...theorem that says that the optimal solution of a complex resource allocation problem is isomorphic to a system where things have prices.
This is true, if we interpret “optimal solution” as the set of Pareto efficient allocations and make some very general assumptions about monotonic utility and demand independence![1] The idea of the proof is to take the dual of the resource allocation problem to turn it into a resource valuation problem, and by strong duality theorem these have the same optima (the solution to the dual of a resource allocation problem is a vector of prices). This is an A-D model, so it is known that markets will clear at these prices. By Welfare Thm I, we know that markets that clear result in a Pareto efficient outcome, and by Welfare Thm II, we know that any Pareto efficient outcome can be supported by some set of prices produced by this process. Any process optimizing an allocation of scarce resources is isomorphic to a set of utility-maximizing agents exchanging those resources in a free market. Pretty cool!
I’m pretty sure they need to be at least locally true to guarantee what I said. Are there specific scenarios you’re imagining?
For what it’s worth, in most cases you’ll probably be able to reframe the resource allocation problem so that these assumptions hold. For example, you can account for negative utility goods by just taking the complement and giving it a positive price. If I want to account for agents having a “negative” price for pollution, you can instead model how much agents value clean air and give it a positive price to satisfy monotonic utility.
One of the scenarios I’m imagining is a scenario where network effects exist, such that you don’t want to have something if you are the only person having it, but you do want it if others have it.
Arguably, a lot of public goods/social media are like this, where there’s 0 demand at a limited size, but have lots of demand when the size starts increasing beyond a threshold.
In essence, I’m asking if we can remove the demand independence assumption and still get an isomorphism between optimal solutions to scarce resources and a system having prices.
Ah, well if there are interdependent demand functions then there is no guarantee of a (general) equilibrium, so the starting resource allocation problem may have 0 or many solutions. So I would say no, the isomorphism doesn’t hold in that case.
Hm, I think that’s more of a supply independence thing, what economists would call “non-excludable”. If the government funds a police force, it’s not as if they protect some citizens but not others. But that’s not a violation of the demand independence assumption because I care about living in a country with a strong police force regardless of whether you want that or not.
Goods with demand independence, from Ferraris to Facebook, generally do get provided by markets in real life, they just don’t have stable prices. It breaks some of the equilibrium models because it can cause divergences or circularity in your demand function, and then there’s no fixed point in positive price/demand space.
Luxury is a good example of this that happens in real life. Here’s an intuition-prompting setup:
Suppose I’m rich and I buy a Gucci bag
You’re poor, but you want to look rich so you also buy a Gucci handbag
Now I don’t think the bag is as exclusive, so I don’t want mine any more
Now that the rich guy isn’t wearing it anymore, you don’t want yours either
But now no one has it, so it seems exclusive again, so now I want it
Repeat
This doesn’t mean markets won’t provide Gucci bags (obviously, they do), but there isn’t a price equilibrium, it will fluctuate forever. In terms of the original point, the Gucci bag allocation problem isn’t isomorphic to a market equilibrium, because there is no such equilibrium.
Sure, I think social media is probably the best example of this. Suppose there are two platforms, A and B, and social media sites are worth more when more people are on it. Our “resource allocation” problem is to maximize utility, so we want to get everyone on the same site. There are two equilibria here; we can either set the price for A much higher than B and everyone will move to B, or vice versa.
If the demand functions weren’t interdependent and every agent just got some amount of utility from A and some amount of utility from B, there would be exactly one equilibrium price.
This is true, if we interpret “optimal solution” as the set of Pareto efficient allocations and make some very general assumptions about monotonic utility and demand independence![1] The idea of the proof is to take the dual of the resource allocation problem to turn it into a resource valuation problem, and by strong duality theorem these have the same optima (the solution to the dual of a resource allocation problem is a vector of prices). This is an A-D model, so it is known that markets will clear at these prices. By Welfare Thm I, we know that markets that clear result in a Pareto efficient outcome, and by Welfare Thm II, we know that any Pareto efficient outcome can be supported by some set of prices produced by this process. Any process optimizing an allocation of scarce resources is isomorphic to a set of utility-maximizing agents exchanging those resources in a free market. Pretty cool!
Agents weakly prefer more of a good to less of it, and agents’ demand for a good doesn’t depend on other people having it.
Can we remove one of the assumptions, or are both assumptions necessary to get the result stated?
I’m pretty sure they need to be at least locally true to guarantee what I said. Are there specific scenarios you’re imagining?
For what it’s worth, in most cases you’ll probably be able to reframe the resource allocation problem so that these assumptions hold. For example, you can account for negative utility goods by just taking the complement and giving it a positive price. If I want to account for agents having a “negative” price for pollution, you can instead model how much agents value clean air and give it a positive price to satisfy monotonic utility.
One of the scenarios I’m imagining is a scenario where network effects exist, such that you don’t want to have something if you are the only person having it, but you do want it if others have it.
Arguably, a lot of public goods/social media are like this, where there’s 0 demand at a limited size, but have lots of demand when the size starts increasing beyond a threshold.
In essence, I’m asking if we can remove the demand independence assumption and still get an isomorphism between optimal solutions to scarce resources and a system having prices.
Ah, well if there are interdependent demand functions then there is no guarantee of a (general) equilibrium, so the starting resource allocation problem may have 0 or many solutions. So I would say no, the isomorphism doesn’t hold in that case.
And this is why some goods, like public safety can’t be provided by markets, because the assumption of demand independence is violated.
Hm, I think that’s more of a supply independence thing, what economists would call “non-excludable”. If the government funds a police force, it’s not as if they protect some citizens but not others. But that’s not a violation of the demand independence assumption because I care about living in a country with a strong police force regardless of whether you want that or not.
Goods with demand independence, from Ferraris to Facebook, generally do get provided by markets in real life, they just don’t have stable prices. It breaks some of the equilibrium models because it can cause divergences or circularity in your demand function, and then there’s no fixed point in positive price/demand space.
Luxury is a good example of this that happens in real life. Here’s an intuition-prompting setup:
Suppose I’m rich and I buy a Gucci bag
You’re poor, but you want to look rich so you also buy a Gucci handbag
Now I don’t think the bag is as exclusive, so I don’t want mine any more
Now that the rich guy isn’t wearing it anymore, you don’t want yours either
But now no one has it, so it seems exclusive again, so now I want it
Repeat
This doesn’t mean markets won’t provide Gucci bags (obviously, they do), but there isn’t a price equilibrium, it will fluctuate forever. In terms of the original point, the Gucci bag allocation problem isn’t isomorphic to a market equilibrium, because there is no such equilibrium.
Any examples of where the starting resource allocation has more than 1 solution/equilibrium, assuming demand independence is violated?
Sure, I think social media is probably the best example of this. Suppose there are two platforms, A and B, and social media sites are worth more when more people are on it. Our “resource allocation” problem is to maximize utility, so we want to get everyone on the same site. There are two equilibria here; we can either set the price for A much higher than B and everyone will move to B, or vice versa.
If the demand functions weren’t interdependent and every agent just got some amount of utility from A and some amount of utility from B, there would be exactly one equilibrium price.