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2x2x3 Modified Ricardian (Factor Specific) Model

Unlike in the simple Ricardian model, in the modified (factor specific) model trade is based on differences in

factor endowments and not on differences in technologies (although like in the simple Ricardian model trade

in the factor specific model is also related to differences in supply conditions and not differences in

consumer preferences).

Model Assumptions

Two countries: Home (England) and Foreign (Portugal)

Two goods: Manufactures (cloth) and food (wine)

Three factors of production: Labor (L), Capital (K), Land (T)

Both countries have the same technologies (production functions), the same supplies of non-specific factor -

labor (L), but have different endowments of capital and land. In particular, England has more capital than

Portugal, while Portugal has more land than England.

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Manufactures are produced using capital and labor (but not land). The output of manufactures depends on

how much capital and labor are used in that sector. This relationship is summarized by the production

function for manufactures:

1),( MMMM LKLKQQ

Food is produced using land and labor (but not capital). Similarly, the output of food depends on how much

land and labor are used in that sector.

1),( FFFF LTLTQQ

Labor is a mobile factor which can move freely between sectors, while land and capital are both specific

factor that can be used only in the production of one good. For the economy as a whole, the labor employed

must equal the total supply L:

LM + LF = L

Production Possibilities

To analyze the economy’s production possibilities we need to ask how economy’s output mix changes as

labor is shifted from one sector to the other. Therefore, start with the graphical representation of production

functions for particular sectors and then derive the production possibility frontier.

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Figure 1 . Production function in the manufacturing sector.

ML

MQ MLKQ ,

4

Figure 2. Marginal product of labor in the manufacturing sector

ML

MMPL

),( MM LKMPL

5

Figure 3. Production possibility frontier.

QM LF

LM

QF

PPF

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The slope of QM(K,LM) represents the marginal product of labor. However, in contrast to a simple Ricardian

model, if labor input is increased without increasing capital, there will be diminishing returns: each

successive increment of labor will add less to production than the last. Diminishing returns are reflected in

the shape of the production function which gets flatter as more labor is used.

In contrast to a simple Ricardian model where the production possibility frontier was a straight line because

the opportunity cost was constant now the production possibility frontier is a curve due to the addition of the

other factors of production. The curvature of PPF reflects diminishing returns to labor in each sector. If we

shift one unit of labor from production of food to production of manufactures this extra input will increase

output in that extra sector by the marginal product of labor in manufactures.

If we shift one unit of labor from production of food to production of manufactures this extra input will

increase output by the marginal product of labor in manufactures MPLM (and lower the output of food by

MPLF). Therefore, if we want to increase the output of manufactures by one unit we have to increase labor

input in production of manufactures by 1/MPLM units of labor. Hence, to increase output of manufactures by

one unit we must decrease the output of food by MPLF/MPLM units.

Thus, the slope of the production possibility frontier reflects the opportunity cost of manufactures expressed

in terms of food. However, unlike in the simple Ricardian model, in the factor specific model the opportunity

cost is not constant. If we decrease output of food then the value of MPLF will increase and MPLM will

decrease. Hence, the opportunity cost of manufactures expressed in terms of food will rise and the slope of

PPF will increase.

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Labor Allocation

Until now we have shown how output in each sector is determined given the allocation of labor. Now let’s

see how labor is allocated across sectors.

The demand for labor in each sector depends on the price of output (pM, pF) and the nominal wage rate (w).

In each sector profit maximizing firms demand labor up to the point where the value produced by an

additional unit of labor equal the cost of labor (wage).

Numerical Example.

Suppose that wage rate is $ 4 per worker, the marginal product of labor is 3 gallons of wine when the firm

employs 10 workers, and the price of wine is $ 2 per gallon. Will the firm employ an additional worker?

Compare gains and costs. By employing an additional worker the firm can earn $ 6 of additional revenue (it

can produce 3 more gallons of wine and sells them for $ 2 each). The cost to the firm will be $ 4 paid in the

worker’s wage. Thus, the firm can increase its profit by $ 2. Therefore, the firm will hire an eleventh worker.

However, when the firm does that it reduces the land-labor ratio, and the marginal product of labor falls to 2

gallons. Now, if the firm were to hire an additional worker it could earn only $ 4 so the firm cannot increase

its profits by employing an additional worker as it would have to pay $ 4 in the worker’s wage.

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Figure 4. Equilibrium employment given the wage rate.

ML

MM MPLp

),( MMM LKMPLp

w

L*M

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In equilibrium in each industry the value of marginal product of labor must equal the wage rate:

wPMPL

wPMPL

FF

MM

Labor market equilibrium requires equalization of the values of marginal products across industries

(due to mobility of workers between industries).

