The 2 × 2 Heckscher-Ohlin-Samuelson Model

The 2×2 Heckscher-Ohlin-Samuelson (HOS) model of production assumes that two commodities, X and ₁ X are produced using two factors, labour (₂ L) and capital (K).

The production function exhibits constant returns to scale with diminishing marginal returns to each factor. The factors are fully employed and are mobile between the two sectors. Commodities can be classified in terms of relative factor intensities and factor intensities are irreversible. The wage rate and the return to capital are denoted by W and r respectively. It is assumed that perfect competition prevails everywhere so that commodity prices P and ₁ P reflect unit costs of production. Commodity prices and ₂ factor endowments are given exogenously.

The production functions are given by the following two equations:

( , )

i i i i

X F L K for i 1, 2 (2.1) & (2.2)

Page | 24 where L_i and K_i denote employment of labour and use of capital in the ith sector. Since

the production functions exhibit CRS the equations of unit isoquants are obtained as follows.

1 f a_i( _Li,a_Ki) (2.3)

where a_Li and a_Ki denote respectively the labour and capital requirement per unit of X . _i

Now given the output level, profit maximisation means minimisation of costs. In other words, the producers minimize cost along the unit isoquant. At the point of cost minimisation, the iso-cost line, with slope(W r/ ), is tangent to the unit isoquant with slope (da_Ki/da_Li). Thus cost minimisation with respect to both the commodities implies

1 1 0

The above two equations are called the ‘envelope conditions’.

Rewriting the equations (2.4) and (2.5) in accordance with the ‘^’ notation implying

These are the alternative expressions of the ‘envelope conditions’.

The competitive profit conditions (equality between price and unit cost) in each sector are represented as

Now, the full employment conditions of labour and capital are given as

1 1 2 2

Page | 25 The model consists of four independent equations (2.6) – (2.9). There are four variables,

1 2

, , ,

W r X X and four parameters P P L and ₁, ₂, K. Thus the system is determinate and each variable can be uniquely determined. Given the commodity prices, the factor prices can be determined from the price system alone consisting of equations (2.6) and (2.7).

Thus, any changes in factor endowments cannot affect factor prices. This kind of a system where factor prices are independent of factor endowments is called a decomposable system.

Equations of Change

To examine the comparative static properties of the model, that is, to determine the effects of a change in the parameters on the variables of the model, let us transform the equations of the system into equations of change. It is convenient to express these changes in terms of the rate of change, denoted by ‘^’, for example, L^ˆ(dL L/ ).

Totally differentiating the equations (2.6) and (2.7) in the price system and using the ‘^’

notation, we obtain

1 ˆ 1ˆ ˆ1 [ 1ˆ 1 1ˆ 1]

LW K r P LaL K aK

      (2.10)

2 ˆ 2ˆ ˆ2 [ 2ˆ 2 2ˆ 2]

L W K r P L aL K aK

      (2.11)

where  is the relative share of the j th input in the total value of the _ji ith commodity, 1, 2

i  and jL K, , for example, _L1(Wa_L1/P₁).

Using the above ‘envelope conditions’ as given by (2.4.1) and (2.5.1), equations (2.10) and (2.11) can be reduced to

1 ˆ  1ˆ ˆ1

LW K r P

 

(2.10.1)

2 ˆ  2ˆ ˆ2

LW K r P

 

(2.11.1)

Page | 26 Equations (2.10.1) and (2.11.1) imply that for each commodity, the distributive-share

weighted average of proportional factor-price changes equals the proportional commodity price change. The above two equations are written in the matrix form as follows.

1 1 1

where  is the determinant of coefficient matrix in (2.12) and is given by

 = ^{1 1} _{1 2 1 2} to unity. Therefore,  can also be expressed as

 _L1_L2 _K2_K1 It is evident that ˆW and ˆr can be determined if both the commodities are produced and

  . 0

Page | 27 intensities for production of the commodities must differ. Hence,  , if the production 0 of X is labour-intensive i.e. ₁ ^{1 2} wage rate and raises the return to its intensive factor, capital.

These results are summarized in the Stolper-Samuelson theorem¹⁹, which states that a rise in the price of a commodity raises the real reward of its intensive factor and lowers the real reward of its un-intensive factor.

