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Walrasian discontinuity -> monetary solution -> monetary disequilibrium

Hayek made this point about the impossibility of predicting the growth of knowledge independently of and prior to his friend Karl Poppper -- and used the point in arguments about social coordination and the advance of knowledge. Whether Popper took idea from Hayek -- who greatly influenced him -- I can't say.

Nick wrote,

" It's a bit like Karl Popper on predicting inventions. If we could predict inventions, we could already have invented the inventions we predicted."

Excellent.

One point.

People don't calculate "Walras" like equations.

They judge relative profits and loses -- these are the signals that guide there adjustments.

See Hayek, _Monetary Theory and the Trade Cycle_, 1929/1933.

"But it does matter if real people are solving the equations in real time. Because the economy we see is the economy created by their repeated attempts to solve those equations."

Beautiful!

But I would separate out the issues of time from the issues of barter. When you add factor markets and production to the CE framework, then you are saying that production also occurs out of time, and a lot of people have problems with that.

However, you can construct barter models in which production takes time -- i.e. without a walrassian auctioneer operating in meta-time -- and you can construct monetary models with a walrassian auctioneer. So the two issues are separate.

Of the two issues -- timelessness and barter -- I have no problem at all with barter, but do have problems with timelessness.

Barter just means that anything is accepted as payment, because anything can be exchanged for anything else.

And in particular, if I write an IOU to deliver 3 pounds of corn next year, and I want 2 pounds of bread today. The bread seller, even though he doesn't want corn, and may not even want to defer consumption of output, will nevertheless accept my IOU as payment because delivery of future corn has value today. The bread seller can trade my IOU for his present consumption just as I sold the IOU for my present consumption. Then the person who bought the IOU can also save some of it (if he wants) and will turn around exchange the remainder of my IOU for his present consumption.

During this process of exchange, relative price of goods in terms of the IOU increases, and demand increases, up until the rest of the economy does want to save (the equivalent of) 3 pounds of corn, at which point the IOU will be distributed among the rest of the population as their savings.

This, in a nutshell, is the endogenous money view.

It works precisely because anything can be exchanged for anything else, and therefore third party IOUs can be used as a means of payment just like pieces of bread. IMO, credit money is fundamentally a barter view. So for me, barter = OK, but timelessness is not OK.

If you assume production of output requires time and is therefore slow to adjust, whereas the production of IOUs is costless and can adjust instantly, then the "exchange rate" of produced goods and non-produced goods can fluctuate; As the rate of production of IOUs falls more rapidly than the rate of production of output, you will get a general glut of newly produced goods, in the sense that is harder to trade all newly produced output in exchange for the IOU, and it is easier to buy all newly produced output with IOUs.

That would be a general glut in the barter+time model, and you can build a case that this will cause production to decrease, just as the increase of IOUs relative to goods will cause production to increase.

So there is an alternative view that credit is what is key, not money, and this view also rejects the notion of meta-time auctions, and this view can also account for general gluts.

The adjustment process is a learning and judgment process, using price and profit signals, which cannot be mathematically modeled. This is the causal explanatory element in economics.

FYI, delong has a post:

"From my perspective, the technocratic economists by 1829 had figured out why these semi-periodic grand mal seizures happened [...]
Semi-periodically in market economies, wealth holders collectively come to the conclusion that their holdings of some kind or kinds of financial assets are too low [...] They thus cut back on their spending on currently-produced goods and services in an attempt to build up their asset holdings. This cutback creates deficient demand not just for one or a few categories of currently-produced goods and services but for pretty much all of them. Businesses seeing slack demand fire workers. And depression results."

And the interesting question would be exactly why the walrassian general equilibrium view covers up or erases this explanation, which is a fairly obvious explanation. But debates about where the walrassian model goes wrong are not the same as debates about where the economy goes wrong.

" It's a bit like Karl Popper on predicting inventions. If we could predict inventions, we could already have invented the inventions we predicted."

Yep and if you can synthetically make a human being in all its detail you have not a robot but a human being. Happy St Valentine.

"Real people take time to solve the equations."

Real people do not solve equations. 90% of humanity cannot navigate two dimensional space. I know, I know neither can pool players.

"There is a point at which all models must stop."

No there is a point at which you have a model and you have reality and you realize that the model is a good enough guide in a well specified range of circumstances and outside of that the model is well a toy. Moreover there are models and there are models.

On Hayek

Hayek's model is loose and poorly specified but intuitively correct. Where he errors is when he argues that the aggregations of information made by markets is superior to aggregations made by a central planner. Both are likely to be wrong.

When I strip Hayek of his ideological preferences I just see a naked man making a fair comment on what the limits of rationality with respect to the future are. But hey I can get that from Orthodox Christian thought too, or Islam or Judaism or Atheism. That is, Hayek knew a thing or two about the existential conditions of existence. Ok I have to go take a shower.

I think Walras belongs to exponential time complexity class. This paper by Robert Axtell, shows that decentralized exchange, is much more computationally efficient. He also provides a non-walras 1st and 2nd welfare theorem.

Also I'm going to bottle this discipline up yet :)

I regret my previous comment, I remember getting smacked around by Barkley Rosser, last year, for bringing up computational complexity.

Hey can you delete my comments? because I also regret the previous comment, I was smacked around for insinuating that Prof Rosser didn't care about computational complexity, which I've done again.

Nick:
The real question is why does the aggregate preference to hold cash balances change. Arnold actually has an answer for that: uncertainty about the future distribution of value and income--even if the aggregate value and income is known. If you accept that, then his model does not fail your test in the slightest; nor do the tools you discuss have the power to disprove the PSST narrative.

