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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.

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.

Level sets, isoquants and indifference curves define equivalence relations, see the definition and properties here.

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