<blockquote>
  <p><em>Are convex preferences needed for the first welfare theorem?</em></p>
</blockquote>

<p>No, convexity of preferences is imposed for other reasons. A general sufficient condition is local non-satiation, which says the agent can be made better off by an arbitrary small perturbation of his consumption bundle.
This can hold without the preference being convex.</p>

<blockquote>
  <p><em>It would see so. For example...</em></p>
</blockquote>

<p>As already pointed out by @Shomak, the example of an allocation you suggest is not an equilibrium. It also has nothing to do with convexity. Crossing of indifference curves at an allocation can occur for convex or non-convex preferences. Therefore it does not speak to the role of convexity in the theorem one way or another. </p>

<p>Assuming preferences are represented by differentiable utility functions (as per usual), standard marginalist reasoning tells you the allocation you describe is not an equilibrium.</p>

<p>Regardless of whether utility function is quasi-concave (i.e. whether the underlying preference is convex), the first order condition is always a necessary condition for optimality. Therefore agents must have the same marginal rate of substitution between commodities in equilibrium. When indifference curves cross, they don't have the same MRS.</p>

CC BY-SA 4.0
2020-06-14T05:54:45.153Z
‘When indifference curves cross, they don't have the same MRS’. Yes, at the point of crossing, but they might be tangent elsewhere. And the problem then would be that, at that tangency, one person can be made better off without the other person being made worse off, despite the optimality condition being met.
George
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2020-06-14T06:30:56.153Z
@George "...they might be tangent elsewhere"---What is "elsewhere"? At your supposed "equilibrium" allocation, do they cross or not? If they cross, it's not an equilibrium. At any equilibrium, they must be tangent. Also, convexity of preferences plays no role in these statements.
Michael
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2020-06-14T07:44:14.870Z
Please refer to the image I have now attached to my post. X is a competitive equilibrium but movement in the bottom-left direction can make the person with the convex IC better-off, while making the person with the non-convex IC no worse-off. Therefore, we have provided a counterexample to the first welfare theorem, and demonstrated that it does, in fact, presuppose convexity of preferences.
George
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2020-06-14T08:18:08.937Z
@George Answer covers situation described by image, $x$ is not an equilibrium, given price vector the non-convex agent can attain higher utility.
Giskard
CC BY-SA 4.0
2020-06-13T16:47:09.880Z
2020-06-13T19:21:09.733Z
2020-06-13T19:21:09.733Z
Michael
<p>Here is an answer that is probably confusing, but adds some advanced perspective.</p>

<p>The first general proof of the first welfare theorem (due to Kenneth Arrow) that did not rely on calculus used the assumption of strict convexity. Tjalling Koopmans later introduced the assumption of local-nonsatiation, which has become the standard assumption in textbooks for proving the first welfare theorem.</p>

<p>But the first welfare theorem holds under a weaker assumption that is at the same time implied by strict convexity: Local-nonsation can fail at most at one consumption bundle that is the best bundle for that consumer. A great source for all the details around how far one can generalize the welfare theorems is <a href="https://link.springer.com/chapter/10.1007/978-3-642-51602-3_2" rel="nofollow noreferrer">Pareto optima and equilibria: the finite dimensional case</a> (paywall, sorry) by Andreu Mas-Colell.</p>

<p>The situation is often somewhat different when one works with infinite-dimensional commodity spaces (important for macro and finance). There, local non-satiation is often not a useful assumption for somewhat technical reasons and one uses strong monotonicity assumptions or nonsation plus strong convexity instead.</p>

