<p>Here are some examples of preferences that satisfy (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves, along with the possibility of corner solutions in the standard utility maximisation problems:</p>
<p>Preferences represented by utility functions <span class="math-container">$u:\mathbb{R}^2_+\rightarrow\mathbb{R}$</span>:</p>
<ul>
<li><span class="math-container">$u(x_1,x_2)=x_1+x_2^\beta$</span>, where <span class="math-container">$\beta\in (0,1)$</span> is given. Link: <a href="https://economics.stackexchange.com/a/58445/11824">https://economics.stackexchange.com/a/58445/11824</a></li>
<li><span class="math-container">$u(x_1,x_2)=(x_1+a)(x_2+b)$</span>, where <span class="math-container">$a&gt;0$</span>, <span class="math-container">$b&gt;0$</span> are given. Link: <a href="https://economics.stackexchange.com/a/58316/11824">https://economics.stackexchange.com/a/58316/11824</a></li>
<li><span class="math-container">$u(x_1,x_2)=x_2-x_1^2$</span> (Example of non-monotonic preference satisfying all the conditions above)</li>
</ul>

CC BY-SA 4.0
2024-06-08T04:03:36.213Z
Thank you! This is very helpful!
Santiago Valdivieso
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2024-06-07T23:12:32.777Z
2024-06-08T03:33:20.407Z
2024-06-08T03:33:20.407Z
Amit
<p>A textbook example of a utility function with frequent corner solutions is perfect substitution, e.g. <span class="math-container">$U(x,y) = x + 2y$</span>. This, however is not strictly convex. But if you think about it, the corner solution just needs that the slopes of the indifference curve and the budget line are unequal in the right way at the corner, it does not care about strict convexity in the interior. Thus you can just 'bend the indifference curve' a little</p>
<p><a href="https://i.sstatic.net/JfxLMeD2.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JfxLMeD2.png" alt="Image of a corner solution" /></a></p>
<p>and still get a corner solution. One can produce formulas for this as well, e.g. this happens with the utility function
<span class="math-container">$$U(x,y) = \sqrt{x+1} + \sqrt{y+1}$$</span>
and the budget line
<span class="math-container">$$x + 3y = 3.$$</span></p>
<p>A textbook example of a utility function resulting in a corner solution would be a quasilinear utility function:
<span class="math-container">$$U(x,y) = \ln{x} + y$$</span>
Here the income offer curve is vertical, a consumer with 'small enough' income will spend only on <span class="math-container">$x$</span>.</p>
<p>You can rule out corner solutions by requiring something like the Inada conditions, i.e.
<span class="math-container">$$
\lim_{x\to 0 } \frac{\partial U(x,y) } {\partial x} = \infty \hskip 20pt
\lim_{y\to 0 } \frac{\partial U(x,y) } {\partial y} = \infty
$$</span>
This makes the indifference curve tangential to an axis it approaches, hence a budget line cannot intersect it in the way displayed in the first graph.</p>

CC BY-SA 4.0
2024-06-07T13:34:05.613Z
I thought about the indifference curve given by $y=1-ln(x)$. However, it crosses the x axis, and when it does, more x produces a reduction in utility, therefore violating the non satiation axiom ($dU/dx >= 0$ for all x). I thought about restricting the support to $x>=0$ and $y>=0$, but wouldn't that violate the discontinuity principle? Finally, if we impose the "Inada conditions" for both x and y, then a corner solution is impossible because there is no point where x or y is 0.
Santiago Valdivieso
CC BY-SA 4.0
2024-06-07T13:40:20.860Z
I was thinking about an indifference curve that has a point at (0,y), but is not defined for negative values of x (as $y=1-ln(x)$) AND that has something like the inada conditions for x. I haven't found a function that fulfills those conditions.
Santiago Valdivieso
CC BY-SA 4.0
2024-06-07T14:16:37.483Z
$\ln x + y$ does not violate non satiation, I don't know why you think that. Give me any point, like (1,0) or something and I will increase the function's value by increasing $x$ by $\epsilon$. If you are uncertain about what some of the axioms mean, please post a new question where you explain your new question.
Giskard
CC BY-SA 4.0
2024-06-07T19:48:33.787Z
I was mixing up the concepst and graphs of indifference and utility curves in my head. Thank you!
Santiago Valdivieso
CC BY-SA 4.0
2024-06-07T07:20:40.907Z
2024-06-07T07:20:40.907Z
Giskard
<p>By standard consumer theory axioms I mean (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves.</p>
<p>If these axioms are not sufficient to rule out a corner solution, can you give me an example of a utility function that satisfies them and still results in a corner solution?</p>

<microeconomics><utility><consumer-theory><optimization><consumption>
Do standard consumer theory axioms rule out corner solutions?
Santiago Valdivieso
# Do standard consumer theory axioms rule out corner solutions?

