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How can you rigorously show that Hicksian demand for an inferior good will decrease when utility increases?

Thanks,

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    $\begingroup$ You make the assumptions that guarantee that Hicksian demand equals Marshallian demand at the expenditure function and for the expenditure function to be increasing in utility depending on whether you want strict decreases. The rest is the definition of an inferior good. $\endgroup$ Commented Nov 9, 2023 at 23:09

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For the good in question, let $d^m(p,m)$ denote the Marshallian demand and let $d^h(p,u)$ denote the Hicksian demand, where $p$ is the price vector, $m$ is income, and $u$ is utility. Since the good is inferior $d^m$ is decreasing in $m$.

We have the identity

$$d^h(p,u)=d^m\big(p,e(p,u)\big) \tag{X}$$

where $e$ is the (minimum) expenditure function. The expenditure function must be (weakly) increasing (i.e. nondecreasing) in $u$. If not, then there exist $u_1$ and $u_2$ with $u_2>u_1$ such that $e(p,u_2)<e(p,u_1)$. But this violates the definition of the expenditure function. So, an increase in $u$ increases $e(p,u)$, and since the good is inferior, this decreases the RHS of (X), i.e. the Marhsallian demand. Thus the LHS of (X) (the Hicksian demand) is (weakly) decreasing in $u$.

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