Reverse-engineering David Andolfatto's and Stephen Williamson's Neo-Fisherian paper

Stephen has a new post defending the Neo-Fisherian perspective. He links to a new paper he and David wrote (pdf). It is not an easy paper. This is what I think is the intuition behind the results.

Start with the Fiscal Theory of the Price Level, where B is the nominal stock of government "bonds", and P is the price level:

(B/P) = PV(primary surpluses)

Government "bonds" are valued like a corporation's stocks, except they are the unit of account for the economy. Suppose the government does a 2-for-1 stock split, holding the Present Value of primary surpluses constant. B doubles, P doubles, and each "bond" is worth half as much in real terms.

Now suppose the government does a continuous stock split, by printing a flow of new "bonds" and using them to pay interest on the stock of existing "bonds". That increase in the nominal interest rate on "bonds" increases the growth rate in the stock of "bonds", and increases the inflation rate, all by an equal amount. We get the Neo-Fisherian result, just like in John Cochrane's paper. And notice that this policy has no fiscal consequences for the rest of the government budget constraint, just like a 2-for-1 stock split has no consequences for a firm's balance sheet.

Now suppose that the "bonds" are also used as the medium of exchange. We re-interpret "B" as "money", but money that pays interest. Because B is used as the medium of exchange, people will hold a stock of B even if the real rate of return on B is less than the rate of return on other assets. This means the government can "tax" the holding of B (collect seigniorage revenue) by paying a real rate of return less than the rate of return on other assets (or even a negative real rate of return). Which in turn means the government can run a smaller primary surplus (or run a primary deficit). Suppose the government does this.

Because holders of B are "taxed", (B/P) will now be smaller than socially optimal; there will be a liquidity premium on B, and so some otherwise mutually advantageous trades will not be made. The PV(primary surplus) is too small, or the PV(primary deficit) is too big, for the socially optimal allocation of resources.

Now let us introduce a second type of money, called "M", that is also issued by the government. There are two differences between M and B:

1. B can only be used in a subset of markets (called market 2). M can be used in all markets (both market 1 and market 2).

2. While nominal interest can be paid on B, the nominal interest rate on M is always 0%. So the real rate of return on M is minus the inflation rate, and the real rate of return on B is the nominal interest rate i minus the inflation rate.

So think of B as like a chequing account at the central bank, and M as central bank currency, and some markets (market 1) do not accept cheques. And people can swap M and B at the central bank, and so hold their desired mix of the two monies.

Suppose (though this is not what David and Stephen assume) the fiscal authority decides the total amount of real seigniorage revenue from M and B, but the central bank decides the composition of that seigniorage revenue between M and B. It can tax M more and tax B less, or vice versa, provided the total (real) tax revenue stays the same. And it varies the composition of the tax by varying the rate at which it prints M and B, using the newly-printed M and B to pay interest on the existing stock of B.

The Friedman rule says that M and B should yield the same real rate of return for optimality, which means that the central bank should set the nominal rate of interest on B equal to 0%. But the Ramsey rule of optimal taxation in a second best world where distorting taxes are needed to collect revenue says that if two goods are taxed, the one with the less elastic demand should pay the higher tax rate.

Empirically, the demand for government-issued bonds seems to me to be much more interest-elastic than the demand for government-issued currency. So the Ramsey rule says that central bank should set a higher tax rate on currency than on bonds, which means it should violate the Friedman rule, and ensure a strictly positive nominal interest rate on bonds. But that is not why the Friedman rule is violated in David and Stephen's model; because the demands for M and B are perfectly symmetric in their model, once the nominal interest rate on B is strictly positive.

It took me some time, plus some help from David, to figure out why their model violates the Friedman rule. They assume that B is a one-period Treasury bill, and the fiscal authority holds [M+B/(1+i)]/P constant when the central bank increases the nominal interest rate i, which means that [M+B]/P must increase, purely from the arithmetic. And we are in a second-best world, where M and B are taxed, so the amount of goods bought is sub-optimal. And the quantity of goods bought with M and B is equal to [M+B]/P, and so that quantity increases, and welfare increases, when the central bank increases i above 0%, purely as a matter of arithmetic. But if the central bank increases i too much, this reduces shopping in market 1 (where M is used) relative to market 2 (where B is used) and welfare decreases.

