The Encyclopedia of Economic Principals

Chapter 56

Comparative Advantage, Factor Endowments, and Distribution

Trace goods prices and endowments into wages, rents and outputs.

Four of the chapter's worked examples, made interactive: Stolper-Samuelson magnification, the Rybczynski output effect, winners and losers with specific factors, and factor-price equalization. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

A price rise magnified in wages

When the labor-intensive good gets dearer, what happens to wages and rentals?

Both zero-profit conditions must hold at once. Raising cloth's price lifts the reward of the factor cloth uses intensively by more than the price, and the food equation then forces the other factor's reward down: the magnification effect.

Equation, written in LaTeX: 2w+r=30, w+2r=30.

Equation, written in LaTeX: 2w+r=36, w+2r=30.

Equation, written in LaTeX: \frac{14-10}{10}=40\%,

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Cloth uses 2 labor and 1 capital per unit, food 1 labor and 2 capital. With competition each price equals unit cost, so 2w + r = pC and w + 2r = pF, where w is the wage and r the capital rental.

Predict first. Cloth's price rises 20 percent, from 30 to 36, with food at 30. Does the wage rise by more or less than 20 percent?

Your prediction

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Figure: A price rise magnified in wages. Unit-cost lines for cloth and food in the wage-rental plane. The baseline lines cross at (10, 10); at prices 36 and 30 they cross at w = 14.00, r = 8.00.
Cloth price: 36, Food price: 30
Constructed example: the chapter's hypothetical economy (cloth 2L + 1K, food 1L + 2K; prices 30 and 30, then cloth 36); cloth prices 33 and 39 and food prices 27 and 33 are added for comparison.

Calculated values

Wage w
14.00
Rental r
8.00
Wage change
+40.00%
Cloth price change
+20.00%
Food price change
0.00%
Real wage in cloth w/pC
0.389
Real wage in food w/pF
0.467
Real rental in cloth r/pC
0.222
Real rental in food r/pF
0.267

Subtracting 2 x (w + 2r = 30) from 2w + r = 36 gives -3r = 36 - 60, so r = 8.00 and w = (36 - 8.00) / 2 = 14.00. The wage rises 40.00 percent against a cloth price change of 20.00 percent and a food price change of 0.00 percent. Starting from 10 / 30 = 0.333 of either good, the wage buys 14.00 / 36 = 0.389 cloth and 14.00 / 30 = 0.467 food; the rental buys 0.222 cloth and 0.267 food. Labor gains in terms of both goods, and capital loses in terms of both goods.

Worked steps

  1. 2w + r = 36 and w + 2r = 30
  2. 3r = 2 x 30 - 36 = 24, so r = 8.00
  3. w = (36 - 8.00) / 2 = 14.00
  4. Wage change = (14.00 - 10) / 10 = +40.00%
  5. Real wage: 14.00 / 36 = 0.389 cloth, 14.00 / 30 = 0.467 food

Use the idea

To see who gains from a tariff or a world price change, find the factor used intensively in the good whose price rises: its real reward rises in terms of every good.

Where the conclusion applies

Fixed coefficients, both goods produced, perfect competition and mobile factors. If an economy stops producing a good, its cost equation no longer binds and the result changes.

Check your understanding: At pC = 36 and pF = 30, what is capital's food purchasing power?
r = 8, so 8 / 30 = 0.267 units of food, down from 10 / 30 = 0.333.

Chapter 56 source: section "Stolper-Samuelson theorem".

Demonstration 2 of 4

More workers, fewer machines

At fixed goods prices, what does extra labor do to the output of each good?

At fixed prices factor proportions in each industry are fixed, so the only way to employ the extra labor is to expand the labor-intensive good. That expansion needs capital, which can only come from shrinking the capital-intensive good.

Equation, written in LaTeX: 2X+Y=300, X+2Y=300.

Equation, written in LaTeX: 2X+Y=360, X+2Y=300.

Equation, written in LaTeX: \frac{140-100}{100}=40\%,

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Cloth X uses 2 labor and 1 capital per unit; machines Y use 1 labor and 2 capital. With both factors fully employed, 2X + Y = L and X + 2Y = K.

