The Intuition behind Wallace Neutrality

[PDF version available here.]

Why not "Attempt 4"? After all, I had my original in 2012, part 2  in 2013, and part 3 last August.

It's because here I think I've really nailed it, really gotten at the heart of it, in a way that's relatively quickly accessible to professional economists (and maybe to well-educated laypeople too, but probably only with a great deal of side googling and reading). So, I think this one really stands out from the others, and is the one that I'd like to make my, "the explanation", or at least my relatively concise and accessible explanation.

It's not that I think my earlier attempts were wrong (by and large). It's just that there are many valuable intuitions you can get from Wallace '81, but what I'm about to present to you, I think, really gets at the heart of it, and thoroughly, from the start of the process through to the finish.

Now, as you can quickly see, this is very long for a blog post. But I think it's well worth the time to be one of the very rare people to really understand (and not misunderstand) this crucial model in the quantitative easing, and just whole monetary policy, debate. However, I will next, in a future post, try to condense this a lot, and perhaps make it more accessible for well-educated laypeople.

But really, for economists and other professionals, this is not that long at all to be one of the few people in the world to really understand this intuitively. And the depth it contains is really worthwhile for professionals. It may sound egotistical for me to say that, but I honestly think it’s true, and important to make clear. It's still short for an academic work (with some blogging humor, informality, and commentary), and I think it's well worth the relatively short time.  

O.k., so let's get to it:

Suppose the government decides to do a quantitative easing (QE) where it creates, and sells, one more dollar ("unit of money"). In the Wallace model, it sells dollars for the single composite consumption good, which I have nicknamed C's.

C's are the real deal, real goods, what everyone ultimately wants, and the one and only argument in their utility functions. And, C's are what people sell their financial assets for, when they eventually sell them.

So, that dollar is sold to, let's just say in this first scenario, one person, even though everything is infinitely divisible in this model.

What's the cost, the price in C's, of that dollar?

Well, first, we're trying to see if this QE can be done without changing the price of any asset in the economy at all, whether financial or real. And that includes money, so with no inflation. And this is what Wallace claims in his Irrelevance Proposition.

Thus, we're going to assume that all prices remain unchanged; they are as before. And then we will see if equilibrium still holds after the QE if we still have those original prices.

The previous regime we call the "_" regime. So, the previous state prices at time t were s_(t). The previous money supply in circulation was M_(t), and so on. And the previous price of a unit of money in C's was p_(t), all consistent with Wallace's notation.

So, if p_(t) is 4, then it takes 4 C's to buy one dollar. If all you have is 1 C, then you can only buy a quarter.

Now, we started in equilibrium in the "_" regime. This means, by the rules of this model, and almost all modern macroeconomic models, that all people were perfectly optimizing at the current, and forecasted, prices. All people had, with their perfect, super-human, foresight, knowledge, expertise, and calculating ability, figured out the 100% maximum utility consumption plan for now and the rest of their lives. And they we're going to follow it perfectly, with their perfect self-discipline; saving and investing exactly as necessary to accomplish it.

And also, markets are assumed to be complete (which is actually crucial), and frictionless, in Wallace’s model.

So, any given person h, of generation t, had a consumption path over his two period life, ch(t), that perfectly optimized his utility, given his budget constraint, and given the ability to buy and sell anything perfectly and frictionlessly in the complete markets with state prices s_(t) .

So, as I started to say, before I was so rudely interrupted, by myself, suppose the government decides to do a QE where it creates and sells one more dollar.

Some person, we'll call him person z, decides to buy that dollar for p_(t) C's. And, the government decides to, in turn, promise to buy it back from him next period for the perfectly foreseen, at least given the state, market price of p_(t+1).

I'll later get to why we assume the government will promise to do this, and will honor its promise with 100% certainty.

Remember, in standard macroeconomic models today, including Wallace '81, everyone has perfect foresight about, basically everything, including the state-dependent future path of all prices. They don't know what state will occur, but they do know what any price will be perfectly given a state, or path of states, occurring.

And, the government promises more:

When they sell that dollar they will store/invest the C's they get for it, earning a state-dependent return of x(t+1). The state-dependent return vector for storing C’s at time t is specified in the Wallace model. It’s considered exogenous, and very interestingly, and consequentially I think, it does not depend on quantity.

The demand to put your C’s into this x investment, no matter how big it gets, never pushes down the return. It is essentially like continuous-returns-to-scale technology. You never run out of x type investments or projects. Their supply curve is perfectly flat, at least for quantities as high as could ever occur in the world of the model.

The important results of the model depend on this. And I really found it interesting because for years I’ve had an explanation of the equity premium puzzle that was based on this kind of supply.

So, please bear with my digression here:

Basically, my idea is that equity gives managers a lot more flexibility in how they invest funds, especially long-term, compared to much more constrained and difficult debt. As a result, the firm can invest the funds more efficiently and productively over the long run with this increased flexibility. Therefore, they can put the funds in higher, even risk-adjusted, return projects than they can with debt-raised funds – There’s no worry about potential disaster from having to make short run interest payments so you decide to pass up better much higher expected return, but longer run and somewhat more risky projects, that, again, to be very clear, are higher expected return even when adjusting for their increased risk.

If the supply curve of these good, equity-based, flexible long-term projects in the world is very long and flat, at about the historical high average equity return (approximately 7% real), then even if the demand for equity did get really high, because it’s such a good deal, with such a long flat supply curve of projects at a rate of 7%, the equilibrium rate still won’t get pushed down below 7%.

Essentially, with this explanation there’s always going to be a really big equity premium, not because of something puzzling about people’s utility functions, or behavioral factors, or there's some hidden source of risk we're not seeing, but just because you will always be able to find equity projects that have a much higher return than the average debt project – as many as you need. Equity just increases the efficiency and productivity greatly by not having all of the hassles and constraints that come with debt, and the result is projects that produce a lot more, especially over the long run, even when considering any increased real risk (although the question still remains, why don't investors jump on this more; why is there not more stock in their portfolios).

You hear a concern with the equity premium puzzle that once people realize that the risk-adjusted return on equity is so high, the demand for equity will shoot up, and its expected return will come down. But this will not be the case if equity-based projects just have a very real and very substantial flexibility and efficiency advantage. And the number of such possible projects is easily high enough to meet even a great increase in demand.