CC BY-SA 4.0
2020-06-14T01:18:21.697Z
Since the answer is already going down this road, would be nice to have a brief description on the reason(s) why local non-satiation is not a convenient assumption in an infinite dimensional setting. (Presumably they're functional analytic). Also, what about the point, brought up by @brunosalcedo, regarding convexity and the second welfare theorem in an infinite dimensional setting? Seems that one can't get away from convexity there.
Michael
CC BY-SA 4.0
2020-06-14T07:38:39.893Z
There are usually many topologies compatible with a vector space structure, so the answer is less universal. But consumption sets may not have any interior points, which makes directly using local-nonsation useless, though one can use the relative topology on consumption sets. More substantial, preferences need not be continuous with respect to a topology under which prices are continuous, and the usual argument requires that the set of points not on the budget line is open. For the second welfare theorem, convexity is of course still needed, and there things are even more complicated.
Michael Greinecker
CC BY-SA 4.0
2020-06-13T23:07:03.477Z
2020-06-13T23:32:49.700Z
2020-06-13T23:32:49.700Z
Michael Greinecker
<p>I would add to Michael's answer that convexity is important because it is needed to prove that there exists at least one competitive equilibria (and also to prove the second welfare theorem).</p>

CC BY-SA 4.0
2020-06-13T18:49:06.190Z
Right. Should have pointed out that "other reasons" in "convexity...is imposed for other reasons" include existence so that the theorem is not vacuously true in the sense that the set of CE is empty.
Michael
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2020-06-14T08:17:10.360Z
Which Michael's answer? :)
Giskard
CC BY-SA 4.0
2020-06-13T18:27:13.203Z
2020-06-15T17:55:07.380Z
2020-06-15T17:55:07.380Z
brunosalcedo
<p><a href="https://i.sstatic.net/6g0JD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6g0JD.png" alt="enter image description here"></a>Are convex preferences needed for the first welfare theorem? It would see so. For example, we could have a situation in which B's indifference curve is non-convex, such that, at the equilibrium X, it crosses A's indifference curve. Therefore, a movement to the left (if B's consumption is measured from the bottom left) will make A better off, and B no worse-off. Therefore, X is an equilibrium which is not pareto efficient. </p>

CC BY-SA 4.0
2020-06-13T06:43:39.550Z
@Shomak But then have we not presupposed the pareto optimality of competitive equilibria, as opposed to having proved it?
George
CC BY-SA 4.0
2020-06-14T12:37:38.480Z
The first welfare theorem is stated under a variety of assumptions, more or less strong (up to a point you can trade off some strictness in one assumption for looseness in another). So you could make a version of the first welfare theorem that would be very general in one way and very special in another. And convex preferences could be dispensed with in some of these versions. Be careful with use of the word "NEEDED": some assumptions can be removed if other assumptions are stricter. And last remark: with weird preferences like that, the solution could be at a corner, not interior.
PatrickT
<microeconomics><general-equilibrium>
First welfare theorem and convexity
George
# First welfare theorem and convexity

(Asked by: username_0 on 2020-06-13)

[![enter image description here](https://i.sstatic.net/6g0JD.png)](https://i.sstatic.net/6g0JD.png)Are convex preferences needed for the first welfare theorem? It would see so. For example, we could have a situation in which B's indifference curve is non-convex, such that, at the equilibrium X, it crosses A's indifference curve. Therefore, a movement to the left (if B's consumption is measured from the bottom left) will make A better off, and B no worse-off. Therefore, X is an equilibrium which is not pareto efficient.

**Tags:** microeconomics, general-equilibrium
> **Comments:**
> **username_0**: @Shomak But then have we not presupposed the pareto optimality of competitive equilibria, as opposed to having proved it?  
> **username_1**: The first welfare theorem is stated under a variety of assumptions, more or less strong (up to a point you can trade off some strictness in one assumption for looseness in another). So you could make a version of the first welfare theorem that would be very general in one way and very special in another. And convex preferences could be dispensed with in some of these versions. Be careful with use of the word "NEEDED": some assumptions can be removed if other assumptions are stricter. And last remark: with weird preferences like that, the solution could be at a corner, not interior.  

---

## 3 answers

### Answer by username_2 on 2020-06-13. Score: 4

> *Are convex preferences needed for the first welfare theorem?*

No, convexity of preferences is imposed for other reasons. A general sufficient condition is local non-satiation, which says the agent can be made better off by an arbitrary small perturbation of his consumption bundle.
This can hold without the preference being convex.

