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.

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