The Keynesian story is that the demand gap causes the change in aggregate preferences. The NK story is that expected demand causes the change in aggregate preferences.

The NK version has some truth but in that end that must be secondary phenomena. We've seen too many recessions go by where nominal income has held to trend and people expected the recession to end soon--yes some equivocation about that last point, but fits nicely with the spectrum of rightness:

Keynesian, NK, PSST: Wrong, a bit right, a lot right.

What everyone gets right: these couple into the preference for cash balances, and therefore a general 'glut' follows.

I can't even talk about it being harder to buy things and easier to sell things without talking about money. In a recession it's easier to sell money, and harder to buy money. That sure looks like a monetary phenomenon to me.

Aren't you just playing with words here? Picture a plausible real-life barter economy, which is nothing like the Walrasian model. Some things will be easier to dispose of than others. Cigarettes and boiled sweets will do for small change, precious stones and suchlike will be required for purchasing more valuable items. A recession is easy to visualise as a consequence of a surge in demand for such liquid assets, coupled with a reluctance to cut the prices of produced goods. Of course you can call the demand for rubies a monetary phenomenon, but as I say that's just playing with words.

I like Travis Fast's comment. Very droll. I have no idea what Jon is trying to say.

But I think there is something missing here. The Walrusian auctioneer is misleading - because he is not bound by sunk cost. He is completely free to explore the whole solution space. But I think history matters. And I think, that from a recession, the solution is almost impossible to find. Because the things that people will buy in a recession, are not the same as the things people will buy in good times. It is like being faced with navigating a built street system, but thinking you can bush bash.

P.S.
In general in case people wonder, I'm a forget equilibrium, concentrate on the dynamics here and now man. Especially in this case.

And for Jon, I think you building up straw men to choose between.

I think the change in liquidity is hardly surprising, the general level of uncertainty changes. And this only needs to be short term - if you worry more about whether you will have a job next month, and whether your investments will lose value next month, you are more like to want to be holding enough cash to pay the rent and buy food next month. The long term equilibrium position is not really of pressing concern (and is basically unknowable given my view that the solution is not smooth and moves away from local maxima can, but won't necessarily, mean sudden and wrenching changes).

"nor do the tools you discuss have the power to disprove the PSST narrative"

Isn't this backwards? I read Nick as objecting to the characterization of "standard and Austrian economics" as being in some way dependent upon a Walrasian auctioneer, his point being that you cannot discriminate between PSST and other schools of economics on this basis. In fact, given the relatively weak claims being made here, I don't see that any party is doing anything to exclude the models of any other. After all, Nick says "many shocks hit the economy that have nothing to do with money", and Kling says "slumps can occur because discovery takes time." My emphasis - he does not claim that all slumps have his cause.

anon: you lost me there.

Greg: I got it from Popper. You may well be right that Popper got it from Hayek.

"People don't calculate "Walras" like equations. They judge relative profits and loses -- these are the signals that guide there adjustments."

Agreed. Except I would now quote Hayek back at you, to explain what I meant. Something like "people may do things that are not part of their intentions." "Solving the equations is done by human action, but not by human design". Something like that. Hayek could say it better.

You didn't pick up on my "(I will let the Austrians speak for themselves, but my reading of Austrian economics says that they are among the least likely to accept the Walrasian auctioneer.)"

I thought you would chime in on that. I'm right, amn't I?

RSJ: "However, you can construct barter models in which production takes time -- i.e. without a walrassian auctioneer operating in meta-time -- and you can construct monetary models with a walrassian auctioneer. So the two issues are separate."

Woah!

1. A Walrasian auction is very different from barter, as we normally understand it. In normal barter, each trade is 2 agents and 2 goods. This auction is one big trade with all agents and all goods.

3. You can't really construct a Walrasian model of a monetary economy. At least, not without faking it (Patinkin faked it well, but knew he was faking it). A monetary economy has to have different markets for different goods, and they don't just open once at the beginning of time.

3. Production can still take time in Walras, and happen in real time. The easiest way to think of the Walrasian (or Arrow-Debreu version) is of one big auction at the beginning of time, then the clock starts, and production takes place, but nobody ever buys or sells goods thereafter. All they do is deliver goods that have already been sold in advance, at the beginning of time.

The relation between the equations and what people really do doesn't have simple answers -- and Hayek's view on the matter evolved over time, and was never exactly clear. Basically, in mature Hayek the GE is a template for perceiving overall order in the economy, but it is not an causal explanatory element. I.e. the helps us see the thing which needs explaining, and gives us a sense of the structure within that thing we see, but it does not provide a causal explanation for that order. He sees entrepreneurial learning and adaptive judgment in the context of changing price and profit signals as something different from the solution to equations in an economic model.

And note well, Hayek seems to have come to see that much of the most important structure within the economy is not mathematically tractable in a GE form, e.g. the structure of the coordination of variable time-taking production processes, esp. in the context of constantly changing technologies.

I really can't make sense of what modern Austrians think of the Walrasian thought experiment -- they certainly don't accept it -- but my sense is they depend on it more than they will admit. Some of them seem to want to do economics with only crude choice theory & price theory and without GE constructs. Something like that seems to be what Boettke and Caldwell are talking about (Caldwell might not consider himself an "Austrian").

Nick writes,

"Agreed. Except I would now quote Hayek back at you, to explain what I meant. Something like "people may do things that are not part of their intentions." "Solving the equations is done by human action, but not by human design". Something like that. Hayek could say it better.