Predict first. Labor grows 20 percent, from 300 to 360, with capital at 300. Does machine output rise, fall or stay put?

Your prediction

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Figure: More workers, fewer machines. Full-employment lines for labor and capital in output space. The baseline lines cross at (100, 100); with labor 360 and capital 300 they cross at X = 140.00, Y = 80.00.
Labor endowment: 360, Capital endowment: 300
Constructed example: the chapter's hypothetical economy (cloth 2L + 1K, machines 1L + 2K; L = K = 300, then L = 360); labor of 330 and 390 and capital of 270 and 330 are added for comparison.

Calculated values

Cloth X
140.00
Machines Y
80.00
Cloth change
+40.00%
Machine change
-20.00%
Labor change
+20.00%
Capital change
0.00%
Both goods produced
yes

From 2X + Y = 360 and X + 2Y = 300: 3X = 2 x 360 - 300 = 420, so X = 140.00, and Y = (300 - 140.00) / 2 = 80.00. Cloth output rises 40.00 percent and machine output falls 20.00 percent, while labor changed +20.00 percent and capital 0.00 percent. Labor grows relative to capital, so cloth output, which uses labor intensively, changes by more than either endowment (+40.00 percent) and machine output by less than either (-20.00 percent): the magnification effect.

Worked steps

  1. 2X + Y = 360 and X + 2Y = 300
  2. 3X = 2 x 360 - 300 = 420, so X = 140.00
  3. Y = (300 - 140.00) / 2 = 80.00
  4. Cloth change = (140.00 - 100) / 100 = +40.00%
  5. Machine change = (80.00 - 100) / 100 = -20.00%

Use the idea

When one factor grows, such as through immigration, expect output of the goods that use it intensively to grow by more than the factor and the other goods to shrink, if prices stay fixed.

Where the conclusion applies

Fixed world prices and coefficients and both goods produced. If the endowment leaves the diversification cone, a solved output turns negative and the economy specializes.

Check your understanding: With L = 360 and K = 300, why can cloth not simply rise by 30 with machines at 100?
130 cloth and 100 machines need 130 + 200 = 330 capital units, 30 more than exist; the equilibrium is X = 140, Y = 80.

Chapter 56 source: section "Rybczynski theorem".

Demonstration 3 of 4

Who wins with specific factors

When the export price rises, who gains: mobile workers or the factors stuck in each sector?

Labor moves toward the sector whose price rose until wages are equal again. Diminishing returns mean the wage rises by less than the price, so labor's real gain is ambiguous, while the fixed factors are the clear winner and loser.

Equation, written in LaTeX: Q_X=10\sqrt{L_X}, Q_M=10\sqrt{L_M}.

Equation, written in LaTeX: w_0=\frac{5}{\sqrt{50}}\approx0.7071.

Equation, written in LaTeX: \frac{7.5}{\sqrt{L_X}}=\frac{5}{\sqrt{L_M}}, L_X+L_M=100.

Equation, written in LaTeX: R_{X,1}=1.5(5\sqrt{69.23})\approx62.40,

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100 workers move freely between the export sector X and the import sector M; each sector also has a fixed specific factor. Output is Q = 10 sqrt(L), so the marginal product of labor is 5 / sqrt(L). pM = 1; the wage equals each sector's price times its marginal product. R is a specific factor's rent.

Predict first. The export price rises from 1 to 1.5. Do workers gain in terms of both goods?

Your prediction

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Figure: Who wins with specific factors. Two value-of-marginal-product curves across 100 workers. They cross at L_X = 69.23 with wage 0.9014; the shaded areas above the wage are the two sector-specific rents, 62.40 and 27.74.
Export good price: 1.50
Constructed example: the chapter's hypothetical economy (100 workers, Q = 10 sqrt(L), pM = 1, pX = 1 then 1.5); export prices 1.25 and 2.0 are added for comparison.

