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The Heckscher-Ohlin Model (Or, How to Build a Lerner Diagram)

Amit Khandelwal & Peter K. Schott

Spring 2008

1. Intro

The Heckscher-Ohlin (HO) model is designed primarily to answer the question “What are the

distributional consequences of globalization?” To see how model answers these questions, we will make

use of a very neat diagram developed by Abba Lerner in 1952.1 This chapter is designed to walk you

through the construction and use of a Lerner diagram, and to show you how useful it can be for

understanding

How trade liberalization benefits some factors of production and hurts others (i.e., why there

are “winners” and “losers” from free trade);

How relative wages respond to movements in relative prices (i.e., “price-wage” arbitrage); and

How factor accumulation influences countries’ comparative advantage and the mix of goods

they produce and trade over time (i.e., “development paths”).

2. Assumptions

Before we begin it’s useful to first be clear about the assumptions we need to make to build the 2x2x2

Heckscher-Ohlin model and construct the Lerner Diagram. We will assume

Two countries, two industries and two factors (i.e., 2x2x2-ness)

Factors (e.g., capital and labor) are mobile across industries within a country but not across

countries and that factors are fully employed;

Technology is identical across countries; and

Industries are perfectly competitive.

1 See also Findlay and Grubert (1959), Leamer (1984) and the many posts of Alan Deardorff on this subject at his

University of Michigan website, http://www-personal.umich.edu/~alandear/.

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3. Constructing the Lerner Diagram

The Lerner Diagram is drawn in factor space. (Recall that the diagram we used to illustrate the Ricardian

model was drawn in goods space.) For our discussion here, we’ll put labor (L) on the x-axis and capital

(K) on the y-axis.

Figure 1: Factor Space

To add two industries to this factor space, we need to assume production functions, that is, some

mapping of inputs into output. Suppose production in the apparel industry can be modeled with the

Cobb-Douglas production function

QApparel = AKL1-, (1)

where QApparel is the quantity of apparel produced and is a number between zero and 1. For now we

will ignore the A in the production function, but later on we will use it to think about the impact of

productivity growth on country and worker outcomes. To get some intuition about the production

function, note that if = 0.5, then QApparel = AKL0.5, that is, the quantity of apparel produced equals the

square root of the amount of capital used times the amount of labor used, all times A.

In order to represent this production function on the diagram, we need to solve for capital as a function

of labor, that is K=f(L). To do this, we re-arrange the production function to get

K = L(-1)/ / (QApparel)-1/. (2)

This expression is a bit messy, but note that if =0.5 it simplifies to K = q2/L. An important feature of our

production function is that different combinations of capital and labor can be used to produce the same

quantity of apparel. (Stare at the two equations above until you can see this statement is true.)

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Now, suppose we wanted to know all possible combinations of K and L that produce 1 unit of apparel,

that is, where QApparel =1. Equation (2) above would then simplify to K = L(-1)/. If =0.5, this would

mean we could get one unit of apparel from the following combinations of (K,L)=(1,1), (K,L)=(2,0.5),

(K,L)=(3,0.3), etc. We can represent this function graphically in Figure 2. Note that in Figure 2 we have

not assumed that =0.5. If we did, the curve would be exactly in the middle of the two axes. In the

figure is more like two-thirds.

Figure 2: Representing an industry with an isoquant

The curve in Figure 2 is known as an “isoquant” (“iso” for equal and “quant” for quantity). In general,

isoquants trace out all combinations of inputs that yield the same level of output. What happens to the

curve if we want two units of apparel? The curve must shift out, as shown in Figure 3, since the firm now

needs more capital and labor to obtain an additional unit of apparel.

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Figure 3: Isoquant shifts out as quantity produced grows

In using the Lerner diagram, it will be convenient to define industry isoquants in terms of the amount of

output that earns one dollar. (The reason for this will be clear as we go along.) How much apparel earns

a dollar? To answer that question, we need to know the price of apparel, PApparel. Knowing that, we can

compute the quantity of apparel that earns one dollar by diving $1 by the price, i.e.,

PApparel QApparel=1 (3a)

QApparel=1/PApparel (3b)

Then, we can redefine equation (2) to be

K = L(-1)/ / (1/PApparel)-1/. (4)

With equation (4) defining the isoquant in Figure 2, it now represents all combinations of capital and

labor that yield enough apparel output to earn one dollar given the price of apparel, PApparel. What

happens to the curve if the price of apparel falls? If the price of apparel falls, you need a greater quantity

of apparel to earn one dollar, and this extra quantity requires the use of more capital and labor. As a

result, the dollar-value isoquant shifts out, just like it did in Figure 3. If the price of apparel increases, on

the other hand, the curve would shift towards the origin, because now less K and L are needed to

produce a quantity of apparel worth one dollar.

