If r < g, bond-finance is like currency-finance

A world where the interest rate on government bonds is (permanently) less than the growth rate of GDP ("r<g") is a weird world. The government can run a Ponzi scheme, where it borrows (sells more bonds) to pay for the interest on the existing bonds, so the stocks of bonds grows at the rate of interest. But since r<g, the economy is growing faster than the stock of bonds, so the debt/GDP ratio is falling over time. So unlike the real Mr Ponzi's scheme, it's sustainable. The government would actually need to borrow more than is needed to pay the interest on the bonds, if it wanted to keep the debt/GDP ratio constant over time.

In a weird world, where r<g, government bonds aren't really a government "liability" in the normal sense of the word. Having a positive debt/GDP ratio, and keeping it constant over time as the economy grows, is instead a source of revenue for the government. It's like the government owning an asset, not owing a liability.

But if you think about the currency in your pocket, there's nothing weird about it at all. It's all very familiar. If r<g, financing government deficits by selling bonds is just like financing government deficits by printing currency. Because currency pays the owner 0% nominal interest (and minus 2% real interest if the central bank targets 2% inflation), which is less than the growth rate of the economy. So r<g for currency.

In our familiar world, where r<g for currency, currency isn't really a government "liability" in the normal sense of the word. Having a positive currency/GDP ratio, and keeping it constant over time as the economy grows, is instead a source of revenue for the government. It's like the government owning an asset, not owing a liability. The government-owned central bank is a profit-centre for the government. Printing currency is a profitable business. We even have a special name for those profits: "seigniorage".

Revenue from printing bonds is like revenue from printing currency, if r<g for bonds. But there's a limit to how much revenue the government can sustainably earn by doing them.

All economics students are familiar with the idea that printing money is a source of government revenue. But slightly more advanced students also know there is a limit to how much revenue, in real terms (adjusted for inflation), the government can sustainably get by printing money (and Zimbabwe is what happens when the government tries to go past that sustainable limit). Because the faster the government prints money and spends it, the higher the rate of inflation. And the higher the rate of inflation the more negative the real rate of interest on money (the higher the opportunity cost of holding money), and the quicker people will get rid of money they earn, and so the smaller their money/income ratio.

Here's some simple math to illustrate: let Mdot be the amount of money printed per year. Let NGDP be nominal GDP. So Mdot/NGDP is the government's revenue from printing money as a fraction of NGDP. Multiply and divide by the stock of money M, and this equation becomes:

Government revenue from printing money as fraction of NGDP = (Mdot/M)x(M/NGDP) = growth rate of money x (money/NGDP) ratio.

Money demand curves slope down. The higher the money growth rate, the higher the inflation rate, and the smaller is M/NGDP. The money growth rate that maximises government revenue (as a fraction of NGDP) is where the elasticity of the demand curve is exactly one, just like for any revenue-maximising monopolist.

The same idea should apply to government bonds. Government bonds are not as special as currency, but they are still special. They are (usually) safer and more liquid than other assets, so it is no surprise if they pay a lower rate of interest than other assets. And that interest rate differential is the opportunity cost of holding government bonds instead of other assets, and the demand curve for government bonds should slope down as a function of that interest rate differential, just like the demand curve for money slopes down. The only difference is that government money (currency) pays 0% nominal interest, while government bonds (normally) pay a positive nominal rate of interest. But if that nominal rate of interest is less than the growth rate of NGDP, the principle is the same.

If r<g for government bonds, then issuing bonds and printing currency are both sources of government revenue. But there's a limit to how much revenue the government can collect from those sources. A higher Bond/NGDP ratio requires a higher rate of interest paid on bonds to persuade people to hold a higher ratio. There is some Bond/GDP ratio (and associated rate of interest on bonds) that maximises sustainable government revenue from printing bonds. And if the Bond/GDP ratio gets too high, then the bond rate of interest rises to equal or exceed the growth rate of the economy, and that source of government revenue disappears.