FFMM PMPLPMPL

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Figure 5. Equilibrium in the labor market

ML

MMPL

),( MMM LKMPLp

),( FFF LTMPLp

L*M L LF

w

MPLF

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Determination of Prices

Let’s now examine how costs and demand for factors are related to the prices of factors when producers

employ two factors.

In a perfectly competitive economy the price of each good must equal unit cost of production (perfect

competition conditions). The unit cost of production equals the sum of the cost of capital and labor inputs.

waracp

waracp

LFTTFFF

LMKKMMM

where: )/( KKM rwa and )/( KLM rwa are unit factor requirements that have been chosen to minimize unit cost

in production of manufactures. Hence, costs cannot be reduced by increasing aKM or reducing aLM (or vice

versa).

Consider the production of the manufacturing good that requires capital and labor as factors of production

(alternatively consider the production of the agricultural good that requires land and labor as factors of

production). The manufacturing good is produced with constant returns to scale. The production technology

may be summarized in terms of a unit isoquant (i.e. a curve showing all the combinations of capital and labor

that can be used to produce one unit of the manufacturing good).

The unit isoquant shows all combinations of capital and labor that can be used to produce one unit of the

manufacturing good. The unit isoquant shows that there is a tradeoff between the quantity of capital used per

unit of output aKM and the quantity of labor per unit of output aLM. The shape of the unit isoquant reflects the

assumption that it becomes increasingly difficult to substitute capital for labor as the capital-labor ratio

increases (and vice versa).

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Figure 6. Unit isoquant.

aKM

a*KM E

E’ a**KM

a**LM

Kr

w

a*LM aLM

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In a competitive economy producers will choose the capital-labor ratio that minimizes their costs.

dCM = 0 = daKMrK + daLMw

Slope KLM

KM

r

w

da

da

An infinitesimal change in the capital-labor ratio from the cost minimizing choice must have no effect on

cost.

“HAT ALGEBRA”

Consider what happens when the factor prices w and rK change. There will be two effects:

i) a change in the choice of aKM and aLM

ii) a change in the cost of production.

Let’s differentiate totally the unit production cost

0

LMKMKLMKKMM wdadardwadradC

Let’s write it in a different form (divide both sides by CM)

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wr

w

dw

C

wa

r

dr

C

ra

C

dCC LMKKM

w

M

LM

r

K

M

KKM

thrateofgrowchange

M

MM

LMKM

ˆˆˆ

ˆˆ%

Where ΘKM – share of capital in total production cost of M, ΘLM – share of labor in total production cost of

M ΘKM + ΘLM = 1.

The cost minimizing labor-capital ratio depends on the ratio of price of labor to price capital:

The relationship between factor prices and capital-labor ratio in the manufacturing sector that results from a

1% change in the ratio of factor prices is known as the elasticity of substitution σM.

)//()/(

)//()/(

KK

LMKMLMKMM

rwrwd

aaaad

KLM

KM

r

w

a

a

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LMKM

LM

LM

KM

KM

LMKM

LMKM aaa

da

a

da

aa

aadˆˆ

)/(

)/(

K

K

K rwr

dr

w

dw

rw

rwdˆˆ

)/(

)/(

Hence,

)ˆˆ(ˆˆ

)ˆˆ(ˆˆ

rwaa

rwaa

FLFTF

MLMKM

Using hat notation we can rewrite our pricing conditions as:

wrp

wrp

LFTTFF

LMKKMM

ˆˆˆ

ˆˆˆ

These equations allow us to derive changes in capital and land rentals given the changes in the prices of

manufactures pM, food pF and labor w.

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)ˆˆ(ˆ)ˆˆ(1

ˆ

)ˆˆ(ˆ)ˆˆ(1

ˆ

wppwpr

wppwpr

F

TF

LFFLFF

TF

T

M

KM

LMMLMM

KM

K

Now let’s determine the change in the wage rate w by examining the demand and supply for labor. Demand

for labor comes from both sectors in which labor is used, while output is determined by supply of the

specific factor. Hence,

TF

F

KM

M

a

TQ

a

KQ

Therefore,

Ta

aQaL

Ka

aQaL

TF

LFFLFF

KM

LMMLMM

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Concentrate on the manufacturing sector and notice that the supply of capital is fixed and employment of

labor in production of manufactures can change only through changes in the capital-labor ratio (unit factor

requirements). Using “hat algebra” we have:

)ˆˆ())ˆˆ(1

ˆ()ˆˆ(ˆˆˆ wpwpwrwaaL M

KM

MLMM

KM

MKMKMLMM

In the same manner we obtain:

)ˆˆ())ˆˆ(1

ˆ()ˆˆ(ˆˆˆ wpwpwrwaaL F

TF

FLFF

TF

FTFTFLFF

Now turn to the full employment condition in the labor market. Note that the labor supply is fixed. If total

employment is to remain, an increase in one sector’s employment must be offset by a decline in the other

sector.

