On Walrasian Arrow-Debreu General Equilibrium and Rational Expectation (1)
Sometimes an obvious point is often forgotten, and importance of Walrasian Arrow-Debreu general equilibrium is one of them. (From now on, I will simply refer to Arrow-Debreu general equilibrium as simply as general equilibrium.) I will first talk of logic of general equilibrium and highlight its importance, and then connect them to rational expectation for this post.
General equilibrium, to its core concept, is that agent decisions really only depend on their own endowment/budget, production capacity, utility and price vector. Thus, if one does believe that agents make (mechanically) best use of their budgets, agents simply look at price vector to decide, and markets do clear, then there is pretty much no way one escapes general equilibrium frameworks. But note that the powerful assumption here is “market clearing.” For if this is not assumed then by limited information capacity of general equilibrium, there are many disequilibrium choices involving excess demand/supply. (Now it is true that agents really do not get to optimize in disequilibrium situation, and I will talk about this later in the post.) Later in the post I will connect disequilibrium choices to the equilibrium approximation computation problem, and show why it was perceived that market clearing assumption is hard to throw out, at least in general equilibrium context.
The above shows that usual reasoning that general equilibrium assumes too much capability for agents is, in some way, wrong. It is true that agents optimizing best for themselves is nearly impossible, and in this regard general equilibrium may assume too much. But it is good to note that general equilibrium does not assume perfect foresight or complete information.
The problem of time complicates the matter though. In Arrow-Debreu contexts, people trade future state-contingent Arrow-Debreu (AD) securities, which are essentially those offering some payoff based on what happens in the future (or simply future state). This no longer purely involves agents just looking at their own utility and making optimal decisions. Agents now have to think about their subjective probabilistic evaluation of future outcomes.
Thus general equilibrium now faces an issue of how agents form their expectations. Arrow-Debreu general equilibrium, in its “pure” original form, ignores this question and simply asks us to provide subjective probability of agents, “effectively” converted to utility for AD securities. But this only provides a descriptive explanation. Thus, Arrow-Debreu general equilibrium can explain the decisions made by the people, given information that agents had, but for those trying to predict an economy, Arrow-Debreu does not provide much of a clue.
This is where rational expectation does come handy. Rational Expectation (more of economics version than what the word expects us to think about) asserts that agent expectations are, on time average, accurate. This often is “upscaled” to the following: agent expectations are accurate, and all errors are purely stochastic. This is what we do see in some of dynamic stochastic general equilibrium (DSGE) models often used in macroeconomics. Of course it is not true that DSGE models do not allow for people’s expectations being systemically wrong. It just requires that systemic errors induced by agents in the models are consistent with actual equilibrium, and agents do try to correct their expectations according to their expectation formation rules in the models. One often says this as “rational expectation being imposed in estimation sense,” or simply internal consistency of the models.
One can look at rational expectation from a different perspective too. If we can assume that different allocation outcomes do not drastically affect macro outcomes, then one can reasonably assume that the market clearing assumption of general equilibrium provides approximate internal consistency requirements, which here is rational expectation.
Though there is another meaning for rational expectation, which more fits the normal non-economics use of the word: that agents, based on their information, form expectations “rationally.” What one means by agents forming expectation rationally must be considered, but for now I will not consider this part.
To summarize what I wrote so far, the following point have been raised:
- General equilibrium is actually a simple “limited information” concept. While it is fundamentally an exchange (production incorporated too) economy model, as opposed to a monetary economy, it is in some way monetary too. Agents do not bargain or trade — instead they make decisions by assigned price vector, just like how many people shop in Walmart, Amazon, Costco or what. There are markets like eBay, but that is an exception rather than a norm.
- Market clearing is a powerful assumption. If it is dropped, then economic analysis becomes increasing difficult. While other issues will also be discussed, computation problems of equilibrium are connected to the problems caused by dropping the market clearing assumption. I will discuss more in depth on this point.
- General equilibrium, in its original Arrow-Debreu form, is silent about how agents form expectations. (“Time” complicates the matter) Thus in order to actually use general equilibrium analysis, one needs to talk about how agents form expectations. Because there are so many possible ways on how agents do form expectations, rational expectation has been used. If agents can learn “positively” by systemic errors they made in the past, then one will approach the “correct” expectation. Or, if one can suppress distributional effects on macro variables (and if everyone stays afloat), then at least for macro analysis, then rational expectation is an internal consistency requirement for macro general equilibrium analysis. I will discuss more on this point, and think of the roles game theory and behavioral studies play in modern (specifically macro) economics analysis.
- ( It is obvious that future expectation affects present outcomes not involving future states. (Though it can be noted that if agents spend every income and wealth they have in the present period, and if this choice can be guaranteed to be optimal, then future expectations play limited role in determining present outcomes, if supply-side expectation issues can be muted.) And the summary point 3 pointed out why rational expectation is useful. As said before though, general equilibrium by itself does not require rational expectation.)