If the model had instead assumed that the fiscal authority held [M+B]/P constant (instead of holding  [M+B/(1+i)]/P constant) when the central bank changes i, the results would be totally different. The Friedman rule would be optimal.

What we are talking about here is the optimal inflation target. Should the central bank target 2% inflation, or 4%, or 0%? Or should it target a negative inflation rate, low enough so that the nominal interest rate on bonds is typically 0%? I can think of many reasons for arguing why one inflation target is better than another. But the last thought on my mind would be whether the fiscal authority holds [M+B/(1+i)]/P constant or [M+B]/P constant when the central bank changes the inflation target.

I get uneasy when some of the key results of a model are extremely sensitive to very small changes in apparently arbitrary assumptions about government behaviour. (If the government issued 2-period Tbills, or God forbid perpetuities, the results would change a lot). (David worries about this too.) I would want to examine those assumptions very carefully indeed.

The Bank of Canada is a Crown Corporation. It operates mostly independently, but hands its profits over to the government. In my view, if the Bank of Canada did something that caused its profits to increase, the government would spend those extra profits, sooner or later, by increasing government expenditure or cutting taxes. I can imagine a world (Zimbabwe?) in which the government tells the central bank that it must deliver a fixed amount of profits each year, and adjust monetary policy to ensure it hits that profit target; but that world is not Canada. The Bank of Canada targets 2% inflation, not $2 billion in inflation-adjusted annual profits. But I find it very hard to imagine a world in which the government tells the central bank it can do what it likes provided the real market value of currency plus government bonds equals some annual target set by the government.

I am coming down on the same side as David in his recent post on the "dirty little secret".

"The starting point for answering the question of how a policy affects the economy is to be very clear what one means by policy. Most people do not get this very important point: a policy is not just an action, it is a set of rules. And because monetary and fiscal policy are tied together through a consolidated government budget constraint, a monetary policy is not completely specified without a corresponding (and consistent) fiscal policy."

I am of the view that the Bank of Canada targets 2% inflation (or NGDP or whatever), and it adjusts the nominal interest rate (or base money or whatever) to hit that target, and its actions affect its profits, and those profits affect the government's spending and taxation decisions. In the long run, the government adjusts its budget to be consistent with the Bank of Canada's actions. Not vice versa. We saw that adjustment in 1995.

Zimbabwe, under Mugabe, is different. The guy with the AK47 chooses how much currency you print for him to spend, even though he does give you a bond in return. And the higher the price level the more currency he wants you to print for him. Raising the nominal rate of interest on bonds lets you sell more bonds and withdraw currency from circulation, but also means you must print currency even faster to pay the higher interest. Because the guy with the AK47 is going to make you print enough additional currency to pay the interest on the bonds he issued. That's a Neo-Fisherite world.

Comments

[JKH]

“They assume that B is a one-period Treasury bill, and the fiscal authority holds [M+B/(1+i)]/P constant when the central bank increases the nominal interest rate i, which means that [M+B]/P must increase, purely from the arithmetic.”

Shot in the dark:

Consider P at the end of the period in which the central bank increases the nominal rate i

I.e. P multiplied by the inflation factor for the period

I.e. P (1 + p)

And if prices are flexible, the increase in i equals the increase in (expected) p

So while [M+B]/P must increase, [M+B]/[P (1 +p)] is constant

[bill woolsey]

Suppose we say that to set a nominal interest rate at R*, that means that we set the money growth rate at m* = R*-rn + growth yp - growth velocity.

All three of the things that determine the target for the growth rate of the quantity of money are not observable. But if you want a 5% nominal interest rate, and you estimate rn = growth yp = 3% and the growth velocity is zero, then you set the target growth rate of the quantity of money at 5%.

With those assumptions, if you want a higher nominal interest rate you raise the growth rate of the quantity of money and if you want a lower nominal interest rate we lower the growth rate of the quantity of money.

Simple enough.

But when the Fed wanted a lower target for "the" nominal interest rate, it raised the growth rate of the quantity of money.