The Rybczynski theorem

Now let us consider the output system comprising of equations (2.8) and (2.9). Totally differentiating the equations and using the ‘^’ notation, we obtain

1 ˆ1 2 ˆ2 ˆ ( 1ˆ 1 2ˆ 2)

Page | 28 between labour and capital in the production of X . ₁

By definition, the elasticity of substitution in sector 1 is given by

1 1

By using equation (2.4.1) the above expression may be written as

1 1 1 ˆ

Using (2.18) and (2.19) equations (2.8.1) and (2.9.1) can be rewritten as

1 ˆ1 2 ˆ2 ˆ ( ˆ ˆ)

The changes in output levels can be determined by solving equations (2.8.2) and (2.9.2) that may be expressed in matrix notation as

Page | 29

where  is the determinant of the coefficient matrix and is given by

1 2 to unity. Therefore,  can also be expressed as

1 1 2 2

Page | 30 From equation (2.25) it follows that if X is labour-intensive, an increase in the labour ₁

endowment raises X by a magnified amount and lowers ₁ X . If ₂ L^ˆ exceeds K^ˆ then

1 2

ˆ ˆ ˆ ˆ

X LK  X

But if X is capital-intensive, ₁   . In this case, an increase in 0 L leads to higher production of X and a decline in that of ₂ X . ₁

This result entails the Rybczynski theorem²⁰, which states that a rise in the endowment of a factor at constant commodity prices leads to the expansion of the commodity that uses the factor intensively and contraction of the other commodity.

Responses of outputs to changes in the commodity prices

Equation (2.25) shows the relationship of changes in outputs with changes in factor endowments and factor prices. The output response to changes in factor endowment is captured by the Rybczynski theorem. To examine the effects of commodity prices on the outputs, let us substitute the link between factor prices and commodity prices depicted in equation (2.17) to obtain

1 2

ˆ ˆ

(X X ) (Lˆ Kˆ) ( ^{L K})(ˆ₁ ˆ₂)

P P

 

  



    (2.25.1)

As already stated,  and _L  are both positive. Now, _K  and  must have the same sign. If X is labour-intensive, both ₁  and  are positive, whereas if X is assumed to ₁ be capital-intensive, both  and  are negative, so that the product   is always positive.

Thus, if Pˆ₁ Pˆ₂ then Xˆ₁ Xˆ₂. In particular, from (2.21) and (2.22) it follows that X  ˆ₁ 0 and X  . If ˆ₂ 0 Pˆ₁Pˆ₂  then 0 Wˆ Pˆ₁Pˆ₂   so that rˆ 0 (W^ˆ rˆ)0 and Xˆ₁Xˆ₂  . 0

20 See Rybczynski (1955).

Page | 31 Therefore an increase in the price of a commodity leads to rise in production of that

commodity and fall in that of the other commodity. If both the commodity prices change at the same rate, the production of both commodities remains unchanged.

If technologies of production are of fixed-coefficient type i.e. ₁₂ 0 then and _L

 are equal to zero. Then from (2.21) and (2.22) it follows thatK Xˆ₁ Xˆ₂ 0.

So any changes in commodity prices have no effect on the composition of outputs.

2.1.1. An alternative presentation

Using (2.18.1) and (2.19.1) equation (2.8.1) can be written as

1 1 2 2

1 ˆ1 1( ˆ ˆ) 2 ˆ2 2( ˆ ˆ) ˆ

L X L S WLL S rLK L X L S WLL S rLK L

      

21 Both types of notations have been used in the subsequent chapters of the book. This alternative version is particularly useful when three factors are used to produce a commodity so that equations of changes involve partial elasticities as well, making the analysis complicated if the earlier version is used.

Page | 32 factor of production in any sector with respect to factor prices must be zero. For example, in sector 1, with respect to labour, we have (S¹_LLS¹_LK)0 while with respect to capital,

1 1

(S_KLS_KK)0. Similarly, in sector 2, (S_LL² S_LK² )0 and (S_KL² S_KK² )0.

Equations (2.8.3) and (2.9.3) can be expressed in the form of a matrix

1 2

Page | 33 ˆ2

X (1 / )[(_L1Kˆ _K1Lˆ) {  _{L2 K1}S_LK²  _{L1 K2}S_KL²  _{L1 K1}(S¹_LK S¹_KL)}(Wˆ rˆ)]

(2.22.1) Subtracting (2.22.1) from (2.21.1) and using (2.17) one gets

1 2

ˆ ˆ

(X X ) (Lˆ Kˆ) ( ^{L K})(ˆ₁ ˆ₂)

P P

 

  



    (2.25.2)

where _L  _{L2 K1}S¹_KL _{L1 K2}S¹_LK  _{L2 K2}(S_KL² S_LK²)0 _K  _{L2 K1}S_LK²  _{L1 K2}S_KL²  _{L1 K1}(S¹_LK S_KL¹ )0

Equation (2.25.2) is analogous to equation (2.25.1). Hence the results pertaining to the Rybczynski theorem and output responses to changes in commodity prices obtained in the preceding section follow.

In document Revisiting the Informal Sector: A General Equilibrium Approach (Page 24-34)