You didn't pick up on my "(I will let the Austrians speak for themselves, but my reading of Austrian economics says that they are among the least likely to accept the Walrasian auctioneer.)""

Make that:

The GE helps us see the thing (overall market order) which needs explaining, and gives us a sense of the structure within that thing we see, but it does not provide a causal explanation for that order -- it doesn't include the causal elements that generate that order.

"The easiest way to think of the Walrasian (or Arrow-Debreu version) is of one big auction at the beginning of time, then the clock starts, and production takes place, but nobody ever buys or sells goods thereafter."

So in this version of "time", there are contingencies uncertain at time zero but every contingency is priced and contracted for? I don't just contract to buy X apples at price P at time T, I contract to buy X apples at price P_A if the state of the world at time T is A, and Y at P_B if the state is B? Otherwise, in what material sense would there be time in the model?

I should add that Boettke sometimes seems to accept my argument for the "template" view of the role of GE constructs -- a view which can also be found in Marshall and Mises and Knight.

And he seems at least a bit troubled by the Sonnenschein-Mantel-Debreu results.

To be clear, on Hayek's view, the tautological equations of the GE expose the causal explanatory elements in economics by revealing those things which have been left out by the logical / math construct -- the things which are "screened out" by the template, and reveal themselves by their exclusion -- e.g. learning, judgment, property rights and other institutional rule following, etc.

See Hayek's "Economics and Knowledge" and the opening chapters of _The Pure Theory of Capital_.

To cut off misunderstanding -- a math formula for "learning" isn't an adequate stand in for the actual causal element of real people updating their understandings and judgments in unique local environments using unique price and profit signals. No more than a math formula or math variable which gets labeled "phenotype" or "geneotype" or "fitness" is an adequate stand in for the open-ended and multiply instantiated real world causal pathways in unique biological environments. Indeed, the compounded levels of multi-instantiated and functionally defined elements involved in changes of understanding -- and the local and unique embedding of this, is dimensionally greater than the complexity in biology, and involves the further irreducible element of the rivalry of understandings between different individuals -- i.e. non-identify of understandings and judgments of things between different individuals.

Jon: I interpret you as asking: "what causes the excess demand for money?"

Almost anything. Could be supply or demand. Could be a financial crisis. Could be some real shock. But I interpret Arnold as saying that money, and monetary AD and AS play no important causal role in the story of why recessions happen. They would happen about the same way even if the supply of money was perfectly and instantaneously adjusted to demand.

Kevin: "Of course you can call the demand for rubies a monetary phenomenon, but as I say that's just playing with words."

If people only buy and sell (expensive) goods for rubies, then rubies *are* the medium of exchange. They are money. That's not playing with words. Sure, the ultimate cause of the increased demand for rubies might be for non-monetary uses. But that only matters because it reduces the monetary supply of rubies. Like the "industrial" demand for gold.

Phil: "So in this version of "time", there are contingencies uncertain at time zero but every contingency is priced and contracted for? I don't just contract to buy X apples at price P at time T, I contract to buy X apples at price P_A if the state of the world at time T is A, and Y at P_B if the state is B? Otherwise, in what material sense would there be time in the model?"

Yep. An "Arrow-Debreu commodity" is a promise to deliver apples at time t in state s. All exchange (buying and selling) takes place at time 0 in one big futures market. All the economics takes place at time 0. From then on, it's all just production, consumption, and deliveries.

The problem is that the ex-ante pareto optimality does not guarantee ex-post pareto optimality when some information about the state of the world is revealed.

And that means that each time some information is revealed, people will update their (imperfect) estimated probability distributions of future events occurring, and there will be (potentially) new gains from making additional trades on the second day the market opens.

But here "information" can be anything -- prices, the strategies of others, messages, etc. Each time more information is provided, there will be more trades, so you can never argue that it is "done", and from this point on, it's just production and consumption.

RSJ: "The problem is that the ex-ante pareto optimality does not guarantee ex-post pareto optimality when some information about the state of the world is revealed."

Suppose it didn't guarantee ex post optimality conditional on the arrival of information i at time t. Then, at time 0, why wouldn't agents do a deal conditional on information i arriving at date t? Since this would make both better off ex post, the probability of information i arriving at time t would make both better off ex ante.

Nick,

Each agent has their own (subjective) probability, and the heterogeneity of this causes non-optimality.

http://cowles.econ.yale.edu/P/cp/p03b/p0381.pdf

"If subjective probabilities lack universal similarity for all states of the world, then no matter what state of the world occurs, the resulting distribution will not be an ex-post Pareto optimum. "

"Pareto optimum results not from the accuracy of traders' beliefs, but from their unanimity."

Most of what is important for the entrepreneur doesn't come in the form of boxed "information" with a bow on top

And we can't predict in advance the growth on understanding and knowledge -- we can't make numerical or "boxed information" predictions of what the next invention will be or what will sell and what won't.

This is why study after study shows that entrepreneurs at the cutting edge don't believe much in planning and don't waste much time on business models and business plans. The are too busy adapting and discovering in real time.

Again, what it looks like you are seeking to understand, Nick, is not the real world, what you are spending your time seeking to understand is a math model, valued in large part for its formal mathematical properties, above all else.

Nick wrote,

"Suppose it didn't guarantee ex post optimality conditional on the arrival of information i at time t. Then, at time 0, why wouldn't agents do a deal conditional on information i arriving at date t? Since this would make both better off ex post, the probability of information i arriving at time t would make both better off ex ante."