I think this math is right (don't trust me):

Sustainable government revenue from printing bonds, as a fraction of NGDP = (g-r)(B/NGDP) where B is the stock of bonds

A higher debt/NGDP ratio means a higher r, so a lower (g-r), so there are two offsetting effects. Sustainable government revenue is maximised where the bond demand curve has an elasticity of one.

The only difference between the bond formula and the money formula is that the nominal rate of interest on money is assumed to be 0%. And if we assume that the growth rate of NGDP equals the money growth rate (which it must be if the M/NGDP ratio is constant over time), the two formulas become equivalent.

Think of the government like a price-discriminating monopolist, that sells two financial instruments (currency and bonds), paying different rates of interest, making a profit on each. (Of course, it's really a lot more than just two, given the whole term-structure.)

[The other important difference between government money and government bonds is that the first is unit of account and medium of exchange and the second isn't. But that doesn't affect the point in this post about their similarity as sources of government revenue when r<g.]

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Comments

[Matthew Young]

r < g (properly leaving spaces, I hope) is the general condition of the currency system. We borrow because we know that we have access to a g that is greater than the current r.

r < g is hidden information, unobservable. The fact that someone can measure an r < g means they already have the steps needed to make the gain, and the currency banker never sees it until the loans are made, after the fact.

Creating a government monopoly that enforces an r < g via regulation simply equalizes by wedging the cost of regulation in the system. Currency banking, by definition, about currency risk, the currency banker shares it in the process of discovering recent growth.

[JKH]

I’ll take a shot at this, quick and dirty:

Currency pays 0 interest
Bonds pay r < g

Interpretation:

Currency:

Issue new currency X and “spend” it on some quantity G. (yes … the actual flow of funds is a little more complicated than that - but that’s the effective result). This can be interpreted as revenue in the following sense. There is no liability in respect of a future interest cost. While the quantity of currency appears on the balance sheet as a notional liability, and while the principal value liability can be actual in terms of contingent balance sheet operations (standard currency redemption), the absence of a future interest liability can be viewed as a benefit that reverses the economic value of the nominal balance sheet liability and makes the money net “free”. That reversal could be interpreted (very loosely) as revenue. It’s also the capitalized value of the running seigniorage interest margin on the typical central bank balance sheet, roughly speaking. But that really is an artificial accounting construct, used for internal transfer pricing on the central bank balance sheet as much as anything. And that sort of thing is typically tracked by periodic accrual accounting of a net interest margin effect, rather than upfront “market value” or “fair value” or “economic value” (or “Rowe” value accounting (often enlightening) for that matter). A given interpretation always requires some chosen mode of accounting, even if economists don’t like to acknowledge that’s what they’re doing.

Bonds:

Issue a new bond B in the same amount as was done in the case of currency X and “spend” it on the same quantity G. (yes … the actual flow of funds is a little more complicated than that - but that’s the effective result). The principal notional value of the bond is B. But the bond is not “free”, because there is a liability in the form of future interest payable. Some would say that the present value of that liability is the value of the bond – at least at the time of issuance. It may fluctuate over time. Nevertheless, there is not the same absolute absence of a future interest liability to reverse the effective economic value of the nominal liability as there was suggested in the case of currency. So the same interpretation of reversing revenue cannot be applicable as it was for currency.

However:

Assume as in the post that r < g, but that the debt to GDP ratio is kept constant. This can only be done by running a primary deficit and issuing more bonds. Now, consider (g – r) as a particular type of interest rate in itself. It is the incremental interest rate that would have to be paid in addition to r, in order to allow the debt to GDP ratio to stabilize without running a primary deficit. That hypothetical incremental rate of interest in fact isn’t paid, which is what allows for the issuance of more debt.

