Deardorff's Glossary of International Economics

Edgeworth Production Box

Factor Reallocation
along Efficiency Locus:

into Y into X

Factor Proportions
and Factor Prices
along Efficiency Locus:

-2 -1 0 +1 +2

Changes in Endowments

*Less K*

*More K*

*Less L*

*More L*

Less K&L (in proportion)

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

Explanation:

The basic Edgeworth Production Box, shown above, has dimensions equal to the factor endowments of a country, and it shows all of the ways that those endowments can be allocated across two industries for producing two goods, X and Y. The lower-left corner serves as the origin for measuring allocations of factors to industry X, and the upper-right corner as the origin for allocations to industry Y. Production functions for each industry are represented by isoquants drawn relative to these two origins, that for industry Y being rotated 180 degrees from the usual orientation. Thus the outputs of the two goods are represented by the two isoquants, one for each industry, passing through any particular point in the box.

An efficient allocation is one for which these two isoquants are tangent, as shown at E₀ for X₀ and Y₀. The set of all such tangencies is the efficiency locus, the curve O_XE₀O_Y. Under the usual assumption of constant returns to scale in both industries, the efficiency locus cannot cross the upward-sloping diagonal of the box, and it must therefore lie either wholly above it, as shown, if industry X is capital intensive compared to industry Y, or wholly below it in the opposite case.

The diagram can be used to illustrate the effects of factor reallocations along the efficiency locus on outputs and factor prices, where the latter appear as via isocost lines tangent to the isoquants. It can also be used to show the effects of changing factor endowments, under the assumption that prices of goods (and therefore factors, due to FPE) are held constant.

[Begin revising here.]
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 needed to be 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 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. In order 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 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 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 reallocated 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 contract, employing a smaller amount of both factors. The capital-intensive industry Y, on the other hand, employs more of both factors and expands. Thus, for a country within the cone, the output of Y rises and the output of X 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 could not 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 needed to be 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 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. In order 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 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 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 reallocated 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 contract, employing a smaller amount of both factors. The capital-intensive industry Y, on the other hand, employs more of both factors and expands. Thus, for a country within the cone, the output of Y rises and the output of X falls.