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Cobb-Douglas utility functions assume price elasticity of $-1$ and income elasticity of $1$.

Are there any utility functions with two goods that lead to a demand function, where you have the choice of changing the price and income elasticities?

Say you wanna test a model that has a demand function, where the price elasticity is $-0.5$ and income elasticity is $2$. Or $-0.25$ and $0.75$.

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  • $\begingroup$ Are you talking about point or arc elasticities? $\endgroup$
    – Giskard
    Feb 5, 2022 at 5:17
  • $\begingroup$ Do you want the utility function to result in an isoelastic demand function, or should it just take the chosen elasticity value locally? It seems like you want the former, but you can't have a demand function with constant income elasticity $\mu > 1$. $\endgroup$
    – Giskard
    Feb 5, 2022 at 5:21
  • $\begingroup$ To be completely honest, I'm not sure how to answer those questions. Maybe it would help if I could explain how I plan to use it? $\endgroup$ Feb 5, 2022 at 15:22
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    $\begingroup$ Maybe! Please edit the additional information into your question. Keep it clear, focused and do not solicit opinions. $\endgroup$
    – Giskard
    Feb 5, 2022 at 18:00

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Among the simplest demand functions allowing for arbitrary price and income elasticities there is: $$ x_1^M(p_1,p_2,y) = \alpha + \beta\ln(\frac{p_1}{p_2}) + \gamma\ln(\frac{y}{p_2}), $$ or its log version: $$ \ln( x_1^M(p_1,p_2,y) ) = \alpha + \beta\ln(\frac{p_1}{p_2}) + \gamma\ln(\frac{y}{p_2}). $$

The elasticity parameters wrt own-price and income are denoted by $\beta$ and $\gamma$ and can take (almost) arbitrary values. Note that the elasticity wrt $p_2$ cannot be arbitrary as a consequence of homogeneity of degree zero in $(p_1,p_2,y).$ The expression for $x^M_2$ is obtained from the budget constraint.

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  • $\begingroup$ This looks like what I'm looking for. A few questions. Is it still possible to normalise p2 as the price of "all other goods"? Is it possible to interpret alpha as something meaningful here? In cobb-douglas it's possible to say something about budget share for each good, I assume that is replaced by being determined with the elasticities? $\endgroup$ Feb 7, 2022 at 14:21
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    $\begingroup$ @Victor Nielsen: yes, normalization is possible. The budget share is easily obtained using its definition $p_1x^M_1/y$ and it is not constant but depends upon $(p,y)$. Regarding the utility function generating such demands, I am afraid that it is not easy (not possible ?) to give its expression. $\endgroup$
    – Bertrand
    Feb 7, 2022 at 15:37
  • $\begingroup$ Quick question. What values can alpha take and what does it tell us? $\endgroup$ Feb 15, 2022 at 11:34
  • $\begingroup$ @Victor Nielsen: it depends which commodity you consider. Both $\beta$ and $\gamma$ can be positive or negative depending on whether the good is normal/inferior, Giffen, luxury, etc. $\endgroup$
    – Bertrand
    Feb 15, 2022 at 13:05
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    $\begingroup$ The constant part of demand $\alpha$ may be zero for several goods, but this is not necessarily the case for commodities with a subsistence level, or for which the State guarantees a minimum consumption level (medicaid, housing). $\endgroup$
    – Bertrand
    Feb 15, 2022 at 18:08

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