Producer Surplus Changes When Price Changes

Why Separate the Two Effects?

When market price changes, producer surplus changes for two related reasons. Sellers receive a different price for units they continue to sell, and the quantity supplied may change. Separating those effects makes the calculation easier to interpret and helps prevent counting the same units twice.

This tutorial considers a price change while the supply curve is unchanged. The new quantity is determined by that same supply curve at the new price. The focus is the change in producer surplus, not the cause of the price change.

Definition: In a competitive market with price-taking sellers, producer surplus at price $P$ and quantity $Q$ is the area between the market price and the supply curve, from zero to $Q$. If $S(q)$ is the inverse supply curve, so that $S(q)$ gives the price required to supply quantity $q$, then $$ PS(P)=\int_0^{Q(P)}[P-S(q)]\,dq. $$ This measure is revenue less variable production cost under the usual interpretation of supply as marginal cost. It is not necessarily accounting profit: fixed costs are not subtracted.

Assume the market is competitive, the supply curve does not shift between the two observations, and the quantities supplied at the two prices can be read from that curve. Write $P_0$ and $Q_0$ for the initial price and quantity, and $P_1$ and $Q_1$ for the new price and quantity. For a smooth supply curve, the surplus areas can be calculated by integration. For a discrete schedule of units, add the surplus on each unit instead.

The Common-Quantity Decomposition

The units sold in both periods number $\min(Q_0,Q_1)$. The price effect on those common units is the price change multiplied by that quantity. What remains is the surplus gained or lost on units whose production changes.

Theorem: If price rises, so $P_1>P_0$ and $Q_1\geq Q_0$, then $$ PS_1-PS_0=(P_1-P_0)Q_0+\int_{Q_0}^{Q_1}[P_1-S(q)]\,dq. $$ The first term is the price effect on units sold in both periods; the integral is the surplus on additional units at the new price. If price falls, so $P_1<P_0$ and $Q_1\leq Q_0$, then $$ PS_1-PS_0=(P_1-P_0)Q_1-\int_{Q_1}^{Q_0}[P_0-S(q)]\,dq. $$ The first term is the price effect on units still sold; the subtracted integral is the old-price surplus lost on units no longer supplied.

The signs follow the economic interpretation. A price rise makes the price effect positive and, for an upward-sloping supply curve, added units earn nonnegative surplus at the new price. A price fall makes the price effect negative; the lost units had earned nonnegative surplus at the old price, so their removal reduces surplus further.

Why the Decomposition Holds

The result follows by splitting the producer-surplus areas at the quantity sold in both periods. For a price rise, split the new area at $Q_0$. On the first $Q_0$ units, the supply terms cancel when the old surplus is subtracted, leaving only the price difference. The remaining area belongs to additional units:

\begin{aligned} PS_1-PS_0 &=\int_0^{Q_0}[P_1-S(q)]\,dq +\int_{Q_0}^{Q_1}[P_1-S(q)]\,dq -\int_0^{Q_0}[P_0-S(q)]\,dq\\ &=(P_1-P_0)Q_0+\int_{Q_0}^{Q_1}[P_1-S(q)]\,dq. \end{aligned}

For a price fall, split the old area at $Q_1$. The common units contribute the new price minus the old price, while the units between $Q_1$ and $Q_0$ appear only in the old area:

\begin{aligned} PS_1-PS_0 &=\int_0^{Q_1}[P_1-S(q)]\,dq -\int_0^{Q_1}[P_0-S(q)]\,dq -\int_{Q_1}^{Q_0}[P_0-S(q)]\,dq\\ &=(P_1-P_0)Q_1-\int_{Q_1}^{Q_0}[P_0-S(q)]\,dq. \end{aligned}

This argument also explains why the price effect uses the quantity sold in both periods, not the larger quantity. The additional or removed units are accounted for separately.