So, having waded through some impenetrable math, trying to get the intuition of that horrible paper you linked, I come to the bottom of page 89, and I extract from it this intuition:

2 guys make a bet on the roll of a coin. They are both risk averse, but one thinks the coin is biased to heads, and the other thinks the coin is biased to tails, so both are willing to bet.

Just before the coin is rolled, they both get new information: the coin is in fact fair. They both want to call off the bet.

The bet was ex ante optimal but ex post sub-optimal.

What's missing from the paper is this. And it's something I mentioned in my comment above: both have an incentive to do a deal conditional on the information received. The receipt of information that the coin is fair is itself a state.

No more math econ please. I don't like reading stuff like that.

"both have an incentive to do a deal conditional on the information received. The receipt of information that the coin is fair is itself a state."

And that's the point, not only is it a state but under complete markets it has an Arrow-Debreu security to trade so they can insure themselves. With the insurance they'll get back to an ex-post pareto optimal allocation.

The counter-example is not a counter-example at all because it implicitly took away an assumption.

Greg: "Again, what it looks like you are seeking to understand, Nick, is not the real world, what you are spending your time seeking to understand is a math model, valued in large part for its formal mathematical properties, above all else."

It's ironic that our last comments crossed in the post! There I was, wailing about the math in that paper that RSJ sent me to!

I'm one of the last economists alive who can't do math. I've been faking it for the last 35 years, in words and diagrams. Trying to get the intuition of papers whose equations I don't get. Reverse-engineering the model, to try to understand what must be going on.

Adam: we are agreeing, right? Because what I think you are saying is what I was trying to say.

yes, I was backing you up. You're absolutely right.

Must be why I appreciate what you do here.

And note well, this sort of thing is exactly what upended the formalist project in logic, the philosophy of science, and in the foundations of mathematics -- Wittgenstein and Popper and Kuhn were trying to get the intuitions behind the projects of Frege, Russell, Hilbert & Carnap, etc. And the result was the exposure of deep pathology right at the heart of those intuitions.

Mises and Hayek did something _directly_ parallel to what Wittgenstein did -- both Wittgenstein and Hayek showed that there was _NO_ god's eye perspective for interpreting a formal "model" of language / the economy that could make sense of the actual phenomena. Wittgenstein uses the language of "bird's eye view", Hayek uses the language of "synoptic view", but they are talking about the same thing.

(Short version, the socialist calculation problem is directly parallel to the private language argument -- both expose pathologies in the conceit of "given" knowledge or "given" meaning, and the God's eye view interpretations of social phenomena from based on formal "models" more generally).

Nick writes,

"I'm one of the last economists alive who can't do math. I've been faking it for the last 35 years, in words and diagrams. Trying to get the intuition of papers whose equations I don't get. Reverse-engineering the model, to try to understand what must be going on."

NIck, the paper was very general. It did not specify what "information" was in the state of the world. It just used any set, S.

Arguing that if you enlarge S, then ex-post optimality is achieved is false, because the theorem applies to any set S, including your enlarged set.

The only way you are guaranteed optimality is if everyone has unanimity of probability estimates ex-ante, for each of the states occurring, whatever those states happen to be. (And if you add production, the situation becomes worse (or rather the requirements for ex-post efficiency become stricter than just unanimity of preferences. Roughly speaking, the production technologies must be equally efficient independent of which outcome occurs)

So in your toin coss example, the problem arises because A thinks the coin is fair, but B thinks that the coin is biased. It doesn't actually matter whether the coin is biased or not. The divergence of beliefs is what creates the inefficiency.

Adding new states -- "the coin is fair and the coin is heads" and "the coin is biased and the coin is heads" doesn't help, because they will still apply different probability estimates to the states occurring, and so the resulting allocations will not be ex-post pareto optimal, even if they are pareto optimal ex-ante.

RSJ: this is why I dislike math-econ. It forces the reader to try to interpret the results. The author probably doesn't understand the results either. Why didn't Starr just give an example, to show what he meant? Otherwise, it's just a snow-job.

I gave an example, in which two risk-averse agents think the coin is biased in different directions. I showed how it can be resolved. As Adam said, if you allow the two agents to hedge their bets by trading contracts on the state of the world in which information is received, showing whether the coin is fair or biased, they would trade that contract, and resolve the problem. It does not matter if they each have different probabilities of the information revealing the coin to be fair. If you offer both a deal in which they make the opposite, hedging, bet, conditional on the state of the world in which the coin is fair, both will accept that deal.

Give an example.

Here's an example with production: Robinson Crusoe plants wheat, because he thinks it will be a dry Summer. The central planner knows it will be a wet Summer, and so Robinson Crusoe should plant barley. Ex post Robinson Crusoe's choice was inefficient. Therefore the market fails.

That is not a good example of market failure.

A 3 sentence disproof of Starr:

1. Assume that if and when state st occurs agents would want to re-open the market and do a deal D because it would increase both agent's utilities (Starr's assertion).

2. Then, at time 0, if they make the same deal D, *conditional on state st*, it will increase both agents' utilities *conditional on state st occurring*, and will leave their utilities unchanged if state st does not occur.

3. Therefore it must increase both their expected utilities at time 0, regardless of the probabilities each attaches to state st occurring, so they will do the deal D at time 0.

What Starr was presumably *trying* to get at, or should have been getting at, is that sometimes it is not possible for all agents to observe all states. So A-D markets cannot be complete. It's in that case that re-opening markets might reveal private information, and might make a difference.

