Does the Sonnenschein-Mantel-Debreu theorem fundamentally undermine Mises' Economic Calculation Argument?

Question

So recently I have been thinking a lot about this fundamental question: Does the Sonnenschein-Mantel-Debreu theorem disprove the "Law of Demand"?

Basically, the Sonnenschein-Mantel-Debreu theorem (SMD from here on out) states that you cannot simply horizontally adding up demand curves and get a downward sloping demand curve, as income effects & consumer preferences get in the way. Basically, what that means is that the demand curve could take the form of any polynomial function. The result of this seems to indicate that the "Law of Demand" cannot hold for markets as a whole all the time, and fundamentally undermines the General Equilibrium Theory. At least that is the argument presented by Steve Keen.

I got some responses to that claim, namely that we have observed downward sloping demand curves and that we recognize that you cannot necessarily aggregate all consumer demand curves to create a downward sloping demand curve. This is just an assumption used cause it makes the math easier, much like how a physicist uses 0 friction and no air resistance as assumptions even though that isn't technically true.

I replied that the only reason that you can do this is because those effects are near 0. And that we cannot necessarily say that is the case for aggregate demand functions.

Then a lot of links were provided to very complicated technical papers I don't have the background to understand which claimed to show empirical evidence of a downward sloping demand curve. Like I said, I don't have the background to understand all those papers, so I will take it on the commenter's word that it shows that there is a downward sloping demand curve. If anyone has an easier paper/solid demonstration I would love to take a look (I'm not a professional economist, i'm literally just a guy on the internet trying to get a better grasp of real economics).

I was also told Steve Keen was a hack and that I shouldn't read his stuff. Maybe that is true, idk the guy so I can't say. But if he is a hack then there must be flaws in what he is saying right? If something is true then i must neccesairly be able to answer criticism, so we shouldn't shy away from reading criticism. It's why I have been reading Marx, and it's why I have been reading Keen.

And that's basically where I left that argument

Now, what I want to examine is a more fundamental question that really drives at the heart of why I was asking about demand functions so much: does the lack of a downward sloping demand curve mean that markets don't tend to produce socially optimal quantities? And does this undermine the ECP?

The Economic calculation problem (https://en.wikipedia.org/wiki/Economic_calculation_problem). Basically, the idea that price conveys demand/production information and that markets are effectively a way of communicating this information for production. This provides a pretty neat and tidy explanation of why central planning has so often failed and resulted in shortages and surpluses. Now, there are a lot of reasons central planning failed, and it did fail, but I think the largest is probably the ECP.

But that was all based on the understanding of supply and demand, and how those price signals push towards equilibrium with socially optimal amounts.
But, if, as the SMD seems to indicate, demand curves can really be any shape, and supply curves don't have to be upwards sloping (as Keen illustrates) then we can effectively have multiple different equilibriums and there is no gurantee we reach the socially optimal one. That kinda fundamentally undermines the ECP no?

So ultimately, if we accept the SMD and accept that demand curves, in aggregate, can be any shape, does that mean that markets themselves are inefficient, or don't produce the socially optimal amount and if that is the case I'd there a way to push them to the "right equilibrium ", i.e. the socially optimal amount?

Or is the ECP more fundamental? Does it even need downward sloping demand?

Answer 1

1. So recently I have been thinking a lot about this fundamental question: Does the Sonnenschein-Mantel-Debreu theorem disprove the "Law of Demand"?

It contradicts law of demand as a general law but it is worth noting that the law of demand is for over a century not considered actual general law but just a special case that is simply applicable to most cases. In fact this is nowadays plainly stated even in 101 economic textbooks (see Mankiw Principles of Economics 5th ed pp 471), which mentions that not all demand curves are downward sloping.

Economists were even aware of the Giffen goods (goods for which demand can be upward sloping) for over a century. In fact if Mises ever read Marshal's Principles of Economics (published in 1890) he would be aware of Giffen goods as well, and given popularity of Marshal's principles I don't think there was any economist living around that time who did not read it. This being said such goods are extremely rare but they can exist.

2. I was also told Steve Keen was a hack and that I shouldn't read his stuff.

Steve Keen was shown not to even understand basic high-school/bachelor level mathematics (see this criticism of Keen or this old SE answer as well).

