Free-Trade Benchmark for an Importing Economy

Introduction

A country can import a good even when its domestic producers continue to supply part of the market. If the world price is below the price that would balance domestic supply and demand in isolation, buyers want more of the good while domestic firms are willing to produce less. Imports fill the difference.

The free-trade benchmark makes this comparison precise. It identifies domestic consumption, domestic production, and imports at the world price, then measures how consumers and producers fare relative to autarky, the situation with no trade. This benchmark is useful for separating the basic effects of trade from the effects of policies that may later alter the domestic price.

The Importing-Economy Model

Assume a single good and a competitive domestic market. Write $Q_d(P)$ for domestic quantity demanded and $Q_s(P)$ for domestic quantity supplied at price $P$. In autarky, with no trade, the market-clearing price $P_A$ satisfies $Q_d(P_A)=Q_s(P_A)$; the corresponding quantity is $Q_A$. Under free trade, assume the economy is small enough to take the world price $P_w$ as given, and that the good can be traded without transport costs, tariffs, or other barriers. The domestic price then equals $P_w$.

Definition: If $P_w<P_A$, the economy is an importing economy in this model. At the free-trade price, domestic consumption is $Q_d(P_w)$, domestic production is $Q_s(P_w)$, and imports equal the excess of consumption over production: $$ M=Q_d(P_w)-Q_s(P_w). $$ For a positive quantity of imports, $Q_d(P_w)>Q_s(P_w)$. The model assumes no inventory changes or re-exports, so imports supply the gap between current domestic demand and domestic production.

For numerical calculations, suppose domestic demand and supply are linear over the relevant price range:

Q_d(P)=a-bP,\qquad Q_s(P)=c+dP,\qquad b>0,\ d>0.

Here $a,b,c,d$ are constants chosen so the calculated prices and quantities are economically meaningful. Solve $Q_d(P_A)=Q_s(P_A)$ first. Then, if $P_w<P_A$, substitute the world price into each curve and subtract domestic production from domestic consumption to find imports. The lower price increases quantity demanded and decreases quantity supplied, which is why the difference is positive.

Changes in Consumer and Producer Surplus

Consumer surplus is the area below the demand curve and above the market price, up to the quantity purchased. Producer surplus is the area above the supply curve and below the market price, up to the quantity sold. These measures compare willingness to pay or willingness to supply with the price actually received or paid. With linear curves, the changes between two prices are trapezoid areas.

Result: When a linear-demand, linear-supply market moves from autarky price $P_A$ to a lower free-trade price $P_w$, the consumer-surplus gain and producer-surplus loss are $$ \Delta CS=\frac{P_A-P_w}{2}\left[Q_d(P_A)+Q_d(P_w)\right], $$ $$ \text{Producer-surplus loss}=\frac{P_A-P_w}{2}\left[Q_s(P_A)+Q_s(P_w)\right]. $$ These formulas measure changes relative to autarky, not the total levels of consumer or producer surplus after trade.

The net surplus gain in this model is the consumer-surplus gain less the producer-surplus loss. To see why it equals a simple expression involving imports, subtract the two trapezoid areas. At the autarky price, demand and supply are equal: $Q_d(P_A)=Q_s(P_A)=Q_A$. Those autarky quantities cancel, leaving

\begin{aligned} \Delta CS-\text{Producer-surplus loss} &=\frac{P_A-P_w}{2}\left[Q_d(P_A)+Q_d(P_w)-Q_s(P_A)-Q_s(P_w)\right]\\ &=\frac{P_A-P_w}{2}\left[Q_d(P_w)-Q_s(P_w)\right]\\ &=\frac{P_A-P_w}{2}M. \end{aligned}

This is a short derivation of the net gain: at the autarky price, the two domestic quantities match; at the lower world price, their difference is imports. With linear curves, the result is the area of the triangle between the autarky price and the world price over the import quantity. It is positive when the economy imports and the price falls. This conclusion relies on the model’s assumptions, including competitive markets and no trade costs or external effects. If curves are nonlinear, the trapezoid formulas need not apply, though surplus changes can still be measured as areas under or above the relevant curves.