Oh God, how did I ever get dragged way off-topic like this?

This post is not about Arrow-Debreu vs Hicksian temporary equilibria. Which are just different versions of Walrasian economics. It's about Walrasian vs non-Walrasian economics.

Nick: "this is why I dislike math-econ. It forces the reader to try to interpret the results. The author probably doesn't understand the results either." I wonder if you're making an interesting methodological point here.

Frances: Dunno. I might be saying the same thing that Greg Ransom is saying. I remember hearing about Wittgenstein's "Private Language" argument in undergrad, but never really understood it. Will have to go back and try to figure it out.

Nick - it is true that Greg's point is very elegantly expressed: T"he GE helps us see the thing (overall market order) which needs explaining, and gives us a sense of the structure within that thing we see, but it does not provide a causal explanation for that order." I think you're making a slightly different point, i.e. that a math econ model is a metaphor for the real world, but it is up to the reader to interpret that metaphor. E.g. Life is a highway. Love is a battlefield. Youth is like diamonds. Tomato is a fruit.

Baffling.

Nick, you are way off on Starr, who is not snowing anyone. He is a good guy.

This has *nothing* to do with incompleteness of markets. First, we are not talking about the securities version at all, Starr is working with vanilla state-commodites Arrow Debreu and you have all the markets you need, namely S*(N+1) for a two period model with S states and N commodities.

The point is, no one knows the "true" probability -- it's not even defined as part of the model -- each agent assigns their own subjective probability to each state of the world occurring, and each agent assigns their own utility to the consumption of each dated commodity in each period of time.

That's the framework. Whether or not the real world has a "true" probability distribution that is not subjective to the person estimating it is left as an exercise to the philosopher or the macro model builder. If you want to wander into Lucas' swamp, then you can assume that the agents know this true distribution in your models.

Now often it is assumed that the probabilities are shared among agents, and the idiosyncratic differences relate to risk tolerance (e.g. utility), not to each individual's different estimate of the likelihood of the state being obtained.

And if, as per your example, you get them to somehow coordinate their subjective probabilities by telling them whether the coin is fair or not, then yes, problem solved, and ex-ante optimality is the same as ex-post.

That's a restatement of Starr's result, it's not a counter-example to his result. Starr is not the one confused here.

Moreover the issue is not that whoever is "right" is better off than the loser. Being ex-post pareto dominated means that the loser can be made better off without making the winner worse off, and this will (in general) occur regardless of the state.

It is not that the social planner knows the future, or knows the "true" probability distribution, but he knows that A believes that there is a 50% chance of rain, and B believes that there is a 25% chance of rain, and so the social planner knows that A is overpaying for rain insurance relative to B, and that B is underpaying for rain insurance relative to A, therefore in all states of the world, utility will be thrown away.

OK, having said all of that, you are right! A simple example is called for here, and I will post one later in this thread. However, all you really need to know is that the MRS FOC doesn't hold, as was described in the paper. If that doesn't sink in, then I'm not sure if a numerical example will help --- but if that's what you want, a numerical example you shall have.

just fyi, the stuff I'm talking about is motivated by the same positivist failure as Greg's, just a different discipline. After Godel's incompleteness theorem -> the halting problem(computational complexity)-> algorithmic information theory. That's why I was confused when he was harping on me for not reading Wittgenstein, I think rhetoric is fuzzy & models are brittle.

And since you've let my pseudonym rot up there. Barkley Rosser's father was the man. Proving a stronger version of Godel's theorem. Barkley Rosser jr, is strong in his own discipline, that's why I was reading through his papers, texts, and used 'slapped around' to signify academic betaness. He thought dynamic complexity was a more useful approach, than algorithmic.

OK, here is a simple example.

Working through it really clarified my thinking.

Suppose we have two agents, A and B.

A two period model, today and tomorrow.

The state of the world is "does it rain tomorrow " or "does it not rain tomorrow".

The consumption good is iced tea, which is less enjoyable when it is raining.

Both A and B have the same time separable utility, with intertemporal discount factor of 1.

They obtain log(c) utility from consuming iced tea today and when it is dry, and they obtain 1/2log(c) utility from consuming iced tea if it rains.

Both A and B have the exact same endowment -- they will receive two units of iced tea today and two units tomorrow.

In terms of state-commodities, their endowment is (2, 2, 2), for (today's iced tea, iced tea tomorrow if rain, and iced tea tomorrow if dry).

Now consider the following:

A and B have the same preferences and the same endowment.

Therefore there are no gains available from trade, and any trade will be welfare reducing.

However, they will trade if their subjective probabilities of rain occurring tomorrow are not exactly the same, and this will create a welfare loss from trade.

Notice that it doesn't matter what the "true" probability is. As long as A and B agree on the probability of rain, then they will not trade, even if they are both wrong.

Trade only occurs, and therefore ex-post welfare loss only occurs in case they disagree, and the ex-post welfare loss occurs in all states of nature obtained. The amount of trade is proportional to the amount of disagreement in subjective probability.

The social planner, if he realized that they have the same utility and endowment, would improve social welfare by banning trade.

OK, here are the numbers. Suppose that A believes the odds of rain is 80%, and B believes the odds of rain is 20%, so the expected utility of A, given a consumption vector of (x_1, x_2, x_3) is

ln(x_1) + .4*ln(x_2) + .2*ln(x_3)

and the expected utility of B, given a consumption vector y, is

ln(x_1) + .1*ln(x_2) + .8*ln(x_3)

The equilibrium allocation for A will be (2.17, 3.3, .92) and the equilibrium allocation for B will be (1.83, 0.7, 3.08).

