The First Fundamental Theorem of Welfare Economics belongs to the moment when Economics became a mathematical science (sorry for the redundancy).

Kenneth J. Arrow’s path already contained the mixture of Mathematics and Social Science. He graduated from the City College of New York in 1940 with a degree in Social Science but with a major in Mathematics. He then went to Columbia, received an M.A. in Mathematics in 1941, and moved toward economics under the influence of Harold Hotelling.

Gérard Debreu arrived at economics from an even more mathematical route. He studied Mathematics at the École Normale Supérieure and after World War II, he became interested in economics through the theory of general economic equilibrium, presented to him through Maurice Allais’s work. He later described the following years as his “conversion from mathematics to economics”.

Cowles Commission was an important point for both economists. Debreu described it as a focal point for mathematical economics, where recent developments were discussed in weekly meetings, seminars, and many informal conversations (a dream environment, isn’t it?). In that environment, his work on Pareto optima, the existence of general economic equilibrium, and utility theory made quick progress. Arrow, for his part, also described Cowles as a formative influence and links that period directly to his work on Pareto efficiency.

In their work, Arrow and Debreu proved, under certain assumptions, that every competitive equilibrium is Pareto optimal. That is the First Fundamental Welfare Theorem.

This is the theorem’s proof that can be found in their joint paper ‘Existence of an Equilibrium for a Competitive Economy’, published in Econometrica in 1954, even though both of them had gotten interesting results in general equilibrium separately before that.

The theorem remains one of the most important results in Economic Theory and has been useful to use competitive equilibrium as a benchmark to compare different settings such as imperfect competition. Many subfields in Economics wouldn’t be as strong as they are had it not been for this theorem by Arrow and Debreu, who were, in my opinion, two of the best economists in history.

References

Arrow, K. J., & Debreu, G. (1954). Existence of an Equilibrium for a Competitive Economy. Econometrica, 22(3), 265–290. https://doi.org/10.2307/1907353

Nash equilibrium is usually defined as a strategy profile in which each player’s strategy is optimal given the strategies chosen by the others. In some games, however, no such profile exists if players are restricted to pure strategies, that is, to choosing a single action with probability one. A standard example is Matching Pennies. Two players simultaneously choose either Heads or Tails. If the choices match, Player 1 wins; if they differ, Player 2 wins. The names Heads and Tails are only labels for actions: the players are not required to toss coins. Consider each of the four possible pure-strategy profiles.

If both players choose the same action, Player 2 can switch to the opposite action and improve their payoff. If they choose different actions, Player 1 can switch to match Player 2's action and improve their payoff. Consequently, from every pure-strategy profile at least one player has a profitable unilateral deviation. Since a Nash equilibrium is a strategy profile from which no player wishes to deviate unilaterally, no pure-strategy Nash equilibrium exists.

Nash’s contribution was to make the strategy space bigger. Instead of choosing only Heads or only Tails, he introduced the possibility that a player chooses a probability distribution over the available actions: for example, Heads with probability (p) and Tails with probability (1-p). These probability distributions are called mixed strategies. Nash proved that when every player in a finite game is allowed to use mixed strategies, an equilibrium always exists. In this article, I will try to explain his one page proof. It’s a great example that a impact is not necessarily correlated with length.

The proof

The proof can be summarised, explaining each concept he uses, as follows:

The essence of Nash’s argument is to transform the problem of finding an equilibrium into a fixed-point problem. Rather than attempting to construct an equilibrium directly, he studies the geometry of the mixed-strategy space and the mathematical properties of the best-response correspondence. Once these objects are shown to satisfy the hypotheses of Kakutani’s Fixed Point Theorem, the existence of a fixed point follows. This fixed point has a natural game-theoretic interpretation: it is a strategy profile in which every player’s chosen mixed strategy is a best response to the strategies of the others. Consequently, no player can unilaterally increase their expected payoff by deviating, which is precisely the definition of a Nash equilibrium. Notice that the proof is an existence argument, it does not provide any guess or computation about what must be the equilibrium(s).

References

Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United States of America, 36(1), 48–49. http://www.jstor.org/stable/88031

Mamdani’s rent freeze has passed for one million housing units. Many voters assume that this policy will truly benefit them. After all, if rents don’t increase, what could go wrong? Many things, actually.

