5 ms·

The Sonnenschein-Mantel-Debreu results after thirty years (2006) [pdf]

33 points by gone35 2y ago

troelsSteegin 2y ago
It's an economic theory for characterizing market dynamics. The paper's conclusion: "Thus many of the problematic outcomes from SMD theory remain entrenched."
- 1oooqooq 2y ago
  like every single social science therory dressed as economic research truth, the slight harder look at it crumbles all assumptions. so tiring. best line on that paper: Much of the intuition that would seem to follow from the SMD results is still intact. Rationality, as Arrow said, does not have aggregate implica- tions, but now, we have to add, if only aggregate data are available.
senkora 2y ago
I didn't know anything about this and found the article hard to understand at first, but thankfully the Wikipedia articles on the subject are pretty okay: https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2%80%93Debreu_theorem https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2... The very abbreviated TL;DR of this post is: 1. We are taught that markets look like this: https://upload.wikimedia.org/wikipedia/commons/8/8c/Supply-demand-equilibrium.svg https://upload.wikimedia.org/wikipedia/commons/8/8c/Supply-d... 2. By applying rigorous math ("general equilibrium theory"), it turns out that there is no reason that markets can't look like this instead: https://upload.wikimedia.org/wikipedia/commons/a/ab/SMD_Demand_Curve.svg https://upload.wikimedia.org/wikipedia/commons/a/ab/SMD_Dema... 3. OP's article summarizes how economists have reacted to this problem.
- hackeraccount 2y ago
  For other reasons I'm reading Adam Smith's Wealth of Nations and this seems like nothing that Smith would object to. He'd say there's a natural price that that the market price gravitates around - very much like that second graphic.
  - bckr 2y ago
    The point is there are multiple "natural prices" that the market price may gravitate around. If Adam Smith thinks there's only one "natural price", then he's wrong, and you probably want to read with that idea in mind.
    - deleted 2y ago
      [deleted]
cs702 2y ago
The Sonnenschein-Mantel-Debreu Theorem states that the excess demand curve for a market economy of rational agents can take the shape of any function, subject only to three fairly general conditions: the function must be continuous, must be homogeneous of degree zero (https://en.wikipedia.org/wiki/Homogeneous_function https://en.wikipedia.org/wiki/Homogeneous_function), and must satisfy Walras's law (https://en.wikipedia.org/wiki/Walras%27s_law https://en.wikipedia.org/wiki/Walras%27s_law). Inconveniently for all economic theories, the implication of this mathematical proof is that excess demand and supply curves can take any shape that meets those three conditions. The nicely sloped supply and demand curves we've all been shown in Economics textbooks are basically figments of economists' imaginations. Moreover, the textbook supply-demand curves we've always been shown are for only one good. The curves for all goods in a market economy are high-dimensional. High-dimensional supply and demand curves can have multiple equilibrium points, and there is no guarantee that those points will be optimal or even good. In other words, for a long time we've had proof -- proof! -- that "free markets" are not guaranteed to converge toward good outcomes. Markets can get stuck in crappy equilibria.
- shermantanktop 2y ago
  Ooh, now do the Laffer curve!
  - cs702 2y ago
    I can't. There's basically no theory behind the "Laffer curve." IIRC, it relies on two assumptions. Assumption #1: If the income tax rate is 0%, the government will collect zero income taxes. This assumption is basically a tautology. No one can disagree with it. Assumption #2: If the income tax rate is the maximum possible, 100%, the Laffer curve assumes the government will collect zero income taxes, because taxpayers will find ways to avoid paying all income taxes. This second assumption is questionable, because taxpayers might successfully avoid paying only some taxes. Even if you agree with both assumptions, there's no agreement as to what tax rate would maximize government tax collections.
    - shermantanktop 2y ago
twic 2y ago
It would be really helpful to see an example of one of these naughty excess demand curves, along with the utility function for each agent. For an economy with a small enough number of commodities and agents that it might be comprehensible, say two or three.
kelseyfrog 2y ago
So the nicely sloped supply and demand curves are economic equivalent of spherical cows and as much as we shouldn't design a milk industry around spherical cows, we shouldn't design an economy around a nicely sloped supply and demand curves.
- cs702 2y ago
  For anyone here who doesn't get the reference to spherical cows, it's from an old joke: > Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer, "I have the solution, but it works only in the case of spherical cows in a vacuum." Source: https://en.wikipedia.org/wiki/Spherical_cow https://en.wikipedia.org/wiki/Spherical_cow

33 points by gone35 2y ago

It's an economic theory for characterizing market dynamics. The paper's conclusion: "Thus many of the problematic outcomes from SMD theory remain entrenched."