And, of course, they aren't really targeting the nominal interest rate--at least, not as a goal. They wanted to lower the unemployment rate without raising the inflation rate above target.

So, is this a theory of what the Fed is doing explaining what has happened?

Or is this a theory of what the Fed should do?

[Market Fiscalist]

'That increase in the nominal interest rate on "bonds" increases the growth rate in the stock of "bonds", and increases the inflation rate, all by an equal amount. We get the Neo-Fisherian result'

A bit of an off-topic question but I will ask it anyway.

In some post-Keynesian models the interest rate is set by the CB and is treated as a cost by businesses and has little impact on economic activity. For example assume that business operate cost + markup pricing and borrow the entire costs of operations against future sales. They decide how much to produce purely based upon last years sales. If the rate of interest increases then this is an increased business cost that will be added to sale price. In this simple model a 5% increase in the interest rate will lead to the same level of output but with a 5% increase in the price level. I think (but am not sure) that you could weave into this model the concept of a "natural" rate that kept the price level constant.

Could you not use a cost + markup model like this to drive Neo-Fisherite results with no need for such tortuous assumptions as in these papers ?

[Nick Rowe]

JKH: I'm afraid you lost me there. But think of it this way: the decisions of the central bank affect M/P and B/P, and so do the actions of the fiscal authority. So if we want to talk about the affect of central bank actions, we need to make an assumption about how the fiscal authority will respond. And the assumption in the model is that the fiscal authority will do whatever it needs to do to keep [M+B/(1+i)]/P constant. So if the central bank increases i, we don't need to know any economics at all to figure out that [M+B]/P must increase. It's just arithmetic.

Bill: remember that prices are perfectly flexible in this model.

"So, is this a theory of what the Fed is doing explaining what has happened?

Or is this a theory of what the Fed should do?"

I think it's a bit of both.

MF: Short answer: no.

The cost of production theory of relative prices died in 1871 (after the marginal revolution).

The cost push theory of nominal prices (inflation) died around 1971 (after Friedman). If we take nominal wages W as exogenous, then saying P=(1+r)W seems to work OK. But as soon as we recognise that W is proportional to P, because both firms and workers care about real wages, not nominal, the whole theory falls apart. Wages determine prices, and prices determine wages. So what determines prices-and-wages?

Now off course a whole bunch of PK trolls may want to wade into this debate. Don't; it's off-topic. I've got enough on my plate with the Neo-Fisherites.

[JKH]

Nick,

Just one more comment on that same point:

"And the assumption in the model is that the fiscal authority will do whatever it needs to do to keep [M+B/(1+i)]/P constant. So if the central bank increases i, we don't need to know any economics at all to figure out that [M+B]/P must increase. It's just arithmetic."

I don't know if its relevant to the Williamson paper you reference, but it looks to me like Cochrane achieves this by allowing the face value of the bills issued to increase in proportion to the interest rate increase. I.e. (M + B)/(1 + i) stays the same because (M + B)increases. And so the discounted present value of the face value of bills stays the same whatever the interest rate because the face value (i.e. the future maturity value) increases to make that happen. And it makes sense for the discounted present value to remain the same whatever the central bank does with interest rates in order to roll over the maturity value of bills maturing from the previous period. So with that, it doesn't seem to have to follow that (M + B)/P must increase.

That looks to me like what Cochrane does, but right or wrong, I don't know if its relevant to the Williamson paper you reference.

[W. Peden]

"The cost push theory of nominal prices (inflation) died around 1971 (after Friedman)."

In North America, maybe. It took a very long time to fade away in the UK, and I imagine even longer in some places. The early 1970s did cause huge problems for non-accelerationist Phillips Curves, though.

In the UK, the key question by 1983 was: why did cost-push addressing methods of reducing inflation (tax cuts, incomes policies, price controls etc.) fail in the UK in the 1970s, and when this approach was abandoned in 1979, why did inflation then subsequently fall dramatically from early 1980 to 1983, BEFORE the really important trade union reforms took effect or were introduced? If you pull a lever and it does nothing, then that lever presumably isn't what's driving the mechanism.

