<p>Take <span class="math-container">$u(x,y)=x^2+y^2$</span> on <span class="math-container">$\mathbb R_+^2$</span>. The function is strictly increasing in both <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, but the indifference curves are concave to the origin. Hence the preference it represents cannot be convex. For example, <span class="math-container">$u(1,0)=u(0,1)=1$</span>, but the average of the two bundles is less preferred:
<span class="math-container">\begin{equation}
u\left(\frac12,\frac12\right)=\frac12&lt;1=\frac12u(1,0)+\frac12u(0,1).
\end{equation}</span></p>
<p>The indifference curves for perfect substitutes are straight lines. They still represent a convex preference, although not a strictly convex one.</p>

CC BY-SA 4.0
2020-09-24T16:12:12.483Z
2020-09-24T16:17:40.463Z
2020-09-24T16:17:40.463Z
Herr K.
<p>Is it possible to have preferences that is strictly increasing but not convex?</p>
<p>Will perfect substitutes indifference curves show strictly increasing but not convex preferences? I am confused, as won't perfect substitutes be considered as convex?</p>

<microeconomics><utility><preferences>
Strictly increasing but not convex preferences
Ben Stoud
# Strictly increasing but not convex preferences

(Asked by: username_0 on 2020-09-24)

Is it possible to have preferences that is strictly increasing but not convex?

Will perfect substitutes indifference curves show strictly increasing but not convex preferences? I am confused, as won't perfect substitutes be considered as convex?

**Tags:** microeconomics, utility, preferences

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## 1 answers

### Answer by username_1 on 2020-09-24. Score: 1

Take $u(x,y)=x^2+y^2$ on $\mathbb R\_+^2$. The function is strictly increasing in both $x$ and $y$, but the indifference curves are concave to the origin. Hence the preference it represents cannot be convex. For example, $u(1,0)=u(0,1)=1$, but the average of the two bundles is less preferred:
\begin{equation}
u\left(\frac12,\frac12\right)=\frac12<1=\frac12u(1,0)+\frac12u(0,1).
\end{equation}

The indifference curves for perfect substitutes are straight lines. They still represent a convex preference, although not a strictly convex one.


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