>The D in SMD was actually awarded the nobel prize in economics.
Debreu's work on general equilibrium, which was the basis of the award, had to skirt around the SMD theorem:
>Prize motivation: "for having incorporated new analytical methods into economic theory and for his rigorous reformulation of the theory of general equilibrium."[0]
In order for the Arrow-Debreu model[1] to give a unique equilibrium, which is usually desired, some strong (arguably, unrealistic) conditions must be assumed, otherwise there can be multiple equilibria, as guaranteed by the SMD theorem.[2]
In particular, the SMD theorem asserts that any polynomial function can be a market demand function, which fundamentally breaks the so-called "law" of demand in neoclassical economics. This is because most polynomial functions are not monotonically decreasing[3], as required by the "law" of demand, but give rise to demand curves that can curve up or down arbitrarily. As an example of such a function, just think of the curve generated by a generic cubic polynomial.[4]
This "anything goes" situation then leads to non-unique equilibria, since a supply curve can now intersect the market demand curve at more than one point. But since an equilibrium point is supposed to be where welfare is maximised in a society, the existence of non-unique equilibria means that there can be multiple economic arrangements under which social welfare is maximised. Then it's a matter of taste (i.e. "ideology") to decide which equilibrium point a society wants to be in.
> Debreu's work on general equilibrium, which was the basis of the award, had to skirt around the SMD theorem:
Watch out for moving the goalposts. He was awarded the nobel prize after all. "Rigorous reformulation" does not sound like skirting at all.
> This "anything goes" situation then leads to non-unique equilibria, since a supply curve can now intersect the market demand curve at more than one point. But since an equilibrium point is supposed to be where welfare is maximised in a society, the existence of non-unique equilibria means that there can be multiple economic arrangements under which social welfare is maximised.
Economy (or climate or orbits) exhibit chatoic and complex behaviour and it is not surprising that they can have multiple stable points (look up bifurcation diagrams for a simple representation of chatoic solutions). If I am not mistaken, one can hear such things routinely in university economy classrooms today especially when giving caveats about models being wrong in general. Even when you include game theory and Nash equilibria with respect to economic solutions, there might not be efficient ways to reach the equilibrium points at all!
> Then it's a matter of taste (i.e. "ideology") to decide which equilibrium point a society wants to be in.
That is to assume that every equilibrium solution you get is tied to a specific policy or a set of them (or even more, an ideology!) or that points can be reached easily which are very strong assumptions. Maybe some ideologies don't exhibit any reachable equilibrium points or pertain only to unstable ones.
"Then it's a matter of taste (i.e. "ideology") to decide which equilibrium point a society wants to be in."
I think this statement is quite representative of the world we live in, don't you?
My personal experience is, strongest adherents to an ideology are the least productive people .
morningseagulls|6 years ago
Debreu's work on general equilibrium, which was the basis of the award, had to skirt around the SMD theorem:
>Prize motivation: "for having incorporated new analytical methods into economic theory and for his rigorous reformulation of the theory of general equilibrium."[0]
In order for the Arrow-Debreu model[1] to give a unique equilibrium, which is usually desired, some strong (arguably, unrealistic) conditions must be assumed, otherwise there can be multiple equilibria, as guaranteed by the SMD theorem.[2]
In particular, the SMD theorem asserts that any polynomial function can be a market demand function, which fundamentally breaks the so-called "law" of demand in neoclassical economics. This is because most polynomial functions are not monotonically decreasing[3], as required by the "law" of demand, but give rise to demand curves that can curve up or down arbitrarily. As an example of such a function, just think of the curve generated by a generic cubic polynomial.[4]
This "anything goes" situation then leads to non-unique equilibria, since a supply curve can now intersect the market demand curve at more than one point. But since an equilibrium point is supposed to be where welfare is maximised in a society, the existence of non-unique equilibria means that there can be multiple economic arrangements under which social welfare is maximised. Then it's a matter of taste (i.e. "ideology") to decide which equilibrium point a society wants to be in.
[0] https://www.nobelprize.org/prizes/economic-sciences/1983/deb...
[1] https://en.wikipedia.org/wiki/Arrow%E2%80%93Debreu_model
[2] https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2...
[3] https://en.wikipedia.org/wiki/Monotonic_function
[4] A graph of such a curve is shown here: https://en.wikipedia.org/wiki/File:Monotonicity_example3.png
vkreso|6 years ago
Watch out for moving the goalposts. He was awarded the nobel prize after all. "Rigorous reformulation" does not sound like skirting at all.
> This "anything goes" situation then leads to non-unique equilibria, since a supply curve can now intersect the market demand curve at more than one point. But since an equilibrium point is supposed to be where welfare is maximised in a society, the existence of non-unique equilibria means that there can be multiple economic arrangements under which social welfare is maximised.
Economy (or climate or orbits) exhibit chatoic and complex behaviour and it is not surprising that they can have multiple stable points (look up bifurcation diagrams for a simple representation of chatoic solutions). If I am not mistaken, one can hear such things routinely in university economy classrooms today especially when giving caveats about models being wrong in general. Even when you include game theory and Nash equilibria with respect to economic solutions, there might not be efficient ways to reach the equilibrium points at all!
> Then it's a matter of taste (i.e. "ideology") to decide which equilibrium point a society wants to be in.
That is to assume that every equilibrium solution you get is tied to a specific policy or a set of them (or even more, an ideology!) or that points can be reached easily which are very strong assumptions. Maybe some ideologies don't exhibit any reachable equilibrium points or pertain only to unstable ones.
ClumsyPilot|6 years ago
I think this statement is quite representative of the world we live in, don't you? My personal experience is, strongest adherents to an ideology are the least productive people .