A recent literature review about government interventions on the housing market, which analyses 1077 papers in 96 different countries, shows that, while rent controls can indeed stop the increase in the units subject to the policy, secondary effects include less new supply, lower quality of the actual supply given that owners have less incentives for it (they face an increasing demand with a lower supply, so they will find someone that will rent the unit either way), less mobility given that the tenant will very difficultly find another unit with such a low rental and higher rentals for not controlled units (the expected result when the demand going to the controlled units moves to the non-controlled ones).

On the other hand, land-use restrictions that limit how much and where housing can be built create artificial scarcity and reduce supply elasticity of supply, therefore causing any demand shock to translate into higher prices rather than more housing. In this context, demand-side policies such as subsidies, cheap credit and tax breaks end up generating higher prices and rents (more demand vs less supply) instead of increasing the housing stock as would happen in a more competitive market. The adjustment is thus via prices, not quantities, as it should be to benefit tenants.

Land-use restrictions that limit how much and where housing can be built create artificial scarcity and reduce supply elasticity of supply, therefore causing any demand shock to translate into higher prices rather than more housing.

What works to reduce prices, then? Increasing housing supply. It’s no surprise since we’re talking about a textbook case policy and it’s also a result of the vast majority of papers about it. Here’s a list of some studies about it that I posted on X:

https://x.com/frannunesecon/status/2071579023667282423?s=46&t=XIt-VeqC4t8IkJ76vtjTPg

To conclude: No, rent control won’t solve the housing problem. It will actually make it worse. But not everything is lost, we know about a policy that does work: increasing housing supply. It might not be the most popular one among voters, but it’s the one that delivers the best results. The evidence is there. Let’s be reasonable and follow it.

It’s very likely that you have, at some point, debated about what the perfect election system would look like. You probably believe that a fair system should respect, for example, the absence of a dictator. There are some other characteristics many people would agree a fair election system should have. The economist Kenneth J. Arrow (1921-2017) axiomatized some of these characteristics in his famous impossibility theorem (that he called “possibility” theorem) in his book ‘Social Choice and Individual Values’ and showed that no election system could satisfy all these fairly general and easy to agree principles. Let’s see what axioms he imposed.

Description of the axioms:

Universal Domain

All individuals i ∈ G have rational preferences over the set of alternatives X, but beyond that, they can have any set of rational orderings.

Weak Pareto Principle

If every member of G prefers a to b, then the social decision rule must prefer a to b.

Formally, if a ≿_i b ∀i ⇒ a ≿ b.

Independence of Irrelevant Alternatives

Suppose we have two different societies G and G’, but within G and G’ everyone has the same orderings of alternatives an and b. Then the social ordering between an and b in both societies must be the same, even if members of G and G’ have different rankings of other alternatives. This is the assumption relaxed in many real life voting systems.

No dictatorship

There is no particular individual i* ∈ G such that the preferences of i* determine the social ranking, regardless of other group members.

Arrow’s Impossibility Theorem

Suppose there are at least three alternatives. There is no social ranking function such that for any group G whose members all have rational preferences, the social ranking is a rational (complete and transitive) ranking and satisfies the Universal Domain, Weak Pareto Principle, Independence of Irrelevant Alternatives and No Dictatorship axioms.

Here’s an image about it with an example and intuition.

And here’s an image with a more formal treatment of the theorem.

Proof:

Does the result still hold if we relax some axioms?

Here’s where Gibbard-Satterthwaite Theorem enters the game. The theorem states the following.

Gibbard-Satterthwaite Theorem:

In every deterministic ordinal electoral system, at least one of the following conditions must hold:

1. The rule is dictatorial.

2. The rule limits the possible outcomes to only two alternatives.

3. The rule is not strategyproof (concept from Mechanism Design that means that players of a game have incentives to report their true preference).

4. The rule does not yield a single winner.

Muller-Satterthwaite Theorem:

Assume at least three alternatives. No resolute voting rule (voting rule that selects a single winner) satisfies strong monotonicity (which means that if a is elected, and then some voters like an even more than before, a should not lose, i. e., improving the winner’s position doesn’t hurt the winner), non-imposition (no alternative is ruled out a priori), and non-dictatorship. Equivalently, there is no social choice function that satisfies these properties.

Bad news for ranked electoral systems lovers, but at the same time it’s amazingly beautiful that we can prove these things mathematically. What do you think about it?

