Specific Factor Model & Labor Allocation

1. Learning Unit 6
Specific Factor Model and Labor Allocation
ECON452
International Economics

2. Objectives
1. Derive a typical PPF from production functions under diminishing
marginal return
2. Determine labor market equilibrium and equilibrium allocation of
labor
3. Understand how mobile factor will respond to price changes

3. Income Distributional Effect of Trade
• On the Ricardian model of trade, labor is only factor of production, so not only
that all countries gain from trade, but also that every individuals is made better
off as a result of international trade.
• In reality, trade creates winners and losers in an economy – Income distribution
effect of trade.
• Two main reasons why international trade has strong effects on the distribution
of income within a country:
– Short-run effect: Resources cannot move immediately or costlessly from one
industry to another. (Learning Unit 6 & 7).
– Long-run effect: Industries differ in the factors of production they demand.
(Learning Unit 8 & 9).

4. The Specific Factors Model
• Specific factors model: One factor is mobile among industries, while other factors
are not (immobile or specific to a particular industry).
– It allows trade to affect income distribution.
• Assumptions of the model:
– Two goods: cloth and food.
– Three factors of production: labor (L), capital (K) and land (T for terrain).
Cloth produced using capital and labor (but not land).
Food produced using land and labor (but not capital).
Labor is a mobile factor that can move between sectors.
Land and capital are both specific factors used only in the production of
one good.
– Perfect competition prevails in all markets.

5. Production Functions
• The production function for cloth gives the quantity of cloth that can be produced
given any input of capital and labor:
QC = QC(K, LC)
– QC is the output of cloth, K is the capital stock, LC is the labor force employed
in cloth
• The production function for food gives the quantity of food that can be
produced given any input of land and labor:
QF = QF(T, LF)
– QF is the output of food, T is the supply of land, LF is the labor force
employed in food

6. Production Function - Diagram
• Output (Qc) is an increasing function of
variable input (Lc).
– The more labor employed in the
production of cloth, the larger the
output.
• The curve gets flatter as level of
employment increases.
– each successive person-hour
increases output by less than the
previous one

7. Marginal Product of Labor
• Marginal product of labor: the increase in output that corresponds to an extra
unit of labor.
• Law of diminishing marginal returns
– Marginal produce of labor decreases as level of employment increases.
– Adding one worker to the production process (without increasing the amount
of capital) means that each worker has less capital to work with.
– Therefore, each additional unit of labor adds less output than the last.

8. Marginal Product of Labor - Diagram
• The marginal product of labor (MPLc) is a
decreasing function of labor input (Lc).
• The marginal product of labor is equal to
the slope of the production function.
MPLc = ΔQc/ ΔLc
ΔQc is a change in Qc and ΔLc is a change
in Lc.
Diminishing marginal return means the
production function getting flatter as labor
input increases

9. Mobile Resource and Production Possibilities
• Budget Constraint: For the economy as a whole, the total labor employed in cloth
and food must equal the total labor supply:
LC + LF = L
• The economy’s mix of output change as labor is shifted from one sector to the
other.
– When labor moves from food to cloth, food production falls while output of
cloth rises.
– Using all resources available and fully allocated to two industries, all possible
production mixes (Production possibilities) can be derived.

10. Four-Quadrant Diagram
• Use a four-quadrant diagram to
construct production possibilities
frontier.
– 1st (Upper right) quadrant indicates the
combinations of cloth and food that can be
produced.
– 2nd (Upper left) quadrant shows the
corresponding production function for food.
– 3rd (Lower left) quadrant indicates the
allocation of labor.
– 4th (Lower right) quadrant shows the
production function for cloth.

11. Deriving Production Possibilities
1. Choose an allocation of labor between cloth and
food industries along AA line in 3rd quadrant (AA2)
and determine quantities of labor in each industry
(LF
2 and LC
2).
2. Find a corresponding cloth output along the
production function of cloth in 4th quadrant (QC
2).
3. Find a corresponding food output along the
production function of food in 2nd quadrant (QF
2).
4. Find a combination of two outputs in 1st quadrant
(PP2) – This is one combination of outputs that the
economy can produce when it employs all labor.
5. Repeat Step 1 through 4 for various combinations
of labor in two industries (AA1 and AA3).

