First welfare theorem and convexity

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.

• In that case, $X$ is not an equilibrium in the first place. A can offer the improved trade to B, and B will accept it. (or A may share an arbitrarily small amount of the gains from trade with B to shift the equilibrium). – user28372 Jun 13 '20 at 6:38
• @Shomak But then have we not presupposed the pareto optimality of competitive equilibria, as opposed to having proved it? – George Jun 13 '20 at 6:43
• 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 Jun 14 '20 at 12:37

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.

• ‘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 Jun 14 '20 at 5:54
• @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 Jun 14 '20 at 6:30
• 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 Jun 14 '20 at 7:44
• @George Answer covers situation described by image, $x$ is not an equilibrium, given price vector the non-convex agent can attain higher utility. – Giskard Jun 14 '20 at 8:18
• Exactly. The non-convex dude, at the given price vector, would optimally go all the way down to the bottom border. So there's definitely excess supply of the goods on the horizontal axis. @george, understand that MRS1=MRS2 is not the definition of equilibrium. The condition denotes the FOC for each agent, which is only necessary for equilibrium and certainly not Sufficient. When the utility fn is wonky (like for non-convex prefs), you need other ways to solve for equilibrium. You can go the KKT way, but pictures do express a 1000 words for GE problems. – user28372 Jun 14 '20 at 9:27

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 (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.

• 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 Jun 14 '20 at 1:18
• 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 Jun 14 '20 at 7:38

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).

• 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 Jun 13 '20 at 18:49
• Which Michael's answer? :) – Giskard Jun 14 '20 at 8:17