International Trade Theory and Policy
by Steven M. Suranovic

Trade 115-1

The Heckscher-Ohlin (Factor-Proportions) Model

This section presents the mathematical formulation of the standard two good, two factor Heckscher-Ohlin (H-O) model. We will present the key assumptions of the model only as they are needed. In this way it may be clearer which assumptions are needed for each result. Comparative statics exercises are then conducted to derive two of the main theorems of the model: the Rybczynski theorem and the Stolper-Samuelson theorem. The following derivations are based on Silberberg (1978).

Model Assumptions

Production

Assume there are two goods, y₁ and y₂, produced within a country using two factors of production, labor L and capital K. Assume the markets are perfectly competitive. Let industry production be described by the following functions:

y₁ = f¹ (L₁ , K₁)

y₂ = f² (L₂ , K₂)

where L_i and K_i are the quantities of labor and capital used in the production of good y_i, respectively, Assume the production functions exhibit positive but decreasing returns to each factor. This means that

,, , and .

The production functions are assumed characterized by constant returns to scale (CRS). This means that if the factor inputs are increased by some proportion, , then output will increase by that same proportion, i.e., f ⁱ(L_i, K_i) = y_i .

Resource Constraints

Assume that the country has a fixed endowment of labor, L, and capital, K, and that these resources can be used only in the production of goods y₁ and y₂. The labor and capital are each homogeneous and are assumed to be freely and costlessly mobile between industries. The following resource constraints must then be satisfied.

Since production exhibits CRS, i.e., is homogeneous of degree one, by definition,

Setting implies and the production functions can be

rewritten as

(1)

where a_Lj and a_Kj represent the unit-labor and unit-capital requirements, respectively, in the production of good j.

We can also rewrite the resource constraints in terms of these unit-factor requirements. The labor constraint becomes,

(2a)

Similarly the capital constraint can be written,

(2b)

Factor Intensity: Definition

Factor intensity is used to compare relative factor usage between industries. Thus, we would say that good one is capital-intensive compared with good two if,

That is, if good one uses more capital per worker in production than the amount of capital used per worker in the production of good two, then good one is capital-intensive. Note that we can rewrite the above condition in terms of the unit-factor requirements. Thus, good one is capital-intensive if,

Note also that if then, by rearranging, . This means that good two uses more labor per unit of capital in production than the amount of labor used per unit of capital in the production of good one. In other words, good two is labor-intensive. Thus, if good one is capital-intensive, it follows that good two is labor-intensive, and vice versa.

Production Choices

We assume that all markets are perfectly competitive. This implies that each industry is comprised of numerous small firms, each which is too small to influence the final price in the market. Each firm, then, takes price as exogenous and chooses its own output to maximize profit. Under these assumptions, the aggregated choices of many small firms will be equivalent to the solution obtained by maximizing national output subject to the industry production relationships and the national resource constraints. This problem is presented below. Note that because the production relationships are rewritten in terms of the unit-factor requirements, these a_ijs become the choice variables rather than the L_is and K_is

Objective

Maximize wrt. y₁, y₂, a_L1, a_L2, a_K1, a_K2
subject to:

where z represents the value of national output or national income, and p_i is the output price of good y_i. The exogenous variables in the problem are p₁, p₂, L, and K.

The Lagrangian for this problem, which for future reference we will call problem A, is written as,

where w, r, lambda₁ , and lambda₂ are the Lagrange multipliers. w and r are also commonly referred to as the shadow prices of labor and capital respectively, and will work out to be the equilibrium wage and rental rate.

The Kuhn-Tucker first-order conditions for problem A are as follows,

(3a)		If < , then y₁ = 0
(3b)		If < , then y₂ = 0
(3c)		If < , then a_L1 = 0
(3d)		If < , then a_K1 = 0
(3e)		If < , then a_L2 = 0
(3f)		If < , then a_K2 = 0
(3g)		If >, then w = 0
(3h)		If >, then r = 0
(3i)		If >, then ₁ = 0
(3j)		If >, then ₂= 0

The first two FOCs, (3a) and (3b), represent non-positive profit conditions. When these hold with equality profit is equal to zero since the price received for each unit sold, p_i, equals the sum of the wages paid to workers per unit produced, a_Li w, plus the rents paid on capital per unit produced, a_Ki r. If these expressions are less than zero then profit is negative which can only be sustained in equilibrium if output is zero, i.e. y_i = 0. Profits can never be positive in equilibrium because perfect competition assumes free entry of new firms whenever there are positive economic profits in an industry.

The next four FOCs, (3c), (3d), (3e) and (3f) represent marginal factor-utilization conditions. If these hold with equality then (3c) and (3e) can be rewritten which implies that the wage equals the value of the marginal product of labor in each industry.⁽¹⁾ Likewise, conditions (3d) and (3f) imply that the rental rate, r, must equal the value of the marginal product of capital in each industry.

The next two conditions, (3g) and (3h) imply that the resource constraints must be satisfied. If (3g) does not hold with equality, then the shadow price of labor (the wage rate) must equal zero at the equilibrium. Similarly if (3h) does not hold with equality then the shadow price of capital, the rental rate, must equal zero at the equilibrium.