12. Marginal Rate of Transformation
• Marginal Rate of Transformation (MRTxy) of X for Y: The amount of Y
that a country must give up to produce each additional unit of X.
– MRTCF is an amount of food that a country must give up to produce each
additional unit of cloth.
• MRTxy is an opportunity cost of X.
– MRTCF is an opportunity cost of producing one more unit of cloth (amount of
food forgiven).
• MRTxy is a slope of PPF, when quantity of X is measured along the
horizontal axis and quantity of Y measured along the vertical axis.

13. Marginal Rate of Transformation - Formula
• Marginal rate of transformation is equal to a ratio of marginal product
of labor in two industries.
MRTCF = MPLF/MPLC
MPLC = ΔQC/ΔLC
MPLF = ΔQF/ΔLF
ΔLF = - ΔLC  (LF + LC =L)
MRTCF = ΔQF/ΔQC = (ΔQF/ΔLF)/(ΔQC/ΔLC) = MPLF/MPLC

14. Marginal Rate of Transformation and
Opportunity Cost
• An opportunity cost of producing one more unit is how many units of the other
goods given up to produce one more unit of output.
– To produce one more unit of cloth, cloth industry needs additional labor of
1/MPLC (= ΔLC/ΔQC: Unit labor requirement).
– This labor resource must be taken from the food industry.
– For each labor, its output will decrease by MPLF (= ΔQF/ΔLF).
– So, with 1/MPLC less labor, an output of food decreases by MPLF/MPLC (= MRTCF)
in order to produce one more unit of cloth (opportunity cost).

15. MRT, Unit Labor Requirement, and
Opportunity Cost
• MPL is a productivity of labor.
• MPL is an inverse of unit labor requirement (MPLC = 1/aLC).
• MRTCF = MPLF/MPLC = (1/aLF)/(1/aLC) = aLC/aLF.
• aLC/aLF is an opportunity cost of cloth in Ricardian model.

16. Specific Factors Model vs. Ricardian Model
• Ricardian model: PPF is a straight line because the unit labor
requirement is constant
– A straight line PPF means a constant opportunity cost.
• Specific factors model: PPF is curved because the unit labor requirement
increases as production increases due the diminishing marginal return.
– As it requires more labor to produce an additional output, it must take more
labor input from the other industry and the production of the other goods
decrease at increasing rate, that means, increasing opportunity cost.
– Opportunity cost of cloth in terms of food is the slope of the production
possibilities frontier – the slope becomes steeper as an economy produces more
cloth.

17. Slope of PPF
• PPF of Specific Factor model is bowed out.
– As its production of cloth increases, PPF becomes steeper, and MRTCF
increases.
• Because of diminishing marginal return,
– QC  LC & LF↓  MPLC↓ & MPLF  MRTCF

18. Labor Allocation
• How much labor is employed in each sector?
– Need to look at supply and demand in the labor market.
• Demand for labor:
– In each sector, employers will maximize profits by demanding labor up to the
point where the value produced by an additional hour equals the marginal
cost of employing a worker for that hour.

19. Marginal Revenue Product
• Marginal revenue product (MRP): additional revenue brought by
hiring an additional hour of labor
MRP = MPL x P
Where MPL is a marginal product of labor and P is a price of output.
– MPL measures a quantity of output produced by one more unit of labor
hired.
– By selling MPL units of output at price of P, a firm will earn MPL x P of
revenue.

20. Labor Demand by Profit-Maximizing Firm
• Marginal revenue of labor: additional revenue brought by hiring an
additional hour of labor
MRP
• Marginal cost of labor: additional cost of hiring an additional hour of
labor
w (wage rate)
• A profit maximizing firm hires up to labor hours where MR = MC
MPL x P = w

21. Demand for Labor
• Marginal Revenue Product (MRP)
curve is a derived demand curve for
labor input.
– MRPC = MPLC x PC is a demand for
labor in Cloth sector where PC is a
price of cloth
– MRPF = MPLF x PF is a demand for
labor in food sector where PF is a
price food
• At wage rate of w1, a profit –
maximizing firm hires (demands) Lc
1
hours of labor.

22. Demand for Labor in Two Markets
• A country has a fixed quantity of labor (L) and
must allocate in two sectors.
LC + LF = L
• The horizontal axis represents the total labor
supply L.
• The demand for labor in the cloth sector and a
quantity of labor allocated in cloth sector (LC) are
measured from the left.
• The demand for labor in the food sector and a
quantity of labor allocated in food sector (LF) are
measured from the right.

