The Fisher Paradox

November 24, 2014 2 comments

There is a bit of a paradox underlying much of monetary economics. If real rates are independent of monetary factors, then a reduction in the nominal rate should be accompanied by a reduction in the expected rate of inflation (or vice-versa). Yet we typically observe, at least in the short run, that if the central bank lowers its interest rate target, it causes a higher rate of inflation. Of course, both old monetarists and market monetarists reconcile this by saying “never reason from a price change” (always good advice) and instead, reason from a change in the money supply (and expected future money supply), assuming sticky prices in the short run and then separate the effects on interest rates into the well-known liquidity, income and Fisher effects which allows for the real rate to change in the short run and for the nominal rate to go either way.

That’s all perfectly reasonable but lately there has been a school of thought emerging known as “neo Fisherites” who are bringing this issue back into the discussion. Nick Rowe (for one) has recently been taking them to task(here, here and here).

Now let me say for starters that I suspect everything Nick says about these papers is correct, and I’m not trying to defend them. I agree that denying that lowering rates raises inflation is contrary to all observations, and I suspect (though I haven’t read them yet) that his analysis of the specific papers as lacking in economic intuition and relying on strange assumptions to “rig” the results in favor of their prior beliefs is most likely spot on. That is how I feel about most modern economic papers I read, sadly. However, I think beneath the snow job and the tiny pebble of wrongness, there is actually a kernel of insight (or at least the pebble started out as a kernel before it got all mangled and turned to the dark side) and it is closely related to the stuff I have been trying to say. So I will try to flesh it out a little bit in a way that does not contradict everything we know about how monetary policy actually works.

Note that this actually began as a discussion of monetary and “fiscal” policy, which I intend to get to but I will put that off for a future post since just dealing with this Fisher paradox will be enough to fill a lengthy post by itself, but keep in mind that adding that piece in will be important for making this model look like the real world. (And also keep in mind that I don’t mean what other people mean when I say “fiscal policy.” Frankly, it’s almost tongue-in-cheek. All macro is monetary.) Read more…

A Reply To Nick Rowe on Robustness

November 22, 2014 2 comments

 

This is a reply to Nick Rowe’s post on the fragility/robustness of equilibria. For the record, I agree entirely with his overarching, macroeconomic point. I’m just nit-picking the technical details here (which I believe is what he’s looking for).

Here are Nick’s definitions.

Let G be a game, let S be a set of strategies in that game (one for each player), and let S* be a Nash equilibrium in that game. Assume a large number of players, and a continuous strategy space, if it helps (because that’s what I have in my mind).

Suppose that a small fraction n of the players deviate by a small amount e from S* (their hands tremble slightly), and that the remaining players know this. Let S*’ (if it exists) be a Nash equilibrium in the modified game.

  1. If S*’ does not exist, then S* is a fragile Nash equilibrium.

  2. If S*’ does not approach S* in the limit as n approaches zero, then S* is a fragile Nash equilibrium.

  3. If S*’ does not approach S* in the limit as e approaches zero, then S* is a fragile Nash equilibrium.

  4. But if S*’ does exist, and S*’ approaches S* in the limit as n or e approaches zero, then S* is a robust Nash equilibrium.

[This began as a comment on the original post so I will proceed in the second person]

Nick,

I think the wheel you are reinventing is basically the idea of trembling hand perfection. I’m not quite an expert on that but I think I know enough game theory to go out on a limb here. So taking the definition from Wikipedia.

First we define a perturbed game. A perturbed game is a copy of a base game, with the restriction that only totally mixed strategies are allowed to be played. A totally mixed strategy is a mixed strategy where every pure strategy is played with non-zero probability. This is the “trembling hands” of the players; they sometimes play a different strategy than the one they intended to play. Then we define a strategy set S (in a base game) as being trembling hand perfect if there is a sequence of perturbed games that converge to the base game in which there is a series of Nash equilibria that converge to S.

I think the main difference between what you are doing and the TH concept is that you are limiting the errors to a “small fraction” of the players whereas the TH definition above assumes that all players have some probability of making a mistake.(also it assumes that all players know this not only the “non-trembling” players, which is only natural since there aren’t any such players.)

Now, I believe your game will pass both the traditional trembling-hand perfect criteria and your modified “robustnest/fragility” criteria for the same reasons, but let’s work with the standard modification since we don’t have to deal with two types. So let us assume that everyone chooses a “target” speed St and let each individual’s actual speed be Sti+ei where ei is an error with some distribution and assume that everyone’s errors are identically distributed and everyone knows the distribution.

Now there are two issues here. First, there is the issue of the number of players. If it is finite, I believe (though I haven’t done the math) that the game will break down even in its original form because when everyone is going the speed limit, any individual driver will be able to change the average slightly by changing their own speed and therefore be able to get paid by doing so and so everyone will want to do this. (Although, there might (in fact I bet there would) be an equilibrium where half of them drive over the speed limit and half drive under and the average speed is S*.)

However, if we assume an infinite number of players, then this won’t be a problem and the equilibrium (the one in question that is) to the base game will be as you say. However, now we have another issue to deal with.

First of all, let me say that the thing which makes TH difficult to deal with is the bit “there is a sequence of perturbed games that converge to the base game” which could mean a lot of different things. But let us assume that the sequence we are interested in is e converging to zero. But the problem here is that the thing that matters to each individual’s payoff is the average speed. And if the mean of e is zero, and everyone is choosing St=S* and there are an infinite number of them, then the average speed Sbar will always be S* and the equilibrium will work no matter what the distribution of e (so long as it is mean zero). This is because the distribution of sample means converges to the population mean as the sample size approaches infinity.  (And note that if the mean of e is not zero, I’m pretty sure they can all just adjust their target to account for it and you will still have an equilibrium.)

I believe this will be the case in your formulation as well since a fraction of the infinite number of players will still be an infinite number and the distribution of the mean of their errors will still be degenerate. So essentially we have an equilibrium that doesn’t work under any circumstances with a finite number of drivers and is not ruled out under any circumstances with our proposed refinements.