FM

FM

dLdL

dLdLdL

0

This expression can also be transformed into one that uses the hat algebra:

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0ˆˆˆ

0

ˆˆ

LLL

L

dL

L

L

L

dL

L

L

L

dL

FFMM

F

L

F

FM

L

M

M

FF

MM

Where M is the share of labor employed in production of manufactures in the economy’s total labor supply.

Finally, let’s substitute the labor demand equations into the transformed labor market equilibrium condition:

TF

FF

KM

MM

F

TF

FFM

KM

MM

F

TF

FFM

KM

MM

pp

w

wpwp

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ˆ1

ˆ1

ˆ

0)ˆˆ(1

)ˆˆ(1

Note that the change in the wage rate is a weighted average of the changes in the prices of manufactures and

food.

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EFFECTS OF CHANGES IN RELATIVE PRICES

Suppose that the price of manufactures increases relative to that of food, i.e. FM pp ˆˆ

We can notice that the change in the wage rate will be smaller than the change in Mp̂ but bigger than the

change in Fp̂ because the change in the wage rate is a weighted average of the change in the two goods

prices:

FM pwp ˆˆˆ

The effect of the allocation of labor is apparent from labor demand equations. Since wpM ˆˆ , 0ˆ ML

employment in the production of manufactures increases and employment in the production of food falls,

.0ˆ FL

The effects on the prices of capital and land may be seen from equations describing changes in unit costs:

FF

TF

LFFT

MM

KM

LMMK

pwppr

pwppr

ˆ)ˆˆ(ˆˆ

ˆ)ˆˆ(ˆˆ

0

0

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The overall description of the relation of the goods prices and factor prices is:

TFMK rpwpr ˆˆˆˆˆ

The price of capital rises in terms of both goods. Therefore, someone who derives income entirely from

capital would be unambiguously better-off.

The price of land falls relative to both goods. Therefore, someone who derives income entirely from land

would be unambiguously worse-off.

Someone deriving income from labor would find that the purchasing power of income increases in terms of

food and falls in terms of manufactures.

These findings can be summarized in the Haberler Theorem:

A CHANGE IN RELATIVE PRICES RAISES THE REAL EARNINGS OF THE FACTOR USED

SPECIFICALLY IN THE INDUSTRY WHOSE OUTPUT PRICE HAS RISEN, AND REDUCES THE

REAL EARNINGS OF THE FACTOR USED SPECIFICALLY IN THE INDUSTRY WHOSE OUTPUT

PRICE HAS FALLEN. THE REAL EARNINGS OF THE MOBILE FACTOR (LABOR) FALL IN TERMS

OF THE GOOD WHOSE PRICE HAS RISEN AND RISE IN TERMS OF THE GOOD WHOSE PRICE

HAS FALLEN.

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INTERNATIONAL TRADE

Having studied the economy of one country we are ready to study the effects of international trade.

Differences in factor endowments lead to differences in transformation curves between countries. When

England has more capital than Portugal and Portugal has more land than England relative price of

manufactures (expressed in terms of agricultural goods) under autarky is higher in Portugal than in England.

A

EnglandF

M

T

F

M

A

PortugalF

M

p

p

p

p

p

p

With free trade the relative price of manufactures is the same in two countries which means that the relative

price of manufactures increases in England and falls in Portugal. As a result of the change in relative prices

of manufactures output of manufactures increases in England and falls in Portugal and output of agricultural

goods increases in Portugal and falls in England.

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Figure 7. Trade in factor specific model

C

UT UA

QF

QM

TPQ

AP

AP CQ

TC

AE

AE CQ

TEQ

AP

AP CQ

TPQ

TC AE

AE CQ

TEQ

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Figure 8. The effect of change in relative prices due to opening to international trade in England.

ML

MMPL

),()/( MMFM LKMPLpp

),( FF LTMPL

L*M L LF

Fp

w

MPLF

L**M

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Trade and Distribution of Income

The Haberler theorem can be used to show how trade affects the real earnings of land, capital and labor.

Knowing that the relative price of manufactures increases in England and falls in Portugal the impact of

changes in real factor rewards in both countries can be summarized in the following tables:

England (capital abundant country)

Sector Employment Output Real wage Real reward to specific factor

Manufactures LM↑ QM↑ MPLM(LM)↓ MPKM(LM)↑

Food LF↓ QF↓ MPLF(LF)↑ MPTF(LF)↓

Portugal (land abundant country)

Sector Employment Output Real wage Real reward to specific factor

Manufactures LM↓ QM↓ MPLM(LM)↑ MPKM(LM)↓

Food LF↑ QF↑ MPLF(LF)↓ MPTF(LF)↑