Look, economics is about how people (the word “agents” is itself a kind of tribal marker) are motivated to take actions, and how those actions interact. Equilibrium is often a very convenient way to think through all of that, and all of us sometimes use wording about what the economy “needs” or “requires” as shorthand. When I talk about the Dornbusch overshooting model of the exchange rate, for example, I might say something like “the currency has to overshoot its long-run value, so that investors expect appreciation that offsets the interest differential.” But behind that verbal shorthand is a story about people doing stuff: investors selling the currency because yields are down, the currency falling until it’s so low that people figure it has nowhere to go but up.Let me put that in other terms, to show you what I think he is trying to say (this relates to a conversation with Nick Rowe in the comments section). I want to model the market for apples, so I draw a downward sloping demand curve and and upward sloping supply curve. There are particular factors that shift the supply and demand curves. If income increases, the demand curve shifts to the right, the price of apples goes up, and the quantity traded goes up.

I haven't looked at any principles of economics textbooks lately, but my guess is that in some of them you will find something like the following: When income goes up, consumers go out to buy more goods. At current prices, some of them are unsuccessful in finding goods to purchase, and firms may have to draw down their inventories of finished goods. As a result, firms increase their prices, and produce more output to restock inventories.

Of course, in the basic static model, the consumers and firms are all price takers. The consumers act under the belief that they can buy all the apples they want, subject to their budget constraints, and the firms act under the belief that they can sell all the apples they want at the market price. The model does not tell us what the consumers will do if they go to purchase goods and cannot find willing sellers. There are no inventories in the model. So what's that story about? That's something that comes from outside the model. If we thought that rationing, inventories, and price setting were important, we should have put those things in the model.

Telling stories outside of the model we have written down opens up the possibility for cheating. If everything is up front - written down in terms of explicit mathematics - then we have to be honest. We're not doing critical theory here - we're doing economics, and we want to be treated seriously by other scientists. I think this quote is to the point:

Academic economics, the stuff that is in the textbooks, is largely based on mathematical reasoning. I hope you think that I am an acceptable writer, but when it comes to economics I speak English as a second language: I think in equations and diagrams, then translate. The opponents of mainstream economics dislike people like me not so much for our conclusions as for our style: They want economics to be what it once was, a field that was comfortable for the basically literary intellectual.That was written by Paul Krugman, in 1996.

I agree with the notion that we have to be able to argue our ideas in plain English. That's what writing this blog is about. But the plain English should be restricted, as Krugman says, to a translation of what's in the mathematics. Otherwise, we can be accused of making things up on the spot that suit some end other than good science.

So, I understand that some of the ideas in my recent papers and blog posts might seem counterintuitive or, indeed, radical, especially if you're wedded to a particular way of looking at the world, or are not versed in recent developments in monetary economics. But give it a chance, and don't be so sad, Paul.

This post isn't responsive to Krugman's critique. His problem isn't that you're doing too much math.

ReplyDeleteHis point is that any successful model has to be able to explain how the posited associations are actually realized in the real world, by real people acting in intelligible ways (preferably, in accordance with observed behavior).

To use your example, suppose I construct a model about apples, including all that stuff about inventories, etc. But the model spits out the result that as the price of apples increases, demand for them increases. That's obviously not how the world works. It would be my job to explain why people are buying more apples because the price has increased. Maybe I tell a story about bubbles (not likely with perishable goods like apples), or about status goods (maybe). But if I can't explain the result in terms of sensible human behavior, it means there's something wrong with the model.

You're right that if I was looking to tell a bubble or status good story, it should be in the model. That's why it would be all the more troubling if my plain vanilla model produced that result.

I won't here opine on whether your model suffers from that flaw -- i.e. whether Krugman's "little blue arrows" point is valid. So maybe his objection is invalid on other grounds. But I'm confident that you haven't actually addressed his point here.

"But if I can't explain the result in terms of sensible human behavior, it means there's something wrong with the model."

DeleteExactly.

1. Write down the model, which includes "sensible human behavior."

2. Derive the implications of the model.

Go back to (1) and ask: Now, what bits and pieces in that collection of sensible human behavior do I need to get the results, strip away what I don't need, so that people can understand it, and I'm done.

