Sunday, September 9, 2012

Want to Understand the Intuition for Wallace Neutrality (QE Can't Work), and Why it's Wrong in the Real World?

This refers to Neil Wallace’s 1981 AER paper, “A Modigliani-Miller theorem for open-market operations”. The paper has been influential today, as it has been used as a reason why quantitative easing can’t work. Here are some example quotes:
"No, in a liquidity trap, if the Fed purchases gold, it does not change the price of gold, just as it will not change the prices of Treasury bonds if it purchases them." – Stephen Williamson

"The Fed can buy all the government debt it wants right now, and that will be irrelevant, for inflation or anything else." – Stephen Williamson

"If it were up to me, I would have given Wallace the [Nobel] prize a long time ago, and I think Sargent would say the same. However, not everyone in the profession is aware of Wallace's contributions, and people who are aware don't necessarily get as excited about them as I do." – Stephen Williamson

"...the influence of Wallace neutrality thinking on the Fed is clear from the emphasis the Fed has put on telling the world what it is going to do with interest rates in the future...I have a series of other posts also discussing Wallace neutrality. In fact, essentially all of my posts listed under Monetary Policy in the June 2012 Table of Contents are about Wallace neutrality." – Miles Kimball
In Wallace’s model, when the Fed prints money and buys up an asset with it, this affects no asset’s price, and doesn’t even change inflation! Amazing claims, but they’re mathematically proven to be true – in Wallace’s model, and with the accompanying assumptions. So the big question is, even in a model, how can claims like this make sense? What could be the intuition for that?

For the vast majority of well-educated laypeople, the paper is impenetrable, foreboding math, and I’d say this is, to a large extent, true of economists not specialized in this area. I have a very mathematical economics background (see, for example here), and it took a lot of time and effort for me to really penetrate and understand this paper. It’s extremely terse, with very little explanation and derivation for non-specialists.

I had hoped to find someone who could explain the intuition on the internet, especially in the economics blogosphere. But after a lot of looking, and a lot of asking, I couldn’t find anything that really did it for me (The closest by far was this post from Brad DeLong.) So, I made the decision a couple of months ago to spend whatever time I could come by reading, studying, understanding, decoding, deciphering, this paper. Here are my current conclusions:

In the paper's model, the government's particular printing of more dollars and buying an asset has no effect on the price of any asset, and no effect on inflation either, but let's look at the particulars:

The government prints dollars and buys the single consumption good, which I like to call c's. It holds the consumption good for one period, storing it (investing it, or putting it into production) at the return x, the same return the private sector, or anyone else, gets for storing (investing in production) c's.

Then, at the end of that period, it takes all of those stored (invested) c's, plus whatever return it got for them at x, and uses it all to buy back dollars at the then prevailing rate of dollars for c's.

Now, note that Wallace does not say this explicitly, but if you study the equations and think about what they imply, you can see, and prove, that this is what must be happening. I spent a lot of time doing this (It would have been nice if he wasn't so amazingly terse and had explained/derived this – and a lot more).

Now, in this economy it's very simple. You either consume c's, or you store (invest) them. You can buy (or sell) state-contingent contracts to get a c in a particular state next period (People, who are clones with perfect information, foresight, and rationality, only live two periods, a young period and an old period.), but those contracts are backed-up just by one thing, storage of c's (or next periods economy-wide endowment, which is analogous to GDP not counting savings and their return).

So, in this model it's simply a case that when government prints money, it's just storing c's for one period. People are going to want to store a certain amount of c's anyway, because that's utility maximizing to help smooth consumption. What the government essentially does in this model is say, hey, store your c's with us instead of at the private storage facility. Give us a c, and we'll give you some dollars, which are like a receipt, or bond. We'll then store the c's – we won't consume them, we won't use them for anything (these are crucial assumptions of Wallace, required to get his stunning results) – We will just hold them in storage (implied in the equations, not stated explicitly).

Next period, you give us back those dollars, and we give you back your c's, plus some return (from the dollar per c price changing over that period). In equilibrium, the return from storing c's via the dollar route must be equal to the return from storing c's via the private storage facility route. Or at least the return must be worth the same amount at the equilibrium state prices; so either way you go you can arrange at the same cost in today c's, the same exact next period payoff in any state that can occur.

Note that dollars in this model are just zero-coupon bonds. Wallace assumes no value of dollars in lowering transactions costs, in convenience, or in liquidity. Transactions, liquidity, and convenience costs are zero in his model. People will hold no dollars (buy no dollars with their c's) unless their value appreciation will be equal (at the equilibrium state-prices) to if they stored their c's privately.

So essentially in the Wallace model the "open market operation", the QE, the printing of dollars, is just the government offering storage of c's that's exactly equivalent to what the private sector is offering, at no better a price (or maybe an epsilon better to get people to switch).

So what happens? Private storing (investing or utilizing in production) of c's goes down, and government storing (investing or utilizing in production) of c's goes up by an equivalent amount (and both the private sector and government get the same return from storing: x). No prices change, and people's consumption in youth and old age doesn't change.