I have a brief write up of this here.

Of course, this is an explanation for why the risk-adjusted equity premium is so seemingly high. There's still the "puzzle" of why people then don't jump on this much more. Why then do they invest so relatively little in stocks, because, after all, we know the market is sooooooo efficient, and people are soooooo rational, and knowledgeable, and expert – in everything, in an ultra-complicated world that's light-years from 1810. Otherwise, libertarianism might look a lot worse, and a government role a lot better. Horrors!

But might I just offer a crazy idea. Perhaps the "puzzle" of why people seem to underinvest in stocks when they appear to carry such an abnormally high risk-adjusted return is not because they know some hidden sophisticated kind of risk that finance professors don't and are missing. Perhaps, just perhaps, instead, it's because 65% of people answered incorrectly when asked how many reindeer would remain if Santa had to lay off 25% of his eight reindeer.

Anyway, I’ve never seen my supply-side explanation of the equity premium puzzle in the literature, and I’ve studied this a lot. As a finance PhD student, you can’t help but. And I don’t know why. It seems to make a lot of sense. But assuming there’s not something wrong with it that I don’t see, it’s likely going to take someone with a name, and/or at a name, writing it very formally, full of math, and completely, to get it considered.

Anywho,

Ok, end of digression, back to Wallace's model.

Recapping where we left off in our QE:

1) The government prints one more dollar. It sells it to a person for the price of p_(t) C's.

And p_(t) is part of the previous regime of prices, the "_" regime, which had us in equilibrium before the QE.

2) The government takes the p_(t) C's it gets and stores/invests them for the state-dependent return it will get next period, x(t+1).

3) Next period – time t+1 – the government will buy back that extra dollar that it printed in its QE.

So, that's where we left off. Now let's continue.

The government sold a dollar for p_(t) C's. It stored/invested those C's for a return of x(t+1) to get p_(t)x(t+1) C's at time t+1.

Then, also at time t+1, the government will buy back that dollar it printed in the QE for p_(t+1) (That will be the price, if the price path that we started with before the QE stays the same afterwards. And we will see if equilibrium can still hold if it does.)

All of this is implied by the equations in the model, primarily requirement (b) of the Irrelevance Proposition.

Therefore, the government will make a profit/loss from the QE, at time t+1, of:

p_(t)x(t+1) – p_(t+1)  

What does the government do with this profit or loss?

Equation (b) requires that the government foist it on the people. The government gives it to one or more members of the citizenry by adding it to their taxes net of transfers. If it's a profit, hey, tax cut! and/or increase in transfer payments! If it's a loss, tax increase, and/or decrease in transfer payments. But due to the perfect foresight and public information that Wallace's model assumes, the people know exactly how the government is going to do it beforehand, and to which specific citizens. And they act accordingly in an optimal way.

Now, WLOG (without loss of generality; this will still be true in general, even though now we will restrict our attention to a specific case.), for simple clarity, let's, for now, assume the case where the government announces its plan to foist this profit/loss on just one person, our intrepid person z. They're not going to split it 50-50 between persons z and w, or 71-29, or split it evenly among every person in the country. They could, again WLOG, as I'll go into later, but here the whole profit loss that will occur at time t+1 will go just to person z.

How will person z, an optimizing mother–shut your mouth! (Shaft reference for you young'ns), react to this?

Well, person z was optimizing before the QE. The amount of saving he chose, and how he chose to invest that saving, was optimal given the prices (and state dependent price paths) at the time, the "_" ones.

If those prices don't change, as we posit, but now person z finds out he will be getting, p_(t)x(t+1) – p_(t+1), at time t+1, how will his optimal strategy change?

Will he save more in his two period life? Will he invest his savings differently? Buy a different set of state-price contracts, store/invest more for the return x(t+1)? What?

Well, the first thing to notice in answering this question is that the profit/loss, p_(t)x(t+1) – p_(t+1), at the existing "_" state prices, is, at time t, worth,…

Zero

You're giving person z, at time t, something that's worth zero at time t. Its net present value at the current ("_") state prices is zero. It's not a cost, and it's not a benefit.

Why?

We started in equilibrium, so there were no arbitrages, and Wallace explicitly requires this with equations (3) and (4).

If p_(t)x(t+1) is worth more than p_(t+1) at the time t state-prices, then there would be an arbitrage. You would just:

1) Sell short a dollar to someone, and get p_(t) C's. You now owe them a dollar at time t+1, which will cost you p_(t+1) C's at time t+1.

2) You take your p_(t) C's and store/invest them at x(t+1) which will give you p_(t)x(t+1) C's at time t+1. You use those C's to pay off the person you owe from the short sale a dollar, which costs you p(t+1) C's to buy.

So, this strategy cost you nothing at time t, and at time t+1 you get:

p_(t)x(t+1) – p_(t+1)

If this expression is not zero, then an arbitrage would exist. So, given you assume that the economy starts in equilibrium, you assume that this expression is equal to zero.

I'm not going to show the details of every arbitrage in this post to keep it from really getting long, but if you'd like to see them, just email me. And, I'll eventually do an article version of this which will at least be at my academic site, which will go through all claimed arbitrages in end notes or appendices.

So, the profit/loss that the government foists upon person z (or some combination of citizens in the economy) is worth zero at time t. So person z can completely rid himself of it for free. He can sell it in the markets for nothing. And, in fact, that's just what he will do!

How can I be so sure of that? Well, whatever his utility function was, he was optimizing perfectly before the QE at the market prices (and their state-dependent paths) that existed before the QE. If those same state prices still exist after the QE (as we're assuming, and then seeing what happens), then he will choose the same consumption and investment path as before. He won't change anything. Give him some new investment, worth zero, that changes his state-dependent consumption and investment paths, and he will sell it.

A good way to look at it is this: There are perfect complete frictionless markets, and perfect people; a person has a lifetime path of income and transfers net of taxes. And in optimizing it, what he essentially does is say, what's the net present value of all of that at birth. People are supermen right out of the womb! Or time t. Then, with that net present value lump of wealth, he plans out completely the consumption and investments he's going to buy over the course of his life, to perfectly optimize his expected utility function.

