Worked Export Subsidy Diagrams

Reading an Export-Subsidy Diagram

An export subsidy encourages domestic firms to sell abroad by paying them for each unit exported. In a small-country model, the country takes the world price as given: its policy does not change that price. If exports are possible both before and after the subsidy, the subsidy raises the domestic price above the world price. That price change affects domestic production and consumption, while the government pays for the subsidized exports.

The essential diagram-reading task is to distinguish production from consumption. At a given domestic price, domestic supply is production and domestic demand is consumption. Exports are the difference between them when supply exceeds demand. A positive subsidy does not guarantee that exports exist: the active-export condition must be checked at the relevant prices.

Definition: In the small-country, competitive-market model, let $P_w$ be the fixed world price and $s$ a per-unit subsidy paid only on exports. If the country exports both before and after the policy, the initial domestic price is $P_0=P_w$, and the post-subsidy domestic price is $P_1=P_w+s$. At each price $P_j$, production is $Q_s(P_j)$, consumption is $Q_d(P_j)$, and exports are $X_j=Q_s(P_j)-Q_d(P_j)$. The active-export condition is $X_j>0$.

This result assumes competitive domestic markets, no transport costs or other trade barriers, a fixed world price, and a subsidy paid only on units exported. It also assumes domestic firms can sell at the domestic price or export and receive the world price plus the subsidy. If the country imports instead, an export-only subsidy does not make its import price rise according to this formula; the active-export case must not be applied automatically.

A Reliable Calculation Sequence

For a standard diagram problem, calculate the initial and final market outcomes first, then evaluate the government budget and surplus areas. Keeping these steps separate helps prevent a common error: calculating a subsidy bill before confirming the number of exports that qualify for payment.

Find the initial quantities.
Set the domestic price equal to $P_w$. Evaluate demand and supply, then check that initial exports $X_0=Q_{s0}-Q_{d0}$ are positive.

Apply the subsidy to the price.
For the active-export case, set $P_1=P_w+s$. Evaluate post-policy supply and demand and check that $X_1=Q_{s1}-Q_{d1}>0$.

Calculate the fiscal cost.
The government pays $s$ on each exported unit, so its expenditure is $sX_1$, not $sQ_{s1}$.

Calculate welfare changes.
Use the price change and the initial and final quantities to find consumer- and producer-surplus changes. Subtract the subsidy expenditure when calculating the change in national welfare.

Surplus Accounting and Deadweight Loss

For linear demand and supply curves, a change in price changes surplus by the area of a trapezoid. Consumers face a price increase of $s$, so their surplus falls. Producers receive a higher price, so their surplus rises. The government’s outlay is a separate budget cost; it must not be omitted from national welfare accounting.

\Delta CS=-\frac{s}{2}(Q_{d0}+Q_{d1}),\qquad \Delta PS=\frac{s}{2}(Q_{s0}+Q_{s1}),\qquad G=sX_1

Here $G$ is government expenditure, $\Delta CS$ is the change in consumer surplus, and $\Delta PS$ is the change in producer surplus. The change in national surplus in this partial-equilibrium model is $\Delta W=\Delta CS+\Delta PS-G$. These trapezoid formulas require linear curves over the quantity ranges being considered. With nonlinear curves, the exact changes are areas under or above those curves, rather than trapezoids based only on endpoint quantities.

Derivation: On a linear demand curve, the loss in consumer surplus is the area between the initial and final prices over the quantities consumers demand at those prices. That area is a trapezoid with height $s$ and parallel sides $Q_{d0}$ and $Q_{d1}$, giving $-s(Q_{d0}+Q_{d1})/2$. The producer-surplus gain is likewise a trapezoid with height $s$ and sides $Q_{s0}$ and $Q_{s1}$, giving $s(Q_{s0}+Q_{s1})/2$. Subtracting the final subsidy payment $s(Q_{s1}-Q_{d1})$ gives $$ \Delta W =-\frac{s}{2}(Q_{d0}+Q_{d1}) +\frac{s}{2}(Q_{s0}+Q_{s1}) -s(Q_{s1}-Q_{d1}) =-\frac{s}{2}\left[(Q_{d0}-Q_{d1})+(Q_{s1}-Q_{s0})\right]. $$ Thus the welfare loss equals the sum of the consumption-distortion and production-distortion triangles. The consumption triangle has area $s(Q_{d0}-Q_{d1})/2$, and the production triangle has area $s(Q_{s1}-Q_{s0})/2$. This result presumes the fixed-world-price, active-export setup and counts the government outlay dollar-for-dollar; it does not include any additional cost of raising public funds.