So, I did all that. I wrote down the model after I cleaned it up for you, so you can see what's going on. For anyone versed in standard economics at the PhD level, it's a piece of cake. Then I added some words to help you along, and tell you what is important, and to suggest how you could modify it and still retain the essence of the results.

I'm not sure what else Krugman is looking for.

A couple of thoughts:

ReplyDelete1. That's a bit like saying: "Unless you can build me a formal physics model of the instability of this equilibrium, I'm going to assume that all eggs on a flat table rest pointy-side down"

2. "If we thought that rationing, inventories, and price setting were important, we should have put those things in the model."

There used to be an approach called "disequilibrium macro" that did explicitly put rationing into the model. To my mind, that was the right way to do (macro) stability analysis. (And Walrasian tatonnement stability is the wrong way.) But equilibrium theorists weren't very interested in it, so it died out. Very few remember it now, or have even heard of it.

The egg analogy actually works in favor of Steve's approach. The same differential equation that says the egg can - theoretically - stand on its pointy head also tells you that the egg will not end up like that from any other position. Everything you need to know is in that model. You don't need any additional story (formal or informal). That is exactly what Steve is trying to say.

ReplyDeleteAnonymous: "That is exactly what Steve is trying to say."

DeleteDoes that mean Steve actually prefers a model with an unstable equilibrium? I didn't read him like that.

OK, suppose we have an egg standing on its pointy end. Now we hang a small weight on the north side of the egg. The egg would need to lean south to restore equilibrium. That's what the pointy-end equilibrium model predicts. But we know the egg will fall over to the north.

"Telling stories outside of the model we have written down opens up the possibility for cheating."

ReplyDeleteBut if the model doesn't tell you how something is supposed to occur, and prove why, wouldn't that also be cheating?

If it's "cheating" to give a story of why something occurs without the math. Isn't it at least as much cheating to give neither the story nor the math of why it should occur?

And, in a sense the assumptions you make can also be cheating. Why did it occur? Because I made ridiculous assumptions to make it occur. Of course, unrealistic assumptions can be fine, if at the end you interpret well to reality, for example you ask what happens if I relax the assumptions.

So sayeth the guy who knows nothing.

DeleteYou ask questions in the hope of learning more. It might actually be useful to answer the questions, and explain why the criticism implied is wrong. And especially if you can do it in a way that doesn't require already being an expert in the sub-area. You want to be useful, try actually giving an answer.

DeleteWho says I want to be useful? I just want you to go away.

Delete"Does that mean Steve actually prefers a model with an unstable equilibrium?"

ReplyDeleteNo! It means that if you want to talk about out of equilibrium dynamics (whatever the hell that means!) you should include them in the model.

"The egg would need to lean south to restore equilibrium." No! The egg moves according to differential equation that describes its motion. The same happens in Steve's model. Policy changes and the economy moves along the *equilibrium path* to the new steady state.

If you want to know about richer "disequilibrium dynamics", you have to add them. You talk as if they are known facts!

A part of judging assumptions is always to assess their plausibility, and the equilibrium concept is just another assumption.

ReplyDeleteFor apples, we believe that tattonement is a reasonable assumption, and that a model with tattonement will yield the same conclusions as Walrasian GE. So we believe GE is a good equilibrium concept.

If people criticize you for having a weird equilibrium concept, it is not clear that they bear the burden of proof. It does not mean that they need to model a learning dynamics model to show that your equilibrium is non-robust.If intuition says that the adjustment dynamics might be unstable, it is up to you as a modeler to suggest why they are not. No proof is actually necessary -- if there is a reasonable verbal mechanism, we can probably reach a joint conclusion that it will work if one spells it out.

Question then: Is it possible to point out roughly how a reasonable adjustment mechanism out of equilibrium will look, enough so that an impartial, economically educated, third person observer would agree that it would probably work out mathematically if one tried?

Stephen,

ReplyDeleteEvery solution needs to be robust against basic perturbation analysis. If, for example, the representative agent is wrong by however little about a parameter or variable in the model does the solution converge back to the ratex equilibrium? If instead it diverges then you don't have an equilibrium that can exist in the real world (which may or may not be interesting to you).