At this point it may be better to consider whether market clearing assumption can be justified. (← Summary point 2) Instead of going directly to reasons for throwing away the assumption or justifying it, I will instead consider what happens in case market clearing is thrown out. One reason to consider disequilibrium is that agents usually do not get price vector right. For example, firms selling a product set price vector wrong — that is, there is excess demand or supply. Equilibrium never outright establishes itself — as long as we assume that agents do not perform bilateral trading, which is what we do see in many markets (though this tends to be slightly flexible), one really has to consider how equilibrium can be reached. And in the end, we will come to see one of the defects of traditional general equilibrium analysis.
As said in the above, general equilibrium provides the right answer(s)/equilibria for any expectation in case agents behave as specified and required by the general equilibrium theory. But it is also true that general equilibrium can “talk” (note quotation) about what happens in case market clearing assumption is dropped. What we get is demand and supply vector of individuals/products. These two will not coincide, and there will be no allocation solution for disequilibrium price vector. The products that have excess supply is of less direct problem, as one may say that firms “chose” the wrong price vector given wrong expectation (now this is present-time expectation. So expectation is not just limited to future-time expectation. About this in the last parts of this post), and carries on with (or accepts) the loss. (However, this loss does require concept of debt and default, and in this way this may be more problematic or actually be more realistic, depending on the view), and If the good is partly durable, then it will be able to take unsold products in inventories and sell them when possible. However, note that even here this requires recursions of fixes into Arrow-Debreu analysis, as quantity of supplied products now has to change (label changes, as goods in different time in Arrow-Debreu mean different goods), which eventually break apart. Excess demand case is serious directly — consumers, realizing that they will not get what they want will reshuffle their choices so that they get additional utility.
Thus general equilibrium analysis, in its final form, requires market clearing assumption, though there are other reasons why this is so, and some of these will be discussed in the later part of this post. However, this does not mean that general equilibrium analysis requires replacement by game theoretic models or so. Game-theoretic models will face the same problem, if they assume that consumers are price takers and optimize their decisions best based on price vector. Thus, eventually they resurrect general equilibrium models while performing analysis.
Effectively for this disequilibrium problem, game-theoretic models carrying some assumptions imposed in general equilibrium analysis, if they are ever to provide something more to what we already have in general equilibrium analysis, must provide how expectations eventually come to settle on a point very close to market clearing. This effectively involves across-time/intertemporal games on expectations played repeatedly, especially involving how firms set price and prepare for expected supply. (By nature of general equilibrium, instead of ordinary assumption that Walrasian auctioneers set price vector, one can instead assume, in case of pure production economies, that firms set price vector and wait for consumer responses.) Meanwhile, one must model how consumers decide when they are faced with excess demand issue, though there is a straight way out — agents re-calculate their optimal choices by eliminating those that cannot be bought. This process has to be repeated, and ensuring that at least some outcome comes to stay at some time period is the role that game theory would have to provide. This may involve dividing a single time period into multiple times that agents get turns, or denying some agents market access on some time period for the model.
Or one may reject ultra-optimality-given-price-vector behavior of consumers, and adopt for more fair version of optimality, such as satisfaction with an allocation even if they cannot buy the goods they desired under the current price vector. (In this case, supply rules — allocation is determined by supply price and vector. One may give psychological microfoundation by asserting that agents simply do not have time to re-calculate their decisions, so they are stuck with the original optimizing choice they had when they faced the price vector.) Or agents can just decide based on their chosen particular subset of products available and calculate. (This direction introduces more complications, though this is much more realistic.)
In case what I want to say via these points slipped, what I am saying is that as long as price taking assumption and some optimality conditions are in place, one still is doing general equilibrium analysis, at least as part of analysis, even if one uses game theoretic models. The problems need to be tackled and cannot be avoided.
All these points are somewhat obvious. Yet we often forget these points. For example, many contemporary theoretical macro models, deviating from the root of general equilibrium analysis, sometimes forget that they are trying to model the actual economy, not some weird economy. General equilibrium analysis has different serious issues, as seen above and as I will point out more later in the post, but it often tends to be a better approximation to reality than many ad-hoc reasons proposed. Does anyone really think “one trades at day in some decentralized market, with some probability of double coincidence, and participates at night in some Walrasian general equilibrium market” is more realistic than simple general equilibrium story? It is true that their are markets resembling auction or bilateral trades out there — for example, mergers and acquisitions, (partly) stock markets, procurement, intermediate good markets and so on. But what some of contemporary macro models seem to do is exaggerate importance of these markets in actual market economy — conflating the fact that non-Walrasian markets exist with the fact that they exist in many parts of our economy. Again, how likely is it for ordinary consumers to participate in the day market, in the above quote? But this is what some classes of these monetary economics models assume. And this is not really a good way to go ahead.
(What I want to say here is importance of really separating non-final-goods markets (intermediate goods, stock markets, etc.) from final goods markets which tend to be fairly Walrasian in that consumers are price takers. I write this in parentheses and bold, because this is not the main point of this post, but as important as the main point.)
(Again, the above points show how important behavioral/game-theoretical understanding is, but also show that “behavioral/game-theoretical understanding” replaces general equilibrium analysis is simply not true. They get together, if we are to study the most common form of markets in modern times.)