23. Allocation of Labor in Two Markets
• A profit maximizing firm hires up to labor hours
in the cloth sector where
MPLC x PC = wC
• A profit maximizing firm hires up to labor hours
in the food sector where
MPLF x PF = wF
• At equilibrium, the wage rates are same in the
two sectors:
MPLC x PC = MPLF x PF = w
• Where the labor demand curves intersect gives
the equilibrium wage and allocation of labor
between the two sectors.

24. Adjustment of Labor Allocation
• The two sectors must pay the same wage because
labor can move between sectors.
• If an economy allocate labor too few in cloth sector
and too much in food sector,
• then MPLC x PC > MPLF x PF,
• the wage is higher in the cloth sector than in food
sector (wC > wF), so workers will move from food
sector to cloth sector.
• As labor moves from food sector to cloth sector, MPL
in cloth sector decreases and MPL of food sector
increases, so the wage rate decreases in cloth sector
and increases in food sector until the wages become
equal.
wC
wF

25. Relative Prices at Equilibrium
• Equilibrium condition in labor market:
MPLC x PC = MPLF x PF
implies that the marginal rate of transformation must be equal to
relative prices:
MRTCF = MPLF/MPLC = PC/PF
• At equilibrium, the slope of production possibility frontier must be
equal to relative price of cloth.

26. Production Combination and Labor Allocation
• The economy produces at the point
on its production possibility frontier
(PP) where the slope of that frontier
equals minus the relative price of
cloth.
• The point of PPF corresponds to the
equilibrium point in labor market
(equilibrium allocation of labor).

27. Change in Prices
• When a country engages in trade, a relative price of goods will
change.
• What happens to the allocation of labor and the distribution of
income when the prices of food and cloth change?
• Two cases:
1. An equal proportional change in prices (No change in relative price)
2. A change in relative prices

28. Equal Proportional Change in Prices
When both prices change in the same proportion,
no real changes occur.
• The labor demand curves in cloth and food both
shift up in proportion to the rise in PC from PC
1 to
PC
2 and the rise in PF from PF
1 to PF
2.
• The wage rate rises in the same proportion, from
w1 to w2, but the allocation of labor between the
two sectors does not change.
• The wage rate (w) rises in the same proportion
as the prices, so real wages (i.e., the ratios of the
wage rate to the prices of goods) are unaffected.

29. Change in Relative Price on Labor Allocation
• When only PC rises, labor shifts from the food
sector to the cloth sector and the output of
cloth rises while that of food falls.
• The wage rate (w) does not rise as much as PC.
When PC rises by 7%, the cloth labor demand
curve (MRP = PC x MPLC) rises in proportion to
the 7 percent increase in PC, but the wage rate
rises less than 7% since cloth employment
increases and thus the marginal product of
labor in that sector falls, so MRP increases by
less than PC increase.

30. Change in Relative Price on Production
When only PC rises,
• The relative price (PC/PF) rises – PPF
becomes steeper.
• Production of cloth (QC) increases, while
production of food (QF) decreases as a
country allocate more labor in cloth sector
and less in food sector.
• The production point moves from 1 to 2
along the PPF corresponding to movement
from Equilibrium point 1 to Equilibrium
point 2 in labor market.

31. Relative Supply in Specific Factors Model
• As relative price of cloth (PC/PF)
increases, the output of cloth (QC) rises
and the output of food (QF) falls.
• The relative quantity of cloth (QC/QF) also
increases.
• The relative supply curve of cloth is
upward-sloping.
• Since outputs of two goods change
gradually along PPF, the relative supply
curve is a smooth upward-sloping curve.

32. Determination of Relative Prices
• The equilibrium relative quantity
of cloth (QC/QF) and equilibrium
relative price of cloth (PC/PF) are
determined by an interaction of
relative supply (RS) and relative
demand (RD).

33. Disclaimer
Please do not copy, modify, or distribute
this presentation
without author’s consent.
This presentation was created and owned
by
Dr. Ryoichi Sakano
North Carolina A&T State University
Disclaimer
Please do not copy, modify, or distribute
this presentation
without author’s consent.
This presentation was created and owned
by
Dr. Ryoichi Sakano
North Carolina A&T State University