As long as the net present value lump he's born with is worth the same amount, and as long as the state dependent price paths are the same, he's going to have the same possibility set to choose from. And he will thus will chose the same optimum (Wallace restricts the possible utility functions hardly at all, but he does say they're well behaved so that the optimum is unique.)

Foisting on a citizen a profit/loss from a QE that has a net present value of zero at the prices in a frictionless and complete market doesn’t change the possibility set at all for that citizen. So he will optimally chose the same exact consumption/investment path as before. And to get to that path he just sells this QE profit/loss for zero. In other words, he will engage in transactions to 100% undo it.

Now, next question: How exactly does he undo it, and who takes the other side of those transactions if market prices stay the same as before the QE.

The answer is, he does the opposite of what the government does in its QE transactions, and so the government is taking the other side of the transactions.

Specifically:

When the government sells that extra dollar in its QE, person z buys it from his private storage of C's. Thus, his saving in stored C's goes down by p_(t) C's, and his saving in stored dollars goes up by one dollar.

And, at time t+1, when the government offers to buy back that dollar at the same price path as before the QE, he buys it back.

What does all of this get person z? How does this alter his wealth at time t+1?

The forgone p_(t) stored C's used to buy the dollar at time t, means that he won't get a return of x(t+1) on those C's now. So the loss is: –p_(t)x(t+1).

But, he will now be able to sell a dollar for p(t+1) at time t+1.

So, at time t+1, his wealth will go up/down by:

–p_(t)x(t+1) + p_(t+1)

And the government will be foisting the QE profit/loss on him of:

p_(t)x(t+1) – p_(t+1)

So, the two perfectly cancel each other out, and he's left with the same exact consumption/investment path possibility set as before, and so he will do exactly the same thing as before. And because he will voluntarily take the other side of the government's QE transactions at the old market prices, the old market prices will have no pressure to move. They'll stay the same.

So, boys and girls, if you start in equilibrium with a certain set of market prices and optimizing behavior of your citizens, you will stay in equilibrium after this QE, and with no change in prices, of any asset, including money, and no change in consumption and investment decisions of any person! Viola!

But! A lot of things to note.

So, it works in the model. But, important notes:

Wallace's requirements for a QE are really strong and unrealistic. This is really sensitive to and dependent on perfectly complete and frictionless markets, which we are far from. Let alone perfect expertise, perfect public information, perfect self-discipline, perfect liquidity, superhumans.

But still, even given these things, you might ask in the just completed example, what if our intrepid person z, who the QE profit is to be foisted on, doesn't have any saved/stored C's with which to buy the government's newly printed QE dollar?

Well, complete and frictionless markets! Person z borrows p_(t) C's from someone, and uses them to buy the QE dollar.

At time t+1 person z owes that person what that person would have gotten had he stored/invested those C's as originally planned: p_(t)x(t+1).

But, person z will get at time t+1, p_(t+1), from selling the dollar. So, the net at time t+1 is:

–p_(t)x(t+1) + p_(t+1)

Just as before. Which will neutralize completely the QE profit the government will foist upon person z at time t+1.

O.k., so what else could go wrong?

What if, in foisting the government's QE profit/loss on the citizens, they decide they will give it to person z only if it's a profit, and person w only if it's a loss. That would certainly cause a change in those two people's consumption and investment plans, which would then put pressure on prices to change.

Well, uh-uh. Wallace has that covered. He just doesn't allow it. Any QE, or other monetary/fiscal operations, are required to not change the net present value of any person's lifetime wealth. So, every person's consumption/investment path possibility set must remain 100% unchanged by the QE. This is what requirement (a) of the Irrelevance Proposition means.

Note, of course, that in the real world any profit/loss the government has from a QE will not be distributed so that there are no winners and losers. If the QE ends up taking money away from the government, some people will lose, and others won't be affected, or won't be affected as much. If the QE gives money to the government, some people will get tax cuts/transfers; others won't, or will get smaller tax cuts/transfers.

So really, honestly, in this model it works because Wallace rigged the game, with his extremely unrealistic requirements for the QE, and assumptions about the people and markets. This is not to say that the model is still not good and useful, that it still cannot give us intuition, but it does show the folly of interpreting it literally to reality.

The Natural State in the World of Wallace – Hold on to your seat!

Now, important and interesting note: People voluntarily hold money in this model even though no liquidity/convenience benefit is included. They do so for the financial return. It's an interesting (or funny) thing about this model, and it makes it so that deflation – and the zero lower bound – are the natural state!

Why? We're assuming we start in equilibrium, so all assets are held unless they are worthless. But a monetary model where money is always worthless is not very useful, so Wallace mathematically requires that this cannot happen with equation (4), which makes it so money is worth something: p(t) > 0, all t. But the only way it can be worth something in this model is its financial return, since the model gives it no convenience for making transactions benefit, or ease of wealth storage benefit. The model gives it no benefits at all other than its return, no different from any other financial asset.

And if you were to calibrate this model to typical historical conditions in an advanced economy, the expected return on all financial assets would be positive (other than those which act as insurance). Thus, typically, money appreciates, i.e. deflation! And people voluntarily hold money without getting interest, just for the appreciation.

And in this model there's not a reason to pay interest when borrowing money. First, if someone borrows a dollar, there is no default risk. The model does not specify one, and it includes no frictions. And dollars, unlike C's, are not productive. They don't produce anything over time. It's C's, the real goods, that produce something, and give you that return the model specifies of x(t). Papers with dead presidents just sit in a vault, or as electrons on bank computers. You only get a positive return from them if their price in real goods, C's, appreciates, which it does under normal circumstances.

So people are going to hold these papers with dead presidents collecting dust and doing nothing anyway. It's no cost to them to loan them to someone during the time they were going to just sit there anyway. And the markets are frictionless, so there's no extra fee for selling short any asset, including dollars. You might say the perfect competition in the model, no monopoly power, pushes the short selling fee to the actual costs of the short-seller, which are zero.

But finally, in the model, equation (4) makes it so the price appreciation alone makes the return on dollars fair at the current state prices. If there were interest on top, there would be an arbitrage. You would just construct a synthetic dollar, like with state price contracts, sell it short, buy an actual dollar, and collect the interest for free. So the arbitrage pressure will push the interest rate on dollars to zero.