[W. Peden]

(Sorry to wander off topic, though I don't think I'm a PK troll.)

I find it very hard to understand what the Neo-Fisherites are trying to explain.

[Left Coast Bernard]

"The starting point for answering the question of how a policy affects the economy is to be very clear what one means by policy. Most people do not get this very important point: a policy is not just an action, it is a set of rules. And because monetary and fiscal policy are tied together through a consolidated government budget constraint, a monetary policy is not completely specified without a corresponding (and consistent) fiscal policy."

Dr. Rowe,

I asked Dr. Andolfatto and you a question on his blog about this quotation, which you cite. My question was that considering Uncle Sam holding unified accounts, with the Treasury his left pocket and the Fed his right pocket, what is the "consolidated budget constraint?" After all, while Uncle Sam may keep separate compartments in his wallet, or keep accounts of his left and right pockets, he can put whatever he wants in those compartments (because he issues the money)? Dr. Andolfatto answered that I was correct that Uncle Sam can issue as much money as he wished, but it would be a bad thing if he issued an infinite amount. You agreed with this.

This, however, does not answer my question, which was, what is the consolidated budget constraint? The word "constraint" implies to me some numeric limit beyond which Uncle Sam cannot go or should not go. Now (I think, but I'm an amateur) that the U. S. Treasury is "constrained" by US law to match its spending to its income from taxes and borrowing. But the Fed is not. Thus consolidated Uncle Sam is not constrained to match his spending to his income.

Also, while you mention that the Fed (and the Bank of Canada does similar things) gives its profits to the Treasury, moving money in accounts from Uncle Sam's right pocket to his left pocket, you neglect to mention that those profits arise, in part, because the Treasury pays interest on Uncle Sam's bonds that the Fed purchases from the private sector and holds in its accounts. Thus, the Treasury transfers money from Uncle Sam's left pocket to his right pocket, which the Fed then transfers from his right pocket to his left pocket. I wonder why, when the Fed purchases one of Uncle Sam's bonds, it doesn't just stamp it "Paid In Full", and drive it from its headquarters on Constitution Ave. to the Treasury Department on 15th Street to be shredded. Then Uncle Sam wouldn't have to worry himself about transferring money back and forth between his pockets. Since the Fed has paid for the bond, the Treasury doesn't have to pay for it again. Why isn't the outstanding US government debt, net between the total issued by the Treasury and paid back by the Treasury and issued and paid back by the Fed the relevant US debt number?

[Market Fiscalist]

If
- the central bank accommodates all demand for money at its chosen rate
- people come to associate that chosen rate with higher expected future inflation

Then you will get something very like cost-push inflation won't you ?

In the short term high rates will lead people to have lowered inflation expectations, but (and I know I am probably foolish to ignore Nick's above comment, and I am definitely NOT trying to annoy or troll) , I can't get away from the idea that if higher interest rates contribute at the margin to higher prices then in the long term we will end with the neo-fisherite result (or at the very least there will be a counterforce to the deflationary black-hole theory).

[Edward Lambert]

Plot nominal rate on the y-axis. Plot real rate on x-axis. You get up-sloping lines of constant inflation rates with slope = 1, according to the Fisher equation. Go northwest, inflation rises. Go southeast, inflation decreases.
Now constrain the CB. The CB cannot change the nominal rate. Where does inflation go? Northwest or southeast? We do not know. The CB cannot react to changes in inflation. We could say that eventually, real rates will go to the Long-run natural real rate. But there are so many paths to get there. There is not one. It depends on economic conditions.
Inflation becomes untethered like a boat from the dock and can float either way depending on the economic winds. The tortuous assumptions of Williamson and Andolfatto are defining certain economic winds that can move that untethered inflation boat.
The ZLB does not always have to be sub-optimal like one might conclude from their paper.
What we need to explain is how the nominal rate can be raised in order to resuscitate inflation in the current economic context. And over how much time would that increase in inflation occur. Is the time period relevant?