(Asked by: username_0 on 2024-06-06)

By standard consumer theory axioms I mean (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves.

If these axioms are not sufficient to rule out a corner solution, can you give me an example of a utility function that satisfies them and still results in a corner solution?

**Tags:** microeconomics, utility, consumer-theory, optimization, consumption


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## 2 answers

### Answer by username_2 on 2024-06-07. Score: 3 | Accepted answer

A textbook example of a utility function with frequent corner solutions is perfect substitution, e.g. $U(x,y) = x + 2y$. This, however is not strictly convex. But if you think about it, the corner solution just needs that the slopes of the indifference curve and the budget line are unequal in the right way at the corner, it does not care about strict convexity in the interior. Thus you can just 'bend the indifference curve' a little

[![Image of a corner solution](https://i.sstatic.net/JfxLMeD2.png)](https://i.sstatic.net/JfxLMeD2.png)

and still get a corner solution. One can produce formulas for this as well, e.g. this happens with the utility function
$$U(x,y) = \sqrt{x+1} + \sqrt{y+1}$$
and the budget line
$$x + 3y = 3.$$

A textbook example of a utility function resulting in a corner solution would be a quasilinear utility function:
$$U(x,y) = \ln{x} + y$$
Here the income offer curve is vertical, a consumer with 'small enough' income will spend only on $x$.

You can rule out corner solutions by requiring something like the Inada conditions, i.e.
$$
\lim\_{x\to 0 } \frac{\partial U(x,y) } {\partial x} = \infty \hskip 20pt
\lim\_{y\to 0 } \frac{\partial U(x,y) } {\partial y} = \infty
$$
This makes the indifference curve tangential to an axis it approaches, hence a budget line cannot intersect it in the way displayed in the first graph.

> **Comments:**
> **username_0**: I thought about the indifference curve given by $y=1-ln(x)$. However, it crosses the x axis, and when it does, more x produces a reduction in utility, therefore violating the non satiation axiom ($dU/dx >= 0$ for all x). I thought about restricting the support to $x>=0$ and $y>=0$, but wouldn't that violate the discontinuity principle? Finally, if we impose the "Inada conditions" for both x and y, then a corner solution is impossible because there is no point where x or y is 0.  
> **username_0**: I was thinking about an indifference curve that has a point at (0,y), but is not defined for negative values of x (as $y=1-ln(x)$) AND that has something like the inada conditions for x. I haven't found a function that fulfills those conditions.  
> **username_2**: $\ln x + y$ does not violate non satiation, I don't know why you think that. Give me any point, like (1,0) or something and I will increase the function's value by increasing $x$ by $\epsilon$. If you are uncertain about what some of the axioms mean, please post a new question where you explain your new question.  
> **username_0**: I was mixing up the concepst and graphs of indifference and utility curves in my head. Thank you!  

---

### Answer by username_1 on 2024-06-07. Score: 3

Here are some examples of preferences that satisfy (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves, along with the possibility of corner solutions in the standard utility maximisation problems:

Preferences represented by utility functions $u:\mathbb{R}^2\_+\rightarrow\mathbb{R}$:

* $u(x\_1,x\_2)=x\_1+x\_2^\beta$, where $\beta\in (0,1)$ is given. Link: <https://economics.stackexchange.com/a/58445/11824>
* $u(x\_1,x\_2)=(x\_1+a)(x\_2+b)$, where $a>0$, $b>0$ are given. Link: <https://economics.stackexchange.com/a/58316/11824>
* $u(x\_1,x\_2)=x\_2-x\_1^2$ (Example of non-monotonic preference satisfying all the conditions above)

> **Comments:**
> **username_0**: Thank you! This is very helpful!  

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