He also often makes false claims about economics that are borderline lies (e.g. see this answer about his claims on Nordhaus work).

Even if you are into heterodox economics (i.e. economics most economists at top universities think is wrong), reading Keen is likely a waste of the time if you want to genuinely pursue the subject.

If you want to casually study economics I would recommend to take any 101 Econ textbook, to read just the text without going deep into oversimplified models that are there just for didactic reasons.

does the lack of a downward sloping demand curve mean that markets don't tend to produce socially optimal quantities?

Not by itself a market with upward sloping demand can still produce socially optimal quantity at socially optimal price. Also note downward sloping demand curve does not guarantee that quantity and price produced will be socially optimal either.

Whether quantity and price is socially optimal or not has little to do with shape of demand curve. It depends more broadly on whether markets are competitive, whether there are market failures such as externalities, public goods etc (see discussion of this in Mankiw Principles of Economics 5th ed pp 203-238). Furthermore, it depends on the choice of social welfare function, e.g. Rawlsian, utilitarian etc (see Mas-Colell et al Microeconomic theory pp 825).

I am personally not aware of any economist that would claim that downward shape of demand curve is what leads to market efficiency.

3. And does this undermine the ECP?

No this in itself does not. Economic calculation problem was never about whether markets are always efficient or optimal. Economic calculation problem was about problem of making economic calculation under socialism.

Market economy might not be 100% efficient, but that has no bearing on whether you can even do economic calculation under socialism or not. Mises and Hayek argument was that without genuine market prices it is impossible to make economic calculation.

The argument, at least the later developed version was not that market prices are sufficient for rational economic calculation, rather the argument was that prices are necessary for rational economic calculation (Hayek 1935). Moreover, Hayek even criticized the supply-demand equilibrium model to begin with (if I remember my University lectures on history of economic thought correctly he was in favor of agent based modeling).

For example, Hayek (1935) explicitly argued (emphasis mine):

Of course, these adjustments [to changes in market condition] are probably never “perfect” in the sense in which the economist conceives of them in his equilibrium analysis. But I fear that our theoretical habits of approaching the problem with the assumption of more or less perfect knowledge on the part of almost everyone has made us somewhat blind to the true function of the price mechanism and led us to apply rather misleading standards in judging its efficiency. The marvel is that in a case like that of a scarcity of one raw material, without an order being issued, without more than perhaps a handful of people knowing the cause, tens of thousands of people whose identity could not be ascertained by months of investigation, are made to use the material or its products more sparingly; i.e., they move in the right direction. This is enough of a marvel even if, in a constantly changing world, not all will hit it off so perfectly that their profit rates will always be maintained at the same constant or “normal” level.

...

Professor Schumpeter is, I believe, also the original author of the myth that Pareto and Barone have “solved” the problem of socialist calculation. What they, and many others, did was merely to state the conditions which a rational allocation of resources would have to satisfy and to point out that these were essentially the same as the conditions of equilibrium of a competitive market. This is something altogether different from knowing how the allocation of resources satisfying these conditions can be found in practice. Pareto himself (from whom Barone has taken practically everything he has to say), far from claiming to have solved the practical problem, in fact explicitly denies that it can be solved without the help of the market.

4. But, if, as the SMD seems to indicate, demand curves can really be any shape, and supply curves don't have to be upwards sloping (as Keen illustrates) then we can effectively have multiple different equilibriums and there is no gurantee we reach the socially optimal one. That kinda fundamentally undermines the ECP no?

As already mentioned before Keen was shown not to understand basic calculus so I would not take any proof of his at a face value.

This being said there could be multiple equilibria in an economy even with all demands being downward sloping and all supplies upward sloping. Downward sloping demand and upward sloping supply does not guarantee existence of unique equilibria in complex general equilibrium models (see Wickens Macroeconomic theory).

This however does not contradict the problem of economic calculation, because problem of economic calculation is literally about problem of doing economic calculations under socialism (economic calculation problem gets its name from Mises' work Economic Calculation in the Socialist Commonwealth, the economic calculation here refers to that particular economic calculation not to economic calculation in any system, even thought the authors did often compare it to market economy). Sure you can apply the corollaries about price mechanism to argue in favor of market economies, and I am aware that both Mises and Hayek did that but it is not the main focus of an argument. Basically the problem of economic calculation boils down to asking how can one perform economic calculation under socialism, and Hayek and Mises answer was that it was impossible.

