Example: A Subsidy That Encourages Extra Trade

Worked Example: A Subsidy That Encourages Extra Trade

Worked Example: A Different Supply Slope

Worked Example: A Different Supply Slope

Common Calculation Traps

Check Your Understanding

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