Utility function for a combination of a normal good and necessary good

I am trying to formulate a decision problem for an agent involving their heating-energy consumption $$c$$. Let $$x$$ denote all other consumption. What would be a reasonable utility function to employ here? I want to capture the intuition that heating energy is a necessary good, whereas everything else might be modelled as a normal good.

Something like Cobb-Douglas doesn't work since $$x$$ and $$c$$ would then both be normal goods. Is CES utility an option? Would be grateful for any references that consider similar setups.

• Hi! Do you know what a homothetic function is? Also, have you studied microeconomics? I am guessing yes, but I want to make sure the answer is digestible. Commented Dec 15, 2023 at 10:17
• Hi @Giskard I'm coming more from an optimisation background, so the more straightforward the explanation the better! Thanks. Commented Dec 15, 2023 at 10:29
• Also see this: x.com/amit_k_goyal/status/1717156615315427769?s=20
– Amit
Commented Dec 16, 2023 at 2:49

A function $$f:\mathbb{R} \to \mathbb{R},(x,y)\to f(x,y)$$ is called homogeneous of the degree $$n\in \mathbb{R}$$, if for all $$(x,y)\in\mathbb{R}^2$$ the following is valid: $$f(kx,ky) = k^n f(x,y) \ \forall k\in\mathbb{R}_0^+$$
You could use something like quasilinear utility, which does not have this property: $$U(x,y) = v(x) + y$$ where $$v()$$ is usually a concave function with an initially steep slope, e.g.; $$\ln()$$. In this case below a certain income level all income is spent on $$x$$, above the income level no income is spent on $$x$$. The amount consumed also depends on the price ratio. Whether such a simple function is useful for your model depends on what you are trying to achieve. This type of function is often used in IO literature, though personally I think its use is not always justified.