Are other 'variables' in demand functions always fixed?

Question

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!

Answer 1

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.