5.6 Appendix A: Deriving the Equation of a PPF

This page derivees the equations for the PPF in this page.

Example 1: A PPF with linear technologies

We showed that the PPF with production functions \(x_1 = f_1(L_1) = 3L_1\) \(x_2 = f_2(L_2) = 2L_2\) and the resource constraint \(L_1 + L_2 = 150\) is a line extending from $(450,0)$ to $(0,300)$:

See interactive graph online here.

To derive the equation for this PPF, we make use of this tight relationship between the resource constraint and the PPF.

If we invert the production functions — that is, solve each for $L$ as a function of $q$, instead of $q$ as a function of $L$ — we get the amount of labor required to produce $x_1$ and $x_2$ units of output: \(\begin{aligned} L_1 &= \tfrac{1}{3}x_1\\ L_2 &= \tfrac{1}{2}x_2 \end{aligned}\) If we plug this into the resource constraint, we get the equation of the PPF: \(\begin{aligned} \tfrac{1}{3}x_1+\tfrac{1}{2}x_2 &=150 \end{aligned}\) This is the equation of the red line in the right-hand panel above.

Example 2: A PPF with diminishing $MP_L$

We also showed that the PPF with production function \(x_1 = f_1(L_1) = 10\sqrt L_1\) \(x_2 = f_2(L_2) = 6\sqrt L_2\) and the resource constraint \(L_1 + L_2 = 100\) is a curve extending from $(100,0)$ to $(0,60)$:

See interactive graph online here.

Following the same method as in the linear case, we invert the production functions to get the labor requirements: \(\begin{aligned} L_1 &= \frac{x_1^2}{100}\\ L_2 &= \frac{x_2^2}{36} \end{aligned}\) Plugging this into the resource constraint gives us the equation of the PPF, \(\begin{aligned} \frac{x_1^2}{100} + \frac{x_2^2}{36} = 100 \end{aligned}\) This is the equation of the red curve in the right-hand diagram above.