Production possibility frontier:

A production possibility frontier (PPF) shows the alternative combinations of two inputs that can be produced with  fixed quantities of inputs when these are employed efficiently. The slope of a PPF depicts the substitution possibilities between two outputs when  the total resources are held constant. The negative of the slope is called  the rate of product transformation (RPT). Thus, RPT of two products X and Y  is given by

We can derive the PPF for the economy which can be obtained by eliminating prices from the equations (a) and (b). This can be done as follows:

Consider 

i.e.,  the supply functions are homogeneous of degree zero in prices. (This can easily be proved).

The concave shape  is  implied  by  sufficient  second-order  equations  for  the model.

It may be useful to remember  the following features  of PPF:

- PPF represents the alternative combinations of two final goods that can be produced with fixed quantities of inputs.

- The frontier of this set corresponds to the technically efficient, feasible combinations of outputs.

- The slope of the PPF not the marginal rate of technical substitution (MRTS) between K and L (i.e., the slope of the production isoquants). Instead, it gives the marginal rate of product transformation (MRTS) between final goods. That is, the rate of technical transformation from one good to the other.

- The cost of one more unit of X therefore is best measured as the rate of product transformation (RPT) of X for Y at the prevailing point on the PPF.