Lecture 4: the Heckscher-Ohlin Model (part I)

International Trade Theory (1/2008) Chulalongkorn University

Lecture 4: the Heckscher-Ohlin Model (part I)

Kornkarun Cheewatrakoolpong, Ph.D.

The Ricardian model uses the difference in technology as a reason for two countries to trade with each other. Now we will consider the Heckscher-Ohlin model in which the technology is the same between two countries. However, there are differences in factors of production. We will use 2-good, 2-factor general equilibrium model to study the Heckscher-Ohlin model.

The basic model

1. two factors: labor (L) and capital (K) 2. two goods: 1,2

3. production functions are defined by Y_i = f_i(L_i,K_i), i= 1,2

4. f_i(L_i,K_i)is assumed to be increasing, concave, and constant return to scale.

5. labor and capital are fully mobiled between two industries. Then the resource constraints can be written as:

L L L₁ + ₂ ≤

K K K₁+ ₂ ≤

With L and K are the total endowment of the country and fixed. For given (K,L), we can write the production possibility frontier (PPF) and with the assumption of concavity of production functions, we can combine technology together with the resource constraints so that Y₂ =h(Y₁,L,K). The function h(Y₁,L,K)is concave so that the production possibility set is convex.

The market structure 1. perfect competition

2. countries are small so prices of 1 and 2 are taken as given.

Y

2

Y

1

Firms’problems:

Consider two goods as if they are from two different firms, we can determine the factor prices and outputs using the unit cost functions which are defined by:

i i K L

i w r wL rK

c

i i

+

=

min

≥ 0 ,

) , (

s.t. f_i(L_i,K_i)=1

Suppose that the optimal choice of L_i is a_iL and the optimal choice of K_i is aiK, then we can write down the solution to the minimization problem as:

iK iL

i w r wa ra

c ( , )= +

As a_iL =a_iL(w,r) and a_iK =a_iK(w,r), then : )

, ( )

, ( )

,

(w r wa w r ra w r c_i = _iL + _iK

The equilibrium conditions:

1. Perfect competition with free entry/exit condition gives us zero profit condition:

) , (w r c

p_i = _i i=1,2 (1)

2. Full employment conditions:

L a Y a

Y_{1 1L} + _{2 2L} = (2)

K a Y a

Y_{1 1K} + _{2 2K} =

(1) and (2) give us 4 equations and 4 unknowns. However, it is not enough to guarantees the unique solutions for (Y₁ ,Y₂,w,r). We need to study in details to determine whether the solutions are unique and strictly positive or not which is our main task here.

Note: We can also use profit maximization to get these conditions.

From the production function, we can depict the isoquant curve as:

} 1 ) , ( ) ,

{(K_i L_i f_i L_i K_i = . K

L

1 ) , ( _{i i} =

i L K

f

We can also draw the unit function is a set of (w,r) s.t. {(w,r) c_i(w,r)=c_min} with c_min= minimum unit cost. This curve is downward sloping as wage has to decrease when the rental rate increases in order to keep the unit cost constant.

Moreover, it is convex because of the concavity of the cost function.

Consider the efficient factor allocations, we represent all possible allocations of factor endowments between two firms (goods) using the edgeworth box diagram.

We represent the isoquants of two firms in this box in this box. The pareto set of factors allocations are the set that isoquants are tangent to each other.

Factor Intensity

The production of good 1 is relatively more intensive in labor than the production of good 2 if:

) , (

) , ( )

, (

) , (

2 2 1

1

r w a

r w a r w a

r w a

K L k

L > for all (w,r)

1 1(w,r) p

c =

r

w

2 2(w,r) p

c =

r

w

) min

, (w r c c_i =

Lemma: So long as both goods are produced and factor intensity reversal (FIR) do not occur, then each price price vector (p₁,p₂)corresponds to unique factor prices (w,r).

To determine the equilibrium factor prices, suppose two things:

1. We have interior solution in which both goods are produced.

2. FIR does not occur.

In the usual case when there is no FIR, the unit cost lines of two goods intersect only once. Therefore, for given (p₁,p₂), there is a unique solution for (w,r) from:

1 1(w,r) p

c =

2 2(w,r) p

c =

However, in some cases, the unit cost lines intersect more than once. Then each price vector (p₁,p₂) may not correspond to unique factor prices (w,r).

Under the factor intensity condition, there is at most a single pair of factor prices that can arise as the equilibrium factor price of interior solution. When the factor prices are known, the equilibrium output levels can be found from the point

) , , ,

(K₁^* K₂^* L^*₁ L^*₂ .

1 1(w,r) p

c =

r

w

2 2(w,r) p

c =

In the equilibrium with no FIR and interior solution, we have the conditions that:

1 1(w,r) p

c =

2 2(w,r) p

c = (3)

and

K L

a a K

L

1 1

* 1

*

1 =

K L

a a K

L

2 2

* 2

*

2 = (4)

Factor price equalization theorem (Samuelson 1949)

Suppose that there is no FIR, what are the implications for the result for the determinants of factor prices under free trade?

Assume that : 1. There is no FIR .

2. Both countries have identical technologies but different in factor endowments.

3. Product prices are the same.

4. Both goods are produced in both countries (interior solution).

Then the equilibrium factor prices should be equal across country. The level of the endowments matter only to the extent that they determine whether the economy specializes.

Factor price equalization states that suppose that two countries are engaged in free trade, having identical technologies but different factor endowment, if both countries produce both goods are FIR does not occur, then the factor prices (w,r) are equalized across countries.