Calculated values

L_X
69.23
L_M
30.77
Wage w
0.9014
Wage change (in M)
+27.5%
Real wage in X
0.6009 (-15.0%)
Rent R_X
62.40 (+76.5%)
Rent R_M
27.74 (-21.6%)

Wage equality gives L_X = 1.50^2 L_M = 2.2500 L_M, so L_M = 100 / 3.2500 = 30.77 and L_X = 69.23. The wage is 5 / sqrt(30.77) = 0.9014, +27.5 percent from 0.7071; measured in X it is 0.9014 / 1.50 = 0.6009. R_X = 1.50 x 5 x sqrt(69.23) = 62.40 and R_M = 5 x sqrt(30.77) = 27.74, against 35.36 each at the start. Workers gain in terms of M but lose in terms of X; the X-specific factor clearly gains and the M-specific factor clearly loses.

Worked steps

  1. L_X / L_M = 1.50 x 1.50 = 2.2500
  2. L_M = 100 / 3.2500 = 30.77; L_X = 100 - 30.77 = 69.23
  3. w = 5 / sqrt(30.77) = 0.9014
  4. Wage in X = 0.9014 / 1.50 = 0.6009
  5. R_X = 1.50 x 5 x sqrt(69.23) = 62.40; R_M = 5 x sqrt(30.77) = 27.74

Use the idea

In the short run, when factors are tied to an industry, expect owners of the export sector's specific assets to gain most and owners of import-competing assets to lose.

Where the conclusion applies

Two sectors, square-root technology, a fixed specific factor in each and perfectly mobile labor. Over a longer horizon the specific factors can move and the Stolper-Samuelson logic takes over.

Check your understanding: What is the wage measured in X at pX = 1.5?
0.9014 / 1.5 = 0.6009, down 15.0 percent from 0.7071.

Chapter 56 source: section "Specific-factors distribution effect".

Demonstration 4 of 4

Trade in goods equalizes factor prices

Can trade in goods alone bring two countries' wages together?

With identical techniques and both goods produced, the two cost equations pin down w and r from goods prices alone. Common goods prices therefore mean common factor prices.

Equation, written in LaTeX: 2w+r=24, w+2r=39,

Equation, written in LaTeX: 2w+r=36, w+2r=33,

Equation, written in LaTeX: 2w+r=30, w+2r=36,

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Both countries use the same techniques: cloth 2 labor and 1 capital, machinery 1 labor and 2 capital. Each price equals unit cost, so 2w + r = pC and w + 2r = pM. In autarky A faces (24, 39) and B (36, 33); with trade both face pC = 30 and the world machinery price.

Predict first. After trade opens at prices 30 and 36, does A's wage rise or fall, and B's?

Your prediction

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Figure: Trade in goods equalizes factor prices. Wage-rental plane with A's autarky point (3, 18) and B's (13, 10); arrows lead to the common free-trade point (8.00, 14.00).
Prices in force: Free trade, World machinery price: 36
Constructed example: the chapter's hypothetical countries (A autarky 24 and 39, B autarky 36 and 33, trade prices 30 and 36); world machinery prices 33 and 39 are added for comparison.

Calculated values

Prices in force
pC = 30, pM = 36 in both countries
Common wage
8.00
Common rental
14.00
A's wage
3 to 8.00
B's wage
13 to 8.00

With common prices pC = 30 and pM = 36, both countries solve 2w + r = 30 and w + 2r = 36. Then 3r = 2 x 36 - 30 = 42, so r = 14.00, and w = 36 - 2 x 14.00 = 8.00. A's wage rises from 3 and its rental falls from 18; B's wage falls from 13 and its rental rises from 10. Factor prices are equal although no factor crosses the border.

Worked steps

  1. 2w + r = 30 and w + 2r = 36
  2. 3r = 2 x 36 - 30 = 42, so r = 14.00
  3. w = 36 - 2 x 14.00 = 8.00
  4. A: (3, 18) to (8.00, 14.00); B: (13, 10) to (8.00, 14.00)

Use the idea

Expect trade to push the reward of a country's scarce factor toward the world level, as long as the country keeps producing both goods with the same techniques.

Where the conclusion applies

Identical fixed techniques, no trade costs and both goods produced in both countries. If B stopped making cloth, one equation would leave many (w, r) pairs and equality would fail.

Check your understanding: What are factor prices if pC = 30 and pM = 36?
3r = 2 x 36 - 30 = 42, so r = 14, and w = 36 - 28 = 8.

Chapter 56 source: section "Factor-price equalization theorem".