How Terms of Trade Divide Gains Between Countries

Introduction

When two countries trade at an exchange ratio between their opportunity costs, both can gain. But a mutually beneficial exchange ratio does not imply that each country receives half of the gains. The position of the ratio within the mutually beneficial range determines how the gains are divided: a ratio closer to one country’s opportunity cost generally gives that country a smaller gain per unit traded.

This tutorial compares exchange ratios while holding the production possibilities and quantity traded fixed. The calculations measure gains in units of one good that each country could have produced instead. These are output-equivalent gains, not a complete measure of well-being.

Set Up the Exchange Ratio

Consider two countries, A and B, and two goods, X and Y. Assume each country has constant opportunity costs: producing an additional unit of either good always requires giving up the same quantity of the other. Let \(c_A\) be the number of units of Y that country A gives up to produce one unit of X, and let \(c_B\) be the corresponding opportunity cost for country B. Suppose \(c_A<c_B\), so A has the lower opportunity cost of producing X.

Define \(r\) as the number of units of Y exchanged for one unit of X. The exchange ratio is quoted in this direction throughout the tutorial. A ratio of \(r=4\), for example, means one unit of X trades for four units of Y. Under these assumptions, the mutually beneficial range is \(c_A<r<c_B\): A receives more Y for each X than it gives up by producing that X, while B gives up fewer units of Y than it would have to give up to produce the X itself.

The quantity exchanged matters as well as the exchange ratio. If \(Q\) units of X are traded, then the output-equivalent gains are

Both expressions are in units of Y. For A, the \(Q\) units of X would have cost \(Qc_A\) units of Y to produce; through trade, A receives \(Qr\) units of Y. For B, the \(Qr\) units of Y given up would have cost \(Qr/c_B\) units of X to produce, while the \(Q\) units of X received would have cost \(Q\) units of X to produce. The equivalent Y-valued gain is \(Q(c_B-r)\). A derivation below makes this comparison explicit.

Why the Exchange Ratio Divides the Gains

The total gain per unit of X traded is the difference between the two countries’ opportunity costs. A gives up \(c_A\) units of Y to produce that X, while B would give up \(c_B\) units of Y to produce it. Trade permits the X to be produced where its opportunity cost is lower. The gap \(c_B-c_A\) is the available output-equivalent surplus per unit traded.

The terms of trade determine how that gap is split. A gains \(r-c_A\) units of Y per X traded, and B gains \(c_B-r\) units of Y per X traded. Adding these amounts shows why the total does not depend on \(r\), provided the quantity traded and the opportunity costs are fixed:

The \(r\) terms cancel: a higher amount of Y received by A per X is exactly a higher amount of Y given up by B per X. In this two-country accounting, changing the exchange ratio reallocates the gains between them; it does not change their total for a fixed quantity traded.

Dividing each country’s gain by the total gives its share:

These shares sum to one. As \(r\) moves upward from \(c_A\) toward \(c_B\), A’s share rises and B’s share falls. Near \(c_A\), A gains little per unit because the exchange ratio barely exceeds its own production cost; B receives a larger share of the gains. Near \(c_B\), B gains little per unit, so A receives a larger share. At either boundary, one country’s gain per unit approaches zero. The strict inequalities describe an exchange that benefits both countries, rather than leaving one exactly indifferent.

Solving for a Desired Share

The share formula can also be rearranged to find the exchange ratio that gives one country a specified fraction of the gains. If A’s target share is \(s_A\), where \(0<s_A<1\), then

Thus the ratio is the lower opportunity cost plus A’s desired fraction of the opportunity-cost gap. This expression also makes the range transparent: for a target share strictly between zero and one, the resulting \(r\) lies strictly between \(c_A\) and \(c_B\).

Common Mistakes and Interpretation

A frequent error is to calculate gains from the gap between the exchange ratio and only one country’s opportunity cost, then assign the entire result to that country. The exchange ratio lies between both costs, so each country has its own comparison.

For example, in the preceding example, the incorrect calculation \(12(7-2.5)=54\) as A’s gain uses B’s opportunity cost. It is actually B’s gain. A’s gain is \(12(2.5-1)=18\). The correction is to match each country with its own cost: subtract \(c_A\) from \(r\) for A, and subtract \(r\) from \(c_B\) for B.

Another trap is reversing the ratio. If a question says one unit of X exchanges for four units of Y, then \(r=4\) in this tutorial’s convention. If instead it says one unit of Y exchanges for one-quarter unit of X, the equivalent ratio is still four units of Y per X; it is not \(r=0.25\). Always convert to the stated direction before applying the formulas.

Finally, a larger share does not necessarily mean a larger absolute gain across different scenarios. The share formulas describe the division for fixed opportunity costs and a fixed traded quantity. If the quantity changes, total gains change in proportion to that quantity. And because these calculations measure output equivalents under constant opportunity costs, they do not by themselves establish how individuals within a country fare or quantify broader welfare.