What we need in order to rule this out is some way of saying that the average speed Sbar might vary for some reason. For instance if there were some error e which were random but applied to every driver (like weather or traffic or something, or “real” shocks in the case of the macroeconomy), that would probably blow it up in a way that would prevent it from converging, although I think you might be able to find one, like I said above, where some people choose a target a bit over and some a bit lower than S* and the amount over/under decreases as the distribution of e collapses to zero, which could be said to be “converging to S.”

This is interesting stuff though, I’m glad you got me thinking about it. There is a sort of fundamental dilemma underlying this I think, which is that much of game theory (and economics) is built around finding conditions under which everyone is indifferent and calling it an equilibrium. For instance, any mixed-strategy equilibrium basically requires the payout function to be flat over some range of strategies. But that ends up looking a lot like the kind of thing you want to rule out when you start looking for some kind of “stability” criteria.

So what we kind of want to do is have a way of determining whether the nature of an equilibrium is such that if you “unflattened” it a little bit, each individual would have a maximum in the general neighborhood of that equilibrium that is somehow qualitatively similar as opposed to “unflattening” it a little and finding a minimum there which is sort of the case we have here. However, this is a highly untechnical way of putting things.

In this case, we only get an equilibrium to the base game there because we made the payoff function flat in that equilibrium by assuming an infinite number of players. But doing that makes other things “flat” in a sense (makes the distribution of the average speed collapse to the target speed) which makes it hard to rule out. What I think you and I would both like to say is something like “let’s assume a ‘large’ number of players such that the effect each of their speeds has on the average is functionally zero but that there is still some random variation in the average.” Then we could say that even a slight variation in the average would torpedo the equilibrium and we would be happy. But man it’s hard to do that rigorously! (I had a similar problem in my dissertation which I never really solved.)

Another thing you could probably do for this particular case is put it in the context of a dynamic game and put some restriction on peoples’ beliefs like: everyone observes the average speed of the previous day and chooses their target speed based on the assumption that it will be the same today. Then ask what would happen if you had one day where the speed were slightly above or below the speed limit. Would it work back toward the equilibrium or would it shoot off to someplace else. Here, I think obviously, it would do the latter. It’s just that with an infinite number of players and an error with mean zero, we can’t get it to depart from the equilibrium in the first place.

Incidentally, I have been working on a bit of an apology for the neo-Fisherites. I agree about the “90 percent snow job with a tiny pebble of wrongness” analysis (great line by the way) but I think there is a kernel of solid intuition in there, it’s just being applied carelessly. I’ll have that soon.

A Modified Gold Standard

October 17, 2014 3 comments

David Gordon has a piece on Mises.org critiquing Steve Forbes’ book Money. The piece is rife with confusion but I don’t want to do my usual Mises.org routine and go line by line pointing out how each point is mistaken. (They never seem to respond when I do that, which is odd because I know they notice, I can see the hits…) For what it’s worth, I haven’t read the book but from what I can gather from the quotes in Gordon’s post, it is also somewhat confused.  This quote, however, got me thinking.

“[Forbes’] gold standard allows the money supply to expand naturally in a vibrant economy. Remember that gold, a measuring rod, is stable in value. It does not restrict the supply of dollars any more than a foot with twelve inches restricts the number of rulers being used in the economy.”

This got me thinking about how gold could be used as a “measuring rod” for money without being “convertible” in the traditional sense of the word and I think that thinking about it this way may help to explain the relationship between money and debt.

Imagine that you have an economy where physical gold is commonly used as money. A bank enters this economy and offers the following deal: You can borrow X “dollars” (a unit which the bank makes up out of thin air).  At some point in the future you must repay the same number of dollars or a given quantity of gold. Let’s say that the exchange rate is one oz. of gold per dollar so if you borrow 100 dollars, you can repay either with 100 dollars or with 100 oz. of gold or any linear combination of the two. The bank has only 1 oz. of gold which it keeps in the vault to act as the “standard gold oz.” like the official meter (or was it the foot?) that the French (or was it the English?) have in a vault somewhere. If you come in to pay off a debt using gold, it is compared to the standard oz. for weight and purity. Otherwise, the bank has no gold, nobody “deposits” gold and the bank does not stand ready to sell gold for dollars or dollars for gold in the traditional sense of “convertibility.”

Furthermore, assume that the contract specifies that, if the borrower does not pay the appointed quantity of dollars and/or gold by the specified date, that the bank (by way of the courts and police) will seize real goods from the borrower which can be traded for the requisite quantity of gold. And assume that only people who can post sufficient collateral are allowed to borrow so that nobody can default.

Now, first question: How much “money” (dollars) can the bank create?

Answer: As much as people are willing to borrow.

Of course, people will only be willing to borrow these dollars if other people are willing to take them in exchange for goods. So does it make sense for people to take these dollars even though they are not “convertible” in the traditional sense into any “real” good?

Yes.

Why?

Because the dollars are convertible. The person who borrows them and spends them today will need to get them back, or else get gold back, or else forfeit some quantity of real goods at some point in the future. They are contractually obligated to do this. So if somebody comes to you and wants to buy seed corn with these dollars and you understand the contract that they signed with the bank and you believe that this contract will be enforced, you can accept the dollars and hold them until the loan comes due and be assured that the borrower will be willing to trade you some portion of his crop (or other goods) to get those dollars back.

Next question: How is the price of a dollar in terms of gold determined?

First of all, let me say that what Forbes seems to mean by the “value” of money is the price of gold and this is what Gordon is erroneously interpreting as the subjective value of money and that is a source of much of the confusion in his criticism. But putting that aside, what forces are acting on the exchange rate between money and gold and how, if at all, is this rate “fixed?”

First of all, it should be fairly obvious that the price of a dollar cannot rise much above 1 oz. of gold. This is because only the borrower has an ultimate use for these dollars (paying off the loan) and he will not be willing to trade more than 1 oz./dollar to get them. If he had to pay 2 oz. (or other goods which he could trade for 2 oz. of gold) he would instead just use the gold to repay the loan.