Worked Examples

Worked Example: A Price Rise with Linear Supply

A hypothetical market has inverse supply $S(q)=20+2q$, with price measured in dollars per unit. The market price rises from $P_0=\$30$ to $P_1=\$38$. Find the initial and new producer surplus, then decompose the change.

Method. Find each quantity by setting the market price equal to the inverse supply price. Then use the common-quantity decomposition for a price rise.

At the initial price, $30=20+2Q_0$, so $Q_0=5$. At the new price, $38=20+2Q_1$, so $Q_1=9$. The price change is $38-30=\$8$.

The initial surplus is $$ PS_0=\int_0^5(30-20-2q)\,dq =\left[10q-q^2\right]_0^5 =50-25=\$25. $$ The new surplus is $$ PS_1=\int_0^9(38-20-2q)\,dq =\left[18q-q^2\right]_0^9 =162-81=\$81. $$ Thus the total change is $81-25=\$56$.

The price effect on the five units sold in both periods is $8\times5=\$40$. The additional-unit effect is $$ \int_5^9(38-20-2q)\,dq =\left[18q-q^2\right]_5^9 =81-65=\$16. $$ Together, $40+16=\$56$, which agrees with the direct subtraction of the surplus totals.

Because inverse supply is linear, the additional-unit area is a triangle with base $9-5=4$ and height $\$8$. Its area is $\frac12(4)(8)=\$16$, matching the integral. The calculation is checked by both methods. No rounding is needed.

Worked Example: A Price Fall with Nonlinear Supply

A hypothetical service market has inverse supply $S(q)=10+q^2$. Price falls from $P_0=\$26$ to $P_1=\$19$. Find the producer-surplus change and its two components.

Method. Solve for quantities at each price, then use the price-fall decomposition. The supply curve is nonlinear, so calculate the quantity-adjustment area by integration rather than assuming it is a triangle.

At the old price, $26=10+Q_0^2$, giving $Q_0=4$. At the new price, $19=10+Q_1^2$, giving $Q_1=3$. The price change is $19-26=-\$7$.

The initial producer surplus is $$ PS_0=\int_0^4(26-10-q^2)\,dq =\left[16q-\frac{q^3}{3}\right]_0^4 =64-\frac{64}{3} =\frac{128}{3} =\$42.67. $$ The new producer surplus is $$ PS_1=\int_0^3(19-10-q^2)\,dq =\left[9q-\frac{q^3}{3}\right]_0^3 =27-9 =\$18.00. $$ The total change is $18-\frac{128}{3}=-\frac{74}{3}$, or $-\$24.67$ rounded to the nearest cent.

The price effect on the three units still supplied is $(-7)(3)=-\$21$. The old surplus on the one unit of quantity removed is $$ \int_3^4[26-(10+q^2)]\,dq =\left[16q-\frac{q^3}{3}\right]_3^4 =\frac{128}{3}-39 =\frac{11}{3}. $$ That surplus is lost, so the quantity-adjustment effect is $-\frac{11}{3}$, or $-\$3.67$ to the nearest cent. The components sum to $-21-\frac{11}{3}=-\frac{74}{3}$, agreeing with the direct calculation.

Worked Example: A Price Rise with Discrete Units

A hypothetical producer can supply five successive units at marginal costs of $\$4,\$7,\$9,\$12,$ and $\$16$ per unit. Each unit is supplied if the market price is at least its marginal cost. Price rises from $\$10$ to $\$14$. Find the change in producer surplus and separate its components.

Method. Count the units supplied at each price and sum the price-minus-marginal-cost surplus unit by unit. This discrete calculation is the counterpart of measuring areas under a continuous supply curve.

At $\$10$, the first three units are supplied because their costs are $\$4,\$7,$ and $\$9$; the fourth costs $\$12$, so $Q_0=3$. Initial surplus is $(10-4)+(10-7)+(10-9)=6+3+1=\$10$.

At $\$14$, the first four units are supplied and the fifth is not, so $Q_1=4$. New surplus is $(14-4)+(14-7)+(14-9)+(14-12)=10+7+5+2=\$24$. The total change is $24-10=\$14$.