What Krugman is saying is that your solution is not stable, and as I mentioned on Noah's blog, David tried to explain this a few years back, and you said it wasn't interesting. It is interesting.

I think this question has been answered, that it is stable. The issue has become whether Steve should also provide a story that is consistent with the dynamics of his model even though the story involves behavior not explicitly modeled. As an empirical economist I have often included in my estimation control variables not in the model that seem, however, relevant based on a story. I think this is legitimate. I am still not sure, however, if this is kosher when one does theory.

DeleteI think Rowe pointed it out better than Krugman but of course you are correct, K.

DeleteIn theoretical economics (implicit) assumptions and analyzing the results of a model not directly inside of the model are at least as important as solving a given setup / choosing the techniques to do so.

This is simple. Program it up in python. Perturb it. See what happens. You are a Ph.D. economist, are you not?

ReplyDeleteThere is a lot of confusion here about words "equilibrium", "disequilibrium", "stability", etc. These notions do not have the same meanings in economics as they do it physics.

ReplyDeleteIn physics, a differential equation describes the entire motions of a system. Equilibrium is defined as the rest point of that system (steady state if you like). Therefore, "disequilibrium" is a well-defined notion (means out of steady state if you like). Hence, it is well-defined to ask 'what would happen if the system is perturbed?' The same differential equation that gives you the condition for equilibrium (steady state) also tells you what happens after perturbation.

In economics, equilibrium is a very different concept/notion with very little connection to physics notion of equilibrium. It is just a bad name, unfortunately. In economics equilibrium is THE solution concept. Nothing is well-defined "outside of equilibrium". So 'perturbing something out of equilibrium' does not mean anything.

You can ask, what happens to *steady state* of an equilibrium after perturbation. The answer to that question is in the model. The same dynamic system of equations that gives you *steady state* also tells you what happens *outside steady state*.

If by programing the model in computer you mean programming the system of equations that define the equilibrium that can be done and has been done. But if you mean programming "disequilibrium", first you have to define what heck it means! So here is a challenge for all you stability/instability/disequilibrium guys out there. Write down a formal definition of stability/instability/disequilibrium and specify what exactly means to "perturb" an economy outside equilibrium. Good Luck!

Ummm, since when must a term that has a very specific meaning within a discipline have the same meaning when used in other disciplines? This is a first!

DeleteSecond, where do you get the idea that in economics nothing exists outside equilibrium? Many models include transitional dynamics, that is, differential (or difference if time is discreet) equations that describe the transition of a variable to equilibrium (or steady state). It is up to the modeler to decide if it is important to include such dynamics, or focus on what happens once the economy gets there.

"differential (or difference if time is discreet) equations that describe the transition of a variable to equilibrium (or steady state)."

DeleteAlmost all economic models only model the transition to steady state. All points on the way to that steady state are part of the equilibrium as they are feasible and all agents optimize. So no, modern macroeconomics very rarely exists outside of equilibrium, even though it often exists outside of steady state.

I am not sure what you mean Hannes. e-co-ngineer wrote that in physics "Equilibrium is defined as the rest point of that system (steady state if you like)." Well, also in economics, equilibrium is typically defined as a state at which the value of the differential, or difference, equation describing the motion of the variable of interest is zero. So, using his/her definition, I am arguing that, contrary to his/her claim, plenty of models do study the motion of the variable outside such a state. Moreover, agents may or may not optimize at each point in time. But this is a separate question. If a ball is rolling towards a wall, then the ball is at disequilibrium but the wall is at equilibrium. So is a model that describe the motion of the ball relative to the wall an equilibrium model?

Delete"Ummm, since when must a term that has a very specific meaning within a discipline have the same meaning when used in other disciplines? This is a first! "

DeleteThere are many examples of bad terminology in economics. Equilibrium is one of them.

"where do you get the idea that in economics nothing exists outside equilibrium? "

I did not say it does not exit. I said it is not well-defined. In order to know if something exists, you have to define it first.

"also in economics, equilibrium is typically defined as a state at which the value of the differential, or difference, equation describing the motion of the variable of interest is zero."

