1
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.
2
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.
3
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
6
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.
7
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.
8
Figure 4. Equilibrium employment given the wage rate.
ML
MM MPLp
),( MMM LKMPLp
w
L*M
9
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
10
Figure 5. Equilibrium in the labor market
ML
MMPL
),( MMM LKMPLp
),( FFF LTMPLp
L*M L LF
w
MPLF
11
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).
12
Figure 6. Unit isoquant.
aKM
a*KM E
E’ a**KM
a**LM
Kr
w
a*LM aLM
13
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)
14
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
15
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.
16
)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
17
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:
18
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
11
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.
19
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
20
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.
21
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.
22
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
23
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
24
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)↑