It is analogous to Miller-Modigliani, in that if a corporation increases its debt holding, then shareholders will just decrease their personal debt holding by an equivalent amount, so that their total debt stays exactly where it was, which was the amount they had previously calculated to be utility maximizing for them (And there's a lot of very unrealistic and material assumptions that go with this that have been long acknowledged as such in academic and practitioner finance; when you learn Miller-Modigliani, at the bachelors, masters, and PhD levels – which I have –  they always start by teaching the model and its strong assumptions, and then go into the various reasons why it far from holds in reality. This is long accepted in academic finance; pick up any text that covers MM.)

There is one more powerful intuition that I'd like to note that's buried implicitly in this model:

Suppose dollars are printed and used to buy 10 year T-bonds. Or gold, like in the Stephen Williamson quote at the beginning of this post. And everybody knows (making a Wallace-like assumption) that in five years the T-bonds or gold will be sold back for dollars. We're making all of the perfect assumptions here: For all investors, perfect information, perfect foresight, perfect analysis, perfect rationality, perfect liquidity,...

Now, what is the price of gold? How is it calculated in this world of perfects?

Well, as a financial asset it's worth only what it's future cash flows are. Suppose you are going to hold onto the gold and sell it in one year. Then, what it's worth is its price in one year (which you know at least in every state – perfect foresight) discounted back to the present at the appropriate discount rate.

But suppose this: During that year that you will be holding the gold in your vault, you are told the government will borrow your gold for five minutes, take it out of your vault, and replace it with green slips of paper with dead presidents, then five minutes later they will take back the green slips and replace back your gold in the vault. Do you really care? This doesn't affect how much you will get for the gold when you sell it in a year, and as a financial asset that's all you care about when you decide how much gold is worth today.

If you're going to hold the gold for ten years, and sell it then, then you only care about what the price of gold will be in ten years. And the price of gold in ten years only depends on what the supply and demand for gold is in ten years. If the government takes 100 million ounces of gold out of private vaults, and put it in its vaults, then puts it back in the private vaults three years later, this has no effect on the supply of gold in ten years. So in ten years the price of gold is the same. And if gold will be the same price in ten years, then it will be worth the same price today for someone who's not going to sell for ten years anyway.

But what if you're going to hold the gold for less than ten years, for only one year, say? Here, I could see how you could do like an overlapping generations model kind of thing and say it still doesn't make a difference.

But I think the bottom line intuition is – with these very strong assumptions – if the price of gold, or zero coupon T-bonds, is the discounted value of what their price will be in ten years. And their price in ten years is based on their supply and demand in ten years. Then, if the government just holds it in its vault for a few of those intermediate years and then releases it, there will be the same supply of it in ten years, and thus it will have the same price in ten years. And if it will have the same price in ten years, then it will have the same price today (ceterus paribus, and with the typical assumptions of perfect frictionlessness, rationality, foresight, etc.). 

Here's a concise, and perhaps clearer version of this:

The government buys 100 million ounces of gold in a QE. The assumption is, of perfect foresight, perfect everything investors, that over the next several years, unemployment will go down and the Fed will reverse course, and then sell all of those 100 million ounces back again. Thus, the supply of gold in 10 years will be exactly the same as if the QE had never occurred. The gold just temporarily sits in government vaults (or with government ownership papers), rather than private ones, then goes back to the private vaults – no difference at all in 10 years. So, in 10 years the supply of gold is exactly the same, so the price of gold in 10 years will be exactly the same. If the price of gold in 10 years will be exactly the same, then its price today will be exactly the same, since with prefect foresight, perfect analysis, etc. investors, the today price is just the discounted 10 years from now price.

Next post, part II, I'll get to the problems when thinking if this actually occurs with QE in the real world:


JP Koning said...

Thanks for enlightening us. I've tried to work through that paper but haven't had much luck.

The idea of Wallace neutrality as you've outlined is really quite interesting. But that's all it should be... an interesting bit of counterintuitive puzzling. Folks like Stephen Williamson and John Cochrane immediately move from describing the model to advocating that model as a blueprint for reality. It's odd. Can't they see how unrealistic the assumptions are?

Incidentally, on the topic of assumptions and Wallace neutrality, it's worth checking out the recent Woodford paper:

See page 61 on assumptions:

"The irrelevance result is easiest to derive in the context of a representative household model, but in fact it does not depend on the existence of a representative household, nor upon the existence of a complete set of financial markets. All that one needs for the argument are the assumptions that (i) the assets in question are valued only for their pecuniary returns — they may not be perfect substitutes from the standpoint of investors, owing to different risk characteristics, but not for any other reason — and that (ii) all investors can purchase arbitrary quantities of the same assets at the same (market) prices, with no binding constraints on the positions that any investor can take, other than her overall budget constraint. Under these assumptions, the irrelevance of central-bank open-market operations is essentially a Modigliani-Miller result, as noted by Wallace (1981)."

Anyone who's spent some time in markets knows that assumption ii simply doesn't hold. That's not to say that Wallace neutrality isn't interesting, and that it doesn't succeed in defining some lower limit to the usefulness of QE, but it surely shouldn't be used to deny the relevance of QE outright.

MrIlir said...

It's impossible to understand inflation without having a better understanding of economic growth. Alex Gheg has a new consumer theory that gives us a scale for growth that measures previously hidden thoughts. We can measure utility growth using the body clock. Quantity, quality, variety and convenience in one equation.

Alexander Hamilton said...


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