The first fundamental theorem of welfare economics (also known as the “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 referenced (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 what may be the most important part of human preferences, preferences concerning social status.

Proof: First recall that a theorem must be true in all the cases it claims to cover. So disproof requires only one counter example. Our counter example consists of a larger 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:

- Each person is endowed with 10 units of effort per day.
- One unit of effort produces one banana.
- The size of your house will be H = 10 Eh, where Eh is the daily effort to build and maintain the house.
- Each person’s utility is U = sqrt(H) + B + Pride, where B = bananas/day.
- Pride is the utility from being above average = [H − Average(H)]/22.
- People can trade bananas for house-building efforts.

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 where all your effort is going into repairs. This equilibrium size is given by assumption 2.

Notice that if all houses are size H = 0, 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 = 10 in equilibrium), your utility will be U = sqrt(10) + 9 − 0 = 3.16 + 9 = 12.16. For every Eh, U = sqrt(10 Eh) + (10−Eh) − 0.

The maximum possible individual utility in such a uniform equilibrium is 12.5 when Eh = 2.5, H=25, the utility from the house is 5, and 7.5 bananas are produced and consumed. (You can easily check this with a spreadsheet.)

But is this the competitive equilibrium? No. Because any individual can choose to put more effort into building a bigger house. Say you choose Eh = 5. Your house will be size H=50, twice as big a average. You will get 7.07−5 extra utility from a bigger house, and 25/22 = 2.27 units of utility from Pride in being ahead of the Joneses. Combined, this more than compensates for your reduced banana consumption. 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. And so the race finally comes to an end when extra house utility plus extra Pride just equals the loss in utility from non-house spending (spending on bananas in this model). That occurs at Eh = 8.4, H = 84, B = 2.6, and total utility = 10.77. Of course, since everyone is identical, everyone ends up with the same size house and everyone gets zero utility from pride.

This is obviously inefficient. Everyone just spent half of there wealth (4.9 units of effort) on building larger houses in order to get ahead of the Joneses and no one succeeded. Everyone ended up with less utility (10.77 vs 12.5). Everyone could be better off than they are in the competitive equilibrium (by spending 2.5 instead of 8.4 on housing). This contradicts the first fundamental theorem of welfare economics—the heart of neoclassical economics.

Of course, this only shows that the theorem is wrong on planet Earth. 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.