A utility function (neither perfect substitues nor perfect complements) which stems from a CES f. and leads to gross complements or gross substitutes

Question

So the most prominent preferences are perfect substitutes, perfect complements and cobb-douglas preferences. Perfect complements and perfect substitutes are extreme cases and I was asked whether there are preferences (modelled by a corresponding utility function) which are e.g. complements but not perfect complements or substitutes but not perfect substitutes. I immediately thought of Cobb-Douglas preferences, since a special property of Cobb-Douglas preferences is that convexity is strictly fulfilled here: A bundle of goods formed as a weighted average of two goods on the same indifference curve is always better than any of the extreme bundles of goods. So at first sight, this appeared to be a quite reasonable candidate for "a bit of complements but not perfect complements."

However, the common definition for good 1 to be a gross complement of good 2 is $\frac{\Delta x_{1}}{\Delta p_{2}}<0$ while for substitutes it must be $\frac{\Delta x_{1}}{\Delta p_{2}}>0$ (see e.g. slide 2 and 3 on this ressource: https://web.stanford.edu/~jay/micro_class/lecture7.pdf). When deriving the demand function of good 1 (using Lagrange with the corresponding budget constraint), demand of good 1 only depends on price of good 1 (plus income) and not on the price of good 2. This is a well documented fact and property of the Cobb-Douglas function. The underlying reason is that the substitution and income effect of an increase in price 2 cancel out here (see e.g. page 2 on this ressource: https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/3/352/files/2012/01/Week-3-Lecture2.pdf)

Consequently the goods are neither gross complements nor gross substitutes when applying the definitions of "gross substitutes" or "gross complements" on the resulting demand functions derived from Cobb-Douglas preferences. My questions:

1. Do you agree that this result is somehow contraintuitive (especially from a learner perspective) since Cobb-Douglas preferences imply strict convexity and one would rather think of a gross complementary relation of these goods when solving for the demand functions? How can one make sense of this result?

2. I would be interested to know whether you have some utility function proposals (except for perfect complements and perfect substitutes) which lead to gross complements or gross subsitutes according to the definitions above when solving for the demand functions? For the CES production function shown on page 5 on these slides https://economics.mit.edu/files/9016 it is pointed out that the factors are gross substitutes for sigmas greater than one and gross complements for sigma smaller than one. I am sure that this should also apply for the CES utility function and that one can derive a concrete utility function (based on sigma values smaller or greater than one), solve for demand of good one and two and hence illustrate that the goods are indeed gross substitues or gross complements. I am especially curious to see how the indifference curves will look like. I am not so familiar with the CES function which is the reason why I cannot do it myself at the moment. Do you have a good reading advice for an understandable explanation of the CES function with its properties? Thanks!

Answer 1

We can show if a utility function exhibits complementarity or substitutability just from its function, without having any information about prices. The utility function gives information about the preferences.

Take a utility function and calculate the $MRS = - U’x_1 / δU’x_2 $:

• Perfect Substitutes: $U(x1,x2) = a x_1 + b x_2$ with

  $ MRS = -a/b$. This means that there is a substitutive relation of $a:b$ between the bundles $x_1$ and $x_2$.

• Perfect Complements: $U(x1,x2) = min(a x_1, b x_2)$ with

  $MRS = 0/b= 0$ for the horizontal part $(a x_1 > b x_2)$

  $MRS = a/0 = - ∞$ for the vertical part $(a x_1 < b x_2)$ and

  $MRS= undef.$ at the kink.

• Cobb-Douglas: $U(x_1,x_2) = x_1^{0.5} x_2^{0.5}$ with $MRS = - x_2/ x_1$

  • The MRS is not constant, hence there is substitutability between the two bundles which can vary depending on the values. Thus, the C-D preferences show imperfect substitutes.
  • The $MRS=0$ (or $U(x_1,x_2)=0$) if $x_i=0$. Meaning that both bundles must be consumed for a positive utility, but the ratio can vary. Thus, the C-D preferences show imperfect complements.