There is, though, the question of why in the real world then money normally has a positive interest rate. I would say the answer is that the interest rate is really on the borrowing of real productive goods. The money just facilitates the borrowing of real goods transactions.

But in any case, you can see clearly by arbitrage that in Wallace's model money pays no interest. All you get is appreciation.

Thus, the interest rate on money is zero, i.e., zero lower bound!

So in Wallace ZLB and deflation are not some weird exception; they're the rule!

Wallace Neutrality in the Real World?

Ok, now, we see, hopefully, the intuition for why irrelevance works in the model, why a QE would have no effect, but what about in the real world?

First, of course, the vast majority of people in the real world are very far from having perfect expertise in finance – and every other subject there is – as the Wallace model, and the typical modern macro model, assumes. They're also far from possessing in their brains all of the public information there is, and being able to access it, and analyze it, instantly, perfectly, and costlessly, to find the perfect optimization path for their consumption, and how to invest their savings.

So, of course we don't interpret this kind of modern macro model literally to the real world. We think very carefully about how the big, or comically big, deviations of the real world from the model's assumptions will affect the results, how policy will work, and what the optimal policy is for what you want, and for your values, for what your optimization function is for society, for what your loss function is, etc.

This last one is not a normative statement where I note one's values, loss functions, etc. I am saying – to give an example – if you want to maximize total societal utils, then this is the best policy. I'm not saying you should choose the policy that optimizes total societal utils, just like if I give the policy that's Pareto optimal, I'm not saying that you should choose that option.

And no, you don't do the Pareto option automatically. Just because it's better than the status quo, does not mean it's the option society will prefer the most of any available option. There are other choices besides Pareto and the status quo. One of the biggest ones of interest to most people would be the one which maximizes total societal utils, and since often that one will provide gargantuanly more total societal utils than the Pareto one, or make 99+% of the people better off than the Pareto one, well, call me funny, but that might be something people might want to know about, other than being kept blind to any option but two, Pareto and status quo.

In any case, whatever our values, and if the analysis is being completely positive, and just exploring and giving information on options of interest to the public, with no endorsement, it is not intelligent, or realistic, to not consider carefully how the real world differs, and behaves differently, from the model. And this is especially true with a model as extremely unrealistic in very material ways as Wallace's.

So, let's consider this here.

Wallace works because people see what the government is going to do in every possible state of nature perfectly, and respond with their plans perfectly to the QE. Again, I have to dwell on this, because it's frighteningly, and maddeningly, absurd to see economists at top universities taking this literally, or highly literally – And yes, many of them are not that stupid or detached from reality. For many they say this to make their hard won specialization more valued, or to make their right-wing ideology sound more attractive to the public, to the policy makers and the voters.

But it should be obvious, if you've lived beyond childhood not detached from the world, that almost no human is anywhere close to like this. And the vast majority are very far. Just one example, which should come as little surprise: People were recently surveyed on what percentage of the federal government budget is foreign aid. Now, these are the people who supposedly know government spending so well that they always respond perfectly in their consumption and investment plans to any change; or expected change. On average, they overestimated it by 28-fold! And it's not 28 times a trivial amount. Actual foreign aid spending is just under 1%, and the average estimate is 28%, of government spending!   

And it's not just some outliers skewing the average. Only 4% of those surveyed answered in the correct range, 0-1%. Only 29% gave an answer that was off by less than 10-fold.

Paul Krugman wrote in his 1994 book, Peddling Prosperity:

Does this argument sound convincing? It did (and still does) to many economists. Akerloff pointed out, however, that it depends critically on the assumption that people do something that they are unlikely to do in real life: take account of the implications of current government spending for their future tax liabilities. That is, the claim that deficits don't matter implicitly assumes that ordinary families sit around the dinner table and say, "I read in the paper that President Clinton plans to spend $150 billion on infrastructure over the next five years; he's going to have to raise taxes to pay for that, even though he says he won't, so we're going to have to reduce our monthly budget by $12.36...the truth is that even families of brilliant economists don't have conversations like this. (page 208)

So the vast majority, to the extent they're aware at all of a QE, are not going to explicitly change their consumption and investment plans – to the extent they even have them – to counter the government's QE.

But what about sophisticated investors? What about actively managed funds, which have some of the savings of the unsophisticated?

It is often counter argued that you don't need every investor to be rational. As long as you have some marginal investors who are rational, then they will be enough to push prices all the way to efficiency, all the way to what the model says. My reply to this argument is as follows, with the first points being more general, followed by those more specific to the Wallace model:

Why a minority of savvy investors at the margin is not enough to push prices to efficiency

1) Enormous, Profound, and Widespread Inexpertise and Ignorance – We always hear the issue as being rational vs. irrational. Well, I could be 100% rational and logical, but if you ask me my opinion on the construction of a nuclear power plant, I will give you some extremely sub-optimal advice. Why? Duh, because it's far more than rationality; it's usually far more expertise and information. No matter how rational I am, I'm incredibly inexpert on making decisions on nuclear power plant design, and have comically little of the information important to making those decisions.

Again, should be ridiculously obvious, yet the discussion in academic economics and finance is always about rationality. Is there some Harvard evolutionary theory that shows how people can be tricked to think and act irrationally in some way, sometimes. Well, this may be fancy and intellectual sounding enough that you can get it published in a top journal, and avoid grievous career punishment and get big career rewards, but it's usually nothing compared to the typical person's massive and profound inexpertise, ignorance, and misinformation.

But that's not fancy enough sounding or otherwise acceptable to give as an answer, or put in a paper, if you don't want the massive sticks or to lose the massive carrots those with power in economics and finance academia wield. So we ignore the pink elephant in the room.

