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

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    $\begingroup$ What do you mean by "convex to the origin". Can you give a precise definition. $\endgroup$
    – tdm
    Jan 21 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 at 19:53

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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.

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