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Equation 4 defines all possible combinations of capital and labor than can earn a dollar once they have

been transformed into apparel. To figure out the combination that is actually used, we need more

information. In particular, we need information about how much capital and labor cost. Let’s refer to

the rental rate of capital as r and the rental rate of labor (i.e., the wage) as w. Recall that one of the

assumptions of the HO model is zero economic profit, that is, that revenue equals cost. The dollar-value

isoquant tells us about revenue because it tells us the combinations of K and L that earn one dollar. A

dollar isocost line, on the other hand, defines the combinations of capital and labor that cost a dollar.

The equation for a dollar isocost line is simple – it merely says that the sum of what is paid to labor and

capital equals one dollar, or

wL + rK = 1. (5)

Note that we can rewrite this equation as

K = 1/r – (w/r)L (6)

if we solve it for K in terms of L. Recall from your fourth grade algebra class that the slope and y-interept

of this line are –w/r (that is, the relative price of labor) and 1/r (that is, the reciprocal of the rental rate

of capital), respectively. Solving for the value of labor when capital equals zero, it should be obvious that

the x-intercept of the dollar-value isocost line is 1/w, which is the reciprocal of the wage rate. The fact

that the intercepts of the dollar-value isocost line are 1/r and 1/w should be intuitive: because this line

represents combinations of inputs that equal a dollar, the y- and x-intercepts just record that amount of

capital and labor that can be hired, respectively, if the whole dollar is spent on only that factor. We

introduce the dollar-value isocost line to our Lerner diagram in Figure 4.

As we will see below, the fact that the returns to capital and labor are so easy to see in the Lerner

diagram turns out to be incredibly useful. It allows us to understand exactly how wages will change

whenever the dollar-value isocost line rotates. Note that because the y- and x-intercepts are the

reciprocals of r and w, respectively, movements of the intercepts towards the origin represent increases

in these factor rewards, while movements away from the origin represent declines. (Make sure you can

see this before continuing.)

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Figure 4: Isocost curve

In Figure 4, the dollar-value isocost line and the dollar-value isoquant are tangent at the point (LA, KA).

This tangency represents and defines several quantities of interest. First, it represents zero profit: inputs

LA and KA cost one dollar to hire and they can be combined to create one dollar’s worth of apparel.

Second, the tangency point also defines the optimal capital intensity, KA/LA, by which apparel will be

made. Finally, the slope of the dollar-value isocost line represents the equilibrium relative wage rate,

w/r.

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Figure 5: Apparel becomes more capital intensive if relative wages rise

To gain some intuition for these quantities, consider what would happen if relative wages were to

increase for some reason, that is, if w were to rise, or r were to fall, or both. Intuitively, one would

expect that an increase in the relative price of labor would lead to an optimal production technique that

relies less on labor. As outlined in Figure 5, this is exactly what happens. Likewise, as illustrated in

Figure 6, if relative wages were to all, optimal production would become more labor intensive.

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Figure 6: Apparel becomes more labor intensive if the relative wage falls

In our thought experiments above we acted as if we knew what the returns to capital and labor were –

that is what allowed us to draw in the particular dollar-value isocost line that we did. In the HO model,

however, relative wages are an outcome rather than an input. That is, given assumptions about

preferences, production and endowments, the model tells us what relative wage and relative goods

prices as well as output and trade, will be. To see how that works, we need to continue building the

model.

Thus far we have 2 factors, 1 industry and no countries. Let’s move on by adding another industry,

chemicals, which is relatively capital intensive compared to apparel.

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Figure 7: Two sectors in the Lerner Diagram

Figure 7 now includes a dollar-value chemicals isoquant as well as a dollar-value apparel isoquant. We

place the chemicals isoquant closer to the capital axis that apparel to represent the fact that it is

relatively capital intensive. That is, if the apparel production function we have represented thus far has

equal to approximately one-third, the chemicals production function has an alpha closer to two-thirds.

