Theorems of international trade:
Now we will discuss two important theorems of international trade - namely, Stolper-Sumuelson Theorem (which talks about the relationship between relative commodity prices and factor prices) and Rybcznski Theorem (which talks about output mix and factor endowment) with the above framework.
We will make the assumption that both the industries consist of firms of identical size. Two important conclusions that follow immediately are
a) each firms are factor price takers - on the aggregate, however, they affect factor prices
b) industry production can be assumed to be linear homogeneous, i.e., exhibiting constant returns to scale.
As we have shown in the beginning, the firm's unit isoquants are given by the equations,
where aLj, aKJ are variable input-output coefficients. Let us now rewrite the resource constraints in terms of aij's
For simplicity, we treat the inequality constraints as equality constraints. w and r are imputed value for wages and rent respectively and are used as Lagrange multiplier. The first order conditions for maximisation are,
Solving the equation set ,'(xiv), (xv) (xx) and (xvi), (xvii) (xxi) we can determine the cost minimising input combinations
, along the unit isoquants for each industry. We also get two marginal cost functions
The remaining four variables need to be determined are yf ,y;, w and r. The remaining four first order conditions are (xii), xiii), (xviii) and (xix). Substituting the solution values above, we get,