Ex-ante, they believe that they have gained expected utility from trade, but this is only due to the fact that each agent is valuing their expected utility inconsistently with the other.

A believes that it obtains a gain from trade of 0.13, and B sees a gain from trade of 0.15.

Now, in case of rain, A's utility will be 1.37 and B's utility will be 0.42. Comparing to the situation of no trade, A gains 0.33 and B loses 0.7, for an overall loss from trade ex-post.

In case of no rain, A's utility will be 0.69 and B's utility will be 1.73. Comparing to the situation of no trade, A loses 0.7 and B gains 0.34, for an overall loss from trade ex-post.

In terms of (ex-post) pareto optimality, A consumed 2.17 iced teas in period 1 and 3.3 iced teas in period 2.

B consumed 1.8 and 0.7 in period 1 and 2.

Even though A was clearly the winner, nevertheless A consumed too much in period 2 relative to period 1 (ex-post), and a dominating allocation for both would be consumption of 2.55 units of iced tea in each period for A, and consumption of 1.45 units of iced tea in each period for B.

There is a similar dominating ex-post allocation in the case of no rain.

So the core issue is that if agents have differing probability estimates, then they will engage in excessive trade, even when overall welfare is reduced by doing so.

The greater the variance in subjective probabilities, the greater the welfare loss from trade.

If agents trade because they have different utilities -- e.g. different tolerances for risk -- then there is an overall welfare gain as risk is transferred to those who prefer to bear it.

But if they trade because they have different probability estimates, then there is an overall welfare loss.

These two effects -- e.g. variance of subjective probability estimates versus variance of risk tolerance -- need to be weighed against each other when computing the overall welfare gain or loss from trade. Nevertheless, if there is any variance of subjective probability, the ex-post allocations will not be pareto optimal, due to the winner's curse issue -- the winner will have consumed too much in period 2 relative to period 1, and ex-post would have preferred to shift his consumption to be more equal across both periods.

RSJ: your example is much more instructive than Ross Starr's several pages of math. We can understand it much better.

But it doesn't show what you want it to show. In particular, if the market re-opens in period 2, there is no way A and B can gain from subsequent trade.

Here's a simpler example, that I think gets closer to the root of what's going on. Imagine a lottery which has a 100% payout. Risk-averse agents should not play the lottery, because there exists no set of probabilities under which playing the lottery is Pareto Improving. The central planner should ban the lottery. But if each agent thinks the lottery is biased in his favour, they all play. And, they all consume too much in the period before the lottery, because each thinks he will probably win in the second period. The central planner, who observes all agent's beliefs, or preferences, will know that at least one agent must be overoptimistic.

But that example is isomorphic to an example in which Robinson Crusoe is overoptimistic about nature, and eats too much today because he is overoptimistic about tomorrow's harvest.

None of this shows that an Arrow-Debreu market in period 0 creates "market failure". It does show that agents who have beliefs we know cannot be jointly correct will do stupid things.

Now, where you *can* use examples like this to argue against Arrow-Debreu is if agents start to infer other agents' beliefs from market prices, and update their own beliefs accordingly. Then, re-opening the market at a later time can change the allocation of resources.

I think I meant formalist rather than postivist.

"But it doesn't show what you want it to show. In particular, if the market re-opens in period 2, there is no way A and B can gain from subsequent trade."

OK, hold on here. I am trying to argue several (different points).

1. Arrow-Debreu equiilibria are not (in general) ex-post pareto optimal
2. In a multi-period economy, people will want to keep trading
3. There is market failure (in general)

For #1, the definition of ex-post pareto optimal means that ex-post, there is a dominating allocation, given that the state of the world obtains in which one person would have been better off without making anyone worse off. In my example, if it rains, A's utility with the market allocation is 1.37 and B's utility is 0.42. The dominating allocation would give A utility of 1.40 and B a utility of 0.56.

Ex-post pareto optimal does not mean that they can re-trade ex-post and obtain more utility, as the source of the disutility is that A consumes too much when he wins in period 2, and too little in period 1. The loser consumes too much in period 1 and too little in period 2.

Relative to their time discounts, this is not optimal. This is an economy without production, so 4 teas will be consumed in every period. The Robinson Crusoe analogy does not apply here.

Now you may not agree that ex-post pareto optimality is the right normative approach. I think it is the right approach, because a system whose objective is to allocate consumption across multiple time periods should be judged by its actual efficiency in allocating consumption across time periods. Otherwise come up with a more efficient system. Note that in every state of the world, the Arrow-Debreu allocation is not ex-post efficient.

2. Of course in this model people will not want to keep trading, because they cannot. It is just a 2 period model in which all uncertainty is in period 2. By the time the uncertainty is resolved, then the economy has ended.

But suppose there are two periods of uncertainty, say period 1 and period 2, in which the state commodities are SxSx{tea}. People trade prior to period 1, and they have an opportunity to re-trade after period 1 but prior to period 2. Now suppose that A believes that there is a 25% chance of rain in period 1, and B believes that there is a 75% chance of rain in period 1. A also believes that whatever happens in period 1, there is a 75% chance that the same thing will happen in period 2. B believes that whatever happens in period 1, there is a 75% chance that the opposite thing will happen in period 2.

Now, they will want to keep trading. And as they trade, they keep making themselves more and more worse off, due to the argument #1. At any given time, they will look back (ex-post) and see that their consumption across time was not efficient, and both the winner and the loser will have wished that they had done things differently.

