## 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

For simplicity, in the videos below I assume . 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

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

## Level sets, isoquants and indifference curves

Let be a function of two arguments. For a fixed constant , the level set is the set of all pairs at which the value of the function is the given constant. Geometrically, is a surface in the three-dimensional space. We cut it by a horizontal plane at height . In the intersection of the plane and surface we get a curve. This curve, projected onto the plane, gives the level set.

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

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