The price effect on the three continuing units is $(14-10)\times3=\$12$. The fourth unit is the quantity adjustment and earns $14-12=\$2$ at the new price. Thus $12+2=\$14$, matching the direct calculation. No rounding is needed.

When Is the Quantity Effect a Triangle?

A triangle shortcut is valid under a specific condition: inverse supply must be linear over the quantity interval that changes, and the price endpoints must correspond to $Q_0$ and $Q_1$ on that supply curve. For a price rise, the surplus margin on the added units falls linearly from $P_1-P_0$ at $Q_0$ to zero at $Q_1$. For a price fall, the old-price surplus on the removed units rises linearly from zero at $Q_0$ to $P_0-P_1$ at $Q_1$. In either case, the magnitude of the quantity-adjustment effect is then

\frac{1}{2}|P_1-P_0|\,|Q_1-Q_0|.

For nonlinear supply, the margin need not vary linearly, so the area need not be a triangle. Use the appropriate integral or add the individual unit surpluses. The price effect still uses the common quantity.

Common Mistake: Using the Larger Quantity for the Price Effect

In the linear-supply example, using the new quantity for the price effect gives $8\times9=\$72$. Adding the correct $\$16$ surplus on the extra units would produce $\$88$, not the actual $\$56$. The price effect has then been counted on the four additional units, even though those units are already included in the quantity-adjustment calculation.

The correction is to use $Q_0=5$, the quantity common to both periods, for the price effect: $8\times5=\$40$. Then add the $\$16$ surplus on the four added units. For a price fall, use $Q_1$, the smaller quantity, for the common-unit price effect, and subtract the surplus lost on units no longer supplied.

Key takeaway: First identify the units sold in both periods. Multiply their quantity by the price change. Then separately calculate the surplus gained on added units or lost on removed units, using the price applicable to those units in the decomposition.

Check Your Understanding

For each problem, find the producer-surplus change and identify its price and quantity components.

A hypothetical market has inverse supply $S(q)=6+2q$. Price rises from $\$14$ to $\$18$. Find both quantities and the two components of the surplus change. (Answer: $Q_0=4,\ Q_1=6$; price effect $+\$16$, quantity effect $+\$4$, total change $+\$20$.)
A hypothetical market has inverse supply $S(q)=5+q$. Price falls from $\$17$ to $\$13$. Find the common-quantity price effect and the surplus lost on removed units. (Answer: $Q_0=12,\ Q_1=8$; price effect $-\$32$, quantity effect $-\$8$, total change $-\$40$.)
A hypothetical market has inverse supply $S(q)=4+q^2$. Price rises from $\$13$ to $\$20$. Use integration to find the price effect, the surplus on added units, and the total change. (Answer: $Q_0=3,\ Q_1=4$; price effect $+\$21$, quantity effect $+\frac{11}{3}$ dollars, total change $+\frac{74}{3}$ dollars, or $+\$24.67$ to the nearest cent.)
A producer’s successive units have marginal costs $\$3,\$5,\$8,\$11,$ and $\$15$. Each unit is supplied when price is at least its cost. Price rises from $\$9$ to $\$12$. Compute the initial and new surplus, then decompose the change. (Answer: $Q_0=3,\ Q_1=4$; initial surplus $\$11$, new surplus $\$21$; price effect $+\$9$, quantity effect $+\$1$, total change $+\$10$.)
A hypothetical market has inverse supply $S(q)=4+q^2$. Price rises from $\$13$ to $\$29$. Calculate the initial and new surplus, and verify the decomposition using the common quantity. (Answer: $Q_0=3,\ Q_1=5$; initial surplus $\$18$, new surplus $\frac{250}{3}$ dollars; price effect $+\$48$, quantity effect $+\frac{52}{3}$ dollars, total change $+\frac{196}{3}$ dollars, or $+\$65.33$ to the nearest cent.)

What You'll Learn