2) Undiversification – A savvy investor is limited in how much he will push the price of an asset to efficiency by how undiversified his portfolio becomes as he buys more and more of that asset. This is a point that honestly I have never heard explicitly stated in six years of intensive finance PhD study, and much academic study after that. I got it published in a letter in The Economist's Voice (a journal written to be accessible to policy makers, but edited by Joseph Stigletz and Brad DeLong). Quoting myself:

...One reason which was missing, at least explicitly, and which I have not seen yet in the literature, at least explicitly, is that a smart rational investor is limited in how much of a mispriced stock he will purchase or sell by how undiversified his portfolio will become. For example, suppose IBM is currently selling for $100, but its efficient, or rational informed, price is $110. It must be remembered that the rational informed price is what the stock is worth to the investor when added in the appropriate proportion to his properly diversified portfolio of other assets. Such a savvy investor will purchase more IBM as it only costs $100, but as soon as he purchases more IBM, IBM becomes worth less to him per share, because it becomes increasingly risky to put so much of his money in the IBM basket. By the time this investor has purchased enough IBM that it constitutes 20 percent of his portfolio, the stock may have become so risky that it’s worth less than $100 to him for an additional share. At that point he may have only purchased enough IBM stock to push the price to $100.02, far short of its efficient market price of $110. Thus, if the rational and informed investors do not hold or control enough—a large enough proportion of the wealth invested in the market—they may not be able to come close to pushing prices to the efficient level.

3) The Limits of Arbitrage – This refers to the seminal paper, The Limits of Arbitrage by Andrei Shleifer and Robert W. Vishny. It really involves "arbitrage" (vs. arbitrage, with no quotation marks). True textbook arbitrage, by contrast, involves zero risk and zero upfront money, yet you get money from the transactions involved, either now or sometime in the future. The far more popularly referred to "arbitrage" is something that makes an abnormally good risk-adjusted expected return, or is an abnormally good risk-adjusted gamble, but does involve some risk (sometimes a lot!), and possibly upfront money too. The use of arbitrage for "arbitrage", as you might guess, is something that irritates me, and I think causes a lot of confusion and misunderstanding. We really need a separate term for "arbitrage". I actually like "arbitrage", as the quotation marks give it the needed pejorative connotation for those who use it like it's arbitrage.

In any case, the paper's main point is that if prices move away from their efficient level, then there exists an "arbitrage" (and maybe, but only very rarely, an arbitrage). However, the benefits of this "arbitrage" may take a long time to appear. And, an "arbitrage" involves risk, so even though ex-ante it's the smart move, there's usually a significant risk, maybe even a large risk, that it will go badly or very badly ex-post.

Next, the paper notes that often wealth is managed by an agent, not the principle. It may be a fund manager, an advisor, or a corporate officer, to name a few. And this is usually because the principle has relatively little understanding of, or information on, investments.[1] So, if a principle's agent takes on an "arbitrage", this arbitrage may take a long time to do well. In the short or medium run, it's not that rare for it to do badly, even horribly. The agent can tell the principle, this is a long-term investment. In the long run it will do well. But the principle will not know if he's lying, given the principle's inexpertise and ignorance, so he may fire the agent and sell the investment for a loss. Moreover, even if the principle is trusting and patient, the investment may still ex-post do badly, or even disastrously. It was only a good deal ex-ante.

The agent knows that he certainly faces these risks to his job, and career, so he may play it safe and forgo the "arbitrage" for an investment that he knows has a worse risk-adjusted expected return, but is well respected among laypeople, and thus relatively safe for his job and career. Asymmetric information may not exist in the typical freshwater model, but that won't stop it from killing an investment manager's career – or for that matter, making Americans pay horrifying costs for their healthcare compared to countries that admit this reality (as well as rampant monopoly power, profound externalities,…)

The result is that lots of wealth will not be put into "arbitrage"s by agents, limiting the forces pushing prices towards efficiency. Agents will, of course, still take every bit of arbitrages they can get, as they don't even need the principle for this. They can do it for themselves. Remember arbitrages, as opposed to "arbitrages", require absolutely no upfront money, and are absolutely zero risk.

Now, in the Wallace model, how would this affect the results?

Well, first, Wallace works because every individual is a superhuman perfect expertise, perfect information, perfect foresight optimizer. So, no one would pay to have someone else decide how to invest their money to start with! And, on top of the fact that an agent would be unnecessary, he would also not know your utility function perfectly. You could do it yourself better, in less than a nanosecond, and perfectly, and at zero cost in effort, time, or money.

That is what the Wallace model, and the typical modern macro model, assumes.

But, people do commonly hire agents, and have them manage a substantial portion of their savings. Let's consider a fund, for example. The fund has many participants. When the government announces the QE, the fund manager can't perfectly counter the QE to maintain the same consumption path for all of its participants in any states, because the transactions that perfectly counter the QE for person i will be different than those that counter it for person j.

But, you may say: Well, each person can just go into the perfectly complete markets and counteract the fund manager, by buying and selling the appropriate combination of state-price contracts. Except, of course, we have nothing close to these kinds of assets in actual financial markets. And the markets are far from frictionless – from transactions costs to taxes.

I'll talk more about this agent-principle problem from "The Limits of Arbitrage" later, with regard to other questions and issues.

4) Incomplete and frictioned markets, especially the inability to short sell short at low cost, or at all – In Wallace, like in the typical modern macro model, markets are perfectly complete and frictionless. And, you start in equilibrium, with perfect efficiency. Suppose then, the government decides to do a QE. They start buying a financial asset, and if they push the price up, then it now has an abnormally low risk-adjusted expected return.

So, the savvy investors will have an incentive to sell their holdings until the price goes back down again, so it's no longer a bad deal. But what if the selling of the savvy investors, even completely exhausting all of their holdings, is not equal to the government's buying at an elevated price? So the price is not pushed all the way back to efficiency. Then, the next step the savvy investors might want to take is to sell short. But if they can't, because the asset is not sold short, then the savvy investors can act no further. The price of the asset will remain above efficiency.

And in the real world it is common for assets to not have a short sale market, or for the transactions costs of a short sale to be high. And, savvy investors are still limited by their resources and credit worthiness, even when a moderately-frictioned short-sale market exists. They have to be able to meet the margin calls, for example.

5) Required equations (a) and (b) won't hold in the real world – Equation (a) says that any monetary/fiscal operation, like a QE, must leave every single citizen no better or worse off financially. That is, the net present value, at the initial state prices, of their lifetime income and wealth must not change as a result of the monetary/fiscal operation for Irrelevance to hold. In addition, (b) says that any profit or loss from the monetary operation must be 100% foisted on the public, on the tax payers, by adding to or subtracting from their transfers net of taxes.

