3.6 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 at https://www.econgraphs.org/graphs/scarcity/ppf/resource_constraint_ppf ]

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 at https://www.econgraphs.org/graphs/scarcity/ppf/resource_constraint_ppf_curved ]

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.