Working the Free-Trade Benchmark

Worked Example: Finding Imports and Surplus Changes

A hypothetical domestic market has demand $Q_d(P)=120-2P$ and supply $Q_s(P)=20+2P$. Price is in dollars per unit and quantity is in units. The economy is small and faces a world price of $P_w=\$15$, with no trade costs or barriers. Find the autarky outcome, the free-trade quantities and imports, and the changes in consumer and producer surplus.

Method. First equate domestic demand and supply to find the autarky benchmark. Then evaluate both curves at the world price and use trapezoid areas for the surplus changes.

Autarky. Set demand equal to supply: $120-2P=20+2P$. Thus $100=4P$, so $P_A=\$25$. Substituting into either curve gives $Q_A=120-2(25)=70$ units; supply confirms $20+2(25)=70$.

Free trade. Since $\$15<\$25$, this is an importing economy. Demand is $Q_d(15)=120-2(15)=90$ units. Supply is $Q_s(15)=20+2(15)=50$ units. Therefore imports are $M=90-50=40$ units. The goods available to domestic buyers are 50 domestically produced units plus 40 imported units, or 90 units in total.

Consumer gain. The price falls by $\$25-\$15=\$10$, and demand rises from 70 to 90 units. Hence $\Delta CS=(10/2)(70+90)=5(160)=\$800$.

Producer loss. Domestic supply falls from 70 to 50 units. The producer-surplus loss is $(10/2)(70+50)=5(120)=\$600$.

Check and common mistake. The net gain is $\$800-\$600=\$200$, which agrees with $(10/2)(40)=\$200$. A common error is to count all 90 units consumed as imports. Only the 40-unit gap between consumption and domestic production is imported.

Worked Example: Keeping Demand and Supply Changes Separate

In a hypothetical market, $Q_d(P)=200-4P$ and $Q_s(P)=20+2P$, with price in dollars per unit and quantity in units. A small importing economy can buy at $P_w=\$20$. Calculate the autarky outcome, free-trade imports, the consumer-surplus gain, the producer-surplus loss, and the net gain.

Method. Calculate the autarky equilibrium before using the world price. For the welfare comparison, use the change in demand for consumers and the change in supply for producers; they are not interchangeable.

Autarky. Solve $200-4P=20+2P$. This gives $180=6P$, so $P_A=\$30$. Demand and supply are both $200-4(30)=80$ and $20+2(30)=80$, so $Q_A=80$ units.

Free-trade quantities. The world price is below autarky. At $P_w=\$20$, demand is $200-4(20)=120$ units and domestic supply is $20+2(20)=60$ units. Imports are $120-60=60$ units.

Consumer and producer changes. The price decrease is $\$30-\$20=\$10$. Consumer surplus rises by $(10/2)(80+120)=5(200)=\$1{,}000$. Producer surplus falls by $(10/2)(80+60)=5(140)=\$700$.

Check and common mistake. The net gain is $\$1{,}000-\$700=\$300$, also equal to $(10/2)(60)=\$300$. Do not use the demand increase of 40 units as imports: imports are 60 units because domestic production also falls by 20 units.

Worked Example: Interpreting Quantities Reported in Thousands

A hypothetical economy has $Q_d(P)=150-3P$ and $Q_s(P)=10+P$, where quantities are measured in thousands of units and prices are dollars per unit. Its world price is $P_w=\$25$. Find the autarky price and quantity, free-trade consumption and production, imports, and the surplus changes. Report surplus in thousands of dollars.

Method. Solve for autarky and evaluate each curve at the world price. Since quantity is in thousands, multiplying a price in dollars by a quantity in thousands gives surplus in thousands of dollars.

Autarky. Equating the curves gives $150-3P=10+P$, or $140=4P$, so $P_A=\$35$. The quantity is $150-3(35)=45$ thousand units; supply gives $10+35=45$ thousand units as a check.

Free trade. Since $\$25<\$35$, the economy imports. Consumption is $Q_d(25)=150-3(25)=75$ thousand units. Domestic production is $Q_s(25)=10+25=35$ thousand units. Imports are $75-35=40$ thousand units.

