<p>I have this CES utility function:</p>
<p><span class="math-container">$$U(f, c) = (f^\alpha + c^\alpha)^{1/\alpha},$$</span></p>
<p>with <span class="math-container">$\alpha &gt; 0$</span>.</p>
<p>The problem set asks does it &quot;satisfy the principle of diminishing marginal rate of substitution for any value of <span class="math-container">$\alpha$</span>&quot;.</p>
<p>To which the answer is &quot;False, if <span class="math-container">$\alpha &gt; 1$</span> the indifference curves are concave, so they don’t satisfy the principle of diminishing MRS&quot;</p>
<p>But short of graphing the indifference curves and hoping that it's visually obvious, I'm not sure how to tell this.</p>
<p>How can one tell from a utility function whether it is or is not non-concave?</p>

<utility><consumer-theory>
How can I prove that a utility function does (or does not) satisfy diminishing MRS?
Duncan Marshall
# How can I prove that a utility function does (or does not) satisfy diminishing MRS?

(Asked by: username_0 on 2020-12-19)

I have this CES utility function:

$$U(f, c) = (f^\alpha + c^\alpha)^{1/\alpha},$$

with $\alpha > 0$.

The problem set asks does it "satisfy the principle of diminishing marginal rate of substitution for any value of $\alpha$".

To which the answer is "False, if $\alpha > 1$ the indifference curves are concave, so they don’t satisfy the principle of diminishing MRS"

But short of graphing the indifference curves and hoping that it's visually obvious, I'm not sure how to tell this.

How can one tell from a utility function whether it is or is not non-concave?

**Tags:** utility, consumer-theory


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## 0 answers