But, of course, in the real world the government might not fully return all profits to the citizens in reduced transfers minus taxes, and in such a way that everyone has the same total net-present-value of wealth as before.

The government might take the profits from the QE and give them to certain groups of people, but not others. Or, it may "consume" the profits, to use Wallace's term, by spending them on infrastructure, or basic scientific and medical research. Likewise, any loss from the QE might be recouped by raising taxes predominantly on only some groups of people, like the wealthy through income and estate taxes, or the poor and middle class through payroll and sales taxes.

The main idea is that even if every person in the country is a perfect foresight, perfect optimizing, super-cyborg savvy investor, the QE might not conform to Wallace's requirements (a) and (b), and so their income paths and lifetime net-present-value of wealth will change. Thus, as perfect optimizers, they will then change their consumption and investment plans to accommodate. And this will affect the demand for financial assets, and their supply, thus affecting prices. So, this is another mechanism making Irrelevance not hold.

Of course, in the real world when the Fed does a QE, or any monetary operation, the fiscal branches of government don't say to the public we guarantee that any profit or loss from this QE will be returned to the citizens in increased/decreased transfers net of taxes, and in such a way that no one is any wealthier or poorer.

Clearly, this is far from what happens, and basically no one watches the ultra-complicated government and political system very closely, or accurately, and acts accordingly with their financial plans anyway. The average person thinks the federal government spends 28% of its budget on foreign aid. The actual amount is only about 1%.  

And even the extremely savvy minority of investors won't have this information and ability in their heads. But even if they did, equations (a) and (b) won't hold, so they won't keep their consumption/investment plans unchanged, and so invest perfectly against the QE buying of the government.

And the profits from a QE can be substantial. Since the financial crisis of 2008, the federal government has received over half a trillion dollars in profits remitted by the Fed, and it looks like it could exceed one trillion before it's over. Depending on how this money is used, it could certainly have a substantial impact on demand, and the economy.

6) Investors cannot be sure if and when the QE will be fully reversed – In the Wallace model, our cyborg investors know, with certainty, that all of the dollars that the government is selling for C's, or financial assets, will be purchased back again at a specific time in the future.

The plan is that these dollars will be sold by the government, then you will buy some, and then the government will buy them back from you in the future. And, if the price changes so you get a loss from this (which will be the government's gain), then the government will fully compensate you for this loss with increased transfers minus taxes.

And if the price changes in your favor, so you get a gain, compared to your consumption/investment plan before the QE, then the government will fully take that from you by decreasing your transfers net of taxes.

So, the public will want to buy what the government sells, so as to keep their optimal consumption/investment path unchanged. They know the government will buy it back, and will be compensating them personally for whatever profit or loss this involves.

But of course, if they don't know that, then it's a very different story. If they think that the government might not buy back their dollars, or all of them; if they think that the government might permanently increase the money supply, then they might not buy all of the dollars that the government is selling at the current market price.

And in the real world, unlike in the world of Wallace, the public is not certain that the government will be buying back all of these dollars from a QE at some future time. So they will not act accordingly, as in the Wallace model.

But can Wallace neutrality kind of work, to some substantial extent? Can it be a substantial factor?

The biggest real world factor I can see in the Wallace model is the thinking, at least by some expert investors, or expert agents of investors, that the Fed is likely to reverse the QE at some time in the not too far future.

If the Fed goes out and buys one billion ounces of gold, and expert investors know that tomorrow the Fed will sell all of them back, then if the price of gold rises by a substantial amount, you're going to see these expert investors selling all of the gold they hold. If it rises more, they'll start really short-selling it.

But with a QE, even expert investors can't be so sure when, or even if, the QE will be fully reversed. So they certainly expose themselves to risk by selling, and/or short-selling, into the QE. And the more they do it, the higher the risk, as their portfolios become more and more weighted by their short-positions, and thus, more and more undiversified, more and more exposed to the little-diversified-away idiosyncratic risks in an increasingly undiversified portfolio.

And the idiosyncratic risks can be very substantial, depending on the assets.

Now, compare this one billion ounces of gold example to another example; one where the Fed's QE is a lot more spread out over many financial assets, and so they're only buying just 1% of the world's gold.

And suppose expert investors think that it will probably be at least 10 years before the government sells any of it back again. Furthermore, they think there's a good chance that the government will never sell any of it back, or will sell back only a small fraction of it.

Now, what if expert investors see the price of gold inch up from the government's QE?

Are they going to say, hey, I'll keep short-selling with every resource I have until the price of gold goes 100% back down again because gold is over-priced?

Of course not.

Over a 10 year period having a portfolio composed entirely, or predominantly, of gold-shorts would expose the expert investor to ridiculous idiosyncratic risk! This would totally outweigh any benefit from gold having a somewhat above-average expected return for its beta. And this isn't even mentioning the principle-agent problem of "The Limits of Arbitrage".

So no, if the government does a large and a very unconventional QE, even the expert savvy investors or agents are not going to sell into it nearly hard enough to negate its effects on asset prices and inflation. And this isn't even to mention the vast majority of investors who are not expert, and are just basically oblivious to the QE, or grossly misinterpreting it.

And look at the current QE, or unusual monetary maneuvers. Already it's been over seven years since the Fed began its large and unconventional stimulus, and the balance sheet has only grown, from around $900 billion in 2008 to $4.5 trillion today. And no one thinks it will wind all the way back to "normal" anytime soon.

So I would think that if the QE is very large and unconventional, it will have a substantial effect. And it appears that's what the empirical research shows. Ben Bernanke said in 2014, “Well, the problem with QE is it works in practice, but it doesn’t work in theory”. Roger Farmer also wrote, "A wealth of evidence shows not just that quantitative easing matters, but also that qualitative easing matters. (see for example Krishnamurthy and Vissing-Jorgensen, Hamilton and Wu, Gagnon et al). In other words, QE works in practice but not in theory. Perhaps it's time to jettison the theory."

And this is with QE's that aren't inordinately big relative to the economy. To those who say QE's cannot have an effect due to theory, that aren't convinced by the empirical studies, I would ask do you still think a QE would have no effect if we made it larger and larger. Would a QE of $10 trillion have no effect? $50 trillion? 100?