Note that we have also placed the chemicals isoquant on the same isocost curve as apparel. Why? This

can be answered in two ways. First, it reflects our assumption that factors are freely mobile within an

industry. One way of thinking about this assumption is that it requires the rental return to capital and

the wage rate to be the same across industries within a country. If they weren’t the same, factors would

have an incentive to move to where rewards were higher, and this movement would result in wage

equilibration. If w was higher in chemicals, for example, labor would flow to chemicals until the wages

were the same in the two industries.

A second intuition for why the chemicals isoquant must lie on the same isocost curve as the apparel

isoquant comes from considering what would happen if that were not the case. Suppose we placed the

chemicals isoquant below the isocost curve, as in Figure 8. Can you see that the distance between the

dashed chemicals isoquant and the solid isocost curve represents a violation of the zero profit

condition? Of course you can: the solid isocost curve represents combinations of factors that cost one

dollar. The dashed isoquant represents combinations that can earn one dollar. The distance between

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them, therefore, represents the profits that can be made by using less than one dollar’s worth of inputs

to produce one dollar’s worth of chemicals. In response to such profits, factors will flee apparel to make

chemicals. Note, however, that relatively more labor will leave apparel than can be matched with the

fleeing capital to make chemicals. Thus, relative wages will adjust from the solid to the dashed isocost

line. Relatively scarce capital’s reward will be bid up, while relatively abundant labor’s reward will be bid

down. Rising r means the y-intercept moves toward the origin, while falling w means that the x-

intercept moves away from the origin. (Remember this intuition, it will come up again and again and

again.)

Figure 8: Relative factor rewards (w/r) adjusts so that both sectors earn zero profits

Getting back to the equilibrium depicted in Figure 7, note that the rays emanating from the origin

represent the optimal capital intensities used to produce each industry given the illustrated rewards for

each factor.

Figure 9 completes the Lerner diagram by adding the final ingredient, two countries. We assume that

the U.S. is capital abundant relative to China, that is, that KUS/LUS > KChina/LChina. Verify that you can see

that China is labor abundant relative to the U.S. in the Figure. Notice that we have placed the countries

between optimal chemical and apparel labor intensities for now. We will explore what happens to

countries outside these lines later.

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Figure 9: The 2x2x2 Lerner Diagram

Before continuing, it is worth emphasizing that the tangencies between the Isoquants and the isocost

curve in Figure 9 do not represent production points. These points only tell us how much capital and

labor need to be combined to produce a dollar of each good, and that these amounts cost a dollar to

hire. In fact, if you think carefully about the diagram for a second, it has a pretty strange implication for

the level of GDP in the U.S. and China. Indeed, if the tangencies represent the amount of factors that

can earn a dollar, and the country endowment points represent the total factors that each country

possesses, then GDP in each country would be pretty low, i.e., under ten dollars! (We can easily solve

this by thinking of the isoquants and isocost curve as being billion-dollar-value rather than dollar-value,

but hopefully you can see by now that the purpose of the model is not to predict GDP.)

One of the many useful features of the Lerner diagram is that it is easy to see how much of a country’s

endowments are devoted to each industry, and, therefore, how much each country produces of each

good. Start with the full employment assumption we made earlier. This assumption implies that the

sum of the amount of each factor devoted to each industry must equal the total endowments of those

factors. For the U.S., this implies that

KUSApp+ KUS

Che= KUS and LUSApp + LUSA

Che = KUS, (7)

where KUS and LUS are the total endowments of capital and labor in the U.S., respectively. Similar

relationships govern factor allocation for China.

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To figure out how much capital and labor a country devotes to each industry, we just need to know the

country’s total endowments and the intensity with which factors are used in each industry. Note that

these quantities are easy to read off the Lerner diagram. The dot for each country gives their total

endowments, and the dotted lines emanating from the origin give the optimal production techniques.

We can see how much of each factor devotes to each country by using vectors with the slopes of these

dotted lines to “reach” each country.

