The first fundamental theorem of welfare economics (also known as Adam Smith’s “Invisible-Hand Theorem”) states that every competitive equilibrium is efficient. To be clear, as a mathematical theorem, given its assumptions about human nature, it is correct. But as it is almost always stated, only the assumptions about the market are mentioned (with the word “competitive”). In this form, it is false, even if we grant all of the assumptions of “perfect competition,” because it is not true for all types of economic actors.

In particular, it is not true for humans. In fact, there is no evidence that it is even approximately true for humans because the flaw in the rationality assumption is not due to small mistakes in human rationality. The problem is that the proof of the mathematical theorem assumes away all social preferences, e.g. preferences concerning social status.

Sketch of Proof:

If utility functions contain a concern with social standing, then they can cause a race to see who can display the most wealth. In the case where people are identical, this results in everyone displaying too much wealth but still not getting ahead of each other so they gain nothing in social standing. This is inefficient even though the market is completely competitive — anyone can trade bananas for a bigger house.

Proof: 

First recall that a theorem must be true in all the cases it claims to cover. So disproof requires only one counterexample. Our counterexample consists of a large number of individuals endowed with the ability to expend 10 units of effort per day. This effort can be used to produce bananas or build houses. Assumptions:

1. One unit of effort produces one banana: B = Eb.
2. Each person is endowed with 10 units of effort per day: Eb+Eh = 10.
3. The size of your house will be H = sqrt(10 Eh), where Eh is the daily effort to build and maintain the house.
4. Each person’s utility is U = H + B + Pride.
5. The utility from being above average is Pride = 0.9×[H − Average(H)]. 
6. People can trade bananas for house-building effort.

The bigger a house gets, the more repairs it needs, and so if you keep putting effort into it, it will eventually reach an equilibrium size at which more house effort would decrease your utility because it’s not worth the foregone bananas.

If all houses are of size zero, and you put all your effort into growing bananas, your utility will be U = 0 + 10 + 0 = 10. But if everyone one puts 1 unit of effort per day into their house (so all H = sqrt(10) ), your utility will be U = sqrt(10) + 9 − 0 = 12.16.

Since people are identical, in equilibrium, they will all put the same effort into their house, so all houses will be the same size, and Pride = 0 for everyone.

The neoclassical equilibrium (which ignores pride) will be efficient. So it will maximize individual utility. This happens when Eh = 2.5, H=25, the utility from the house is 5, and 7.5 bananas are produced for a total utility of 12.5. (You can easily check this with a spreadsheet.)

But is this the true competitive equilibrium? No. Because Pride has not been taken into account. Say you choose Eh = 5. Your house will be size H=sqrt(50) which gives you more house and some pride in having a bigger-than-average house.

• U = sqrt(50) + 5 + 0.9 × [sqrt(50) – sqrt(25] = 14.1

By building a bigger house you find you can increase your utility, and so you start adding to your house.

Of course, everyone else has the same idea, and the race is on. But the bigger the houses get, the less utility you get from expanding your house. The race finally comes to an end when spending more effort on your house increases its size by too little and the payoff from more house and a little pride no longer compensates for less to eat.

This happens when Eh reaches 9, and you are only growing one banana per day. At this point, your utility is down from 12.5 to 10.5. This is obviously inefficient.

So the welfare theorem is wrong when utility functions contain social concerns. In this case, the perfectly competitive equilibrium is not efficient.

This only shows that the theorem is wrong for humans. In other words, it is an interesting math theorem, but it is without any scientific merit. Neoclassical economics is not a science, it is not even a social science, it is an ideology with a mathematical formalization.