The Cobb-Douglas Utility Function
The Cobb-Douglas functional form was first proposed as a production function in a macroeconomic setting, but its mathematical properties are also useful as a utility function describing goods which are neither complements nor substitutes.
The general form of a Cobb-Douglas function over two goods is \(u(x_1,x_2) = x_1^a x_2^b\) However, we will often transform this function by taking the natural log, which allows us to transform exponents into coefficients: \(\hat u(x_1,x_2) = \ln(x_1^a x_2^b) = a \ln x_1 + b \ln x_2\) This can be particularly useful when performing linear regressions. It’s also much easier to find the MRS.
Marginal utilities and the MRS
Using the exponential form \(u(x_1,x_2) = x_1^a x_2^b\) the marginal utilities are \(\begin{aligned} MU_1(x_1,x_2) &= ax_1^{a - 1}x_2^b\\ MU_2(x_1,x_2) &= bx_1^ax_2^{b - 1} \end{aligned}\) so the MRS is \(MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = {ax_1^{a - 1}x_2^b \over bx_1^ax_2^{b - 1}} = {ax_2 \over bx_1}\)
Using the log form \(u(x_1,x_2) = a \ln x_1 + b \ln x_2\) the math is even simpler: since the derivative of $k \ln x$ is just $k/x$, the marginal utilities are \(\begin{aligned} MU_1(x_1,x_2) &= a/x_1\\ MU_2(x_1,x_2) &= b/x_2 \end{aligned}\) so the MRS is \(MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = {a/x_1 \over b/x_2} = {ax_2 \over bx_1}\)
Either way, the MRS is increasing in $a$ and decreasing in $b$; the more you like good 1 (or the less you like good 2), the more good 2 you’ll be willing to give up to get more good 1.
Its indifference map is familiar to you by now, as we’ve seen several examples of it. Try changing $a$ and $b$ to see how it affects the indifference map:
As you can see, at any given point, increasing $a$ (or decreasing $b$) causes the MRS to increase and the indifference curve to become steeper at that point. Conversely, decreasing $a$ (or increasing $b$) causes the MRS to decrease and the indifference curve to become flatter at that point.
Try choosing a pair of values for $a$ and $b$ and then doubling both of them: for example, look at $a = 2$ and $b = 3$, and then $a = 4$ and $b = 6$. You’ll see that this generates the same indifference map and MRS. We can use this fact to “normalize” functions of this form, as described in the next section.
Normalizing a Cobb-Douglas utility function
As we discussed earlier, it’s often possible to normalize a utility function by making its relevant coefficients (or in this case, exponents) sum to 1. In this case that means raising the utility function to the power $1/(a+b)$: \(\hat u(x_1,x_2) = \left[x_1^a x_2^b \right]^{1 \over a + b} = x_1^{a \over a + b} x_2^{b \over a+b}\) or \(\hat u(x_1,x_2) = x_1^\alpha x_2^{1 - \alpha}\) where \(\alpha = {a \over a + b}\) Of course, you do this in the log form as well, to get \(u(x_1,x_2) = \alpha \ln x_1 + (1-\alpha) \ln x_2\)
For example, you can normalize the function $u(x_1,x_2) = x_1^3x_2^2$ by raising it to the power of 1/5: \(\hat u(x_1,x_2) = \left[x_1^3 x_2^2\right]^{1 \over 5} = x_1^{3 \over 5} x_2^{2 \over 5}\) which would be $x_1^\alpha x_2^{1 - \alpha}$ for $\alpha = {3 \over 5}$.
We can therefore plot the utility function just using the single parameter $\alpha$ to express the relative weight the agent places on good 1:
This is handy because it allows us to summarize an agent’s preferences over two goods with a single parameter. It’s also particularly important for the Cobb-Douglas utility function, because it will turn out when we analyze market behavior that this normalized $\alpha$ will be the fraction of a consumer’s income they spend on good 1. But we’ve a while to go before we get to that result…
Cobb-Douglas Utility Functions with Many Goods
The Cobb-Douglas utility function can easily be extended to any number of goods; for example, \(u(x_1,x_2,x_3) = x_1^ax_2^bx_3^c\) or \(u(x_1,x_2,x_2) = a \ln x_1 + b \ln x_2 + c \ln x_3\) Again, we can normalize this so that the sum of the exponents or coefficients is 1.