On the other hand, the borrower will always be willing to pay up to 1 oz./dollar because if he can get dollars cheaper than that, then the difference represents a surplus to the borrower. (If you are imagining a kind of hold-up problem, just imagine that there are a hundred borrowers bidding for the dollars.)

So this type of “convertibility” should fix the exchange rate between gold and dollars right around the rate specified in the debt contract. This does not depend on the quantity of dollars that are created this way, the quantity of gold in bank vaults or the quantity of gold relative to other goods.

Now if there is some liquidity preference for gold or dollars relative to the other, the exchange rate might deviate slightly in one direction or the other (and likewise for risk preference). The magnitude of the liquidity preference will likely depend on the quantity of dollars and gold in circulation so these things may have a marginal effect on the exchange rate between dollars and gold and this will factor into the interest rate charged by the bank and how prices change over time in a more complicated model but just ignore all that for now. And obviously, the quantity of gold and other goods affects the price of gold (and therefore dollars) relative to other goods.

The important point is that liquidity preference is not the sole (or even the main) explanation for the value of a dollar. It explains a small deviation from a certain value relative to other less liquid assets but it does not explain the existence of any value in the first place. That depends on the real assets which someone is contractually allowed/obligated (depending on how you look at it) to exchange them for. This means that it is the quantity of money relative to the quantity of debt which is the main anchor holding the “value” of a dollar in place.

“Well that’s all well and good Mike but there are no gold clauses in debt contracts so this isn’t how the real world actually works” I can hear the skeptics reply. But the skeptics are wrong. We no longer have a fixed gold “measuring rod.” But we still have fixed convertibility between dollars and real goods built into the debt contracts that create money. It’s just that the goods and the rate are not the same for everyone.

If you want to borrow money to buy a house, you put the house up as collateral. The contract specifies that if you do not repay a specified number of dollars by a specified date, the bank (via the courts and police) will seize your house (a real good). It’s the same thing.

We all (Keynesians, Austrians, monetarists, whatever) act like when they suspended convertibility of dollars into gold at a fixed rate for everyone, they severed all concrete (read: “contractual”) ties between money and real goods and money just sort of magically behaves as though it were still backed by something even though it isn’t.  That is not what happened.  They only severed one particular kind of convertibility into real goods.  But this does not require everybody to be able to exchange dollars for real assets at a given rate, it just requires somebody to be able to.  And the ability of debtors to “convert” dollars into real assets at a contractually fixed rate remains.

Of course, since this rate (and the particular goods) can vary from one contract to another, it is possible for the price of a dollar, measured in any (and for that matter all) particular real good(s), to drift over time and modelling that is a complicated matter which I have been attempting. But any attempt to model it which ignores debt entirely and assumes either that liquidity preference is all that matters or that there is no reason for money to be valuable at all except for some form of mass delusion is like trying to model the position of a sailboat based on the direction of the wind without realizing that the anchor is down.  The wind matters.  The length of rope and the depth of the water matter. But you can’t really make sense of how or why they matter if you don’t notice that there is an anchor involved.

“Negative Money” (A Variation on Nick Rowe)

October 8, 2014 Leave a comment

As I said recently, I have a bunch of outstanding business with Nick Rowe which I am trying to work through. Foremost on the list is a couple of older posts about negative money (Part I, Part II). This comes remarkably close to my way of looking at things, but let me make a couple amendments.

First, let me address another point on which Nick and I agree. Here is one of his comments on a different post.

Start out be assuming One Big Bank, that is both a central bank and a commercial bank. That issues only one type of money. And it does not matter if that money is paper or electrons. Now make an assumption about what the Bank holds constant: is it r, M, or NGDP, or what? Then ask your question.

I believe that one of the main mistakes people make which causes us to miss some important insights is to separate the central bank from the commercial banks and sort of lump the commercial banks in with the rest of the private economy as a facility that simply matches borrowers with lenders or something like that. The commercial banks play a key role in the functioning of the money supply and they have the special privilege, granted by the central bank, of performing this role. So let’s take the opposite approach and lump the commercial banks in with the central bank and treat it as one big bank.

However, instead of having it issue one kind of money, let’s have it issue two kinds: red and green. Anyone who wishes, can go to the bank and ask for some quantity of green dollars and an equal quantity of red dollars (and assume that the bank just keeps track of this in their records, as in Nick’s model, the actual paper currency is not the important thing here.

Then let us make two changes to the model. First, in Nick’s model, either red or green money can be used in exchange. Let us instead assume that only green money can be used. So instead of this.

. . . if neither the buyer nor seller of $10 worth of apples has any money, each goes to the central bank and asks for 5 green and 5 red notes, the buyer gives 5 green notes to the seller, the seller gives 5 red notes to the buyer, and they do the deal.

We would have the buyer going to the bank and getting ten red notes and ten green notes and trading the ten green notes to the seller. Notice that this difference is not particularly meaningful in terms of the model as in both cases the seller ends up with ten green notes and the buyer ends up with ten red notes. This does however, start to look a lot like how things actually work.

Second, in Nick’s model, the interest rates the bank “pays” on each type of note are constrained to be equal. Instead of assuming that, let us assume that the bank can only “pay” interest on red notes and the rate on green notes is constrained at zero. This means that the quantity of red and green notes will not be equal unless one of two things happens.

1. The rate on red notes is zero at all times.

2. Additional green notes are created and somehow distributed to balance out the red notes which are “paid” out as interest.

Now, if this doesn’t look like what really goes on in a modern economy, just replace “red money” with “debt” and “pay” with “charge” and it should start to look familiar.

This causes several things to start making sense. First, we have the whole issue of why, seemingly worthless bits of paper are stubbornly (and stably) valuable. They aren’t just meaningless bits of paper, they represent one half of a debt contract. Behind those pieces of paper is another half–red money, if you will–and a vast infrastructure dedicated to seizing your property if you hold too much “red money” for too long without producing the requisite green money to cancel it out.

