# Example of a utility function with cross-price effects in demand functions

I am looking for examples of utility functions where the demand function for a good does not depend solely on its own price, but on the price of the other good(s) as well.

For instance, Cobb-Douglas utility does not satisfy this requirement since the demand functions for each good are solely own-price dependent.

CES is the only example that I have been able to locate where the demand function for each good depends on the price of all other goods. To wit, for $$U = \left( \sum_{i=1}^n a_i^{1/\sigma} x_i^{(\sigma-1)/\sigma} \right)^{\sigma/(\sigma-1)}$$ the demand function is given by $$x_i^* = a_i M \frac{\sum_{i=1}^n P_j^{1-\sigma}}{P_i^{\sigma}}\ ,$$ where $$M$$ is the budget constraint (Source).

Are there other examples of commonly-used utility functions where cross-price effects are present in the the demand functions? It would be great to use CES, but it is proving analytically intractable for my use case. References are always appreciated!

If the demand functions do not depend on the prices of other goods, then they looks something like this: $$D_i(I,p_i) = \frac{\alpha_i I}{p_i}$$ were $$\alpha_i$$ is the share of the consumer's total income $$I$$ that is spent on good $$i$$. Where the share of income spent on good $$i$$ not constant w.r.t. $$p_i$$, the total sum of money spent on other goods would change, hence their demand would be affected by $$p_i$$.
No other utility functions have this property, so you can pick any, e.g. the ones used in the textbooks: perfect complements, perfect substitutes, $$U(x_1,x_2) = \sqrt{x_1} + \sqrt{x_2}$$, etc. (To be fair some of these are also CES functions.)
• Is your last example $U(x_1,x_2)=\sqrt{x_1x_2}$ a Cobb-Douglas utility function? Feb 13 at 15:02