Figure 10 does this for China. Start at the origin and move along the ray for apparel. As you move along

this ray, you are increasing the amounts of capital and labor China devotes to apparel. Note that to get

to the China endowment point, you eventually have to leave the apparel ray. Note however, that once

you leave the apparel ray, you can only travel along a line with the slope of the chemicals sector,

because that line devotes China’s capital and labor in that sector in the optimal proportions necessary to

produce it. Getting to the China endowment point, therefore, is just a matter of figuring out how to

combine the lines with the two optimal industry factor-intensity slopes to reach China. Stare at the

diagram to make sure you see this. Then, to make sure you understand, try reaching the China point by

starting with a line along the optimal chemical capital intensity. Demonstrate to yourself that the lines

you travel along that alternate route to China pick out the exact same amounts of capital and labor

devoted to each industry in China as we find using the route in Figure 10. Another way to make sure you

understand this is to plot in the figure the quantities

(LChinaApp,KChina

App) and (LChinaApp + LChina

Chem,KChinaApp+ LChina

Chem).

Figure 11 shows factor usage in both the U.S. and China. [Fix Figure 10]

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Figure 10: Factor Use in China

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Figure 11: Heckscher-Ohlin Theorem

4. Seeng the Important HO implications in a Lerner diagram

The factor allocations displayed in Figure 11 highlight the result that the U.S. devotes more of its

endowment to chemicals that China, while the opposite is true of apparel. A “shortcut” to seeing this is

to note that the U.S. endowment point lies closer to the optimal chemical capital intensity while the

China endowment point lies closer to the optimal apparel capital intensity. More evocatively, U.S. factor

endowments are more similar to (have a greater affinity for) the optimal chemical factor intensity than

the optimal apparel factor intensity. This relationship between countries’ relative endowments and

industries’ relative factor intensities is formally known as the Heckscher-Ohlin Theorem.

Heckscher-Ohlin Theorem: under the assumptions made above, a capital-abundant

country will produce more of, and export, the capital-intensive industry.

Note that In Figure 11, China and the U.S. manufacture both goods. This is an outcome of both countries

lying inside the same cone of diversification, where the word cone refers to the area between the

optimal apparel capital intensity and the optimal chemical factor intensity. (Can you see in the diagram

that the cone is defined as the set of points in factor space picked out by all possible combinations of

vectors with slopes of the optimal production techniques?)

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Suppose for a moment that China and the U.S. were not as close in factor space as Figure 11 implies.

Indeed, suppose, as in Figure 12, China lay outside the cone of diversification. Can you see that in that

case, there is no way that China would produce chemicals? As we will discuss further below, when a

country lies outside the cone, it specializes completely in the good to which it is closest, and it does so

using a capital intensity equal to its capital abundance. (You should have intuition for that last statement

already: if only produces one good, and I can use a good’s capital intensity to reach the endowment

point as a way of figuring out the optimal factor allocation, then it has to be a vector going from the

origin straight to the endowment point.

Figure 12: China's Endowment outside the Cone

The equilibrium depicted in Figure 9 displays another interesting result of the HO model, which is that

the wage rate w and the rental rate of capital r are the same in the two countries. This is known as

factor price equality.

Factor Price Equality (FPE): If countries product both goods, trade equalizes the

goods prices and factor input prices.

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This is a very sharp implication of the model because it says the w and r would be the same in the U.S.

and China! Note that FPE is precisely what many commentators go on and on about when they write

editorials denouncing U.S. free trade agreements with developing economies. They fear that free trade

will lead to wages being equal in the U.S. and developing countries, that is, that wages in the U.S. will

sink to the level they are in developing countries. (Remember the global labor pool chart we showed

you the first day of class?)

You can take FPE with a grain of salt for the time being. Below, we will show how to think about it in a

much more interesting and general setting. Along the way, we’ll develop a lot of intuition for what

makes FPE work. For now, just make sure you realize that FPE occurs both countries produce both

goods, technologies are identical and free trade means that goods prices are the same in both

economies.

Recall that the primary objective of the HO model is to help us understand the distributional

implications of free trade. Let’s start with Figure 9, but forget for a moment about China. Ask yourself

how wages in the U.S. would be affected if the price of apparel were to drop for some reason (perhaps

due to a free trade agreement like NAFTA or a relaxation of apparel import quotas as occurred in 2005).

