<p>Without convexity, we'd lose the convenience of using the first order condition (or the tangency condition) to identify the optimal consumption bundle.</p>

<p>As shown in the following figure, where the red-shaded region is a non-convex budget set, the tangency at point <span class="math-container">$E$</span> is not the optimal consumption point; rather, the optimum is a corner solution, where only <span class="math-container">$x$</span> is consumed. </p>

<p><a href="https://i.sstatic.net/8RJV7.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8RJV7.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2019-09-07T16:14:46.670Z
Thank you for your answer, the visual is very helpful and clearly shows the corner solution implication of when convexity is violated. Both answers provided are very helpful but I accepted the other as it brought up the violation of the Hyperplane Theorem and provided links connected with that. I wish I could accept both as answers. Both are complements and I really appreciate it
Brennan
CC BY-SA 4.0
2019-09-09T20:18:53.253Z
@afreelunch: I guess my point is not that non-convex budget set will lead to corner solutions; optimization with non-convex budget set could very well have interior solutions (e.g. a set with a hole in the middle). Rather, I meant to suggest that we can no longer use the FOC to determine the optimum, even if we have a "nice" utility function.
Herr K.
CC BY-SA 4.0
2019-09-06T23:36:28.040Z
2019-09-06T23:36:28.040Z
Herr K.
<p>Convexity can be a very important assumption as much of economic analysis is built on working with convex sets, which makes things easier. </p>

<p>Very importantly, convexity of sets allows us to work with the Separating Hyperplane and Supporting Hyperplane theorems, which have applications to many results in economics, as partially discussed <a href="https://economics.ubc.ca/files/2014/02/pdf_course_erwin-diewert-ECON594Ch2.pdf" rel="nofollow noreferrer">here</a> and <a href="https://en.wikipedia.org/wiki/Convexity_in_economics" rel="nofollow noreferrer">here</a>.</p>

<p>Here is also a short, concrete example of something for which we need convexity of the budget set:</p>

<p>Suppose you want to prove that Marshallian Demand <span class="math-container">$x(p,m)$</span> is single-valued for every (larger than zero) price (<span class="math-container">$p$</span>) and (non-negative) income (<span class="math-container">$m$</span>) combination. That is to say, a Marshallian Demand Function exists, since without single-valuedness we would have a Marshallian Demand Correspondence (i.e. we'd have multiple good combinations as part of the optimal demand choice for a single given price and income level), which can be harder to work with. This is a pretty nice result that we (almost) always work with in undergraduate texts. For that you need strictly convex preferences and a convex budget set <span class="math-container">$B(p,m)$</span>. The proof is as follows:</p>

<p>If two points <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are both elements of the demand <span class="math-container">$x(p,m)$</span> at given prices <span class="math-container">$p$</span> and income <span class="math-container">$m$</span>, i.e. if <span class="math-container">$x_1, x_2 \in x(p,m)$</span>, then by definition we must have indifference between the two points, that is <span class="math-container">$x_1 \sim x_2$</span>. For a single-valued demand function two different points <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> cannot exist  for single given level of prices and income, i.e. they would have to be the same point. In other words  <span class="math-container">$x_1 = x_2$</span>. We prove it by contradiction. Suppose <span class="math-container">$x_1 \neq x_2$</span>.</p>

<p>For all <span class="math-container">$\lambda \in (0,1)$</span>, we have <span class="math-container">$\lambda *x_1 + (1-\lambda) *x_2 \in B(p,m)$</span> by convexity of <span class="math-container">$B$</span>. By strict convexity of preferences we have <span class="math-container">$\lambda *x_1 + (1-\lambda) *x_2 \succ x_1$</span>. However, this contradicts the fact that <span class="math-container">$x_1 \in x(p,m)$</span> in the first place. Hence, <span class="math-container">$x_1 = x_2$</span>.</p>

<p>So, in words, some combination of <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> is better than just <span class="math-container">$x_1$</span> at the given price and income level. But such a combination is also <strong>attainable</strong> at that given price and income level because of the convexity of the budget set. This is where the budget set convexity assumption is important. Furthermore, if for a given price and income level there is a point (in this case a combination of <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>) that is both strictly better and attainable, then this contradicts the original starting point that <span class="math-container">$x_1$</span> is a point in the demand function for the given price and income level.</p>

<p>As a side note, I think it is a bit misleading to talk about producers in your post when talking about budget constraints. If we are looking at a budget constraint, we should be discussing consumers.</p>

CC BY-SA 4.0
2019-09-07T16:11:53.597Z
Thank you for your answer, and for pointing out that I shouldn’t be talking about producers. I only used the example for somehow bringing a convex (in terms of curvature) budget constraint into question. But yes we are taking about consumer theory here so I agree
Brennan
CC BY-SA 4.0
2019-09-07T00:10:06.653Z
2019-09-07T14:31:37.033Z
2019-09-07T14:31:37.033Z
BB King
<p>In Microeconomic theory, the budget constraint is defined by 4 distinct properties:
It is</p>

<ol>
<li>Bounded</li>
<li>Closed</li>
<li>Convex</li>
<li>Non-empty</li>
</ol>

<p>The 1. 2. and 4. are very straight forward and the benefits in terms of analysis are apparent. My question is with respect to convexity.</p>

<p>Convexity of a set implies that any line between two points is the set is also within the set (i.e. take any two points say <span class="math-container">$q$</span> and <span class="math-container">$q'$</span> that both satisfy the constraint, then any linear combination of the <span class="math-container">$q$</span> and <span class="math-container">$q'$</span> also satisfy the constraint)</p>