The triangles arise because the subsidy changes decisions away from the quantities that would prevail at the world price. Domestic firms produce more, even though the marginal cost of the added output exceeds the world price. Domestic consumers buy less, even though the value they place on the units they stop buying exceeds the world price. The transfers among consumers, producers, and government are not themselves deadweight loss; the two distorted quantity margins are.

Worked Example: A Subsidy with Positive Exports Before and After

A small country has domestic demand $Q_d=100-P$ and domestic supply $Q_s=20+2P$, where $P$ is dollars per unit and quantities are units per period. The world price is $P_w=\$30$ per unit. The government introduces a subsidy of $s=\$4$ per exported unit. Find prices, quantities, exports, fiscal cost, surplus changes, and deadweight loss.

Method and initial check. At the world price, $P_0=30$. Demand is $Q_{d0}=100-30=70$, and supply is $Q_{s0}=20+2(30)=80$. Initial exports are $X_0=80-70=10$, so the active-export assumption holds.

Post-subsidy quantities. The domestic price is $P_1=30+4=\$34$. Demand is $Q_{d1}=100-34=66$; supply is $Q_{s1}=20+2(34)=88$. Exports are $X_1=88-66=22$, which are positive.

Government expenditure. The subsidy is paid on 22 exported units, so $G=4(22)=\$88$.

Surplus changes. The consumer-surplus change is $\Delta CS=-4(70+66)/2=-\$272$. The producer-surplus change is $\Delta PS=4(80+88)/2=\$336$. Therefore, $\Delta W=-272+336-88=-\$24$.

Deadweight-loss check. Production rises by $88-80=8$, giving a production triangle of $4(8)/2=\$16$. Consumption falls by $70-66=4$, giving a consumption triangle of $4(4)/2=\$8$. Total deadweight loss is $16+8=\$24$, matching the surplus calculation. The common mistake to avoid is charging the subsidy on all 88 units of production: only the 22 exported units receive it.

Worked Example: Separating the Two Distortion Triangles

Suppose demand is $Q_d=150-3P$, supply is $Q_s=15+3P$, the world price is $P_w=\$25$ per unit, and the export subsidy is $s=\$3$ per unit. Quantities are units per period. Calculate the post-policy trade flow, budget cost, and welfare loss, and verify the result using surplus changes.

Initial outcome. At $P_0=25$, demand is $Q_{d0}=150-3(25)=75$, while supply is $Q_{s0}=15+3(25)=90$. Initial exports are $90-75=15$, so the country is exporting before the subsidy.

Post-subsidy outcome. The domestic price is $P_1=25+3=\$28$. Demand is $Q_{d1}=150-3(28)=66$, and supply is $Q_{s1}=15+3(28)=99$. Thus exports are $99-66=33$, still positive.

Fiscal cost. The expenditure is $G=3(33)=\$99$.

Surplus changes and check. Consumer surplus changes by $-3(75+66)/2=-\$211.50$. Producer surplus changes by $3(90+99)/2=\$283.50$. Net welfare changes by $-211.50+283.50-99=-\$27$. Supply increases by 9 units, so the production triangle is $3(9)/2=\$13.50$. Demand falls by 9 units, so the consumption triangle is also $3(9)/2=\$13.50$. Their sum is $\$27$, agreeing with the surplus accounting. The method avoids treating the entire subsidy payment as deadweight loss: the payment is a fiscal outlay, but its transfer component is offset in the producer-surplus change.

Worked Example: Checking Exports Before Calculating Welfare

A hypothetical market has demand $Q_d=140-2P$ and supply $Q_s=20+4P$, with price in dollars per unit and quantities in units per period. The world price is $P_w=\$25$ and the export subsidy is $s=\$5$ per exported unit. Compute the outcomes and compare the fiscal cost with the welfare loss.