What chain of events would such a price decline set off, and how would they lead to new factor prices?

They are illustrated in Figure 13 and can be summarized as follows:

1. A decline in the price of apparel results in a shifting out of the apparel dollar-value isoquant to

the dashed isoquant in Figure 13. As noted in our discussion of Figure 3, This shift is due to the

fact that, at a new, lower price, it takes more apparel output, and therefore more capital and

labor, to earn a dollar.

2. Because of the gap between the new apparel isoquant and the old dollar-value isocost line,

producing apparel is no longer profitable. Indeed, the distance between the curve and the line

measures just how negative profits are because it shows how many more inputs are needed to

produce a dollar than can be rented for a dollar. These negative profits induce a reallocation of

resources away from apparel and towards chemicals.

3. Capital and labor flee apparel. Where do they go? To chemicals, where economic profits are still

zero. Note that as labor and capital leave apparel, they do so in proportion to how they were

used in that industry, i.e., apparel sheds relatively more labor than capital as it downsizes. This

is important because it provides intuition for what happens to factor rewards when the price of

apparel drops. As labor and capital leave apparel and show up to work in the chemical industry,

they are in the wrong proportion for that relatively capital intensive industry. From the

perspective of the chemical industry, too little capital per worker showing up, or, alternately,

there too much labor per capital showing up. Since labor is relatively plentiful, its reward gets

bid down. Because capital is relatively scarce, it’s reward gets bid up. These reactions cause a

counter-clockwise rotation in the dollar-value isocost line to the dashed line depicted in Figure

13.

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This link between relative prices and relative wages is known as “price-wage arbitrage”. It is summarized

formally in the Stolper-Samuelson Theorem.

Stolper-Samuelson Theorem (“price-wage arbitrage”): A rise in the price of a good

raises the real return of the more important factor in that industry. It lowers the real

return of the other factor.

Note that the Stolper-Samuelson theorem is expressed in terms of real wages. For capital this should not

be too hard to understand: the nominal return to capital r has gone up, and the price of one of the

goods has fallen, so the increase in the real return to capital is even higher than the nominal increase.

For labor, a bit more thought is required. Note that the nominal return to labor has fallen by the

distance between the x-intercepts of the old and new isocost lines. On the other hand, the price of

apparel has fallen as well. How do we now that the price decline of apparel is not enough to offset the

fall in w? That too, can be seen in the Lerner diagram. Remember that old and new apparel isoquant are

driven apart by the decline in the apparel price. We can see how large this price decline is by considering

a hypothetical new isocost curve parallel to the old one but tangent to the new apparel isoquant. The

distance between the x-intercepts of those two lines measures the change in the apparel price.2 Can you

see that the distance between these two lines will always be less than the distance between the x-

intercepts of the isocost lines? (Neat, huh?)

Figure 13 highlights a related consequence of the declining apparel price that may not have occurred to

you at the outset. Note that the rotation to the new dollar-value isocost line creates new tangency

points that are both more labor-intensive than before. In response to the relative decline in the price of

apparel, both industries substitute towards the now relatively cheaper factor. This should be intuitive.

The new optimal capital intensities for each industry are given by the solid lines emanating from the

origin.

One further test of your intuition before we move on. Can you see how a decline in the apparel price

might be large enough to move the U.S. out of the apparel sector entirely? Is this relevant to the real

world?

[Get rid of China in Figure 13]

2 This outcome is known formally as the Jones (1965) magnification result.

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Figure 13: Stolper-Samuelson, “price-wage arbitrage” and "winners" and "losers" from trade

The final result we will study in the 2x2x2 Lerner diagram relates to how countries’ output mix changes

with its factor endowments. It is known as the Rybczynski Theorem.

Rybczynski Theorem (“development paths”): At constant prices, an increase in the

endowment of one factor increases the output of the industry that uses that factor

intensively and reduces the output of the other industry.

This theorem provides useful intuition for the influence of development, that is, the accumulation of

capital, on country output. It says that as a country accumulates capital it will increase the output of the

industry for which that factor is important (chemicals) and will decrease the output of the industry for

which that factor is unimportant (apparel). Note that this outcome assumes no change in prices as a

results of factor accumulation.