<p>My question is, what is the benefit or added utility to this property holding?
I immediately think of a budget constraint that, rather than exhibiting diminishing marginal returns, exhibits increases returns in a kind of learning sense: As more of one good is produced, the production becomes more efficient as the producer is learning. Whether or not this is a realistic circumstance, it begs the question.</p>

<p>What happens if convexity is violated? or rather, what can we <strong>no longer do</strong>?</p>

<microeconomics><mathematical-economics><consumer-theory>
Set Theory Properties of the Budget Constraint
Brennan
# Set Theory Properties of the Budget Constraint

(Asked by: username_0 on 2019-09-06)

In Microeconomic theory, the budget constraint is defined by 4 distinct properties:
It is

1. Bounded
2. Closed
3. Convex
4. Non-empty

The 1. 2. and 4. are very straight forward and the benefits in terms of analysis are apparent. My question is with respect to convexity.

Convexity of a set implies that any line between two points is the set is also within the set (i.e. take any two points say $q$ and $q'$ that both satisfy the constraint, then any linear combination of the $q$ and $q'$ also satisfy the constraint)

My question is, what is the benefit or added utility to this property holding?
I immediately think of a budget constraint that, rather than exhibiting diminishing marginal returns, exhibits increases returns in a kind of learning sense: As more of one good is produced, the production becomes more efficient as the producer is learning. Whether or not this is a realistic circumstance, it begs the question.

What happens if convexity is violated? or rather, what can we **no longer do**?

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


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## 2 answers

### Answer by username_2 on 2019-09-07. Score: 5 | Accepted answer

Convexity can be a very important assumption as much of economic analysis is built on working with convex sets, which makes things easier.

Very importantly, convexity of sets allows us to work with the Separating Hyperplane and Supporting Hyperplane theorems, which have applications to many results in economics, as partially discussed [here](https://economics.ubc.ca/files/2014/02/pdf_course_erwin-diewert-ECON594Ch2.pdf) and [here](https://en.wikipedia.org/wiki/Convexity_in_economics).

Here is also a short, concrete example of something for which we need convexity of the budget set:

Suppose you want to prove that Marshallian Demand $x(p,m)$ is single-valued for every (larger than zero) price ($p$) and (non-negative) income ($m$) combination. That is to say, a Marshallian Demand Function exists, since without single-valuedness we would have a Marshallian Demand Correspondence (i.e. we'd have multiple good combinations as part of the optimal demand choice for a single given price and income level), which can be harder to work with. This is a pretty nice result that we (almost) always work with in undergraduate texts. For that you need strictly convex preferences and a convex budget set $B(p,m)$. The proof is as follows:

If two points $x\_1$ and $x\_2$ are both elements of the demand $x(p,m)$ at given prices $p$ and income $m$, i.e. if $x\_1, x\_2 \in x(p,m)$, then by definition we must have indifference between the two points, that is $x\_1 \sim x\_2$. For a single-valued demand function two different points $x\_1$ and $x\_2$ cannot exist for single given level of prices and income, i.e. they would have to be the same point. In other words $x\_1 = x\_2$. We prove it by contradiction. Suppose $x\_1 \neq x\_2$.

For all $\lambda \in (0,1)$, we have $\lambda \*x\_1 + (1-\lambda) \*x\_2 \in B(p,m)$ by convexity of $B$. By strict convexity of preferences we have $\lambda \*x\_1 + (1-\lambda) \*x\_2 \succ x\_1$. However, this contradicts the fact that $x\_1 \in x(p,m)$ in the first place. Hence, $x\_1 = x\_2$.

So, in words, some combination of $x\_1$ and $x\_2$ is better than just $x\_1$ at the given price and income level. But such a combination is also **attainable** at that given price and income level because of the convexity of the budget set. This is where the budget set convexity assumption is important. Furthermore, if for a given price and income level there is a point (in this case a combination of $x\_1$ and $x\_2$) that is both strictly better and attainable, then this contradicts the original starting point that $x\_1$ is a point in the demand function for the given price and income level.

As a side note, I think it is a bit misleading to talk about producers in your post when talking about budget constraints. If we are looking at a budget constraint, we should be discussing consumers.

> **Comments:**
> **username_0**: Thank you for your answer, and for pointing out that I shouldn’t be talking about producers. I only used the example for somehow bringing a convex (in terms of curvature) budget constraint into question. But yes we are taking about consumer theory here so I agree  

---

### Answer by username_1 on 2019-09-06. Score: 4

Without convexity, we'd lose the convenience of using the first order condition (or the tangency condition) to identify the optimal consumption bundle.

As shown in the following figure, where the red-shaded region is a non-convex budget set, the tangency at point $E$ is not the optimal consumption point; rather, the optimum is a corner solution, where only $x$ is consumed.

[![enter image description here](https://i.sstatic.net/8RJV7.png)](https://i.sstatic.net/8RJV7.png)

> **Comments:**
> **username_0**: Thank you for your answer, the visual is very helpful and clearly shows the corner solution implication of when convexity is violated. Both answers provided are very helpful but I accepted the other as it brought up the violation of the Hyperplane Theorem and provided links connected with that. I wish I could accept both as answers. Both are complements and I really appreciate it  
> **username_1**: @afreelunch: I guess my point is not that non-convex budget set will lead to corner solutions; optimization with non-convex budget set could very well have interior solutions (e.g. a set with a hole in the middle). Rather, I meant to suggest that we can no longer use the FOC to determine the optimum, even if we have a "nice" utility function.  

---

