I thought this required (quasi-)concavity of the utility, but can this (e.g. declining MRS) occur with fewer assumptions?
2
-
1$\begingroup$ What do you mean by "convex to the origin". Can you give a precise definition. $\endgroup$– tdmJan 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) $\endgroup$– SteveJan 21, 2022 at 19:53
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
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.