But suppose it *was* paid. For ease of illustration, suppose the debt to GDP target is stabilized at the start of the fiscal period. Suppose r < g has been paid on an ongoing basis on all outstanding debt. Consider two cases. One is continuation of r < g. The other is a sudden conversion of all outstanding debt to floating rate, and an increase in the interest rate paid on all debt to r * = g. So in the latter case, the incremental interest rate in question (g – r) is suddenly being paid in addition to r on all outstanding debt. Then that new incremental interest paid amounts to the same quantity of money that would have been raised by issuing new bonds in the (now counterfactual) case where r < g, a primary deficit being necessary in that case in order to stabilize the debt to GDP ratio. But there cannot be a primary deficit in this new case, because r* is exactly the rate of interest that stabilizes the debt to GDP ratio only if the primary deficit is 0. Another way of saying this is that the result of paying r < g on newly issued bonds amounts to a total outstanding interest liability that is equivalent to paying r* = g on previously issued bonds and issuing new bonds B that pay an interest rate of 0 rather than an interest rate of r or r*. So the benefit of avoiding that incremental (g – r) rate of interest on the outstanding debt is equivalent to the benefit of issuing new bonds B or new currency X at a 0 rate of interest while paying the higher r* = g on all the previously outstanding debt. This may appear to be an empty or open ended result, except for the fact that the number chosen for the issuance of 0 interest debt B or money X results in the same total amount of outstanding debt and currency in both cases.

All that said, there is revenue in the case of bonds where r < g, but only in the sense of a comparison of factual to counterfactual. This is an opportunity cost calculation. Whereas in the case where money is actually issued in the normal course, the revenue interpretation corresponds to an absolute economic benefit in the sense that the principal value of money issued can be treated as “free” (apart from redemption complications noted above). It is only when the interest rate is zero that an opportunity cost analysis becomes an absolute calculation as well. A quantity of a future interest liability of 0 is both relatively and absolutely beneficial. 0 is a singular point rate of interest when it comes to capitalizing the value of a liability. But any bond that pays a positive interest rate is only partially “free” or equivalent in some economic sense to revenue relative to alternatives.

Like I said, quick and dirty. And maybe wrong.

But I think it may possibly align (apart from the simplicity of the revenue interpretation) with:

“Sustainable government revenue from printing bonds, as a fraction of NGDP = (g-r)(B/NGDP) where B is the stock of bonds”

[Frank Restly]

Nick,

"A world where the interest rate on government bonds is (permanently) less than the growth rate of GDP ("r The government can run a Ponzi scheme, where it borrows (sells more bonds) to pay for the interest on the existing bonds, so the stocks of bonds grows at the rate of interest."

Yes money is fungible and so it's not possible to say that this dollar received by the government when it taxes is spent on interest and this other dollar that the government borrows is spent on defense. I think we can agree that the government would be performing Ponzi finance when in aggregate the interest expenditures on it's debt exceed it's available tax revenue.

Even when r < g, it remains to be seen whether the government can reduce all tax rates to 0% (all expenditures, interest or otherwise are paid with money borrowed by government).

[Robert Barbera]

"A world of r less than g is a weird world".
Then we must acknowledge that the real world, for at least 125 years, has been very weird.

A careful look at real returns, 1870-2015, led the authors to this:
"Summing up, during the late 19th and 20th centuries, real returns on safe assets have been low, 1% for bills and 2.5% for bonds."
"The Rate of Return on Everything", Jorda et al, JUNE 2017

Why are risk free rates so low? Why are risky rates so high? Two puzzles.
Why do pundits lecturing fiscal authorities insist that r soon and forevermore must equal or exceed g?
Perhaps because to say something different sounds weird.

Bob Barbera

[Nick Rowe]

Bob: was that for the US? I think r < g is not quite so common in all countries. But yes it's not uncommon, but it's still weird, because what looks like a "liability" acts like an asset, which really is weird. But it's a lot less weird if we think about currency, which typically has a negative real return.

[Nick Rowe]

JKH: you lost me somewhere about halfway down. I was following you till then.

