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Sep 17

## The Cobb-Douglas function and level sets

Here and here I started to discuss topics in optimization. This post is a preparatory step to the next topic: the Lagrange method.

The Cobb–Douglas production function made a big splash in economics. It is still used a lot. In case of just two inputs, capital and labor, its definition is

$f(K,L)=AK^\alpha L^\beta.$

For simplicity, in the videos below I assume $A=1$. We discuss the definition, homogeneity and the reason for choosing a multiplicative form.

Video 1. Cobb-Douglas production function

It is important to realize that the same function is used for modeling preferences. That is, the utility of consuming a pair of goods is measured by the function

$f(G_1,G_2)=AG_1^\alpha G_2^\beta.$

Thus, everything that has been said about production functions, applies to utility functions as well.

## Level sets, isoquants and indifference curves

Let $z=f(x,y)$ be a function of two arguments. For a fixed constant $c$, the level set $\{(x,y):\ f(x,y)=c\}$ is the set of all pairs $(x,y)$ at which the value of the function is the given constant. Geometrically, $z=f(x,y)$ is a surface in the three-dimensional space. We cut it by a horizontal plane at height $z=c$. In the intersection of the plane and surface we get a curve. This curve, projected onto the $(x,y)$ plane, gives the level set.

The next video explains that isocosts, isoquants and indifference curves are all level sets for certain functions.

Video 2. Level sets, isoquants and indifference curves