I thought this required (quasi-)concavity of the utility, but can this (e.g. declining MRS) occur with fewer assumptions?

  • 1
    $\begingroup$ What do you mean by "convex to the origin". Can you give a precise definition. $\endgroup$
    – tdm
    Jan 21, 2022 at 15:45
  • $\begingroup$ I just mean diminishing marginal rates of substitution, so for example, of the type of an indifference curve from a cobb- Douglas preference specification (i.e y = u/x) $\endgroup$
    – Steve
    Jan 21, 2022 at 19:53

1 Answer 1


The answer is no. If you want to have decreasing marginal rates of substitution then you need quasi-concavity.

In order for the notion of decreasing marginal rates of substitution to make sense, you need a few assumptions.

  • First you restrict yourself to a two good setting, say with goods $x$ and $y$
  • You assume that the utility function $u(x,y)$ satisfies all assumptions, so you can use the implicit function theorem in order to express indifference curves as $Y(x,U)$ being the amount of the second good that you need when you have $x$ units of the first good and want to reach utility level $U$.
  • Finally, you assume that these indifference curves $Y(x,U)$ are differentiable in $x$.

Now assume that all these conditions are satisfied, then decreasing marginal rates of substitution is the same as requiring that $Y(x,u)$ is a convex function in $x$ for all utility levels $u$. This is equivalent to the convexity of the upper level sets $A_u$, where: $$ \begin{align*} A_u &= \{(x,y)| y \ge Y(x,U)\}\\ &= \{(x,y)| u(x,y) \ge U\} \end{align*} $$ But convexity of these sets is identical to requiring that the function $u$ is quasi-concave.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.