Let me try this: consider provincial governments that can issue bonds at r < g, but not money (to keep it simple). There are two otherwise identical provinces. One has a positive debt/GDP ratio (and keeps it constant); the second has zero debt. Both have the same taxes. The first can have G/NGDP permanently higher than the second, by the amount (g-r)(B/NGDP).

[JKH]

Continuing from where I left off:

“Sustainable government revenue from printing bonds, as a fraction of NGDP = (g-r)(B/NGDP) where B is the stock of bonds”

Or more directly:

“Sustainable government revenue from printing bonds = (g-r)(B) where B is the stock of bonds”

My analysis is consistent with that.

Except it is incorrect to refer to this as revenue. It is only a reduction in interest paid on bonds relative to the counterfactual where g = r.

Similarly, from your example in comment:

“The first can have G/NGDP permanently higher than the second, by the amount (g-r)(B/NGDP).”

Yes. But that’s simply a comparison between two cases. And the difference is due to a lower interest expense, not revenue.

By contrast, some economists seem to like to treat seigniorage "revenue" on currency as the immediate value of the money issued. That I can understand, even though it is inconsistent with actual central bank accounting and with the required integration of central bank accounting with government budget accounting, all of which is done on a running net interest expense basis. Nowhere in real world institutional treatment will you find seigniorage calculated as the immediate capital value of the money issued. This is only done by economists who insist on doing it that way. Still, that I can understand, even to the point of calling that value revenue, notwithstanding the inconsistency of that terminology with how things are actually handled in real world institutions. That can work in an outright sense because the interest rate is 0. But it is a bridge too far to suggest that the interest differential between two positive interest rate cases be capitalized and/or considered as revenue in the same way.

[Robert Barbera]

Nick,
The data was collected for 16 developed economies, 1870-2015, mostly European, plus USA and Japan. Very low r was the rule, not the exception. My WEIRD explanation? The economy throws of an aggregate return. It is disseminated across asset classes. We all scratch our heads but accept the inexplicably large risk premium. If risky and risk free assets must split that return, then inexplicably low risk free rates are the necessary compliment to ...

Bob Barbera

[Nick Rowe]

JKH: If I ran a bank, borrowed at r, and lent all that I have borrowed at g, and then consumed my profits as they came in, you would (I think) have no problem in treating my bank's profits as part of my income (or revenue). Now suppose I borrow, at rate g, against my bank's future profits, and spend it on consumption. In fact, let us assume I borrow from my own bank, so my bank lends only to me, at rate g.

[Nick Rowe]

JKH: I just remembered: we argued about this same(?) thing before, in this old post! Accounting for Central Bank Profits

[JKH]

thanks Nick, I'll have a look back at that

but before doing so:

Apart from the usual standard central bank operations (which operate with respect to currency seigniorage, and even then are a somewhat artificial internal transfer pricing arrangement), there’s no financial intermediation function embedded in (the rest of) government borrowing and spending. So there is no lending function and no profit spread associated with that. The government is not borrowing at r and lending at g, with profit spread then being spent on additional G. There’s only a difference in government borrowing costs for 2 cases, with the lower borrowing cost case accommodating additional, sustainable periodic borrowing and spending before overall sustainability is breached. The spread is between 2 borrowing costs – not between a borrowing cost and a lending return. It is perhaps possible to calculate a capitalized value of that spread between two potential borrowing costs, but that is a capitalized quantity of saving – not revenue.

[JKH]

slightly ironic that I'm the one who seems to be arguing in part that central bank accounting, while in fact quite useful, has the potential to obscure the economics of consolidated government operations and finance

[bill]

How should we react when the Fed freely chooses to pay a higher rate for overnight money (IOR) than the Treasury pays for 30, 90 and 180 day money? The Fed is purposely wasting the asset you've described.

[bill]

I like the example of the two provincial governments. I have a follow-up question. Can we determine which province will have the higher RGDP? ie, is the higher G/NGDP likely to be additional GDP or a tradeoff between G and the private sector? Or a different price level?

