The Production Possibility Frontier (Variable Proportions Case)

The production possibility frontier can be derived in the variable proportions case by using the same labor and capital constraints used in the fixed proportions case but with one important adjustment. Under variable proportions the unit-factor requirements are functions of the wage-rental ratio (w/r). This implies that the capital-labor ratios (which are the ratios of the unit-factor requirements) in each industry are also functions of the wage-rental ratio. If there is a change in the equilibrium (for some reason) such that the wage-rental rate rises, then labor will become relatively more expensive compared to capital. Firms would respond to this change by reducing their demand for labor and raising their demand for capital. In other words firms will substitute capital for labor and the capital-labor ratio will rise in each industry. This adjustment will allow the firm to maintain minimum production cost and thus the highest profit possible.

The labor constraint with full employment can be written as,

where a_LC and a_LW are functions of (w/r).

The capital constraint with full employment becomes,

where a_KC and a_KW are functions of (w/r).

Under variable proportions the production possibility frontier takes the traditional bowed-out shape as shown in the adjoining Figure. All points on the PPF will maintain full employment of both labor and capital resources. The slope of a line tangent to the PPF (such as the line through point A) represents the quantity of steel that must be given up to produce another unit of clothing. As such, the slope of the PPF is the opportunity cost of producing clothing. Since the slope becomes steeper as more and more clothing is produced, (as when moving production from point A to B) we say that there is increasing opportunity cost. This means that more steel must be given up to produce one more unit of clothing at point B than at point A in the Figure. In contrast in the Ricardian model the PPF was a straight line which indicated constant opportunity costs.

International Trade Theory and Policy - Chapter 60-7: Last Updated on 6/8/98