Second is the issue of recessions. Once you look at it this way, it is easy (relatively speaking) to see that there are two separate but related “willingnesses” at play here. There is a willingness to hold red money (debt) and there is a willingness to hold green money (money). People hold green money until their marginal liquidity preference is equal to the foregone interest from lending the money or from “investing” in real goods. People hold red money until the interest rate on red money is equal to the marginal rate of substitution between current and future consumption. These are equilibrium conditions so there are a bunch of different ways to express them.  I tackled it more thoroughly in my model. The important thing is that there is red money and green money and people can hold different quantities of each depending on their situation.  If you only see (and your model only includes) one and not the other, you are missing a very important piece of the puzzle.

But since it is possible for the quantities of these two things in circulation to change relative to each other while they are still “convertible” at a 1:1 ratio, the real value of each type can change differently over time. And since the constraints involved in equilibrium involve expectations about these changes over time, those expectations can be wrong. And the important thing to note is that the expectation of the quantity (and therefore the value) of green money that will exist in the future is tied to the quantity of red money people are willing to hold in the future. In order for the quantity of green money to increase, people must hold more red money. If people decide to reduce their holdings of red money, they must “redeem” green money to get rid of it and this will reduce the quantity of green money.

That is, unless number 2 above happens. Number 2 is required in order to have the type of inflation expectations and interest rates that we have amount to a long-run equilibrium. Number 2 is what I meant before when I said “fiscal policy.” This is not exactly what other people mean when they say “fiscal policy” and that got me into a bit of trouble but the thing that I mean is the relevant thing whatever you want to call it.  (I’m still not entirely clear on what everyone else means by “fiscal policy”…)

If people expect some level of inflation which requires the (green) money supply to keep growing at some rate and we come to a point where the quantity of red money refuses to keep growing at a rate which will make that growth rate of green money possible, everything starts to fall apart unless the bank or the government or somebody finds a way to pump more green money in.

There are a lot of ins and outs and what-have-yous wrapped up in the last four paragraphs here but for a more careful treatment, again, see the model.

Walras with Money

October 2, 2014 2 comments

As I’ve been saying, in the standard Walrasian model you don’t get absolute prices, you get only relative prices and you have to apply an arbitrary restriction in order to make them look like absolute prices (like all prices sum to 1 or something similar) these relative prices can be multiplied by any scalar (“price level”) without changing the solution. So what if, just for fun, we try to add money in, make it an economy where all goods are traded for money, try to get a price level and see if we can characterize a general glut. This is, I suspect, exactly what most economists have in mind when they imagine a general glut and I assume it has been done before but I don’t recall seeing anyone put it explicitly in this context.

Let’s say you have an economy with n “real” goods and you also have money. The quantity of all of the goods produced as well as the quantity of money are determined exogenously. People only care about the quantity of each good they consume as well as their (average) real money balances (m/P) where m is the quantity of money an individual holds and P is the price level somehow defined. (For instance, we might let P be the sum of all nominal prices or the average nominal price or something along those lines such that we can characterize the price vector as a vector of relative prices–somehow defined–multiplied by the price level). So we have utility functions that look like this.

U(X1,X2,….Xn,m/P)

And assume, for ease of exposition, that this function is separable in money so that we can write:

Ux(X1,X2,…Xn)+Um(m/P)= U(X1,X2,….Xn,m/P)

And everyone has a budget constraint that looks like this.

Sum[Pi(Xi-Xi’)Pi]+ m-m’=0

Where Xi is the quantity of good i consumed, Xi’ is the initial endowment of good i, Pi is the price of good i and m’ is the initial endowment of money (nominal).

Now assume that you have a Walrasian auctioneer calling out nominal prices until every market clears. If you take out the money part and just have Ux() and the Xs in the budget constraints, then you will get a vector of relative prices that clears all markets. If you say that one price is fixed too low, then you get excess demand for that good and excess supply of some other good(s). If you then add to the model by saying that people change their demands for other goods in response to the constraint on their ability to purchase the good with the fixed price and you then have the Walrasian auctioneer call out prices for the other goods until those markets all clear conditional on that constraint, then you have what Nick Rowe has been talking about.

But if you have no money and the Walrasian auctioneer calls out prices which are all too high what happens? The answer is: that question doesn’t make any sense. Without money, he is only calling out relative prices. It’s impossible for them to all be too high. If the supposedly “too high” prices are all exactly half of the supposedly correct prices, then they are the same prices and the markets all clear. If the relative prices change, then you have a case where there is excess demand for some good(s) and excess supply for some good(s) and what happens depends on how you alter the model from the original to account for the persistence of this phenomenon.

In order to even consider the possibility of all prices being “too high” or “too low,” we have to change the model. We have to put money in. Luckily I did that already. So return to that formulation.

With money, the solution will be a vector of prices such that the sum of the excess demands for all real goods equals zero and everyone is holding their desired quantity of money. This means that the marginal utility of a dollar will be equal to the marginal utility of one dollars-worth of each good. This allows us to get an actual set of nominal prices (and by extension, a price level).

So let us assume that the relative price vector called out by the Walrasian auctioneer is the “correct” one (the one which would clear all markets in the case with no money). What if the price level is too low? Even if the real goods are allocated efficiently, the marginal utility of a dollar’s-worth of money balances will be higher than the marginal utility of an additional unit of some good for at least some people and they will try to trade dollars for goods. Since the number of dollars is fixed exogenously, they can’t all do this at once. There will be an excess demand for goods and an excess supply of dollars.

The only way to alleviate this situation will be for the Walrasian auctioneer to call out a higher price level. As he dos this, the quantity of real money balances will fall (the nominal value stays the same but the price level rises) and the marginal utility will rise. At some point, the marginal utility of a dollar will be equal to the marginal utility of a dollar’s-worth of any other good (since we are assuming the equilibrium relative prices) and that will be the equilibrium price level—the level at which people are just willing to hold the quantity of dollars that exist.

Conversely, if the Walrasian auctioneer calls out a price level that is too high, people will want to hold more dollars than there are and the only way to alleviate this is for the price level to fall. This is a general glut. If, for instance, the money supply contracts, prices will need to fall to bring things into equilibrium. If they can’t fall because they are “sticky” for some reason, then you may get a general glut in which the excess supply of real goods is offset by an excess demand for money.

