<p>That is not the case. Consider <span class="math-container">$u(x, y) = \lfloor x + y \rfloor$</span> i.e. greatest integer less than or equal to <span class="math-container">$x+y$</span>. It is both quasi-concave and quasi-convex, but no transformation of the kind <span class="math-container">$v(x) + y$</span> exist. This is because <span class="math-container">$v(x) + y$</span> is strictly increasing in <span class="math-container">$y$</span>, but <span class="math-container">$u$</span> is not.</p>
<p>Another much simpler example is <span class="math-container">$u(x, y) = 0$</span>.</p>

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2022-08-24T11:45:50.770Z
I have question on game theory.  I will be appreciate if you help me by sharing your suggestions and hints. Thank you https://economics.stackexchange.com/questions/52468/principal-agent-problem
studentp
CC BY-SA 4.0
2022-08-18T08:34:07.747Z
2022-08-18T08:40:39.620Z
2022-08-18T08:40:39.620Z
Amit
<p>From <a href="https://economics.stackexchange.com/questions/52390/please-check-those-two-production-functions-they-also-seems-to-be-quasiconvex/52392?noredirect=1#comment85225_52392">the discussion</a> under <a href="https://economics.stackexchange.com/questions/52390/please-check-those-two-production-functions-they-also-seems-to-be-quasiconvex/52392">this question</a>:</p>
<p>Can utility function <span class="math-container">$U(x,y)$</span> that is both quasiconcave and quasiconvex always be transformed (via some positive monotonic function) into a quasilinear form <span class="math-container">$v(x) + y$</span>?</p>

<microeconomics><utility>
Quasicon... and quasilinearity
Giskard
# Quasicon... and quasilinearity

(Asked by: username_0 on 2022-08-18)

From [the discussion](https://economics.stackexchange.com/questions/52390/please-check-those-two-production-functions-they-also-seems-to-be-quasiconvex/52392?noredirect=1#comment85225_52392) under [this question](https://economics.stackexchange.com/questions/52390/please-check-those-two-production-functions-they-also-seems-to-be-quasiconvex/52392):

Can utility function $U(x,y)$ that is both quasiconcave and quasiconvex always be transformed (via some positive monotonic function) into a quasilinear form $v(x) + y$?

**Tags:** microeconomics, utility

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## 1 answers

### Answer by username_1 on 2022-08-18. Score: 5 | Accepted answer

That is not the case. Consider $u(x, y) = \lfloor x + y \rfloor$ i.e. greatest integer less than or equal to $x+y$. It is both quasi-concave and quasi-convex, but no transformation of the kind $v(x) + y$ exist. This is because $v(x) + y$ is strictly increasing in $y$, but $u$ is not.

Another much simpler example is $u(x, y) = 0$.

> **Comments:**
> **username_2**: I have question on game theory.  I will be appreciate if you help me by sharing your suggestions and hints. Thank you https://economics.stackexchange.com/questions/52468/principal-agent-problem  

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