Deardorff's Glossary of International Economics

Lerner Diagram

Pass mouse over underlined tags to change figure:

Price Changes

Increase in price of

X Y

Decrease in price of

X Y

Technological Change
(Hicks-Neutral)

Improvement in

X Y

EXTENSIONS

Diversification
Cone

Patterns of
Specialization

Factor Allocations
and Factor Prices:

Endowments

Increase in Price of X:

Key:

K, L		Quantities of factors capital and labor
r, w		Rental price of capital and wage of labor
X, Y		Quantities of goods X and Y
p_X , p_Y		Prices of goods X and Y

K_X , K_Y		Capital employed in production of goods X and Y
L_X , L_Y		Labor employed in production of goods X and Y
k_X , k_Y		Capital-labor ratio employed in diversified production of goods X and Y

Explanation:

The basic Lerner Diagram, shown above, uses unit-value isoquants and unit isocost lines to determine the factor prices that are consistent with positive production of two goods under perfect competition. The unit-value isoquants incorporate both nominal prices and technology, both of which are taken as given for the diagram. The unit isocost line is then determined within the diagram as the straight line tangent to both of the unit-value isoquants -- the "common tangent". The factor prices represented by (the reciprocals of) the intercepts of this isocost line permit both industries to just break even and therefore permit both to produce without a positive profit. If only one such common tangent exists, as it appears above, then these factor prices are the unique ones consistent with positive production at the goods prices for which the unit-value isoquants were drawn.

The diagram can be used as a powerful tool for deriving many of the results of the Heckscher-Ohlin Model. It is especially useful for finding the effects of a change in the prices of goods on factor prices, but with the extensions shown it can also determine other things.

Price Changes
A price change appears in the diagram as a shift of a unit-value isoquant. An increase in the price of good X, for example, means that a smaller quantity of good X is worth one dollar. With linearly homogeneous technologies, the new unit-value isoquant is just a shrunken version of the old, contracted inward toward the origin by the fraction of the price increase.
As shown, an increase in the price of X, then, shifts the X-isoquant inward, causing the common tangent to rotate clockwise. From the intercepts, the wage rises and the rental falls. A fall in price of X has the opposite effects, while a change in the price of Y is analogous.
Technological Change
A technological improvement for producing a good causes the isoquant for the same quantity to be shifted inwards. The shift need not be proportional, but the figure above assumes Hicks-neutral progress, for which it is. Thus the picture looks exactly the same as that for an increase in the price of the good. The effects are correspondingly the same, except that one must be careful in distinguishing changes in quantities from changes in values for a good with a changed technology.
Extensions
The Lerner Diagram provides a convenient starting point for further analysis of the Heckscher-Ohlin Model. The figure can be used to determine, essentially in the following order:

industry factor ratios (techniques) corresponding to diversification;
the diversification cone;
patterns of specialization;
and with addition of a point representing factor endowments,

factor employment in each industry; and
industry outputs.
The effects on all of these variables can therefore be derived, for any initial pattern of specialization, due to changes in anything exogenous to the diagram: prices, technology, and factor endowments.
The Diversification Cone
If factor prices are those given by the common tangent, as they must be for both goods to be produced, then the cost-minimizing techniques of production in the two industries are at the two points of tangency. Therefore, the factor ratios at these points of tangency, and the rays with these slopes labeled k_X and k_Y above, represent these techniques. The diversification cone is the set of all factor endowments lying on or between these rays.
Patterns of Specialization
The reason for calling it the diversification cone, aside from its shape, is that only for factor endowments lying inside the cone -- between the rays k_X and k_Y -- will a country produce both goods. Otherwise it would not be able to employ both factors fully, since it would be using either a higher or lower ratio of factors in both industries than it has in its endowment. Outside the cone, therefore, the country completely specializes, producing only the most labor-intensive good, X, below the cone and only the most capital intensive good, Y, above it.
Factor Allocations and Factor Prices
Outside the cone, the factor allocations are simple. The entire endowment is employed in one industry. For firms to do that willingly, they must face factor prices that induce them to hire factors in the same proportions as the endowment. Therefore, factor prices outside the cone are not given by the common tangent, but rather by the slope of the operating industry's isoquant at the factor ratio of the country's endowment.
Inside the cone, factor prices are given by the common tangent, and the industries employ factors in the ratios k_X and k_Y. With two factors and two goods, there is only one way that factors can be fully employed given this constraint. It can be found by constructing lines parallel to the k_X and k_Y rays from the point representing the country's factor endowment to where each intersects the other industry's ray. The point of intersection is the amount of factors allocated to that industry.
Effects of a Price Change
All of this can be combined to determine the effects of changes in variables that are exogenous to the Lerner diagram, such as prices. The case shown is an increase in the price of good X. The price increase pulls the unit-value isoquant for X in toward the origin. This causes the common tangent to rotate clockwise, becoming steeper and representing a higher relative price of labor compared to capital. In nominal terms, the wage rises and rental falls. These changes cause both industries to substitute toward using less labor and more capital, moving up and to the left along their isoquants to higher ratios of capital to labor. This increase in the capital-labor ratios of diversification means that the diversification cone rotates counterclockwise. Some factor endowments that previously would have involved specialization in the more capital-intensive industry now will accommodate both industries, while other factor endowments that were inside the cone but closer to its bottom edge are now below it, switching from producing both goods to producing only X. For endowments that remain within the cone, factors must be re-allocated in order to keep them fully employed at the new higher capital-labor ratios. The more labor-intensive industry, even though it now employs a higher ratio of capital to labor, also must expand, employing a larger amount of both factors. The capital-intensive industry Y, on the other hand, employs less of both factors and contracts. Thus, for a country within the cone, the output of X rises and the output of Y falls.

