# In simple terms: what are the implications of homothetic and nonhomothetic consumer preferences?

I am looking for a simple explanation of the implication of having homothetic/nonhomethetic preferences in relation to consumers' preferences when consuming goods.

• The income offer curve is linear if and only if consumer preferences are homothetic. – Giskard Jan 19 '16 at 17:20

From a mathematical point of view, if the function $f(x,y)$ is homogeneous (of any degree), and $g()$ is a function whose first derivative is everywhere non-zero, then the function

$$H(x, y) = g[f(x,y)]$$

is homothetic. In economics, we usually impose something more restrictive, namely that $g' >0$. But this makes a homothetic function a monotonic transformation of a homogeneous function. Now, homogeneous functions are a strict subset of homothetic functions: not all homothetic functions are homogeneous.

Therefore, not all monotonic transformations preserve the homogeneity property of a utility function. The simplest example is Cobb-Douglas utility. It is homogeneous of degree one. In an ordinal utility framework, we are ok with monotonic transformations, so we can consider the natural logarithm of it. Fine, but the natural logarithm will not preserve homogeneity. Nevertheless it will be homothetic.

The fundamental property of a homothetic function is that its expansion path is linear (this is a property also of homogeneous functions, and thankfully it proves to be a property of the more general class of homothetic functions).

In consumption theory, this means that, keeping the prices or the price ratio constant, if we vary the income of the consumer, in the $(x,y)$ plane the tangency point of the income constraint with the highest feasible indifference curve will always reflect a fixed ratio $x/y$. This in turn implies that expenditures for each good grow all at the same rate as income, and so expenditure shares remain constant for the whole income range (always for a given price ratio). While this may sound restrictive, in fact, it has been shown that homothetic preferences do not impose any special restrictions on aggregate demand (essentially due to the endowment vector being arbitrary and independent of preferences, which messes, or frees, things up).

• Great response. Can you expand on your last sentence, where you mention that homothetic preferences do not impose any special restriction on AD. With little knowledge on the topic, it seems to me that assuming homothetic preferences goes against the empirical evidence of Engel's Law or that income elasticity of demand varies across goods. By disregarding this evidence, wouldn't we be affecting any reasonable construction of AD? – StatsScared Mar 18 '16 at 17:46
• @StatsScared Thanks. You have posted a question on the matter, economics.stackexchange.com/q/10629/61, which has already generated interesting answers. If I have something to add, I will post it there. – Alecos Papadopoulos Mar 18 '16 at 18:35

Well, if the preferences are homothetic, they will have a bunch of cool features. Formally, if they are homothetic, then you can affirm that: $$U(\alpha x, \alpha y) = \alpha U(x,y), \forall \alpha > 0$$

Something cool about this kind of functions is that they are, as you can see from the definition above, homogeneous functions of degree 1. So, you have the "right" of one normalization of the arguments. That's why when solving for the indirect utility function, you can always set one of the prices as the numeraire and set its price as 1.

Another interesting feature is that the goods demanded by consumers in a competitive model will depend only on the price ratio, not income.

• But you always can define a strictly increasing monothonic transformation that will make the ratio the only thing really. For example, in this case, you can always divide by $\alpha$ and get the same ordering and values. – Raul Guarini Jan 21 '16 at 15:46
• As far as I know, yes. Because $U$ is a homogeneous function of degree 1, isn't it? – Raul Guarini Jan 21 '16 at 17:44
• Nope. I'm claiming that it is equal $\alpha \sqrt(xy)$ – Raul Guarini Jan 21 '16 at 18:38
• This will totally seem like nitpicking but $U(x,y) = x^2 \cdot y^2$ describes homothetic preferences as well. So not all $U$'s describing homothetic preferences are homogeneous functions of degree 1, but all homothetic preferences that have a function representing them also have a representing function that is a homogeneous functions of degree 1. – Giskard Jan 22 '16 at 6:50
• @RaulGuarini I am afraid that this question includes certain inaccuracies. As already mentioned in a comment, homothetic preferences are not necessarily homogeneous. Also, the level of quantity demanded under homothetic preferences still depends on income (how could it not?). What remains fixed is the ratio of the quantities demanded for two goods, which leads to constant expenditure shares, for given price ratio. – Alecos Papadopoulos Feb 21 '16 at 5:13