That is not true. In a dynamic model (almost all macro), equilibrium is described by a differential/difference equations. To be more precise *equilibrium is a probability distribution over sequences of random variables*. In other words it is dynamic and possibly stochastic. What you refer to in your comment is the *steady state* of an equilibrium.

What in physics is called equilibrium, in economics is call steady state. A well-defined concept.

"There are many examples of bad terminology in economics. Equilibrium is one of them"

ReplyDeleteSo suppose I say,

"There are many examples of bad terminology in physics. Relativity is one of them".

This is nonsense, unless I first define what "bad" means. I find your current definition of bad = having a meaning different from that in physics quite bizarre.

"In order to know if something exists, you have to define it first."

Which, like it or not, I did.

"That is not true."

Yes it is. First, equilibrium has nothing to do with probability distributions. For example, growth models are not stochastic. Does this mean there is no equilibrium in growth theory? I don't think so. Second, my definition also applies in dynamic models. For example, in error-correction models two variables may evolve over time, but the relationship between them in the long-run is stationary. In this case, the equilibrium is defined as a situation in which the difference of their difference is zero. Or, in cases where the variable of interest is growing over time, equilibrium is defined as a situation in which the derivative of some transformation of that variable (of its growth rate, of its deviation from trend, etc.) with respect to time is zero.

In fact, equilibrium and steady-state are closely related concepts. Usually, the term equilibrium is used to denote a steady state in a particular market (labor, credit, etc.) while the term steady state applies to the entire system (e.g. when all markets are at equilibrium). For example, one can induce a shock to an exogenous variable. Given the value of the exogenous variable, each market may be at equilibrium, but the system as a whole is not at its steady-state since as the value of the exogenous variable is changing, so are the equilibrium allocations.

"Which, like it or not, I did. "

ReplyDeleteNo you did not. You defined what it means to be out of steady state. You did not define out of equilibrium. Steve's model (like most dynamic macro models) has transition dynamics in it. If this is what you mean by "disequilibrium" we don't have any argument. It is all there. We are just giving different names to the same concept.

"First, equilibrium has nothing to do with probability distributions. For example, growth models are not stochastic."

A deterministic growth model is a special case of a stochastic growth model. I wanted to make a general general statement. Set the variance of stochastic shock to zero and we are in agreement again! :-)

"For example, in error-correction models..."

We are talking about competitive equilibrium here as a solution concept. You seem to like the idea of slapping the label equilibrium on anything you like. Look at Steve's paper (or better yet a graduate macro textbook) to see the definition of equilibrium. Just to make sure we are still talking about the same thing.

"In fact, equilibrium and steady-state are closely related concepts."

True. Steady state is an equilibrium that satisfies certain conditions. But not all equilibria are steady state or stationary. Hint: go back to your growth example and think about transition dynamics. Do you call that disequilibrium? Again, if you do, we agree on almost everything but what to call this thing. I call it pop, you call it soda. It is the same damn thing. Stop arguing with me! :-)

"Given the value of the exogenous variable, each market may be at equilibrium, but the system as a whole is not at its steady-state since as the value of the exogenous variable is changing, so are the equilibrium allocations."

I don't know why you are arguing with me. You seem to agree with me on everything that matters! I am out! :-)

OK, I think I see where the problem is. No, we are not talking necessarily about a competitive equilibrium or a rational expectation equilibrium. Now, in a rational expectations equilibrium people reason all the way through, so the initial allocation is the equilibrium allocation. Put differently, the transition from one equilibrium to another following a change in the environment happens instantaneously. So yes, there is in fact a transition, but it takes place within a unit of time, so it is never observable in the model. As Steve mentions, however, one can introduce disequilibrium dynamics by adding, for example, learning, in which case the transition becomes gradual and hence observable. But this is holding Steve's model at a higher standard than most other models. Nevertheless, the idea of disequilibrium is not ill-defined, as models with learning do exist. So I disagree with your notion that the concept of equilibrium is ill-defined in economics. And to me your criticism matters.

Delete"Put differently, the transition from one equilibrium to another following a change in the environment happens instantaneously."