Notes:

The Lerner Diagram works only for linearly homogeneous technologies. With that assumption, every isoquant for an industry is exactly like every other, just scaled inward or outward, and a single isoquant -- such as the unit-value isoquant -- completely describes the technology. Without linear homogeneity or something equivalent to it, the Lerner Diagram would make no sense and cannot be used.
Only Hicks neutral technological change is illustrated above. However, one of the counter-intuitive messages of the Lerner Diagram is that this neutrality is not very important for the effects of technological change on factor prices. An improvement in the technology for producing X, for example, will pull its unit-value isoquant inward no matter how it may be biased in favor of labor or capital, and as long as the other isoquant does not change, the common tangent must get steeper. Thus if both goods are produced so that the country is (and remains) inside the diversification cone, technological progress in the labor-intensive industry causes a rise in the relative wage regardless of any factor bias in the progress itself.
The isoquants above do not display a factor intensity reversal. If they did, then there could be two common tangents, not one, and with multiple reversals there could be even more. For each common tangent there is a corresponding diversification cone.

Price Changes
	A price change appears in the diagram as a shift of a unit-value isoquant. An increase in the price of good X, for example, means that a smaller quantity of good X is worth one dollar. With linearly homogeneous technologies, the new unit-value isoquant is just a shrunken version of the old, contracted inward toward the origin by the fraction of the price increase. As shown, an increase in the price of X, then, shifts the X-isoquant inward, causing the common tangent to rotate clockwise. From the intercepts, the wage rises and the rental falls. A fall in price of X has the opposite effects, while a change in the price of Y is analogous.
Technological Change
	A technological improvement for producing a good causes the isoquant for the same quantity to be shifted inwards. The shift need not be proportional, but the figure above assumes Hicks-neutral progress, for which it is. Thus the picture looks exactly the same as that for an increase in the price of the good. The effects are correspondingly the same, except that one must be careful in distinguishing changes in quantities from changes in values for a good with a changed technology.
Extensions
	The Lerner Diagram provides a convenient starting point for further analysis of the Heckscher-Ohlin Model. The figure can be used to determine, essentially in the following order: industry factor ratios (techniques) corresponding to diversification; the diversification cone; patterns of specialization; and with addition of a point representing factor endowments, factor employment in each industry; and industry outputs. The effects on all of these variables can therefore be derived, for any initial pattern of specialization, due to changes in anything exogenous to the diagram: prices, technology, and factor endowments. The Diversification Cone If factor prices are those given by the common tangent, as they must be for both goods to be produced, then the cost-minimizing techniques of production in the two industries are at the two points of tangency. Therefore, the factor ratios at these points of tangency, and the rays with these slopes labeled k_X and k_Y above, represent these techniques. The diversification cone is the set of all factor endowments lying on or between these rays. Patterns of Specialization The reason for calling it the diversification cone, aside from its shape, is that only for factor endowments lying inside the cone -- between the rays k_X and k_Y -- will a country produce both goods. Otherwise it would not be able to employ both factors fully, since it would be using either a higher or lower ratio of factors in both industries than it has in its endowment. Outside the cone, therefore, the country completely specializes, producing only the most labor-intensive good, X, below the cone and only the most capital intensive good, Y, above it. Factor Allocations and Factor Prices Outside the cone, the factor allocations are simple. The entire endowment is employed in one industry. For firms to do that willingly, they must face factor prices that induce them to hire factors in the same proportions as the endowment. Therefore, factor prices outside the cone are not given by the common tangent, but rather by the slope of the operating industry's isoquant at the factor ratio of the country's endowment. Inside the cone, factor prices are given by the common tangent, and the industries employ factors in the ratios k_X and k_Y. With two factors and two goods, there is only one way that factors can be fully employed given this constraint. It can be found by constructing lines parallel to the k_X and k_Y rays from the point representing the country's factor endowment to where each intersects the other industry's ray. The point of intersection is the amount of factors allocated to that industry. Effects of a Price Change All of this can be combined to determine the effects of changes in variables that are exogenous to the Lerner diagram, such as prices. The case shown is an increase in the price of good X. The price increase pulls the unit-value isoquant for X in toward the origin. This causes the common tangent to rotate clockwise, becoming steeper and representing a higher relative price of labor compared to capital. In nominal terms, the wage rises and rental falls. These changes cause both industries to substitute toward using less labor and more capital, moving up and to the left along their isoquants to higher ratios of capital to labor. This increase in the capital-labor ratios of diversification means that the diversification cone rotates counterclockwise. Some factor endowments that previously would have involved specialization in the more capital-intensive industry now will accommodate both industries, while other factor endowments that were inside the cone but closer to its bottom edge are now below it, switching from producing both goods to producing only X. For endowments that remain within the cone, factors must be re-allocated in order to keep them fully employed at the new higher capital-labor ratios. The more labor-intensive industry, even though it now employs a higher ratio of capital to labor, also must expand, employing a larger amount of both factors. The capital-intensive industry Y, on the other hand, employs less of both factors and contracts. Thus, for a country within the cone, the output of X rises and the output of Y falls.