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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 o

• 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
      In other words, nobody can draw the curve, but people assume it is a simple shape with a single maximum…which is why I made my silly request. But if it were anything other than a talking point, I’d expect the possibility of complex Laffer curves with multiple local maxima.
• bjourne 2y ago
  Don't you mean homogeneous of degree one?
  • cs702 2y ago
    Sorry, I wrote the equation for homogeneity of degree one by mistake. I edited my post to the link to the Wikipedia page on homogeneity instead. Thanks.
• mturmon 2y ago
  Not a disagreement with your take, which was helpful to me, but just your parenthetical aside. Homogenous of degree 0, which is key to the conditions of the results in OP, means f(tx) = t^0 f(x) = f(x) [https://en.wikipedia.org/wiki/Homogeneous_function https://en.wikipedia.org/wiki/Homogeneous_function]
  • cs702 2y ago
    Thank you. I inadvertently wrote the equation for degree one homogeneity, from memory. I replaced the equation with a link to the Wikipedia page.
• emmelaich 2y ago
  Surely it's one thing to say that "curves can take any shape that meets those three conditions" and yet still hold that any observed or realistic actual curves will only take certain shapes - as long as the consumers and producers have the right information.
  • kibwen 2y ago
    > as long as the consumers and producers have the right information Honest question: what is "the right information"? Unless you want someone choosing what is and is not the right information, which feels like market distortion, then "the right information" becomes "all possible information". And in practice, we as consumers don't even have a thousandth of "all the information", let alone a hundredth, let alone half. (For example, where can I look up the details of the entire supply chain of this phone in my hand, for every single material input it requires, to know what degree I'm culpable in supporting slave labor, which is something that would influence my purchasing decision?) And then even if we gave someone all the information, or even some useful approximation of all the information, are humans even capable of using that to make decisions that are informed enough to allow markets to approach any sort of rationality? Given finite time, isn't it possible that the amount of time required to make a single informed decision is longer than a lifespan? So then the question is, if useful amounts of information are neither practically available nor practically actionable, what might that imply about our absolute faith in market mechanisms that require the right information to operate efficiently?
    • emmelaich 2y ago
      By "right" I mean roughly correct. To your other points I suspect that close to correct is often enough and is more correct over time. But this is where research should be directed I suppose.
    • btilly 2y ago
      I can make it worse. We make decisions based on partial information, processed in a probabilistic manner. The logically right way to reason involves the update of a Bayesian network - each belief ties to each other belief. But updating a Bayesian network - even approximately, is an NP complete problem. So we not only can't make a correct decision, we can't even try to approximate it in reasonable time. We can only apply crappy heuristics. Like, "Are you with me or against me?" Hence political alignment replaces our decision making. Even for apparently pure economic decisions.
• Etheryte 2y ago
  I think it's a pretty crude oversimplification to say it's a proof when the whole idea hinges on the whole rational actors axiom. Sure, it might be a proof with that axiom, but it doesn't say much anything about the real world.
  • cs702 2y ago
    I'm not sure I understand what you're saying. The whole edifice of "markets converge toward general equilibrium" rests on the assumption that people are rational, i.e., maximize utility -- which is economist jargon for "people are trying to do better for themselves." If you take away that, the whole theoretical edifice justifying markets collapses.
• cratermoon 2y ago
  Another inconvenient truth for the theorem is that "a market economy of rational agents" is a mathematical fantasy and doesn't describe any real world market ever.
• Veserv 2y ago
  It has been proven that global utility optimization can be equivalent to solving a arbitrarily complex, arbitrarily non-linear, arbitrarily quickly changing, function. Great. There exists exactly zero techniques for solving all such systems even if the function was not unknown. That is proof that "free markets" are not guaranteed to converge toward good outcomes because it proves that every decision/optimization process that is believed to be even theoretically possible is not guaranteed to converge toward good outcomes. Singling out "free markets" as if they are even somewhat uniquely susceptible is just plain ridiculous. Everything is thoroughly screwed in the presence of arbitrarily complex functions. As we have now seen and proven that everything is doomed in a arbitrarily malicious universe, it is only prudent to re-focus on problems under normal conditions and figure out the practical question of what systems produce good, high quality outcomes in a normal implementation over normal conditions.
  • cs702 2y ago
    As I wrote elsewhere on this thread (https://news.ycombinator.com/item?id=42783130 https://news.ycombinator.com/item?id=42783130), the problem is that a lot of busy people treat the "practical notion" that markets produce good, high-quality outcomes as if it were a fundamental law of the universe. It's not. There are no guarantees. Markets can converge to horrific equilibria. Let's be humble, open-minded, willing to rely on markets when they work, willing to try other things when they don't.
    • kelseyfrog 2y ago
      This fundamentally changes the debate on market interventions. It's now possible to argue that the current equilibrium is only meta-stable and that interventions that change the supply and demand curves can potentially kick us out of one meta-stable equilibrium and into another more efficient one. I wish we had more empirical supply-demand curves so we could refine our priors on supply and demand curve shapes, but that's a topic for another day.
  • cherryteastain 2y ago
    Free market economics is "singled out" because, at least in the Anglosphere, it is the only type of economy considred to reach an (eventual) optimum by itself.
• vtashkov 2y ago
  For even longer time we have a REAL proof (not mathematical wordplay) that whenever, without exception, someone takes a swing at free markets it leads to people dying. IT has always been a disaster and people always have been exiting those societies where that happens towards more free societies. You yourself have plenty of options to go somewhere with less free market - Cuba, Venezuela, North Korea, Russia, even China. Have YOU emigrated to such a country?
  • lern_too_spel 2y ago
    You also have the option to go somewhere with even more of a free market, like Somalia. Have you emigrated to such a country? It turns out that societies can write laws for mutual benefit.
    • _uxvx 2y ago
      If you believe that free market means no laws you have not enough knowledge to participate in this conversation. Somalia is not a free market, it is a lack of state where you are not free at all, but the opposite. It is not that different though than what happens in socialist countries. Venezuela used to be very rich and advanced and now it has more or less the same situation as in Somalia.
      • lern_too_spel 2y ago
        Somalia has a free market in security, water infrastructure, transportation infrastructure, communication infrastructure, power infrastructure, schooling, healthcare, etc. instead of allowing a state to interfere in all of these things. That is the definition of a free market. There is also no border security for free movement of goods and people. You are free to move there and enjoy all of its advantages.
  • cs702 2y ago
    Oh, I'm a capitalist. I believe Churchill was right when he said that capitalism is the worst economic system ever invented, except for all the others that have been tried before. I never said we should do away with a market economy. All I said is that market economies are not guaranteed to converge to optimal or even good outcomes. Markets CAN lead to horrifically bad equilibria (extreme inequality, financial crises, recessions, depressions, etc.). There are no guarantees that they won't. So let's be humble, open-minded, willing to rely on markets when they work, willing to try other things when they don't. Please don't attack a straw-man.