1. Consumer Problem

Choose *X*_{1} and *X*_{2}
to maximize _{} subject to _{} The Lagrangian
function associated with this problem is:

_{}

The FOCs are:

_{}

From the first two FOCs, we
obtain the equilibrium condition, _{} which is marginal
utility of income. This implies _{}and

_{}

2. A Numerical Example of the
Heckscher-Ohlin Model with Cobb-Douglas Production Functions

This example is from Rod Tyers’
lecture notes (http://teaching.fec.anu.edu.au/ECON3054/)

Consider Cobb-Douglas
production functions for two goods,

_{}

where and are the capital shares of the two industries.

Each industry chooses its
inputs to minimize its production cost. For instance, industry 1 chooses *L*_{1} and *K*_{1 }to minimize its cost. The Lagrangian function
associated with this problem is:

_{}

The solution to this problem
yields a pair of expansion paths. The FOCs are:

_{}

where * and **are the shadow prices or marginal costs of good 1 and 2,
respectively, which are equal to their prices in competitive markets. *

From the FOCs, we get

_{} (1)

Let _{}be the relative price of the importable good. Rearranging these
terms, we obtain

_{} (2)

Labor and capital costs in
industry 1 are

_{}

Similarly, labor and capital
costs in industry 2 are:

_{}

The equilibrium conditions
for cost minimization are:

_{}

It follows that

_{} (3)

(Industry 2 is 9 times more
capital intensive than industry 1.)

Note that the Cobb-Douglas
production functions can be written as:

_{}

_{}

Then, the output ratio in (2)
is written as

_{}

It follows that

_{}

Squaring both sides, we have _{} Using the relationship
between the two expansion paths in (3), we get

_{}

Thus, capital-labor ratios
are functions of p.

_{}

From (1), factor prices are:

_{}

which show the
Stolper-Samuelson Theorem that an increase in the price of the capital
intensive good raises the return to capital and lowers the wage rate. Wage-rent
ratio is inversely related to the price of the capital-intensive good.

_{}

To obtain the Rybczynski
Theorem, consider the labor and capital input constraints using factor shares
(θ_{ij}’s):

_{}

The solution to this problem
is:

_{}

Thus, for given commodity
prices, an increase in labor endowment raises the output of the industry 1
which intensively uses that factor:

_{}

For solutions when more
general production functions are used, see Choi, E. Kwan, “Factor Growth and
Equalized Factor Prices,” International Review of Economics and Finance (forthcoming).