How confident are you that this freshwater theory mirrors reality? And how confident would you really be if you had to pay a big price for being wrong, like the 99% who don't have jobs they can never lose, like tenured professors.

So, all other things equal, the longer expert investors think the QE will last before it's substantially unwound, the less they will sell into it. And so the more the effect from the QE. There's just more time to be exposed to idiosyncratic risk from taking on a large short-position.

And, again ceterus paribus of course, the more unconventional the QE, the less investors will sell into it. Because more unconventional assets usually have higher idiosyncratic risk.

And the less sure expert investors are that the Fed will fully unwind the QE, ever; that it won't, to some extent, be a permanent increase in the path of money supply, the less they will sell into the QE.

And finally, with the Fed potentially handing over trillions in profits to the fiscal side of government, it is possible for a QE to result in a large, or much larger, fiscal stimulus. And to expert investors this fiscal stimulus certainly could justify higher asset prices. Thus, the larger the forecast profits to the federal government from a QE, the less expert investors will sell into it

Next, we examine the important issue of reasons why Wallace neutrality works in the model, as opposed to reasons people might incorrectly think are why it works in the model, but in fact are not necessary.

Reasons why Wallace neutrality works in the model, and not reasons why it works in the model

Reasons why Wallace neutrality works in the model

Each of these are necessary, but not sufficient:

1) Perfectly complete and frictionless markets – In the model, the government gives the citizens all the incentive they need to buy what the government sells in a QE. The government says any profits/losses from this QE will be 100% foisted upon you in increased/decreased taxes-minus-transfers. We will be buying back all of these dollars next period, and when we do, any profit or loss from this will come out of your transfers-net-of-taxes.

Now, I've talked about this extensively previously in this post/paper. The government could give the whole profit/loss to just one citizen, or it could spread it around however it wants amongst the population. It won't matter for Irrelevance (Wallace neutrality) to hold.

But, to make some points clear, let's suppose this special case: The government will be printing 1 trillion dollars and using it to buy 2 trillion C's. We'll assume the current market rate is $1 for 2 C's, or 50 cents per C. And suppose that the whole profit/loss from this will be foisted on just one citizen, Mr. Jones, who we will assume has no savings at all.

Well, for irrelevance to work, Mr. Jones will have to be able to go short enough on C's that he can buy every one of those one trillion dollars. And, in the end, when the government reverses the QE, and buys back all of those dollars with its stored C's, any resulting profit will be given to Mr. Jones. And he will use that profit to pay off his shorting contracts.

Any loss to the government from the QE, if one occurs, will also be foisted on Mr. Jones. And he will pay this loss, perfectly, with his profits from his shorting contracts.

Thus, by doing this, Mr. Jones guarantees that his consumption path will not change as a result of the QE. And that's just what he wants, as a perfect-optimizing, perfect-public-information, perfect-foresight, perfect-expertise, instant-calculating, zero-calculation-cost-in-time-or-effort, cyborg. Which, of course, everyone is.

As such, Mr. Jones had already figured out that this consumption path optimizes his expected utility, and he doesn't want it to change. Not when the net-present-value of what he has to work with, his consumption path possibilities set, hasn't changed as a result of the governments plan to do a QE.

And it hasn't. The government is buying and selling at the market prices. It's buying a bunch of stuff at the current market prices, and then selling it all back at the same future market prices given the future state. So the net present value of all this is zero, discounting at the current state prices.

If you tell someone I'm going to take some of the dollars I've promised you, buy an ounce of gold with them for you, then I'll hold your ounce for one period, then sell it, and give you what I get. Then your wealth hasn't changed, if markets are complete and frictionless.

Why? Because if you don't like all this stuff that was done for you at the market rates, then you can just undo it all at those same market rates with shorts. And you end up exactly back to where you were, with the same exact cash flow paths, the same exact consumption/investment possibility set.

And from there you can change your paths and investments anyway you want at the market prices, just the same as you could before the government did this. Your possibility set has not changed. You can do all the same things you could before this intervention, get all the same consumption paths over the states as before. And, I talked about this earlier in the post in a more elaborate way.

But, for you to be able to do this, the markets have to be complete enough for you to reverse what the government has foisted on you. And you have to be able to do it with no transactions costs.

Otherwise, your consumption path possibilities set will change, and so you may choose a different consumption path, and so different investments, and so you will put pressure on the markets in a different way, and so the market prices will not be the same, and so Irrelevance – a.k.a. Wallace neutrality – will not hold.

2) The profit/loss from the QE is 100% returned to the tax-payers – There's zero change in government spending as a result. So, as noted in our example above, Wallace basically requires that we perfectly rig the game. Government has to credibly promise any profit from the QE to the citizens in increased transfers net of taxes. Otherwise, they won't have the incentive to completely take the other side of the government's trades at the current market prices.

If the citizens think they aren't getting those profits, that instead some of them will be going to wasteful "government consumption" – you know, like basic scientific and medical research, Heckman-style early human development investments, and infrastructure – then the net-present-values of their endowments change. And thus, their consumption path possibilities sets change. And, as a result, they will choose different paths, with different investments, which will pressure market prices differently, moving them from where they were.

3) Superhumans, and that means everyone, not just a minority of savvy marginal investors –Everyone has to have perfect public information and be perfectly rational. But it's much more than that. What's more important, but almost always ignored, is perfect expertise. You can be as rational as Spock, and have all of the information in the world, but if you have very little understanding of finance, like almost everyone, then you will be far from optimizing your investing – Or your medical care for that matter. And you must employ this perfect expertise and information instantly; zero cost in effort or time, or you're going to take short-cuts in your analysis, use rules-of-thumb, and so forth. So everyone must be an Ultron. And just having some savvy marginal investors who are Ultrons won't bail you out.

As I've discussed earlier, that an asset get mispriced means that its price is good or bad for how it adds to a diversified portfolio. The price is good, say, for the beta. It's a good deal only when added in the appropriate amount to your portfolio. That makes it an "arbitrage", but it doesn't make it an arbitrage. A minority of savvy investors won't keep buying it without limit, because the more they buy, the more they get exposed to the assets idiosyncratic risk; the more unbalanced their portfolios become. They're going to stop pretty quickly. If the price of gold goes from $1,500/ounce to $1,600/ounce, with the same fundamentals, the savvy investors of the world aren't all going to sell all of their gold, and then if the price only drops to $1,590 short it with every dollar they possibly can. The idiosyncratic risk is just too great. That is not optimizing behavior for them.