Check the initial trade flow. At $P_0=25$, demand is $Q_{d0}=140-2(25)=90$, and supply is $Q_{s0}=20+4(25)=120$. Initial exports are $120-90=30>0$.

Find the post-policy quantities. The domestic price becomes $P_1=25+5=\$30$. Demand is $Q_{d1}=140-2(30)=80$; supply is $Q_{s1}=20+4(30)=140$. Exports are $X_1=140-80=60>0$.

Calculate fiscal and surplus effects. Government expenditure is $G=5(60)=\$300$. Consumer surplus falls by $5(90+80)/2=\$425$, while producer surplus rises by $5(120+140)/2=\$650$. National surplus therefore changes by $-425+650-300=-\$75$.

Verify with triangles. Production increases by 20 units, so the production triangle is $5(20)/2=\$50$. Consumption decreases by 10 units, so the consumption triangle is $5(10)/2=\$25$. Deadweight loss is $\$50+\$25=\$75$, consistent with the surplus calculation. The fiscal outlay is $\$300$, not $\$75$: the first is the government’s payment, while the second is the net efficiency loss after accounting for surplus transfers.

Common Exam Traps and Interpretation

Export-subsidy diagrams are especially vulnerable to mistakes about the direction of trade. Calculate exports as supply minus demand, not demand minus supply. If that difference is negative, the country imports at that price; an export-only subsidy cannot be treated as if it were paid on those imports. In that case, the standard active-export calculation is inapplicable, and the assumed domestic-price rise may not occur.

Using the wrong domestic price. In the active-export small-country case, the post-subsidy domestic price is $P_w+s$, not $P_w-s$.
Paying the subsidy on production. The government pays on exports $X_1=Q_{s1}-Q_{d1}$, not on all domestic production $Q_{s1}$.
Skipping an export check. Verify $X_0>0$ and $X_1>0$ before applying the active-export formulas. If either check fails, reconsider the case rather than forcing the calculation.
Calling the full fiscal bill deadweight loss. Calculate $\Delta CS+\Delta PS-G$, or add the production and consumption triangles. These methods should agree under the stated linear, small-country assumptions.

The same careful separation of prices, quantities, fiscal cost, and surplus areas is useful whenever a policy changes the difference between domestic and world prices. For export subsidies, a final sign and consistency check is particularly valuable: positive exports must agree with domestic supply exceeding domestic demand at the price used in the calculation.

Check Your Understanding

For each problem, use the small-country model and check exports before applying export-subsidy formulas. Prices are dollars per unit and quantities are units per period unless stated otherwise.

A market has $Q_d=90-2P$, $Q_s=10+3P$, $P_w=\$20$, and an export subsidy of $\$2$ per unit. Find initial and final exports and the government’s final expenditure. (Answer: Initial exports $20$, final exports $30$, and expenditure $\$60$.)
Demand is $Q_d=130-2P$, supply is $Q_s=10+2P$, the world price is $\$35$, and the subsidy is $\$5$ per exported unit. Find the final domestic price, production, consumption, exports, and subsidy expenditure. (Answer: Price $\$40$, production $90$, consumption $50$, exports $40$, and expenditure $\$200$.)
Let $Q_d=160-2P$, $Q_s=10+3P$, $P_w=\$35$, and $s=\$4$. Calculate initial and final exports, fiscal cost, and the two deadweight-loss triangles. (Answer: Exports rise from $25$ to $45$, fiscal cost is $\$180$, production-distortion loss is $\$24$, consumption-distortion loss is $\$16$, and total deadweight loss is $\$40$.)
Demand is $Q_d=110-P$, supply is $Q_s=20+2P$, and the world price is $\$32$. What per-unit export subsidy produces exports of 18 units? Then find the post-subsidy quantities and fiscal cost. (Answer: $s=\$4$, domestic price $\$36$, production $92$, consumption $74$, and fiscal cost $\$72$.)
A candidate calculates the final subsidy bill as $sQ_{s1}$ in a problem where the subsidy is paid only on exported units. State the correct expression and identify the quantity difference needed to calculate it. (Answer: $G=sX_1=s(Q_{s1}-Q_{d1})$; subtract domestic consumption from domestic production to find exports.)

What You'll Learn