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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.

Anthony
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  • 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. – Giskard Dec 15 '23 at 10:17
  • Hi @Giskard I'm coming more from an optimisation background, so the more straightforward the explanation the better! Thanks. – Anthony Dec 15 '23 at 10:29
  • Also see this: https://x.com/amit_k_goyal/status/1717156615315427769?s=20 – Amit Dec 16 '23 at 02:49

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From "Dynamic economics An online textbook with dynamic graphics for the introduction to economics" by Prof. Dr. Christian Bauer:

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: $$ f(kx,ky) = k^n f(x,y) \ \forall k\in\mathbb{R}_0^+ $$

A lot of utility functions (all CES, including C-D) have this property. It is also well-known that for functions with this property the demand for the goods is linear in income, thus none will be 'more necessary' than another.

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.

Giskard
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  • Thank you! What is "IO" literature? Also, would you be able to supplement your answer with a brief discussion of Stone-Geary utility? I would just like some intuition on how the quasilinear approach differs from CD with a subsistence level. – Anthony Dec 15 '23 at 11:32
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    IO stands for Industrial Organization. I will defer to others on questions of Stone-Geary utility, I don't know enough about it :) – Giskard Dec 15 '23 at 11:36