Thus, there are two main ways game theory or behavioral understanding helps in general equilibrium analysis. One way is by providing how the market corrects disequilibrium that is inherent in Walrasian (or modern monetary non-bilateral-trade) markets, and the other is how future expectations are pinned down and affect present outcomes.
Now let’s re-discuss market clearing assumption, in terms of computing equilibrium price vector and allocation. There are two main types of approximations for computing general equilibrium. The strong version is calculating the approximation so that we get equilibrium price vector or allocation down to some given precision level. The weak version is calculating the approximation so that the resulting excess demand/supply is less than some given level. If we are to trust some form of Church-Turing thesis, one can think of a hypothetical disequilibrium-to-equilibrium convergence process as a computer that tries to calculate approximately an equilibrium allocation/price vector.
In the above parts, I focused on why non-market-clearing leads to problems, and we cannot avoid the problems if we have faith in some form of price-taking behavior. I discussed how additions of how agents behave in case of disequilibrium may help us complete the analysis. However, if there are ways such that markets can quickly restore back to an equilibrium, then one may still continue with equilibrium analysis. (There are still obvious defects on continuing analysis on this ground, but hypothetically one may assume that while agents rely on price vector to make decisions, for some behaviors, agents do have complete information on firms and agents. This means that they can technically try to calculate where they will be at.) This is the question we ask in this part of the post.
By Walras law, excess supply necessarily means excess demand, so it is not possible to avoid the consumer optimization issue discussed above in case of disequilibrium: that if faced with lack of a product a consumer wants to buy, the consumer will run optimization calculations with different constraints again. In order for us to suffice with the weak approximation, and conclude that our general equilibrium analysis can be conducted with the weak approximation case, consumer behavior must be so that it can tolerate some excess supply/demand, and move on. And this is in some way reasonable. However, excess demand means excess supply (by Walras’ law), and firms can try to sell excess supply they have. Thus this requires both consumer behavior and firm behavior being tolerant, in “no pure exchange” production economies, of some loss they suffer by excess demand/supply. Otherwise, the markets will not stay on the weak approximation outcome.
So can we compute the strong version? The short answer, if we are to trust Vela Velupillai’s result, is no. Again, I do not know how much trust I should place in Velupillai’s result, as not many have discussed the result. However, it is true that so far we really do not have no general way to compute general equilibrium under some precision. We can, however, compute, specific cases, and the question then is whether agents can believe that they are under “nice-behaving” economies, as in the results of Yannakakis.
The weak version case is better in terms of computation, though it still requires some computation power.
One takeaway from this post is that how the models behave under disequilibrium matter for equilibrium analysis. In ordinary intuition, agents will likely first begin with the price vector they already experience, and move based on that price vector. And then price vector will move again based on the new price vector obtained and so on. If this ordinary intuition is correct for analysis, then this brings the case where a new equilibrium that is dictated by new conditions of economies may not be reached because it cannot be reached from an old equilibrium. This especially affects how central banks must do monetary policies, and why “I have existence of equilibrium, so that equilibrium be the end of story” should be dealt with cautions. For example, if there is only one equilibrium consistent with central bank’s choice of its policy interest rate, this does not mean agents will all jump to that equilibrium. Instead, what one may notice instead is disequilibrium.
This again is not really a new point — this is more of a forgotten point. To add, notice how behavioral assumptions change the story. Suppose that agents do have complete information of each other. Then it may be possible to compute at least the weak approximation to the equilibrium. (Though complete information raises the question of why agents even trade in Walrasian markets. This in fact is equivalent to the question of money — what roles money actually plays. Because Walrasian markets require money, but analysis of money is visibly absent from Walrasian analysis.) Or if, in special cases, it can compute the strong approximation to the equilibrium, then one can forget about convergence issues, assuming complete information. Then existence of equilibrium can serve as the end of the story.
Again, the point here should remind that general equilibrium is not really that restrictive relative to popular conception of modern neoclassical economics. Though there is some truth that modern macro went somewhat far away from these roots in general equilibrium, and care mostly about equilibrium itself then how equilibrium may be reached and dynamics that can support equilibria. In this way, general equilibrium project, used in practice, went more restrictive in terms of information requirement. However, general equilibrium by itself is not restrictive in information requirement, though it requires ability to handle completely disequilibrium situations.
In Part 2, I will discuss more of monetary economics, in relation to general equilibrium research. And while discussing monetary economics, infinite-time intertemporal models will be discussed, and often forgotten theoretical foundations behind intertemporal models in terms of original general equilibrium.
Addendum: As this post mentions general equilibrium — Steve Keen has for years notoriously claimed that perfect competition is logically inconsistent. In terms of general equilibrium analysis, there are multiple ways to make sense of perfect competition. An obvious consensus understanding of course is price taking by both firms and consumers, and in this understanding Keen’s analysis breaks down from the beginning, as many have noticed. Though there is worth investigating into Keen’s claim on how one does not recover perfect competition from imperfect competition, and how Keen’s arguments break down. Maybe in a different post, I may demonstrate one example where a repeatedly-played firm decision game does not lead to perfect competition from imperfect competition even when number of firms increases — however, this is perfectly consistent with mainstream understanding.
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