How to prove that if the Engel curves (expenditures as a function of wealth) are linear in wealth, then the indirect utility function has the form $v_{i}(p,a_{i})=\alpha_{i}(p)+\beta(p)a_{i}$ for an agent with utility function $u_{i}$? I think the converse statement is easy by using Roy's Identity, but how about this one? Thanks!
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$\begingroup$ Could you please define the variables in your notation? $\endgroup$– GiskardCommented Mar 7, 2023 at 15:49
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$\begingroup$ Sorry. I forgot to give the notation. p is the price and a is the wealth. alpha and beta is two coefficients depend on p only. $\endgroup$– DRMCommented Mar 8, 2023 at 2:31
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I think this statement is untrue. Gorman form is a sufficient condition for linear Engle curves, but it is not necessary. Consider the following example: $$v(p,a_i)=\beta(p)a_i^2$$ Applying Roy's lemma yield demands which are still linear in $a$.
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$\begingroup$ The reasoning seems reasonable. I would check it. Thanks a lot! $\endgroup$– DRMCommented Mar 8, 2023 at 2:39