Now does this contradict Walras’ Law? Not exactly. Since we changed the model, we have to change the characterization of the law before we can ask a question like that. If what you mean by “Walras’ Law” in this context is that an excess supply in the market for some real good, measured in dollars, must be offset by an excess demand in the market for another real good, measured in dollars, then no. If what you mean is that an excess supply of goods must be offset by an excess demand for something, potentially money, then yes. Is the latter characterization of the law meaningless? Maybe some would say yes but I think that a lot of people out there could benefit from carefully considering in what sense “Walras’ Law” applies in an economy with money and in my book, that makes it pretty useful.

For the record, this is pretty standard stuff, I don’t think I’m saying anything groundbreaking here. I also think there is more to the story but saying groundbreaking things is hard. I’ll get around to it eventually.

 

More on Walras’ Law

October 1, 2014 2 comments

Have taken a hiatus from blogging to deal with moving, new job, weddings, etc. and trying to get back in the habit so I figure I will finish up a post on Walras’ Law that I mostly wrote a while ago.  The topic may be a little stale now but whatever.  After all, this debate seems to have been going on for years.  I have a bunch of outstanding business with Nick Rowe but am having difficulty putting it all together.  After this little warm-up, I will try to work through that backlog.

Following the latest [at the original conception of this post] installment from Nick Rowe, it is pretty clear to me that there are three distinct issues which are all mixing together in the discussion so I want to try to separate them.  I will go through them in increasing order of significance.

1.  Is Walras’ Law useless?

I say no but that’s because I’m a micro guy at heart (and in training).  And for the record, I think I got kind of a weak acquiescence out of Nick on this so I don’t think there is very much room between our views but just for the record, here is my argument.

This is the entry from the index of Mas-Colell, Whinston and Green (the standard graduate micro text).

Walras’ Law: 23, 27, 28, 30-2, 52, 54, 59, 75, 80, 87, 109, 582, 585, 589, 599, 601, 602, 604, 780

Why am I telling you this?  Because I’m trying to demonstrate that if you want to expunge Walras’ Law from the record, you will need to totally rewrite microeconomics.  You can’t solve the Walrasian model without it.  You can bad-mouth the Walrasian model all you want, I’m not saying it perfectly represents every aspect of a real economy but if you want to tear down the pillars of that model (rather than adding on to it) you are essentially taking a wrecking ball to the rock on which our church is built.  Some people will argue for doing that, for sure, but it’s a rather extreme position which I don’t think is what folks like Nick really want.

Now the real issue is some people like to misuse the law by applying it carelessly to other models without doing the necessary work to determine whether it actually makes sense or not in those contexts.  This, I think, is what Nick objects to.  I didn’t carefully go through all of the above sections but I would be willing to bet that nowhere in there does it say that Walras’ Law proves that if we observe a shortage in some market because the price mechanism is not functioning in the way specified in the model, then there must also be a surplus in some other market.

2.  What if some price doesn’t adjust?

The Walrasian model is a model of price adjustment.  If you want to hold some price constant and ration quantity somehow, you are changing the model.  That’s fine, but you can’t take a “Law” from a different model and just try to slap it carelessly onto your new model.  If you fix the price of some good and put a quantity constraint on buyers of that good, you can find a vector of prices for the other goods such that all other markets clear given that constraint.  Whether this “violates” Walras’ Law is a nonsensical question because that law can’t be stated in the same way in the new model.

If you want to have an analogue for Walras’ Law in your new model, you have to redefine things.  The way I would go about doing this would be to treat it as a model of price adjustment in the markets for the n-1 goods, since there is nothing happening endogenously in the other market (at least nothing interesting, you have a kind of “corner solution” where you run into the constraint).  Then you would get a version of the law that applies in the subset of the market where the price mechanism is functioning in the same way that it functions in the original Walrasian model.

Alternatively, if you want to get a bit more esoteric, you can define excess demand for each good in real terms (in quantities of other goods).  This will complicate your model because you will need a lot more prices, but then you can take the price vector to be all prices, including the fixed price, and you will find that even when the remaining markets “clear” given the constraint, there is still some “excess supply” (assuming a shortage in the fixed market) of those goods relative to the good whose price is too high.  This is the sense in which Walras’ Law indicates something about such a market that is true but this phenomenon will not show up if you just look at any one of those markets and see if there is a shortage or surplus at the prevailing money prices (which is another reason to keep it, but only if you use it carefully).

This is all consistent with everything Nick has said but it is worth mentioning that the issue isn’t whether we think of it as one market for n goods or n markets for goods and money.  The issue is what constraints we put on people’s behavior and how we define things like excess demand and Walras’ law in the presence of these constraints.  The original model is set up in such a way that defining this in terms of money is equivalent (at least in equilibrium) to defining it in real terms and makes the model simpler.  But the reason it is equivalent is that when all prices can freely adjust, the marginal rate of substitution between any good and any other good has to be equal to the ratio of their prices in equilibrium so the marginal value of apples measured in dollars worth of bananas has to be equal to the marginal value of apples measured in dollars worth of papayas.  This means that instead of measuring the marginal value of each good in relation to each other good and getting a price of each good in terms of every other good, we can just measure the marginal value of each good in terms of dollars and get a price of each good in terms of dollars and have only n prices rather than n(n-1)/2 prices.  The whole matrix of relative prices in equilibrium can be expressed by this vector of dollar prices because of the equilibrium conditions on all of the marginal rates of substitution.

But once you stick in a price that doesn’t adjust, this will not be the case in equilibrium.  The marginal value of a good will be equal to the same dollar amount of every good whose price is free to adjust but not of the good whose price is fixed.  So how do we define excess demand?  In real terms or nominal terms?  The answer is: it doesn’t matter, it’s just two ways of describing the thing that happens in the model.  The important thing is whether we understand what is going on in the model.  If you just memorized Walras’ Law, without really appreciating what it means and tried to clumsily apply it to every model, then you probably don’t understand.  But by the same token, if you were never taught Walras’ Law at all, then you probably never understood the original model and you still probably don’t understand.  (Neither of these is meant to apply to Nick, who, I think, completely understands what is going on in the model.)

