Cobb Douglas Preferences

Another commonly used utility function is the Cobb-Douglas utility function u(xi,x2) = x{x2l where c and d are positive numbers that describe the preferences of the consumer.2

The Cobb-Douglas utility function will be useful in several examples. The preferences represented by the Cobb-Douglas utility function have the general shape depicted in Figure 4.5. In Figure 4.5A, we have illustrated the indifference curves for c = 1/2, d = 1/2. In Figure 4.5B, we have illustrated the indifference curves for c = 1/5, d = 4/5. Note how different values of the parameters c and d lead to different shapes of the indifference curves.

Cobb-Douglas indifference curves. Panel A shows the case where c = 1/2, d = 1/2 and panel B shows the case where

Cobb-Douglas indifference curves look just like the nice convex mono-tonic indifference curves that we referred to as "well-behaved indifference curves" in Chapter 3. Cobb-Douglas preferences are the standard example of indifference curves that look well-behaved, and in fact the formula describing them is about the simplest algebraic expression that generates well-behaved preferences. We'll find Cobb-Douglas preferences quite useful to present algebraic examples of the economic ideas we'll study later.

Of course a monotonie transformation of the Cobb-Douglas utility function will represent exactly the same preferences, and it is useful to see a couple of examples of these transformations.

2 Paul Douglas was a twentieth-century economist at the University of Chicago who later became a U.S. senator. Charles Cobb was a mathematician at Amherst College. The Cobb-Douglas functional form was originally used to study production behavior.

First, if we take the natural log of utility, the product of the terms will become a sum so that we have v(xi,x2) = \n(x\x2) = clnxi -f dlnx2.

The indifference curves for this utility function will look just like the ones for the first Cobb-Douglas function, since the logarithm is a monotonic transformation. (For a brief review of natural logarithms, see the Mathematical Appendix at the end of the book.)

For the second example, suppose that we start with the Cobb-Douglas form v(Xi,X2) — XiX$. Then raising utility to the l/(c + d) power, we have c d c-hd c + d XI X2

Now define a new number c

We can now write our utility function as v(xi,x2) = Zix\~a.

This means that we can always take a monotonic transformation of the Cobb-Douglas utility function that make the exponents sum to 1. This will turn out to have a useful interpretation later on.

The Cobb-Douglas utility function can be expressed in a variety of ways; you should learn to recognize them, as this family of preferences is very useful for examples.

Project Management Made Easy

Project Management Made Easy

What you need to know about… Project Management Made Easy! Project management consists of more than just a large building project and can encompass small projects as well. No matter what the size of your project, you need to have some sort of project management. How you manage your project has everything to do with its outcome.

Get My Free Ebook

Post a comment