Next, we will consider two comparative static exercises to get two more theorems regarding the Heckscher-Ohlin model.

K^*1 K^*₂

L^*₂

L^*1

O¹

O²

Change in product prices

How does a change in the price of one of the outputs, say p₁, affect the equilibrium factor prices and factor allocations?

From

r a w a r w c

p₁ = ₁( , )= _1L + _1K r a w a r w c

p₂ = ₂( , )= _2L + _2K (5)

Total differentiate the zero profit condition (5), we obtain:

dr a dw a

dp₁ = _1L + _1K dr a dw a

dp₂ = _2L + _2K

Or r

dr c ra w dw c wa p

dp

i iK i

iL i

i = + i=1,2 (6)

(6) allows us to express the variables in terms of percentage changes such as dw/w as well as cost shares.

Let

i iL

iL c

= wa

θ be the cost share of labor in industry i

i iL

iK c

= ra

θ be the cost share of capital in industry i.

With 1θ_iL +θ_iK = . Denote

w wˆ = dw and

r

rˆ= dr. Then we have:

r w

pˆ_i =θ_iL ˆ+θ_iKˆ i=1,2

Express the equation using the matrix to follows Jones (1965).

⎥⎦

⎢ ⎤

⎣

⎥⎡

⎦

⎢ ⎤

⎣

=⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

r w p

p

K K L L

ˆ ˆ ˆ

ˆ

2 1 2 1 2

1

θ θ θ

θ (7)

Given that _⎥

⎦

⎢ ⎤

⎣

=⎡ Θ

K K L L

2 1 2 1

θ θ θ θ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎥⎡

⎦

⎢ ⎤

⎣

⎡

−

−

= Θ

⎥⎦

⎢ ⎤

⎣

⎡

2 1 1 1 2 2

ˆ 1 ˆ

ˆ ˆ

p p r

w

L K L k

θ θ θ

θ (8)

The factor intensity assumption implies that:

2 0

1 2

1_La _K −a_Ka _L >

a

So Θ =θ_1Lθ_2K −θ_1Kθ_2L >0 and then

⎥⎦

⎢ ⎤

⎣

⎡

−

−

= Θ

−

L K L k

1 1 2

1 1 2

θ θ θ

θ θ exists.

Right now we assume that industry 1 is labor intensive and industry 2 is capital intensive.

Now consider when the price of good 1 increases, then pˆ₁ − pˆ₂ >0, we can solve for a change in factor prices. Recall that :

L K K

L 2 1 2

1 θ θ θ

θ −

= Θ

= θ_1L(1−θ_2L)−(1−θ_1L)θ_2L (9) =θ_1L −θ_2L = θ_2K −θ_1K

Use (8) and (9), we have Θ

= ² ˆ¹− ¹ ˆ²

ˆ p p

w θ ^K θ^K =

K K

K K

K p p p

1 2

2 1 1 1 1

2 )ˆ (ˆ ˆ )

(

θ θ

θ θ

θ

−

− +

−

> pˆ₁ since pˆ₁− pˆ₂ >0 And

Θ

= ¹ ˆ² − ² ˆ¹

ˆ p p

r θ ^L θ ^L =

L L

L L

L p p p

2 1

2 1 2 2 2

1 )ˆ (ˆ ˆ )

(

θ θ

θ θ

θ

−

−

−

− < pˆ₂ .

Therefore, we can see that wage increases by more than the price of good 1, i.e. wˆ > pˆ₁ > pˆ₂ >rˆ which means that workers can afford to buy more of good 1 and more of good 2 as real wage has increased.

Consider the rental price, it changes by less than the price of good 2 as p2

r and

p1

r increase which means that the real return to capital has fallen.

Stolper-Samuelson Theorem (1941)

In the 2x2 production model with the factor intensity assumption, if p_i increases, the equilibrium price of factor more intensively used in the production of good i increases while the price of the other factor decreases (assuming interior equilibria both before and after the price changes).

So obviously, an increase in the relative price of a good will increase the real return to the factor intensively used in that good and reduce the real return to the other factor. Therefore, both firms must move to a less intensively use of labor in the new equilibrium. The factor allocation moves to a new point in the pareto set at which the output of good 1 has risen and the output of good 2 has fallen.

Jones has called the set of inequalities “the magnification effect”

r p p

wˆ > ˆ₁ > ˆ₂ > ˆ

Any change in the product prices has a magnified effect on the factor prices.

The magnification effect says that there will be both gainers and losers due to the change in price. At least, trade opportunities have strong distributional consequences making some people worse off and some better off although we over all gain from trade.

Changes in endowments

The second comparative static exercise we consider is a change in factor endowments. Suppose that the total availability of labor increases from L to L’. What is the effect of this change on equilibrium factor prices and output levels?

First consider a change in the factor prices. Because neither the output prices nor technologies have changed, the factor input prices remain unaltered as long as the economy does not specialize.

As a result, factor intensities do not change. The new input allocation is then easily determined in the superimpose edgeworth box. We merely find the new intersection of the two rays associated with the unaltered factor intensity level.

Rybcszynski Theorem

In the 2x2 production model with the factor intensity assumption, if the endowment of a factor increases, then the production of the good that uses this factor relatively more intensively increases and the production of the other good decreases (assuming interior equilibria both before and after the change in endowment).

Note that it is possible tat when L increases to very high level, then the country no longer diversifies, it will produce only good 1. How to determine which good to specialize? Consider the one that gives you higher GDP.