[PDF version available here.]
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…
[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.
