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My question is whether our demand functions e.g. Hicksian (compensated) demand, are ever functions of 3 or more variables, or if the other price variables and utility are always fixed, and hence just parameters.

E.g. The classic problem: Max: $U(x,y) = (xy)^{1/2}$ s.t. $p_xx + p_yy = m$

Hicksian demand for $x$ is: $h_x = U(\frac{p_2}{p_1})^{1/2}$

Should I be writing this as $h_x = \bar{U}(\frac{\bar{p_2}}{p_1})^{1/2}$ to indicate that $p_2$ and $U$ are fixed?

I.e. Is the function $h$ always: $h(p_x, \bar p_y, \bar U)$ or can we write it generally as $h(p_x, p_y, U)$

One reason we might not assume the other variables are always fixed is so that we can account for shifts in the demand curve caused by $∆p_y$ & $∆U$ is that correct?

A brief outline of when is and isn't appropriate to fix the other variables could be most helpful if someone has time. Thanks!

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Generally you would write the hicksian demand $h(p_x,p_y,U)$. But when you graph it is easier to think of it as a single variable function.

What you see on a typical demand curve (assuming it is a hicksian one) is actually $h(p_x,\overline{p_y},\overline{U})$, i.e. you would graph the quantity as a single variable function of its price, for given values of the other variables.

Changes on the other variables $p_y, U$, cause the “curve” to shift. In the context of graphing, the other variables can be thought of as parameters.

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    $\begingroup$ Thanks again @Nicolas for another great answer! This fits my expectation. The Hicksian demand function keeps $P_y$ and $U$ as variables, but when we graph it, we want to show demand given these particularly values, and we so fix them as parameters. $\endgroup$
    – CormJack
    Feb 14, 2023 at 17:18
  • $\begingroup$ Exactly @CormJack, you got it! $\endgroup$ Feb 14, 2023 at 18:04
  • $\begingroup$ Thank you, I also just commented on another post you are involved in, where the answer above yours seems to contradict your answer? Perhaps you can provide clarity. economics.stackexchange.com/questions/54209/… $\endgroup$
    – CormJack
    Feb 14, 2023 at 18:10
  • $\begingroup$ I also find it hilarious how there seems to be a gang of hardcore users here that just go around helping everyone. Of which you seem to be in that gang. Much appreciated! $\endgroup$
    – CormJack
    Feb 14, 2023 at 18:11

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