3.  What is the role of money in all of this? (And is a general glut possible?)

While the most recent rounds of Walras-bashing have centered mainly on the issue above, the original debate (which started years ago) was mostly about general gluts.  Walras’ law seems to imply that such a thing is impossible, yet we seem to observe them.  This is a different question from the one above.  Above the question is can one market be out of equilibrium while all others are in equilibrium?  Here, the question is can all markets be out of equilibrium in the same direction (excess supply) at the same time?

This is where the role of money becomes critical.  The Walrasian model is not a model of money.  Money is used as a rhetorical device to streamline the model.  There is no attempt made in that model to characterize the demand for money, the velocity of money or anything like that.  It is assumed that people don’t care about money, they only care about “real” goods and that money is nothing more than a mechanism which somehow allows the market to work perfectly, eliminating any frictions and allowing the “Walrasian auctioneer” to call out more complicated matrices of relative prices as a relatively simple vector of nominal prices.  (Though it is worth noting that this does restrict the set of possible relative prices.)

So this begs, not the question: does Walras’ Law hold in the real world, but the question: is that really how money works?  And the answer to that is obviously no.  Since the answer is no, it is dangerous, again, to take a simple conclusion from such a model and clumsily try to apply it to the real world.  But, also again, that doesn’t make the model worthless.  Another question one might ask is does money work kind of like that sometimes?  This is sufficiently vague to admit of no concrete answer but there is room to argue in the affirmative I think.  A better question is how does the actual nature of money differ from that assumed in the model and what are the possible consequences of that difference.  It’s questions like this that allow us to climb onto the shoulders of giants like Walras and hopefully see a bit further over the horizon.

Of course, I have a lot of thoughts about that which I will mostly avoid getting into here.  But here is a question that I think is worth pondering.  If a technology were developed tomorrow that allowed barter to be carried out frictionlessly, like with the Walrasian auctioneer, what would happen to the value of money?  Would it go to zero?  (Hint: no.)

 

 

 

Walras’ Law, General Gluts and Zero-Sum Economics

August 29, 2014 13 comments

There is a thread in the blogosphere that seems to have started years ago but which popped back up recently about general gluts and Walras’ law which I have been pondering for a while and have a lot of comments on.  I will take them one at a time.  This will be kind of a deep dive.  Here are the relevant posts going back to 2011.

DeLong [2011]

Dan Kuehn [2011]

DeLong [2014]

Rowe [2014]

[Update: missed the most important one] Rowe

1. Walras’ Law and general gluts

First of all, I think that Nick Rowe, who is becoming my favorite econ blogger, is being a bit too reactionary about Walras’ Law when he says this:

Walras’ Law is the biggest fallacy we are still teaching in economics.

But before I get to why that is, let me agree with what I think are most of his general points.  At the heart of the matter is the question of whether there is such a thing as a general glut.  Some people cite Walras’ Law to argue that such a thing is not possible but this is an abuse of the law.  It reminds me of another similar law (theorem) which is particularly near and dear to my heart: the Coase Theorem .

The Coase Theorem says essentially that if there are no transaction costs, any particular assignment of property rights will result in an efficient outcome.  There are two ways that people commonly abuse this theorem.  One is to say “the Coase theorem proves that property rights don’t matter.”  Another is to say “well we observe inefficient outcomes so clearly the theorem is wrong.”  Both of these entirely miss the point of the theorem.  It is not meant to assert that everything is efficient all the time and property rights don’t matter.  It is to focus our attention on the necessary cause of market inefficiency, namely transaction costs.  If transaction costs are present, you might expect the assignment of property rights to matter.  Similarly, if you have what seems to be a market failure, you aught to be looking for some type of transaction cost (which could be a lot of things) as the culprit.  (Also note, for the record, that there is a big difference between getting an efficient outcome with any distribution of property rights and property rights not mattering.)

The same thing is true of Walras’ Law.  (Though I must say I have never read Walras, so I can’t speak to what he was actually trying to argue but this, I believe, is the proper context in which to view the law.)   Walras’ Law is “true” in the context of the model in which it is derived, the general equilibrium model of an endowment economy with a Walrasian auctioneer, a point which, for the record, Nick acknowledges.  This does not mean that the conclusion reached in the context of that model must be true in the real world, but that doesn’t mean that it is a useless thing to teach.

If your physics teacher told you that the Law of Gravity proves that a bird can’t fly in a vacuum, and you came to the conclusion that therefore, birds can’t fly, you would be mistaken.  However, if you came to the conclusion that, because the Law of Gravity implies that birds can’t fly in a vacuum and birds can, in fact, fly, that the law of Gravity is wrong/useless, you would also be mistaken.  It just so happens that birds don’t exist in a vacuum.  What you should do is notice that air must play some critical role in the process of flight.  This is an important insight into the process which is easier to notice once you have ruled out the possibility of flight in its absence.

Now the “air” in the Walrasian model is money.  In that model, money acts as a “veil over barter.”  If you had a frictionless barter economy, the Law would apply.  And for that matter, if you have an economy where money serves only to replicate a frictionless barter economy, it will apply.  With this in mind, if you think you might be observing a “general glut,” there are two ways of interpreting it.  One is to say that such a thing is not possible so we must actually be observing something else.  This, I think, is the basic idea that Nick is arguing with and he is right to do so.  The other, I think more correct approach, is to say that we don’t appear to live in a frictionless barter economy (or something analogous).  Specifically, I would argue that the role of money is much more important in ways that are not captured by that model.  And this is exactly what Nick is arguing.  So I whole-heartedly agree on that point.  However, I think that rather than proclaiming Walras’ Law “wrong” and saying we shouldn’t teach it, he should be using it to make his point by saying “look, we have this law which says that if money performs its task of making the economy work like a frictionless, barter economy, we wouldn’t have general gluts and yet we have them.  So this should tell us that the root cause of them must be somehow contained in the functioning of the monetary system.”