And, there is the principle-agent problem explained in "The Limits of Arbitrage".

So, everyone has to be an Ultron. Everyone has to instantly respond to the QE saying, hey, the government's doing a QE; of course, I know with certainty that they're going to 100% reverse the QE, and, of course, when they do, 100% of the profit/loss will be remitted to the citizens; and of that profit, I know exactly how much will be foisted on me, and so I'll go out into the perfectly complete and frictionless markets and 100% counteract my chunk, taking the other side of the government's trades, so that I keep my perfectly optimal consumption path for the current market prices.

It's just so obvious the economy works this way. I don't know why it's even a discussion!

Not reasons why Wallace neutrality works in the model, not necessary

These reasons I have seen put forward, or just thought of myself, as possibilities for why Wallace neutrality might work in the model, but not in the real world. But it's important to understand that they, in fact, are not specified in the model, either explicitly or implicitly. And it's interesting that they aren't needed to have Wallace neutrality in the model.

1) Representative Agent – Everyone can be their own person. You're free to be you – as long as you're an Ultron. You can have a unique utility function, and the economy includes as many unique people as you want.

The basic reason is that whatever utility function you have (with some completely mild restrictions), Wallace rigs it so that it will be in your best interests to either sit on the sidelines, or take the other side of the government's trades at the current market prices.

The unique individuals in the economy want to keep their optimizing consumption/investment paths at the current market prices. And Wallace requires the government to hoist the profit/loss from a QE on this population of individuals. And that gives the hoistee's, no matter what their utility functions are, an incentive to take the other side of the government's trades, so as to make sure that their unique utility functions, whatever they are, stay optimized at the current market prices.

2) Simple unrealistic utility functions – Nope again, Wallace only requires very mild restrictions on the utility functions: More income is better, the law of diminishing returns sets in, and twice differentiable. Within that, it can be as complicated as you want. The utility functions are not much of a source of consequential unrealism in the model. That lies elsewhere.

3) Simple unrealistic set of financial assets – Again, no. Markets are 100% complete, so any asset you can imagine is included, or can be produced synthetically at zero transactions cost as markets are also frictionless.

4) Closed economy – Interestingly, it's not necessary. For Wallace neutrality to work the government must credibly promise to foist whatever profits/losses result 100% on investors in increased/decreased taxes net of transfers. This makes it optimal for those investors to take the other side of the govenrment's QE trades.

But the government doesn't have to distribute the QE profits/losses to everyone in the world. They can foist them on even just one of the seven billion people on planet Earth, and it works. That one person takes the other side of all the trades, borrowing and shorting as much as necessary in the perfectly complete and frictionless markets.

So the fact that the markets are international, yet investors in other countries are not affected by U.S. tax and transfer policy makes no difference. You don't need them to get enough money to take the other side of the U.S. govenrment's QE trades.

Extensions – Ideas for new research based on all of this

I hope I've made it obvious at this point that there's no way Wallace neutrality will hold in the real world. But the big question next is, how far from holding will it be?

I could do a theoretical model where we deviate from some of Wallace's assumptions and formally prove that now equilibrium doesn't hold at the current market prices. I could introduce a class of people who are not Ultrons, say, rule of thumb investors; or goldbugs, people with an extreme lack of expertise and/or information; or just normal people, who are people with an extreme lack of financial expertise and information. And/or, I could relax the perfectly complete and frictionless markets assumptions; I could introduce borrowing and short selling constraints and costs.

It wouldn't be hard in these cases to prove formally that equilibrium no longer holds if there's Wallace neutrality. Would this be publishable? It depends on how much I could math it up nicely, and make the math look long and impressive enough. I easily might not be able to. It might just be too fast and simple to prove these things formally, and my sketches of this seem to indicate this. And since I have no name, it's not looking like that could get published.

But aside from that, a big problem is that Wallace's model is really really general and vague. It's basically just, people have utility functions – that's it! I mean, to get some idea of how big an effect real world size deviations from the model have, you need to introduce some kind of specificity, and hopefully calibration.

So, I think what would be most useful would be to do a specific model with some calibration to reality. Give people specific utility functions, say isoelastic, or whatever's best, and calibrate the parameters to the real world. Then, look at real world studies to see what percentage of people are rule-of-thumb savers and investors, and put that in the model. Now, finding closed form impressive looking mathematical solutions is going out the window, but is the goal to impress with our math or be useful. Ok, let me rephrase that, should the goal be to be the most useful to society rather than to most impress with fancy looking math?

If yes, then a long time ago we should have been putting a lot of resources into constructing elaborate and realistic economic computer simulations, where you don't get closed form solutions, but you do get a lot more realism and precision. And I've said this for a long time. So, what I would want to do here is do a computer simulation, where I give the citizens utility functions, and calibrate the parameters to reality as best I can, and put in other specifics, and then just start running it, trying various QE's, with various levels of non-perfection of people and markets, and seeing how it affects things.

So, very largely a programming project.

I have some programming chops, and know some professionals who can help me, so maybe some year, or decade, with my five minutes a week of free time…

As far as other future projects. I think it was important to really explain all of this in detail, and in as clear and easy to understand way as possible (for those with the necessary considerable pre-requisites), so it was important to have a long version. But, of course, few will read the long version. So next I'd like to work on boiling it down, and then linking to the longer and better explanation. So, on the agenda is a perhaps 1,500 word Intuition Behind Wallace Neutrality, then 500 words, and even a two or three hundred word version. And, there's at least a few specific related questions I'd like to discuss, like the long awaited, and very interesting, answer to this. Stay tuned…

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[1] Yes, this is a case where inexpertise and ignorance are actually acknowledged, and, in fact, pivotal, in a paper published in a top academic economics journal. But, it's an exceedingly rare case. And its authors are famous Harvard professors, who used clever math, and referred to a big mathematical model. If you were much short of these things it would have been extremely hard and unlikely to get this same exact insight published in an influential academic journal.