[David Beckworth]

Nick,

After reading David Andolfatto's "Dirty Little Secret Post" and a recent Twitter exchange with him, I think all of are us are not that far apart on the big issue. Specifically, in order for aggregate demand to be spurred by policy the Neo-Fisherites believe it depends on policy adjusting the expected path of the consolidated government balances sheet. As I explained to David, this understanding has been implicit in our calls for level targeting and explicit in our calls for a permanent expansion of the monetary base. The key hang up that I see is that the Neo-Fisherites challenge the assumption that fiscal policy would not offset a CB attempting to permanently expand the monetary base. Your Canada example above--where the government works around the CB's goals in a way that does not offset it--is a good illustration of why I believe our assumptions of no fiscal offset are reasonable.

I do wonder, though, if the 'Great Inflation' experience in the United States--where rapid fiscal spending growth was accommodated by the Fed--would be a counterpoint for them. One could argue, though, that the Fed could have been more vigilant then.

[Nick Rowe]

JKH: my interpretation is that the face value of each bill stays the same, but the quantity of bills increases. But yours is an interesting way of looking at it. Do we double the number of $20 notes, or make each $20 note a $40 note? And does it matter? (I don't think it does). And it would be equally relevant/irrelevant to this paper. (Irrelevance results are important too!).

W. Peden: cost-push kept walking for a long time, zombie like!

Left Coast Bernard: One explanation is that owning a stock of government bonds gives the central bank some independence from the fiscal authority. Like segregating the assets of Social Security. Even though it's symbolic, symbols matter.

MF: Let P=(1+r)W. If r increases, P increases relative to W. Or W decreases relative to P. Cots-push simply assumes the first, not the second. (Plus the sunk cost fallacy).

Edward: "We could say that eventually, real rates will go to the Long-run natural real rate. But there are so many paths to get there."

We could. Or we could say that the economy moves further and further away from long run equilibrium. Think of holding a broomstick balanced upright in the palm of your hand. The only equilibrium is where the broomstick is perfectly vertical. But it's unstable. If you move your hand north, the broomstick will lean south, and fall over, unless you move your hand quickly south again.

Have a look at my collective speed limit game post

David: yes. I thought David A's post encouraging, and converging.

If you think of a firm that has a production arm and a finance arm, then if the finance arm decides to stop paying dividends on shares, it must use share buybacks instead, as long as production and profits are unaffected (why should they be affected?). So the current share price is unchanged, but the price of shares will be rising from now on. And if those shares are used as medium of account, that means deflation.

But if the firm is a charity, and the finance arm is one of its ways of raising revenue, it's not obvious we will get the same result.

[JKH]

Nick,

I don’t think it matters either.

But actually I was thinking of/referring to an increase in the aggregate face value of treasury bills issued – not an increase in the face value of each bill. So for example a face value denomination of 100 per bill would remain the same. But the number of bills and their aggregate face value would increase to allow for the larger aggregate interest rate accrual that would be required from the point of the initial discounted Treasury bill price - as a result of an increase in the nominal interest rate set by the central bank. That said, the present value or price paid in aggregate for the increased number of bills would still remain the same - given the increase in the aggregate interest rate discount from maturity face value that would be required as a result of an increase in the nominal interest rate set by the central bank.

It’s not quite the same in the case for interest accruals on reserves. The analogous face value in that case would be the future combined value of reserve principal balances plus interest accrued – choosing some arbitrary “term to maturity” of reserves, which most naturally would be one day.

My point beyond that was that Cochrane always includes the accrual of interest when he refers to B – i.e. maturity face value for a treasury bill or (although he never states it or refers to it) reserves plus interest on reserves. And therefore it seems he always applies the P deflator to something that already includes an expected interest accrual (as far as I can tell). That seems essential to the whole idea of his paper. For example, when he increases the central bank nominal interest rate under flexible prices but holds real surpluses and the real interest rate constant, that increases B because B includes interest and therefore forces up P to retain the equivalence of the two sides of the valuation equation.

[JKH]

Not easy to express that first point I was trying to make above - the number of bills with the same face value would have to increase to allow for the reduced present value / price of each bill, which in turn results from the increased nominal interest rate and therefore the increased discount from the same face value.

[JKH]

And its the aggregate present value that needs to remain the same when the central bank increases the nominal rate - in order to roll over the previous debt.