> *It would see so. For example...*

As already pointed out by @Shomak, the example of an allocation you suggest is not an equilibrium. It also has nothing to do with convexity. Crossing of indifference curves at an allocation can occur for convex or non-convex preferences. Therefore it does not speak to the role of convexity in the theorem one way or another.

Assuming preferences are represented by differentiable utility functions (as per usual), standard marginalist reasoning tells you the allocation you describe is not an equilibrium.

Regardless of whether utility function is quasi-concave (i.e. whether the underlying preference is convex), the first order condition is always a necessary condition for optimality. Therefore agents must have the same marginal rate of substitution between commodities in equilibrium. When indifference curves cross, they don't have the same MRS.

> **Comments:**
> **username_0**: ‘When indifference curves cross, they don't have the same MRS’. Yes, at the point of crossing, but they might be tangent elsewhere. And the problem then would be that, at that tangency, one person can be made better off without the other person being made worse off, despite the optimality condition being met.  
> **username_2**: @George "...they might be tangent elsewhere"---What is "elsewhere"? At your supposed "equilibrium" allocation, do they cross or not? If they cross, it's not an equilibrium. At any equilibrium, they must be tangent. Also, convexity of preferences plays no role in these statements.  
> **username_0**: Please refer to the image I have now attached to my post. X is a competitive equilibrium but movement in the bottom-left direction can make the person with the convex IC better-off, while making the person with the non-convex IC no worse-off. Therefore, we have provided a counterexample to the first welfare theorem, and demonstrated that it does, in fact, presuppose convexity of preferences.  
> **username_3**: @George Answer covers situation described by image, $x$ is not an equilibrium, given price vector the non-convex agent can attain higher utility.  

---

### Answer by username_4 on 2020-06-13. Score: 3

Here is an answer that is probably confusing, but adds some advanced perspective.

The first general proof of the first welfare theorem (due to Kenneth Arrow) that did not rely on calculus used the assumption of strict convexity. Tjalling Koopmans later introduced the assumption of local-nonsatiation, which has become the standard assumption in textbooks for proving the first welfare theorem.

But the first welfare theorem holds under a weaker assumption that is at the same time implied by strict convexity: Local-nonsation can fail at most at one consumption bundle that is the best bundle for that consumer. A great source for all the details around how far one can generalize the welfare theorems is [Pareto optima and equilibria: the finite dimensional case](https://link.springer.com/chapter/10.1007/978-3-642-51602-3_2) (paywall, sorry) by Andreu Mas-Colell.

The situation is often somewhat different when one works with infinite-dimensional commodity spaces (important for macro and finance). There, local non-satiation is often not a useful assumption for somewhat technical reasons and one uses strong monotonicity assumptions or nonsation plus strong convexity instead.

> **Comments:**
> **username_2**: Since the answer is already going down this road, would be nice to have a brief description on the reason(s) why local non-satiation is not a convenient assumption in an infinite dimensional setting. (Presumably they're functional analytic). Also, what about the point, brought up by @brunosalcedo, regarding convexity and the second welfare theorem in an infinite dimensional setting? Seems that one can't get away from convexity there.  
> **username_4**: There are usually many topologies compatible with a vector space structure, so the answer is less universal. But consumption sets may not have any interior points, which makes directly using local-nonsation useless, though one can use the relative topology on consumption sets. More substantial, preferences need not be continuous with respect to a topology under which prices are continuous, and the usual argument requires that the set of points not on the budget line is open. For the second welfare theorem, convexity is of course still needed, and there things are even more complicated.  

---

### Answer by username_5 on 2020-06-13. Score: 2

I would add to Michael's answer that convexity is important because it is needed to prove that there exists at least one competitive equilibria (and also to prove the second welfare theorem).

> **Comments:**
> **username_2**: Right. Should have pointed out that "other reasons" in "convexity...is imposed for other reasons" include existence so that the theorem is not vacuously true in the sense that the set of CE is empty.  
> **username_3**: Which Michael's answer? :)  

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