Furthermore, the problem of socialist calculation is about socialism not being able to produce the most efficient quantity, rather that rational economic decision making under socialism was not possible (although in a simple models rational economic decision making certainly leads to efficient outcome this is not universally true as discussed previously in this answer).

PS

I put this here because you did not asked about but I think it is relevant.

If anything supply-demand analysis would be solution for the economic calculation problem. In fact as mentioned previously demand and supply analysis was argued to be one of the solution in the socialist calculation debate.

If there is some nice monotonic demand and supply that is fully known to a central planner then economic calculation is trivial. Just equate supply and demand calculate the optimum price and quantity and send commands to the comrades working in factories.

What even more, the more 'exotic' (i.e. not well behaved/defined) supply and demand is the harder it is to estimate what supply and demand is based on observable data. Hence if anything existence of multiple equilibria or some exotic demand or supply curve makes the problem of economic calculation under socialism even worse.

Answer 2

First, the Sonnenschein-Mantel-Debreu theorem has nothing to do with demand functions. It is a result about excess-demand functions, which represent demand minus supply. They are formulated in the context of an exchange economy in which no production happens. The result says that if one only looks at prices that are not too close to zero, any continuous function $f$ from prices to quantities that is homogenous of degree zero and such that $f(p)\cdot p=0$ (Walras' law) is the excess demand function of a group of at least as many gents as there are commodities, all having "well-behaved preferences." If one could observe endowments of agents, one could recover their demand from their excess demand and aggregate demand from aggregate excess demand. The main problem is that demand cannot be negative and this puts more structure on the possibility to disaggregate demand. Nevertheless, there exists a result that locally almost nothing can be said about aggregate demand by [Chiappori, Pierre-André, and Ivar Ekeland. "Aggregation and market demand: an exterior differential calculus viewpoint." Econometrica 67.6 (1999): 1435-1457.]

Of course, Keen would have actually known the difference between aggregating demand and excess demand if he actually read the 1982 survey "Market demand and excess demand functions" by Shafer and Sonnenschein. Or if he read the title, for that matter. There he could also read that what he calls the Mantel-Sonnenschein-Debreu conditions are due to Antonelli (in 1886!) and Gorman. He also clear does not understand the local formulation of Gorman.

That aside, excess demand functions do matter. First, equilibria are the zeros of excess demand functions and thus having little structure on excess demand functions means little can be said about the equilibrium set.

More importantly, it raises big problems for dynamics and this is where one might find a connection to the socialist calculation debate. At equilibrium prices, excess demand is zero. This does not tell us how equilibrium prices come to be, the problem of finding equilibrium prices. One approach were so-called tatonnement dynamics. Here, one considered prices dynamically announced and adjusted until excess demand is zero and trades are finalized. The idea is to increase prices for goods for which there is excess demand and decrease prices of goods for which there is excess supply. One can do this by making price changes proportional to excess demand. If everything is sufficiently smooth, this even gives us a nice ordinary differential equation. Sadly, the Mantel-Sonnenschein-Debreu theorem implies that there are economies for which this dynamical process need not converge. This was already known from a 1960 example of Herbert Scarf, but the the Mantel-Sonnenschein-Debreu theorem kind of killed the quest for finding nice out-of-equilibrium dynamics tat converge to an equilibrium. If we have no good idea about how markets find equilibrium prices, there is not much reason to believe they solve the involved calculation problem. There is, however, a lot of work giving conditions for economies to allow for such dynamics to work and these are actually related to the law of demand. One approach, due to Werner Hildenbrand, derives the law of demand in the aggregate from the statistical assumption that demand is more dispersed the higher income is- rich people have more ways to show eccentricity in demand.

Lastly, it should be mentioned that Mises lost the socialist calculation debate on intellectual grounds. Oskar Lange and Abba Lerner, two "market socialists", pointed out that a socialist government could use markets too. Most economists will probably blame incentive problems more for the historical problems of planned communist economies than the problem of finding the right prices.