(sh*t)

[Majromax]

Isn't all of this a discussion of whether the fiscal authority leads or follows the central bank?

The Fisherite view seems to require the fiscal authority to both follow the central bank and to work in concert with it. That definitely seems to be at odds with reality, where fiscal decisions are made both on a slower time-scale (generally yearly), and they are also made subject to political and other non-economic constraints. Even the Eurozone, where you point out that the fiscal authorities have every incentive to follow the ECB rather than lead, is constrained by its deficit requirements.

> They assume that B is a one-period Treasury bill, and the fiscal authority holds [M+B/(1+i)]/P constant when the central bank increases the nominal interest rate i, which means that [M+B]/P must increase, purely from the arithmetic.

> (B/P) = PV(primary surpluses)

Hmm. I think this is over-constrained if it is market-clearing.

We have four variables, M, B, i, and P. We have three explicit relationships here: i is set by the central bank, the fiscal authority holds [M+B/(1+i)]/P constant, and B/P is given by the fiscal theory of the price level.

That imposes a relationship on the composition of M and B -- liquidity preference. In particular, holding PV(surplus) to be constant and moving from i1 to i2 (with corresponding M1->M2 and B1->B2), we get:

M1/B1 + 1/(1+i1) = M2/B2 + 1/(1+i2)

That gives a downward-sloping LM curve. If the interest rate increases from i1 to i2, 1/(1+i2) < 1/(1+i1), which means that for any given (M,B) composition a greater share has to be in the form of money at a higher interest rate.

The alternative is to presume that the present value of future surpluses is not fixed, but instead it is determined endogenously by the system. That makes my head spin by introducing another seemingly arbitrary relationship, but I think it means that a high CB-set interest rate would force the government to commit to larger, long-term budget deficits.

(Worked: Set PV(surplus) = B1/k1 for i=i1 and B2/k2 for i=i2. That means that the above equation holds with the LHS multiplied by k2 and the RHS multiplied by k1. If we make the sensible assumption that the LM curve is not downward-sloping, this means that k2 < k1 for the equality to hold, so i2>i1 implies a larger long-term primary budget deficit.)

Ironically, this story is very compatible with Paleo-Keynesianism. We *do* see interest rates move in the same direction as budget deficits if monetary policy targets the quantity of money (by itself or via a weakly-responsive standard like precious metals) and the government faces competition and "crowding out" for its debt issues.

[Majromax]

@Me:

> Worked:

This section is wrong. Set PV(surplus) to be k1 for i=i1 and k2 for i=i2, so P1=B1/k1 and P2=B2/k2.

Now we get k1[M1/B1 + 1/(1+i1)] = k2[M2/B2 + 1/(i+i2)], instead of the opposite. That was the algebraic error of my post above.

If the LM curve is not downward-sloping, then the term in brackets on the RHS is smaller than the term in brackets on the LHS. For the equality to still hold, that means that k2 > k1, so the present value of future budget surplus should be greater in the higher interest rate case. Fortunately, that's a reasonably intuitive result.

However, what's tricky here is that the actual, short-term deficit is governed by the net change of (B2+M2)-(B1+M1). Since increasing interest rates in this model imply that the government should issue more (money+bonds), I think this means that we're again working the Wicksellian natural real rate of interest backwards: the new bonds must somehow lead to more productive investment opportunities for the PV of surpluses to increase notwithstanding the new debt.

[Nick Rowe]

Majromax: "Isn't all of this a discussion of whether the fiscal authority leads or follows the central bank?"

Those two discussions seem to be very separate. But I think you are right that they must be related. I was trying to get my head around this, for a post.

There are two (related?) questions:

1. Who moves first/last?

2. Who would win a game of chicken?
2a. If monetary wanted to tighten, and fiscal wanted to loosen, would monetary let fiscal default, or would monetary loosen to bail out fiscal? Dunno, depends.
2b. If monetary wanted to loosen, but fiscal wanted to tighten, would fiscal hoard the cash forever, that monetary keeps giving it???

[Too Much Fed]

JKH, if you see this, I left a reply at:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2014/11/black-holes-and-neo-fisherites-are-a-monetary-phenomenon.html

Sorry it is so late.