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?
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.)