6.1 Overview

Last Friday, we introduced the notion of a level set. Since then, we have seen two examples of level sets:

• An isoquant is the level set of a production function: it’s the set of all combinations of inputs $(L,K)$ which produce some amount of output $q$. Thus the equation of an isoquant is implicitly defined by the equation \(f(L,K) = q\)
• A PPF is the level set of a resource requirement function: it’s the set of all combinations of goods $(x_1,x_2)$ which may be produced with a set of resources, e.g., a total amount of labor $\overline L$. Thus, if labor is the only input, the equation of the PPF is implicitly defined by the equation \(L(x_1,x_2) = \overline L\) where $L(x_1,x_2)$ is the total amount of labor required to produce $x_1$ units of good 1 and $x_2$ units of good 2.

We have also seen that the slopes of these level sets have an economic importance:

• The slope of an isoquant is the marginal rate of technical substitution, or MRTS. It represents the tradeoff between labor and capital: specifically, the amount of capital you could give up per additional unit of labor hired (or conversely, the amount of capital you would need to acquire in order to keep output the same after firing a worker).
• The slope of a PPF is the marginal rate of transformation, or MRT. It represents the opportunity cost of good 1, in terms of good 2: that is, the amount of good 2 you would have to give up in order to produce another unit of good 1.

In this lecture we will derive these slopes using the implicit function theorem, which relates the partial derivatives of a multivariate function at a point to the slope of the level set passing through that point.

Simply, stated, the formula for the implicit function theorem is \(\left.{dy \over dx}\right|_{f(x,y) = z} = - {\partial f/\partial x \over \partial f/\ partial y}\) In this lecture, we will derive this formula mathematically, and then apply it to the cases of isoquants and PPFs.