I know that in the 2-good world it is easy to derive the demand functions from a utility function for imperfect substitutes or complements, but what if I have N goods that include many combinations of substitutes and complements?

For example if you have hot dogs $D$, hamburgers $H$ (substitutes of each other), and mustard $M$, mayo $Y$ (substitutes of each other, complements with hot dogs and hamburgers), I can try something like this:

$$ U(D, H, M, Y) = (D^{0.5} + H^{0.5})(M^{0.5} + Y^{0.5}) $$

Unfortunately the math gets pretty hairy for this one pretty quickly. Before I dive in much further with solving these equations, is there a well-known function that is used in the literature for this type of analysis?

Note: I am interested in the more general N-good case, not just the 4-good one.


The general way to do this would be using a nested CES Function. CES Wikipedia

For your example you could define the utility of sandwiches (S) and condiments (C) to be

$$ U(D,H,M,Y) = (a_1 S^{\frac{s-1}{s}} + a_2 C^{\frac{s-1}{s}})^{\frac{s}{s-1}} $$ then you can define $S$ and $C$ "nests" as $$ S = (b_1 D^{\frac{\rho-1}{\rho}} + b_2 H^{\frac{\rho-1}{\rho}})^{\frac{\rho}{\rho-1}} $$ $$ C = (c_1 M^{\frac{\eta-1}{\eta}} + c_2 Y^{\frac{\eta-1}{\eta}})^{\frac{\eta}{\eta-1}} $$

$s$ determines whether S and C are complements ($s \rightarrow 0$) or substitutes ($s \rightarrow \infty$), or neither ($s=1$).

The same holds for $\rho$ and $\eta$ in the "nests". $a$, $b$, and $c$ establish the relative importance of each item within a nest.

Extending this to $N$ you can have an arbitrary number of nests and an arbitrary number of elements within each nest. This way may look complicated, but it has a lot of flexibility and gives you nice derivatives when solving maximization problems.

  • $\begingroup$ Ah, great idea, thanks! Didn't think to nest the CES functions. I'm fiddling around with the equations now and am having some quirks finding closed-form demand functions. Do you know if it is possible to find them and I should keep trying? Or should I use numerical methods? $\endgroup$
    – robbrit
    Feb 23 '18 at 5:55
  • $\begingroup$ Hm, since using projected gradient ascent is pretty easy, I will go ahead with that one. $\endgroup$
    – robbrit
    Feb 23 '18 at 6:38
  • $\begingroup$ Take a look at section 3.1 of these notes. There are some tricks dealing with CES that aren't obvious at first. $\endgroup$
    – dsmithecon
    Feb 23 '18 at 16:42
  • $\begingroup$ I took a look at those notes, unfortunately the step where they divide the FOCs for two separate goods doesn't work with nested CES. There's a summed term that cancels out in the non-nested case, but not with nested. $\endgroup$
    – robbrit
    Feb 26 '18 at 23:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.