What are the necessary and sufficient assumptions for indifference curves to be convex to the origin?

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

• What do you mean by "convex to the origin". Can you give a precise definition.
– tdm
Jan 21 at 15:45
• 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) Jan 21 at 19:53

• 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.