<p>The answer is no. If you want to have decreasing marginal rates of substitution then you need quasi-concavity.</p>
<p>In order for the notion of decreasing marginal rates of substitution to make sense, you need a few assumptions.</p>
<ul>
<li>First you restrict yourself to a two good setting, say with goods <span class="math-container">$x$</span> and <span class="math-container">$y$</span></li>
<li>You assume that the utility function <span class="math-container">$u(x,y)$</span> satisfies all assumptions, so you can use the implicit function theorem in order to express indifference curves as <span class="math-container">$Y(x,U)$</span> being the amount of the second good that you need when you have <span class="math-container">$x$</span> units of the first good and want to reach utility level <span class="math-container">$U$</span>.</li>
<li>Finally, you assume that these indifference curves <span class="math-container">$Y(x,U)$</span> are differentiable in <span class="math-container">$x$</span>.</li>
</ul>
<p>Now assume that all these conditions are satisfied, then decreasing marginal rates of substitution is the same as requiring that <span class="math-container">$Y(x,u)$</span> is a convex function in <span class="math-container">$x$</span> for all utility levels <span class="math-container">$u$</span>. This is equivalent to the convexity of the upper level sets <span class="math-container">$A_u$</span>, where:
<span class="math-container">$$
\begin{align*}
A_u &amp;= \{(x,y)| y \ge Y(x,U)\}\\
&amp;= \{(x,y)| u(x,y) \ge U\}
\end{align*}
$$</span>
But convexity of these sets is identical to requiring that the function <span class="math-container">$u$</span> is quasi-concave.</p>

CC BY-SA 4.0
2022-01-25T10:52:23.420Z
2022-01-25T10:52:23.420Z
tdm
<p>I thought this required (quasi-)concavity of the utility, but can this (e.g. declining MRS) occur with fewer assumptions?</p>

CC BY-SA 4.0
2022-01-21T15:45:38.790Z
What do you mean by "convex to the origin". Can you give a precise definition.
tdm
CC BY-SA 4.0
2022-01-21T19:53:46.640Z
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)
Steve
<microeconomics><mathematical-economics><utility><consumer-theory>
What are the necessary and sufficient assumptions for indifference curves to be convex to the origin?
Steve
# What are the necessary and sufficient assumptions for indifference curves to be convex to the origin?

(Asked by: username_0 on 2022-01-20)

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

**Tags:** microeconomics, mathematical-economics, utility, consumer-theory

> **Comments:**
> **username_1**: What do you mean by "convex to the origin". Can you give a precise definition.  
> **username_0**: 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)  

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## 1 answers

### Answer by username_1 on 2022-01-25. Score: 3

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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