Surplus changes. The price falls by $\$10$. The consumer-surplus gain is $(10/2)(45+75)=5(120)=600$ thousand dollars. The producer-surplus loss is $(10/2)(45+35)=5(80)=400$ thousand dollars. The net gain is $600-400=200$ thousand dollars.

Check and common mistake. The independent net-gain check is $(10/2)(40)=200$ thousand dollars. Reporting the surplus figures as dollars rather than thousands of dollars would understate them by a factor of 1,000.

Assumptions, Interpretation, and Exam Traps

The benchmark is a partial-equilibrium model: it studies one domestic market while taking other conditions as fixed. Its price-taking assumption means the economy does not change the world price by its own purchases. With frictionless trade, domestic buyers and sellers face that same price. If transport costs, tariffs, quotas, or other barriers create a difference between the world price and the domestic price, this free-trade calculation does not describe the resulting market. Those policies require a separate price or quantity condition.

A positive import quantity is not the same as a fall in domestic production. Both changes occur when the world price is below autarky, but imports equal the rise in consumption plus the fall in production relative to autarky. In symbols, $M=Q_d(P_w)-Q_s(P_w)$, not simply $Q_d(P_w)-Q_A$. This distinction is a frequent source of calculation errors.

Check the price comparison. First find $P_A$. A world price below $P_A$ implies imports in this model; a higher world price does not.
Keep the quantities distinct. Free-trade consumption is demand at $P_w$; domestic production is supply at $P_w$; imports are their difference.
Use the correct endpoints for each area. Consumer surplus uses demand quantities at the two prices. Producer surplus uses supply quantities at those prices.
Respect the units. If quantities are in thousands, surplus calculated in dollars per unit times quantity is in thousands of dollars.
Check the net result. For linear curves, consumer gain minus producer loss should equal $(P_A-P_w)M/2$. A mismatch signals an arithmetic or setup error.

In this benchmark, consumers gain because they pay a lower price and consume more. Domestic producers lose because they receive a lower price and sell less. The consumer gain exceeds the producer loss, leaving a positive net gain under the model’s assumptions. This distributional comparison, aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus, does not imply that every individual benefits or that adjustment costs are absent; it is a market-surplus comparison for the stated model.

Check Your Understanding

For each problem, calculate the requested quantities before checking the answer. Unless stated otherwise, assume a small economy, linear domestic demand and supply, and no trade costs or barriers.

A hypothetical market has $Q_d(P)=90-P$ and $Q_s(P)=10+P$. The world price is $\$30$ per unit. Find the autarky price and quantity, free-trade consumption and production, and imports. (Answer: $P_A=\$40$, $Q_A=50$, consumption $=60$, production $=40$, imports $=20$ units.)
Demand is $Q_d(P)=160-2P$, supply is $Q_s(P)=10+P$, and the world price is $\$30$. Find the autarky outcome, imports, consumer-surplus gain, producer-surplus loss, and net gain. (Answer: $P_A=\$50$, $Q_A=60$, imports $=60$, consumer gain $=\$1{,}600$, producer loss $=\$1{,}000$, net gain $=\$600$.)
A market has $Q_d(P)=240-3P$ and $Q_s(P)=24+P$. At a world price of $\$34$, calculate autarky and free-trade quantities, imports, and both surplus changes. (Answer: $P_A=\$54$, $Q_A=78$, consumption $=138$, production $=58$, imports $=80$, consumer gain $=\$2{,}160$, producer loss $=\$1{,}360$.)
For $Q_d(P)=260-4P$ and $Q_s(P)=20+2P$, the world price is $\$25$. Find the autarky price and quantity, free-trade imports, consumer-surplus gain, producer-surplus loss, and net gain. (Answer: $P_A=\$40$, $Q_A=100$, imports $=90$, consumer gain $=\$1{,}950$, producer loss $=\$1{,}275$, net gain $=\$675$.)
Demand is $Q_d(P)=120-P$ and supply is $Q_s(P)=20+P$. A world price of $\$40$ applies. Find imports and the consumer gain, producer loss, and net gain relative to autarky; then state the highest world price that still results in positive imports. (Answer: imports $=20$ units, consumer gain $=\$750$, producer loss $=\$650$, net gain $=\$100$; positive imports require $P_w<\$50$, with zero imports at $\$50$.)

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