If the economy never ends, then the trading will never stop, either.

The way to think about this is that trading only ends when there is no more disagreement about subjective probability, and this is really the key point, as per #3 below.

3. If you define market failure as an outcome in which the social planner will do better than voluntary exchange, then this is an example of market failure.

I can do better -- just have the social planner allocate resources based on equalizing MRS for all commodities, ignoring subjective probability estimates entirely.

This is equivalent to assigning each state of the world a probability of occurring of 1/|S|, or P, for any P. It doesn't matter as the P cancels out in the maximization problem.

It's the differences in each person's P that causes P to not cancel out, and this causes the problem.

The probability of an event occurring, at the macro level, should never be taken into account in allocating state commodities.

This seems counterintuitive, but there are two sides to every trade, so the issue is not how likely it is to rain, but who enjoys the drinking tea in the rain more.

Whoever enjoys drinking tea in the rain more should be the one to get more of that state commodity, and this is the only welfare-enhancing form of trade possible.

All other forms of trade are welfare reducing.

The problem is that if you are maximizing P_A*utility_A, then a relative increase in A's probability estimate, P_A, has the same effect as an increase in A's utility. So a system (without production) that maximizes Sum {Prob*Utility} will be less efficient that just a system that maximizes SUM {Utility} across all states.

The two situations are indistinguishable ex-ante, but ex-post it is revealed that A in fact does not enjoy drinking tea in the rain more, and he ends up with too much of that state commodity and too little of the other state commodities. Ex-post, he overpaid.

This is a different problem from your example in which people, thinking that they will win the lottery, consume too much in period 1 relative to those who do not believe that they will win. The actual situation is that those who think they will win trade their present consumption for lottery tickets, so they consume too little in period 1, and those more pessimistic about winning consume too much in period 1. Then even if the optimists win the lottery, they will have ended up consuming too little in period 1 and too much in period 2. If the pessimists win the lottery, then they will have ended up consuming too little in period 2 and too much in period 1. In all cases, the allocation is inefficient.

I think you are confusing the situation of production versus pure exchange. In an economy with production, being right is important. I.e. it makes a difference whether you plant a rain resistant crop or not. That decision will affect total consumption.

But in a pure exchange economy, total consumption is fixed, and the actual probabilities of events occurring can only mislead people into overpaying or underpaying.

Later on, I will try to post an example of an economy with production.

Oh, and one more comment (!)

" It does show that agents who have beliefs we know cannot be jointly correct will do stupid things."

The problem is that any time there is even the slightest disagreement about subjective probabilities -- e.g. You believe that there is a 49% chance of rain and I believe that there is a 51% chance of rain, then the beliefs cannot be jointly correct.

So in that case, it is just a statement that "agents will always do stupid things if they trade in arrow-debreu markets".

I.e. it's one thing to argue, as per EMH, that the optimists cancel out with the pessimists, so that the mean expectation of all agents, weighted by their wealth, is more or less correct.

But here, the mean is irrelevant, it is the variance that creates inefficiencies, and arguing that there is zero variance in subjective probability estimates is a much stricter assumption than arguing that the mean is basically right.

Who on earth would create a system that required zero variance in beliefs among a heterogenous population as a pre-requisite for efficiency?

And given that you have this problem, why do people keep focusing on incompleteness of markets as the source of trouble? It could well be that we already have too much trading going on, and that more markets would make things worse. In the 70s and early 80s there was literature about this point, but it seems to have been forgotten, and everyone just moved on to the securities version and worried about incomplete markets, when you have this glaring sore thumb of variance of beliefs staring you in the face. Just as an observer, it seems to me that variance in beliefs is a large motivator for people making trades, rather than just variances in their risk tolerance. I would say that variance in utility is secondary to variance in probability estimates, in terms of why people make trades.

Anyways, sorry if I hijacked the thread. I think this stuff is really fascinating.


RSJ:

1. In your 3 period example (0,1,2, with uncertainty over periods 1 and 2): You can define states like {rain in periods 1 and 2), and they can do all the trades they want in period 0. Intuitively, in period 0 they can place a bet on rain in period 2, conditional on rain in period 1. I.e. the bet is off if there's no rain in period 1. There's no need to re-open trade.

2. What the ex-post sub-optimality means is this: after the outcome of the bet is revealed, the winner will have consumed too little in the first period, and the loser will have consumed too much. Again, this is just the same as a Robinson Crusoe case, where he bets against Nature with a storeable/investable good. If he wins the bet against Nature, ex post he consumed too little in the first period. If he loses the bet against Nature, ex post he consumed too much in the first period.

The only difference between the 2 person bet and the bet against Nature is that when two risk-averse people bet, you know that at least one of them has the wrong probabilities.

3. Ex post, there exists a dominating allocation if A wins the bet. There exists a different dominating allocation if B wins the bet. But the central planner cannot go back in time to impose that allocation after finding out who wins the bet. And since the central planner doesn't know ex ante who will win the bet, he can't impose a dominating allocation before-hand. He doesn't know *which* of the two dominating allocations will make both agents better off ex post.

4. True, the central planner can ban gambling. But then he can only evaluate the results of that ban ex ante. And which probabilities should the central planner use? For example, if A knows the "true" probabilities, and B doesn't, then banning gambling makes A worse off ex ante.

5. If rational agents see that another agent is willing to take the other side of a bet, they will stop and think: "Hmmm he must have different beliefs than me; I wonder if I should change my beliefs?".