2. More on the Walrasian model (skippable)

This is kind of in the weeds and a bit beside my main point but I can’t resist going a little deeper.  Take Nick’s example.

An excess demand for bonds cannot cause demand-deficient unemployment. Remember my three women? The hairdresser, manicurist, and masseuse? Suppose they all have an excess demand for bonds. They want to sell their services for bonds. But they can’t, because none of them wants to sell bonds. So do they suffer deficient-demand unemployment? Not if they can barter their way back to full-employment. And Walras’ Law is supposed to be true in all economies, whether barter or monetary.

To me this is playing fast and loose with the model.  Of course, there is a lot of that going on on all sides here, so this is also a criticism of the excess-demand-for-bonds-causes-a-general-glut theory.  In the model, you can’t have an excess demand for money because there is no demand for money because money is acting only as a veil over barter.  People only actually want goods.  Now if you put bonds in as a good, then people are actually willing to trade hair dressing and massages for bonds.  This means that if the Walrasian auctioneer calls out a certain vector of prices for these goods and there is an excess demand for bonds there will be an excess supply of other goods.

However, I don’t think this model was ever intended to be a model of disequilibrium.  Walras’ Law is meant to show that such a frictionless economy will tend toward a general equilibrium.  It is analogous to saying, in an individual market, that supply is upward sloping and demand is downward sloping.  Saying that doesn’t mean that the model is always in equilibrium.  But it does imply that if people are allowed to bid prices up and down, and there are no other funky constraints, that it aught to tend toward an equilibrium and that equilibrium aught to be stable.  If you had both upward sloping or downward sloping or if the process of bidding prices up/down worked differently than hypothesized, prices and/or quantities might shoot off to zero of infinity or something.

By the way, notice that when people say things like “the Walrasian model does not specify any kind of mechanism for determining prices” they are not really being straight.  The mechanism is the process of bidding prices up when there is a shortage and down where there is a surplus.  This is explained in any intro class but then promptly forgotten by professors who like to pooh-pooh classical economics.  The Walrasian  auctioneer is essentially used to represent this mechanism in a simpler way.  Walras’ Law is analogous to this in a general-equilibrium sense in that it tells you that if one market is out of equilibrium for a given vector of prices, then some other market must also be out of equilibrium in such a way that the hypothesized process of bidding prices toward equilibrium in both markets moves you toward a general equilibrium rather than spiraling off into some kind of black hole.

Now, just like you can add in a constraint in the market for a single good, like a price ceiling and find that the market wont then arrive at the Walrasian equilibrium but rather at some other equilibrium, you could do this in the general equilibrium model.  If you said the price of bonds is X but include a constraint limiting the quantity demanded to some amount less than would be demanded at that price, and let every other market clear, you will arrive at some kind of second-best equilibrium in which the prices of every other good clear the markets for those goods given the constraint on bonds.  But in some sense, there will still be an excess supply of those goods relative to bonds given the price X.  People would be willing to trade hair dressing and/or massages for bonds at that price if they could.  What Nick seems to be saying is that this doesn’t matter because they don’t care about bonds, they only care about real goods and the real goods markets are all in equilibrium.  But this is treating bonds like a real good in one sense and then later treating them as if they don’t matter and are just part of the veil.  If nobody cares how many bonds they have, then there isn’t really an excess demand.

The same thing applies to unobtainium.  If you make a model with a price for unobtainium and a constraint setting the quantity equal to zero and let the rest of the markets find a market-clearing equilibrium, you can say that there is excess supply of other goods relative to unobtainium at those prices but that the rest of the market is in a second-best equilibrium given the constraint.  This does not mean that this “excess demand” for unobtainium or bonds causes a general glut or unemployment of course (technically Nick’s point in this context), but I also don’t think it makes Walras’ Law wrong, it’s just playing with definitions in an esoteric and not very helpful way.  So I feel like Nick is  shadow-boxing with an erroneous expression of Walras’ Law crudely repurposed as a theory of recessions (or I guess a couple different theories….or is one an anti-theory?)

I can see why a macro guy would be annoyed by these but I think we should avoid throwing it out all together.  At most, we should let the micro folks keep it but make them promise not to bring it up in discussions of the business cycle at faculty cocktail parties.

3.  Shortage of bonds

Having said the above, I think the whole shortage-of-bonds thing is a silly basis for a theory of recessions because I don’t think I have ever seen a shortage of bonds.  There is a big difference between high demand and excess demand.  If you can go buy a bond at the market price, it’s not excess demand.  A shortage of money is a different story (again, as Nick notes).

4. Zero-sum economics

Dan Kuehn:

You will occasionally hear people say that general gluts can’t happen. That’s zero-sum economics, and it’s been proven wrong empirically and theoretically time and time again. We need more Smithian and Keynesian economics, but I don’t think that means we need less Hayek or Lucas. It simply means that this paradigm shift needs to be more completely integrated and we’ve got to stop this balkanization of the discipline that is always looking for grand ideological fights.

I just want to point out that, even though I am not saying there cannot be a general glut, such a statement does not amount to zer0-sum economics.  I want to do this because I think the single most important thing that we can learn from economics, and probably the thing most commonly forgotten, is that trade is not a zero-sum game.  But there are two ways in which this is the case.  One way is that we can actually get more stuff.  This is what Adam Smith had in mind when he talked about specialization of labor being limited by the extent of the market.  The other is that people have different subjective values of goods.  This means that even if the quantities of goods are fixed, trade can be mutually beneficial.  It’s important not to forget the second one.

Because of this second consideration, saying that a general glut can’t happen is not adopting a sort of zero-sum economics.  It is possible to have a fixed level of output without having an efficient allocation.  It is similarly possible to be on the production possibility frontier without being at the efficient point.  Because of this, there is a place for models of trade that work from a fixed quantity of goods.  And by the way, if you have a PPF that is shifting around endogenously based on the decisions you are trying to model, you haven’t specified it correctly.  [There Is some room for it to evolve endogenously over time but I don’t think that is what we’re talking about here.]