We demonstrate the Rybczynski Theorem in Figure 14. Suppose China adds capital to its endowments,

so that it moves from the round up to the square endowment point in the Figure. (Note that we are not

modeling where this capital comes from. To understand more about that, we would need a different

model, one of the appropriately named “growth” models that are typically studied in a course on

macroeconomics). To see how China’s production will change in response to this movement we can

make use of the techniques we developed in discussing Figures 10 and 11, that is, trying to find the two

arrows with slopes equal to the optimal industry factor intensities that get us to China’s endowment

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points. In this case, we can see that the amount of capital and labor devoted to apparel falls as the

country adds capital. As a result, the there is an increase in the capital and labor devoted to the chemical

industry. This latter effect is likely obvious, the former perhaps less so.

[Get rid of US in Figure 14]

Figure 14: Capital Deepening in China

What happens when development moves a country from outside of the cone to within the cone? To

answer this question, take anointer look at Figure 12. In our discussion of that figure above, we

concluded that China would not produce any chemicals if it lay outside the cone because its capital was

so rare relative to its labor – the country would be better off sticking to apparel. We also noted that if

China devotes all of its resources to apparel, it will produce apparel at an industry capital intensity that

is equal to its country capital abundance. Recall that if it does not do this, some resources will go

unused, which violates the full employment assumption we made at the beginning of this chapter. In

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Figure 15, we denote the technique by which China produces apparel outside the cone via a .

Figure 15: China outside the Cone

Can you see that the w and r that it will pay its factors is determined by a China-specific isocost line that

will be tangent to the apparel isoquant at this point? Further, can you see that as China accumulates

capital relative to labor outside that cone that the capital intensity of its apparel production and its

relative wages (w/r) will increase until it hits the boundary of the cone? These outcomes are displayed in

Figure 16.

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Figure 16: Factor Price Equalization breaks down in a multiple-cone world

Figure 16 conveys an important message about the factor price equality we noted earlier. It says that

the U.S. and China will offer the same factor rewards only if their endowments are sufficiently similar to

place them in the same cone of diversification. In the next section, we will pursue an even more

meaningful violation of FPE than the one displayed in Figure 16.

Before doing that, however, we note one more property of the HO model, factor price insensitivity,

which is an alternate interpretation of factor price equality.

Factor Price Insensitivity: All else equal, a country’s factor accumulation within a

cone does not lead to any change in factor rewards.

Most people (especially labor economists) find this result very counter-intuitive. Can you see why it is

true? The reason is simple. Even though an increase in the supply of capital in a country puts downward

pressure on the relative return to capital, the concomitant shift in output towards the chemical industry

creates and exactly offsetting increase in the demand for capital, thereby pushing relative wages right

back up to where they started.

5. The Multiple Cone Model

In this section we honor our earlier promise of generalizing the Lerner diagram in a way that avoids the

unsatisfying implication that the returns to capital and labor are the same in the U.S. as in China.

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Figures 15 and 16 provided us with the intuition that countries might offer different factor rewards if

their endowments were insufficiently similar. Here, we push that intuition further by thinking of a world

of more than two goods, i.e., a world with more than one cone of diversification. Figure 17 contains

four goods: chemicals, machinery, textiles and apparel, which decline in capital intensity as one moves

from the capital to the labor axes.

[In Figure 17, add the optimal chemical and apparel capital intensity rays from origin to intersect their

respective tangencies]

Figure 17: Cones and Multiple Industries

In Figure 17, the U.S. is in one cone of diversification while China is in another. As a result of the same

reasoning used above, the two countries offer different relative factor rewards (w/r). Can you see that

the return to labor is higher in labor-scarce U.S. and lower in labor-abundant China? Does this make

sense? What about the difference in their returns to capital?

Now, consider what happens if the price of apparel suddenly falls from some reason outside the model.

As we saw in Figure 13, the dollar-value apparel isoquant shifts away from the origin and China’s relative

factor rewards fall. Note, however, that nominal relative wages in the U.S. are not influenced by the

apparel price change. Why? Good specialization. Because the U.S. specializes in a different subsets of

goods than China, it is insulated from any price changes affecting the Chinese bundle. Furthermore, note

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that the real rewards of both factors rise in the U.S. when the price of apparel falls. Nominal wages are

unaffected, but the price decline in apparel means that real r and real w go up!