Given $$U(x,y)= x^\alpha y^{1-\alpha}$$ $\alpha \in (0,1)$, are Cobb Douglas goods (here $x$ and $y$) complements, substitutes, or neither? Why?

An explanation with mixed partial derivatives would be great. My calculus is better than my economics skills.


2 Answers 2


Start with the more general CES function $U=[a*x^b+(1-a)*y^b]^{1/b}$. Compute the elasticity of substitution of this function. Then compute the functional form of U for $b=0$,$b=1$ and $b=-\infty$. You will find the two extreme (perfect complements / substitute) cases for CES, and the common case C-D for b=0.

  • $\begingroup$ I like that this answer shows how to get the necessary answer and promotes understanding but still leaves work for the questioner to figure it out. $\endgroup$
    – BKay
    Apr 27, 2015 at 10:35
  • $\begingroup$ I actually did exactly this last week, but I don't feel like this told me what it meant to be a substitute. All it told me was there was a range of values the CES could take and that perfect complements were one extreme and perfect substitutes another extreme. Cobb Douglas lies somewhere in the middle (obviously $b=0$) but that doesn't tell me to what degree Cobb Douglas goods are a substitutes or complements, does it? I thought considering Cobb Douglas with CES just tells me they aren't perfect complements or perfect substitutes. $\endgroup$ Apr 27, 2015 at 14:10
  • $\begingroup$ This didn't answer my questions about the partial derivatives but was a really cool way to answer the question. Hence, I accepted :D $\endgroup$ Apr 27, 2015 at 16:40

Try to think what you mean when you ask whether they're complements or substitutes.

You could mean: "Does my marginal utility in $x$ increase when I get more $y$? That would correspond to the cross derivative $\frac{\partial U^2}{\partial x \partial y}$.

You could (and this is the convention) mean the response to a change in prices. Denote with stars the bundle that maximizes utility given a budget constraint.

$$x^* (p_x, p_y), y^*(p_x, p_y) = \arg\max_{x,y} U(x,y) + \lambda(Y - p_x x - p_y y) $$

Then, the goods are complements if you increase the price of one of the goods, and the demand of the other one decreases:

$$ \frac{\partial x^* (p_x, p_y)}{\partial p_y} < 0 \\ \frac{\partial y^* (p_x, p_y)}{\partial p_x} < 0 $$

And substitutes if that is larger than zero. For the case where that partial is exactly zero, the elasticity of substitution is zero. Try out yourself which case is correct for Cobb-Douglas!

  • $\begingroup$ Except in additively separable utility, is utility in x well defined? Your explanation would improve if instead you said "marginal utility of x". $\endgroup$
    – BKay
    Apr 27, 2015 at 12:11
  • $\begingroup$ @BKay thought that was clear from the context (and the partial), but made it explicit. $\endgroup$
    – FooBar
    Apr 27, 2015 at 12:22
  • $\begingroup$ The cross partial one is the one we are discussing. I thought the change in marginal utility with respect to a change in the amount of $x$ would correspond to $$\frac{\partial U}{\partial x}$$ so I got confused when I take the partial of that with respect to $y$. Does this measure the rate MU changes wrt $x$ as we change $y$? How is that related to being a substitute? Maybe I didn't ask this question correctly. Perhaps I should have just asked what a substitute was. $\endgroup$ Apr 27, 2015 at 13:47
  • 1
    $\begingroup$ @StanShunpike perhaps you should open a new question and ask exactly what you put as a comment here :) $\endgroup$
    – FooBar
    Apr 27, 2015 at 13:57
  • $\begingroup$ economics.stackexchange.com/q/5340/2679 $\endgroup$ Apr 27, 2015 at 14:06

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