Homothetic preferences and utility functions

I know that if you have homothetic preferences and a utility function that represents it, then this utility function must present constant Marginal Rate of Substitution (MRS). My question is whether the opposite direction of implication is also true, to be very specific, is it true that a utility function that presets constant MRS always represents a homothetic preference?

• Homothetic preferences do not necessarily have a constant MRS (this is true only for perfect substitutes); what homotheticity implies is that the MRS is homogeneous of degree zero, i.e, that it only depends on the ratio of the amounts of goods. And yes, it is an equivalence result. – Oliv Apr 14 '18 at 6:29

As noted in the comments, it is not true that homothetic preferences must have constant marginal rates of substitution.

To see this, recall that preferences given by the utility function

$$u(x,y) = x^\alpha y^{1-\alpha}$$

are homothetic. (More generally, Cobb-Douglas preferences are homothetic.) However, the marginal rate of substitution is

$$\text{MRS}(x,y) = \frac{\alpha}{1-\alpha}\frac{y}{x},$$

which is not constant. However, the MRS is homogeneous of degree zero, since

$$\text{MRS}(\lambda x, \lambda y) = \text{MRS}(x,y).$$

Homogeneity of degree zero of the MRS is a general property of homothetic preferences. This follows from the fact that (continuous) homothetic preferences have a utility representation that is homogeneous of degree one.

Conversely, when the MRS is homogeneous of degree zero, preferences are homothetic. Hence, preferences that exhibit constant MRS are also homothetic. The proof is a little involved. For this, I refer to you lemma $1$ of "Duality and the Structure of Utility Functions" by Lau (1970). (Note that Lau states a different definition of homotheticity than you do. However, the Lau's definition and yours are equivalent when preferences are continuous -- which they must be for the MRS to be well-defined.)

• Thank you very much for the answers! I certainly forgot in my question to write "constant in proportion expansion along rays", but can you help me with the proof of: if the MRS is homogenous of degree zero, then the preference is homothetic? Thanks again! – Alessandro Rivello Apr 14 '18 at 13:21
• @AlessandroRivello What definition of homothetic preferences are you using? The property regarding MRS is sometimes used as the definition. – Theoretical Economist Apr 14 '18 at 13:23
• I'm using this: Definition 3.B.6: monotone preference relation $\succeq$ on $X=\mathbb{R}^{L}_{+}$ is homothetic if all indifference sets are related by proportional expansion along rays; that is, if $x \sim y$ then $\alpha x \sim \alpha y$ for any $\alpha \ge 0$. Which is the definition in Mas-Collel at al. Sorry, don't know how to put a print screen here. – Alessandro Rivello Apr 14 '18 at 13:41
• @Oliv if you could also help me with that would be very nice. – Alessandro Rivello Apr 14 '18 at 19:13
• @AlessandroRivello Apologies for the delay; it's been a busy couple of days. Please see the edit to my answer. – Theoretical Economist Apr 18 '18 at 5:51