Finally, conditions (3i) and (3j) must hold with equality if factors are being fully utilized within each industry.

A diagrammatic depiction of the solution to this problem is shown in the adjoining diagram. Note that the resource constraints and the production relationships together describe the production possibilities set for this economy depicted as the solid blue area. The objective function is a linear relationship between outputs y₁ and y₂. The line is a plot of the national income objective function when national income is at a maximum. National income is maximized where a national income line is just tangent to the production possibility set as at point A.

Cost Minimization

An alternative method of solving this problem is achieved by reinterpreting the exercise in terms of its cost minimization dual problem. The reformulation is advantageous since it will provide alternative ways of interpreting the results of the model.

Consider the following two-stage process. In the first stage we minimize factor cost in production subject to a fixed level of production. In the second stage, we maximize output subject to the resource constraints.

First, rewrite the Lagrangian from above by reordering the terms to get,

Now maximize by first individually minimizing the two sets of terms in brackets. That is,

Minimize wrt. a_L1 , a_K1	y₁ ( a_L1 w + a_K1 r )	and	Minimize wrt. a_L2 , a_K2	y₂ ( a_L2 w + a_K2 r )
subject to	f¹ ( a_L1 , a_K1 ) = 1		subject to,	f² ( a_L2 , a_K2 ) = 1

Notice that y_i a_Li is simply the total amount of labor used to produce y_i units while y_i a_Ki is the total amount of capital needed to produce y_i units. Thus, y_i ( a_Li w + a_Ki r ) = wL_i + rK_i and represents the total cost of producing y_i units.

The first-order conditions for the first minimization problems are given as,

(4a)		If < , then a_L1 = 0
(4b)		If < , then a_K1 = 0
(4c)		If >, then ₁ = 0

Notice that these conditions are identical to (3c), (3d) & (3i) respectively.

Diagrammatically, this problem is depicted in the adjoining diagram. The isoquant is fixed at the level of production y₁. The exogenously given wage, w, and rental rate, r determine the slope, -(w/r), of the blue isocost line y_i (a_Li w + a_Ki r). The problem, then, is to minimize cost subject to being on the y₁ isoquant. The choice variables a_L1 and a_K1 determine the capital-labor ratio in production, a_K1/a_L1 , depicted as the slope of the red line 0A.

If we assume that only one unit of output is produced such that y₁ = 1 and if the second-order sufficient conditions are satisfied then we can apply the implicit function theorem and write the unit-factor requirements as functions of the wage and rental rates. This means we can write the a_ijs as follows,

and

where the asterisk signifies the optimal value of the variable.

Comparative statics on these variables yields

The first expression means that as the wage rises, the labor output ratio would fall. This is because relatively less expensive capital would be substituted in production for the relatively higher priced labor. Likewise the second expression implies that as the rental rate rises, the capital-output ratio will fall. This is because relatively cheaper labor would be substituted in production for the relatively higher priced capital.

We can simplify these optimal values even more if we then divide (4a) by (4b) to get,

and combine this with (4c), we have 2 equations in 2 unknowns. This implies that we can further reduce the optimal values of the unit factor requirements to functions of the wage-rental ratio (w/r) to get,

and

Conducting the same exercise for good two likewise would yield optimal values,

and

The optimal values of the unit-factor requirements as functions of the wage and rental rate now contain all of the information from the first-order conditions (3c), (3d), (3e), (3f), (3i) and (3j) from the original problem A. If we plug these optimal values into the remaining four first-order conditions, i.e., (3a), (3b), (3g) and (3h), then the equilibrium conditions of the entire H-O model can now be reduced to the following set of four equations in four unknowns, w, r, y₁, and y₂. There are four exogenous parameters: p₁, p₂, L and K. Note, by writing these as equalities we are assuming an interior solution.

(5a)
(5b)
(5c)
(5d)

Notice however, that the first two equations, (5a) and (5b), are two equations in only two unknowns, w and r. Thus the solution to the first two equations would result in optimal values of the wage and rental rates as functions of output prices, i.e., w*( p₁ , p₂ ) and r*( p₁ , p₂ ). This implies a direct relationship between goods prices and factor prices in the model.

When w*( p₁ , p₂ ) and r*( p₁ , p₂ ) are plugged into equations (5c) and (5d), these two can be solved for its two unknowns as functions of the remaining parameters, i.e., and . These two expressions represent output supply functions.

1. Since the production functions are homogeneous of degree one, the first derivatives of the production functions are homogeneous of degree zero. This implies that . The term can be interpreted as the marginal cost of production. Since marginal cost equals price in a perfectly competitive equilibrium, the equilibrium conditions (3c) and (3e) are equivalent to which means that the wage equals the value of the marginal product of labor. Similarly, conditions (3d) and (3f) imply that the rental rate on capital is equal to the value of the marginal product of capital.

International Trade Theory and Policy - Chapter 115-1: Last Updated on 3/10/98