IIRC, there are theorems which imply that agents with common priors and common knowledge of heterogeneous beliefs will not trade, or will adjust their beliefs until they agree. See Milgrom, Stokey (1982). "Information, trade and common knowledge" and Aumann (1976). "Agreeing to Disagree". The resulting literature is quite large.

anon: Yep. That's the literature I was alluding to in my point 5 above. It's important. But RSJ and I are ignoring (setting aside) that literature. We assume people stick to their prior beliefs.

RSJ: BTW:

1. This comment thread is dead anyway, so there's no worry about going off-topic at this point. So keep arguing if you want.

2. I think that I, you, and anyone else reading, will have learned a lot more from us arguing through these examples than reading the original Starr paper.

3. I still think the simplest example for us to argue through is a simple coin toss 2-period example with risk-averse agents. Your example had state-dependent utility (you like ice tea more when it's sunny), but I don't think that's at all essential, and it just complicates the story.

Simplest example is a 2 period coin toss, with risk averse agents, and everything symmetric. If A wins the toss, the dominating allocation has A consume more in period 1, less in period 2, and B doing the opposite. And vice versa if B wins.

Nick,

"True, the central planner can ban gambling. But then he can only evaluate the results of that ban ex ante. And which probabilities should the central planner use? "

In a pure exchange economy, if everyone starts out with the same endowment, and if the social planner just invents some probabilities and then allocates commodities based on solving that maximization problem, the resulting allocation of endowments will have the following properties:

1. It will be ex-post efficient in all states
2. It will be independent of which probabilities were assigned by the social planner
3. It will reduce to the market allocation in that case when everyone agrees on the state probabilities, regardless of whether the market participants share the same set of probabilities as the social planner.
4. It will deliver a greater *total* utility in all states, than would be delivered if everyone started out with the same endowment and engaged in voluntary trade (with equality holding when everyone's beliefs are identical).

Now you are right, it will not ex-post pareto dominate the market outcome when there is disagreement, but that market outcome is ex-post pareto-inefficient, whereas the planner's outcome is ex-post pareto efficient, *and* the planner's outcome has higher social welfare (e.g. total utility). Key to this result is that when endowments are identical, then the probability of a state occurring, if universally believed, has nothing to do with the final allocation of that state-commodity.

The probability estimate of a state occurring always influences price of the state commodity, and so it influences the budget constraint. But if everyone has the same endowment, then everyone has the same budget constraint, and solving the competitive equilibrium problem amounts to solving the FOC

p(i,s)*MRS(i,s) = MRS(j,s)*p(j,s)

where p(i,s) is the probability that agent i believes that state S will occur. and MRS(i,s) is the marginal rate of substitution from giving up 1 unit of the present good for 1 unit of the good that will be obtained in state s. Notice that if the beliefs are shared, then the probabilities cancel out, and play no role in setting the final allocation of goods.

Therefore the social planner does not need to "know" the probabilities.

Any set of probabilities, if shared, will yield the same allocation -- the allocation will be set by MRS.

Now, suppose everyone does not start out with the same endowment -- which is crucial for 1-4 to hold. In that case, first do a transfer to equalize the endowments and then proceed as above.

In terms of whether the utility of consuming the state-commodity should depend on the state -- of course it should!

This is because unlike your example, the agents are not placing bets against nature, they are placing bets against each other.

Your model -- placing bets against someone outside the model -- is a production model in which you decide to plant one kind of seed if you think it will rain or not. It will give you the wrong intuition.

That is why you keep thinking it is important that agents know the "true" probability. But when agents have the same initial endowment and are placing bets against each other, then the true probability is not important. All that is important is whether one agent enjoys the state commodity more than another. Assuming everyone starts out with the same endowment, the fact that probabilities are assigned to states is just an illusion that allows people to confuse each other and get into trouble. The usefulness of the model is that utility differs across states.

And this is how you can achieve idiosyncratic insurance. Suppose the states were "it rains on A" and "it does not rain on A". Or equivalently, suppose B didn't care if he was drinking iced tea in the rain.

Then even though A cannot prevent it raining on him, he can sell some of that state-commodity to B in exchange for more present consumption, or more consumption should it be sunny.

B will be happy to enjoy iced tea if it is raining on A, so that state commodity is more valuable to B than it is to A, and therefore there are gains from trade available. Trade will occur up until those gains are exhausted, and B will end up with "interest" or additional expected consumption of iced tea due to the fact that he is better able to bear the risk of it raining on A.

But again, the equilibrium allocation will not be determined by the probability of it raining on A, but by the relative difference in preference for that state-commodity. The price of the commodity will be affected by the likelihood of the state, in that the value of the rain-on-A commodity goes up and the value of the sun-on-A commodity falls. But the resulting allocation will not change as the probabilities change, if everyone starts out with the same initial endowment.

But what the agents cannot do is mitigate the effects of systemic risk. If it rains on both B and A, and if both B and A have the same preference for tea when it rains, then by trading, they cannot improve their situation, and again, this is also independent of the probability of rain occurring.

On the other hand, when you are betting against nature, and not against each other, then you are not trying to obtain gains from trade, you are trying to increase your overall endowment by making wise production decisions. That's a completely different issue, and it has its own intuition which does not apply here.

Similarly, if you are trying to endogenize the probabilities (or the utility functions), then that's another separate issue. Of course, everything is endogenous, including both beliefs and preferences, and of course maximizing your endowment by making wise investment decisions is important. But those considerations are exogenous to this pure exchange model in which state-commidities, preferences, and beliefs are given, and we are just searching for the optimal allocation of consumption across time.

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