The transformation curve is defined, in international economics, as the maximum amount of commodity X obtainable for any given amount of commodity Y, and vice versa. This concept is basically the same as the production-possibility frontier studied in microeconomics. In this case, however, the transformation curve shows the trade-off (or opportunity cost) for a country when deciding to produce one commodity or good instead of some other.
Transformation curves may be graphically represented in two alternative ways. The first one will show the marginal rate of transformation between good Y and good X. In other words, it shows the amount of X that the economy has to give up to obtain an additional unit of Y (and vice versa). As seen in the adjacent figure, it resembles the graphical representation of the production possibility frontier. The transformation curve can be either concave, convex or both at the same time. However, with constant returns to scale in the production of both commodities, and different capital-labour ratios (in which case it will be linear), it will always be concave to the origin.
All points on the curve represent an efficient production: countries with a production bundle inside the transformation (such as point A) can be better off by producing more of one of the commodities, moving to the transformation curve (for instance, more of good Y, to point E); bundles outside the transformation curve (such as B or C) are unattainable, because either the technology or inputs are insufficient. Since the transformation curve is derived from an optimizing procedure (as when analysing production optimization in a general equilibrium model), the amount of X (or of Y) given up is the minimum possible under the given technical knowledge.
It’s worth mentioning that, under competitive conditions, the economy will always be on the transformation curve, at a point where the marginal rate of transformation equals the relative price of the two commodities, pX/pY. For instance, as seen in the adjacent figure, if the relative price was such as the one shown by the green straight line (the relative price will be equal to tan α), the economy would operate at point E’, where the marginal rate of transformation equals the relative price of the two commodities, thus being at equilibrium.
The other way to represent transformation curves is using an Edgeworth box. In the figure below, we can see the transformation curves of country A (TCA, red) and country B (TCB, green). Note that we consider only one factor of production, and that the technology is fixed, which makes the transformation curves linear. The slope of these transformation curves correspond to the marginal rate of transformation, which in turn represent the comparative cost: tan α corresponds to the comparative cost of country A, while tan β corresponds to the comparative cost of country B. The terms of trade, which in this case have been arbitrarily determined, are given by tan γ.
It’s worth mentioning that this graphical representation is the most common way to represent the Ricardian trade model.