1. Consumer Problem
Choose X1 and X2
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 L1 and K1 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
(θij’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).