1oooqooq 2y ago
like every single social science therory dressed as economic research truth, the slight harder look at it crumbles all assumptions. so tiring. best line on that paper: Much of the intuition that would seem to follow from the SMD results is still intact. Rationality, as Arrow said, does not have aggregate implica- tions, but now, we have to add, if only aggregate data are available.

senkora 2y ago

I didn't know anything about this and found the article hard to understand at first, but thankfully the Wikipedia articles on the subject are pretty okay: https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2%80%93Debreu_theorem https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2... The very abbreviated TL;DR of this post is: 1. We are taught that markets look like this: https://upload.wikimedia.org/wikipedia/commons/8/8c/Supply-demand-equilibrium.svg https://upload.wikimedia.org/wikipedia/commons/8/8c/Supply-d... 2. By applying rigorous math ("general equilibrium theory"), it turns out that there is no reason that markets can't look like this instead: https://upload.wikimedia.org/wikipedia/commons/a/ab/SMD_Demand_Curve.svg https://upload.wikimedia.org/wikipedia/commons/a/ab/SMD_Dema... 3. OP's article summarizes how economists have reacted to this problem.

hackeraccount 2y ago
For other reasons I'm reading Adam Smith's Wealth of Nations and this seems like nothing that Smith would object to. He'd say there's a natural price that that the market price gravitates around - very much like that second graphic.
- bckr 2y ago
  The point is there are multiple "natural prices" that the market price may gravitate around. If Adam Smith thinks there's only one "natural price", then he's wrong, and you probably want to read with that idea in mind.
  - deleted 2y ago
    [deleted]

cs702 2y ago

The Sonnenschein-Mantel-Debreu Theorem states that the excess demand curve for a market economy of rational agents can take the shape of any function, subject only to three fairly general conditions: the function must be continuous, must be homogeneous of degree zero (https://en.wikipedia.org/wiki/Homogeneous_function https://en.wikipedia.org/wiki/Homogeneous_function), and must satisfy Walras's law (https://en.wikipedia.org/wiki/Walras%27s_law https://en.wikipedia.org/wiki/Walras%27s_law). Inconveniently for all economic theories, the implication of this mathematical proof is that excess demand and supply curves can take any shape that meets those three conditions. The nicely sloped supply and demand curves we've all been shown in Economics textbooks are basically figments of economists' imaginations. Moreover, the textbook supply-demand curves we've always been shown are for only one good. The curves for all goods in a market economy are high-dimensional. High-dimensional supply and demand curves can have multiple equilibrium points, and there is no guarantee that those points will be optimal or even good. In other words, for a long time we've had proof -- proof! -- that "free markets" are not guaranteed to converge toward good outcomes. Markets can get stuck in crappy equilibria.

shermantanktop 2y ago
Ooh, now do the Laffer curve!
- cs702 2y ago
  I can't. There's basically no theory behind the "Laffer curve." IIRC, it relies on two assumptions. Assumption #1: If the income tax rate is 0%, the government will collect zero income taxes. This assumption is basically a tautology. No one can disagree with it. Assumption #2: If the income tax rate is the maximum possible, 100%, the Laffer curve assumes the government will collect zero income taxes, because taxpayers will find ways to avoid paying all income taxes. This second assumption is questionable, because taxpayers might successfully avoid paying only some taxes. Even if you agree with both assumptions, there's no agreement as to what tax rate would maximize government tax collections.
  - shermantanktop 2y ago

twic 2y ago

It would be really helpful to see an example of one of these naughty excess demand curves, along with the utility function for each agent. For an economy with a small enough number of commodities and agents that it might be comprehensible, say two or three.

kelseyfrog 2y ago

So the nicely sloped supply and demand curves are economic equivalent of spherical cows and as much as we shouldn't design a milk industry around spherical cows, we shouldn't design an economy around a nicely sloped supply and demand curves.

cs702 2y ago
For anyone here who doesn't get the reference to spherical cows, it's from an old joke: > Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer, "I have the solution, but it works only in the case of spherical cows in a vacuum." Source: https://en.wikipedia.org/wiki/Spherical_cow https://en.wikipedia.org/wiki/Spherical_cow