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
Let be the relative price of the importable good. Rearranging these terms, we obtain
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
(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).