5.  Money: two goods, one price

It’s a notion which is almost as dirty as it sounds.  Here’s Dan Kuehn.

Money is really two products trading at the same price: it’s a medium of exchange and it’s a source of liquidity.

Now here is the most important point relative to the core agenda of this blog.  Money is, in a sense, two things but they are not the two things that DK claims and they do each have their own price (though they are related).  This highlights perfectly the central misconception that nearly everyone seems to be under regarding the nature of money.  Medium of exchange and source of liquidity are not different things.  They are the same thing.  To say that money is “liquid” is only to say that it is a convenient medium of exchange.  Anything can be used as a medium of exchange.  Some things are more convenient than others.  “Liquid” is just a word we use to say that something is particularly convenient for that purpose.

But deep down we know that it makes no sense for something which would be otherwise worthless to be used as a medium of exchange.  We know it has to have two uses.  This becomes paradoxical in the presence of the prevailing theory of fiat money which tells us that money has no other purpose, so we end up saying things that don’t make sense like medium of exchange and source of liquidity are two different things.  To try and sort this out, let’s go back to commodity money where the actual  two things can be easily seen.

If you had a pure barter economy, gold would be valuable.  It has use in various applications, most notably making things look more interesting or impressive than they would otherwise look, which is one of man kind’s greatest aspirations.  In other words, it’s a “real” good.  Of course, barter is complicated and gold is a particularly convenient good to use for trading because it is durable, easily identifiable, divisible, etc. so it makes sense for it to become a common medium of exchange.  This means that it would be highly liquid.

Now, in this case, gold is serving two purposes.  It can be used to make things look cool (and whatever else it can be used for) and it can also be used as a medium of exchange.  If it had no added value as a medium of exchange, it would still be valuable.  It would have some price relative to other goods and that price would change at some rate.  That rate of change in its price would be such that the value of it in terms of other goods would increase as the value of other investments excepting unexpected shocks to the long-run supply and demand.  That rate would be the interest rate (see the Hotelling Rule).

Already, there are two prices of gold, the spot price and the rate of interest (the spot price would really be many prices because there are many other goods but let’s just call it one price).  But of course, the interest rate would not be specific to gold, it would be the same for all assets so it wouldn’t really be the price of holding gold, it would be the price of holding wealth in whatever form and the spot prices of all of those various forms would distinguish their respective values in use.

Now, if we let gold be the common medium of exchange and therefore the most liquid good available to hold as wealth, people will prefer holding it to holding other goods at the same rate of return.  This means that the rate of return on holding it will fall below that of other goods.  And this means that there will be two rates of return.  Let the rate on all other assets (investments), measured in other assets be r and the rate on gold, measured in other assets be d (for “deflation”).  Since this return has to come from the price changing and there is now a differential between the rate at which prices are changing and the real rate of return on other investments, this differential must manifest itself in the form of a third rate which we know as the nominal rate or i.  This nominal rate is the price of liquidity.  (And naturally, the spot price will be higher than it would otherwise be because the price has to fall more slowly, or alternately because the added benefit from liquidity will cause it to be used up at a slower rate.)

So now you have a good with two purposes and two prices.  There is a price paid for the liquidity which is also the price paid for services as a medium of exchange.  This is the nominal rate.  There is also a spot price, let’s call it the price level, which includes both the value of the eventual use and the value of holding for liquidity purposes.  The system is not over-identified.  You just have to look at it as a dynamic system.  Oh, and you have to notice the two reasons that money (gold) is valuable.

Now, if you have gold notes, which are redeemable in gold, you have the same thing.  The notes are valuable because gold is valuable and because they are useful in exchange and gold is valuable because it is useful for other things like making stuff look cool and because it can back notes which are useful in exchange.  That part is pretty straightforward.

Where it goes off the rails is when notes are suddenly not backed by anything, or at least not by anything obvious.  Then we are left trying to say with a straight face that they are valuable only because they can be used for exchange.  The reality, as I have been saying, is that this is not the sole reason.  They still have value for two reasons.  One is that they are nice and liquid and the other is that they are the contractually required payment of debt and defaulting on debt has real consequences.  It’s not that they are not backed by anything, it’s just that the assets backing them are not uniform.  They are our houses, cars, businesses, etc.

Once you realize this, you can start to see why money does not act as a veil over barter, just like gold doesn’t act as a veil over barter.  Gold is a real good and a medium of exchange, money acts as a medium of exchange and a contractual claim on specific real goods.  So in the Walrasian general equilibrium with no money and gold as a good, you can get an excess demand for gold and an excess supply of all other goods.  Similarly, you can get an excess demand for money and an excess supply of all other goods with “fiat” money.  But in order to really grasp what is going on, you have to look at money and credit as a form of trading between periods in a dynamic model.

When you do this, it is clear that the expected rate of inflation and the nominal interest rate make up an important part of the budget constraint.  Since money is “convertible” into peoples’ houses, cars, businesses, etc. at a fixed rate, their willingness to hold money and debt depends on how valuable they expect that money to be in the future.  If they suddenly decide that it might be harder to get money in the future to pay off their debts, they may try to hold less debt and/or more money.  At current pries this is an “excess demand” for money which really represents an “excess demand” for future goods (like keeping their houses) relative to current goods.  But this only happens because their budget constraints across present and future goods shifted inward unexpectedly.  If everyone keeps eating out twice a week, they will all be unable to pay off their mortgages at the new expected rates of interest and inflation so they try to get more money and fewer current goods.  if prices cannot fall, you get a general glut.

So if you want to work at it, I think you can put this into the context of Walras’ law (more or less) if you imagine a dynamic version of it.  But this is only possible once you recognize the true nature of money and abandon the “fiat explanation of value” (it makes sense because we say it makes sense), and the notion of a veil over barter that goes with it.  And in order to do that, it helps to sit around and ponder what Walras’ Law really means and how it relates to what is actually going on for an afternoon or six.  And in order for people to do that, we have to keep teaching the damn law!

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