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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!

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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 $p_i$.

The Cobb-Douglas utility function is the only one which has this property for all goods, for the entire range of the demand functions.

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.)

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    $\begingroup$ Is your last example $U(x_1,x_2)=\sqrt{x_1x_2}$ a Cobb-Douglas utility function? $\endgroup$ Feb 13 at 15:02
  • $\begingroup$ Ooops! I meant to add the square roots, will correct. Thank you! $\endgroup$
    – Giskard
    Feb 13 at 18:48
  • $\begingroup$ Thanks! Would you happen to have a reference for the statement "The Cobb-Douglas utility function is the only one which has this property for all goods..." I would imagine that the proof proceeds by beginning with a demand function that only depends on own-price, and then working backwards to construct the utility function? $\endgroup$
    – Anthony
    Feb 14 at 11:23
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    $\begingroup$ @Anthony I am not sure if you are interested in mathematical nuances, but if you are not: any utility function that would result in the exact same demand functions would be functionally indistinguishable (same consumer behavior under any parameter combination), so we 'might as well' call it Cobb-Douglas. $\endgroup$
    – Giskard
    Feb 14 at 16:46
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    $\begingroup$ @Anthony If you are interested in mathematical nuances: look into the "integrability of demand" or post your new question as a new question. $\endgroup$
    – Giskard
    Feb 14 at 16:48

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