References

Arrow, K. J. (2012). Social Choice and Individual Values. Yale University Press. http://www.jstor.org/stable/j.ctt1nqb90

Geanakoplos, J. (2005). Three Brief Proofs of Arrow’s Impossibility Theorem. Economic Theory, 26(1), 211–215. http://www.jstor.org/stable/25055941

Shoham, Y., & Leyton-Brown, K. (2008). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press.

Monotone Comparative Statics (MCS) deals with how the solutions to an optimization problem change with its parameters. It can deliver interesting results under relatively weak assumptions. One of its main tools is Topkis’ Theorem, which allows us to derive comparative statics results without differentiability assumptions and without explicitly solving the model. In order to understand the theorem, we need to know some concepts first. In every concept I introduce, I will adjunct an image with an example for better understanding. The theorem will also have an image with an example and the intuition behind it. Let a,b ∈ R^n. We define:

Meet and Join

We define the meet and join of an and b as the vectors of minimum and maximum coordinate-wise comparison of an and b, respectively.

Strong Set Order
A set A ⊆ R^n is greater than a set B ⊆ R^n in the strong set order if, for every a ∈ A and b ∈ B, (a ∨ b)∈ A and (a ∧ b) ∈ B.

Lattice
A lattice is a set X ⊆ R^n such that (x ∨ y) ∈ X and (x ∧ y) ∈ X for every x,y ∈ X.

Increasing Differences
A function f has increasing differences in (x,θ) if whenever xᴴ ≥ xᴸ and θᴴ ≥ θᴸ,

f(xᴴ,θᴴ) − f(xᴸ,θᴴ) ≥ f(xᴴ,θᴸ) − f(xᴸ,θᴸ).

Supermodularity
A function f:X×Θ→R is supermodular in x if

f(x∨y,θ) − f(x,θ) ≥ f(y,θ) − f(x∧y,θ).

Topkis’ Theorem (1978)
Let x ∈ X and θ ∈ Θ. If X ⊆ R^n is a lattice, Θ ⊆ R^m and f:X×Θ→R has increasing differences in (x,θ) and is supermodular in x, then the solution set X*(θ) is increasing in the strong set order.

Single Crossing

What’s the minimal (necessary and sufficient) condition for Monotone Comparative Statics? Milgrom and Shannon (1994) answered this question. What we need is the Single-Crossing property:

A function f:X×Θ→ℝ is single-crossing in (x,θ) if whenever

xᴴ ≥ xᴸ and θᴴ ≥ θᴸ, we have

f(xᴴ,θᴸ) ≥ f(xᴸ,θᴸ)  ⇒  f(xᴴ,θᴴ) ≥ f(xᴸ,θᴴ),

and

f(xᴴ,θᴸ) > f(xᴸ,θᴸ)  ⇒  f(xᴴ,θᴴ) > f(xᴸ,θᴴ).

Increasing differences implies single-crossing, but the converse is not true because single-crossing means that the ranking between xᴴ and xᴸ can’t reverse, whereas increasing differences means that the increase in the function from going from xᴸ to xᴴ must weakly increase, which is a stronger condition.

Milgrom and Shannon’s Theorem (1994)

Topkis showed that increasing differences and supermodularity are sufficient conditions for Monotone Comparative Statics. Milgrom and Shannon showed that single-crossing is the minimal condition for Monotone Comparative Statics (it’s necessary and sufficient, if and only if).

If f is single-crossing in (x,θ), then the solution correspondence

X*(θ) = argmax f(x,θ)

ₓ∈X

is increasing in the strong set order.

Conversely, if X*(θ) is increasing in the strong set order for every choice set X⊆ℝ, then f is single-crossing in (x,θ).

Application: Consumer Theory

In the next image, we have an application of Topkis’ Theorem to Consumer Theory in both the nondifferentiable and differentiable function cases.

The differentiable case example comes from Amir, R. (2005).

References

Amir, R. (2005). Supermodularity and Complementarity in Economics: An Elementary Survey. Southern Economic Journal, 71(3), 636–660. https://doi.org/10.2307/20062066

Milgrom, P., & Shannon, C. (1994). Monotone Comparative Statics. Econometrica, 62(1), 157–180. https://doi.org/10.2307/2951479

Topkis, D. M. (1978). Minimizing a Submodular Function on a Lattice. Operations Research, 26(2), 305–321. http://www.jstor.org/stable/169636