DeleteIf by equilibrium you mean steady state, the answer is no. Again, think about your growth model. There is a transition dynamics and it takes many many periods. All that transition happens *along the equilibrium path*.

"as models with learning do exist"

Yes. And they are equilibrium models. Look at Tom Sargent's work on that (or George Evans and Seppo Honkapohja). All those models have a well defined notion of equilibrium. All the learning happens along the equilibrium path. The economy may or may not converge to certain rational expectation equilibrium. But there is always an equilibrium (agents expectations/behavior are clearly specified and stuff add up, i.e., all markets clear in every period).

I still think we are arguing - mostly- over pop vs soda.

"If by equilibrium you mean steady state, the answer is no."

DeleteNo, by equilibrium I mean equilibrium. For example, within a growth model, the wage and interest rate adjust instantaneously to an increase in TFP. I think this is what you are saying. Except you seem to be missing that what adjusts is the entire time path. Along the new time path the allocations may change as the state of the system evolves, but within each unit of time there is no change as agents cannot do any better by re-arranging given the state of the system. So, in a way, it is like shrinking the unit of time enough so that, at the limit, the state of the system is constant (even if the system is not at its steady state), and then define equilibrium as a situation in which, within that unit of time, there is no change in the choice variables.

So then, what would be an example of disequilibrium modeling? Perhaps the most famous one is the cobweb model (http://en.wikipedia.org/wiki/Cobweb_model) whose embarrassing predictions paved the way to rational expectations. For an example of disequilibrium with learning see here http://www.sciencedirect.com/science/article/pii/S0304406802000629

And then, of course, you have the evolutionary framework introduced by Nelson and Winters. These are alternatives to general equilibrium modelling. In them, changes in choice variables are not due to changes in the state of the system, but rather responses to the divergence between what is desired by the agents and what is realized. What form this response takes is up to the modeler to decide, with learning being one of them.

And, once again, equilibrium does not necessarily entail market clearing. For example, with costly search the labor market never clears, yet, there exists a well-defined equilibrium level of unemployment. It entails a situation where agents cannot do any better by re-arranging.

e-co-ngineer, you're an incredibly patient man.

Delete(different anon)

different anon, so you agree with e-co-ngineer that the concept of equilibrium is ill-defined in econ?

DeleteNo, not really. But that's only his opinion. What he's saying is basically this: "in physics, equilibrium is used when we mean some sort of steady state. In economics, however, the term has a different meaning, encompassing not only the steady state, but the entire paths of the variables we're interested in, thus including transition dynamics. People who aren't economists aren't used to this meaning of the word, and thus are confused"

DeleteThis is a fair point, but not because the notion of equilibrium was ill defined in econ, and I think e-co-ngineer recognizes that.

For examples of econ equilibrium (in this case, competitive equilibrium), see Prof. Williamson's macro notes. When discussing a simple overlapping generation model, for instance, he defines the competitive equilibrium as:

"A sequence of quantities and a sequence of prices, which satisfy (i) consumer optimization; (ii) firm optimization; (iii) market clearing; in each period t=0,1,2,... given the initial capital-labour ratio".

Yes, I am familiar with this definition. Similar definitions appear in pretty much every DSGE model. However, the question here is whether one can define equilibrium in terms of its dynamic properties and yet distinguish it from the concept of a steady state. This is what I am trying to do, obviously unsuccessfully, and yes, it has been a while since I have had a chance to talk about this with someone. So let me try one last time.

DeleteThe way I see it, a steady state is a situation where the state variable (e.g. the stock of capital per worker in the neoclassical growth model) is not changing. Equilibrium is a situation where the chosen sequence (which includes current and future allocations) does is not changing. So yes, consumption, for example, may be changing along a saddle path, but it is changing in a manner consistent with what the agent decided previously. So the entire sequence is not changing. Disequilibrium would be a case where, as time goes by, agents keep scraping their old plans and choosing new sequences (regret their previous choice). So there is a definition of equilibrium that involves dynamic stability that is conceptually different from the stability we have in mind when we talk about a steady state. A non-economist may not like it or understand it, but it is certainly a well-defined concept.