[john]

Think of the flows, too.

If r is less than g then interest paid is always less than marginal tax receipts.

On the other hand, I'm sure the math presented is based on solid accepted theory, but it is really difficult to imagine a sovereign getting into trouble substituting printing for tax. Printing likely has a different "tax incidence" than whatever exists, but its still just money. Conceivably, one could design cash taxes to have identical incidence with printing, which puts the burden of proof that printing is somehow "bad" in a different cash flow light.

[Nick Rowe]

JKH: "slightly ironic that I'm the one who seems to be arguing in part that central bank accounting, while in fact quite useful, has the potential to obscure the economics of consolidated government operations and finance"

Yep! :-) We seem to have sorta swapped sides!

bill: "How should we react when the Fed freely chooses to pay a higher rate for overnight money (IOR) than the Treasury pays for 30, 90 and 180 day money? The Fed is purposely wasting the asset you've described."

Yep. Looks pretty daft to me too.

john: "If r is less than g then interest paid is always less than marginal tax receipts."

I don't think that's right. We can imagine a government that collects very low taxes and spending, and a second government with very large taxes and spending, but both have the same debt/GDP ratio and interest rates.

[Nick Edmonds]

This suggests to me that the constraint on expansionary fiscal policy should be about the level of interest rate consistent with the inflation target, and not any arbitrary debt / GDP ratio. As long as fiscal policy is managed to keep the interest rate at a low enough level, the debt level is irrelevant.

[Frank Restly]

Nick E.

Total interest payments are a function of both the interest rate and the total debt. If the interest rate is 2% annual and the debt is growing at 12% annual, then total interest payments on the debt will be growing at roughly 2% + 12% = 14% annual. To avoid the Ponzi finance condition, tax revenue used to make the interest payments would need to grow at that rate or higher. The assumption is that tax revenue will grow at roughly the same rate as the economy, hence a stable debt / GDP ratio could be used as an anchor of sorts.

[JP Koning]

"In our familiar world, where r < g for currency, currency isn't really a government "liability" in the normal sense of the word. Having a positive currency/GDP ratio, and keeping it constant over time as the economy grows, is instead a source of revenue for the government. It's like the government owning an asset, not owing a liability."

Whenever you mention this idea, it always makes me think of Warren Buffett on insurance float. Not sure if you're familiar with the idea or if you see any analogies:

Link to Buffet Here NR

[Nick Edmonds]

Frank,

Why is the debt growing at 12%? If the interest rate is only 2%, then that must be because of a primary deficit. Now if the debt is growing at 12% and GDP is only growing at say 4%, then the ratio of debt to GDP is growing. Which means that if debt continues to grow forever at the same rate, then the primary deficit as a ratio to GDP must also be growing. Eventually it would need to be more than 100% of GDP.

[Frank Restly]

Nick E.

"Why is the debt growing at 12%?"

It was an example. The reason I brought it up is that debt/GDP numbers are not arbitrary and debt levels are not irrelevant.

[Nick Edmonds]

Frank,

Sorry if it was unclear, but that question was rhetorical. My point was that your assumptions implied that the primary deficit would eventually exceed GDP.

[Nick Rowe]

Nick E: The way I see it: an increase in the debt ratio causes r* to increase, but provided the central bank sets a higher interest rate to match the increase in r*, any r* is consistent with hitting the inflation target.

We could say that the government should set the debt ratio high enough that r=g. Which is dynamically efficient (a la Samuelson), and sort of parallel to Friedman's Optimum Quantity of Money, but it means the government gets no revenue from issuing debt (or money).

[Nick Rowe]

JP: Hmm. I wasn't familiar with that idea, but do see a resemblance. Bit like an interest-free loan, that always gets rolled over. Which is like currency.

[Nick Edmonds]

Nick,

Yes, I probably wasn't very clear, but what I meant was setting fiscal policy so that the inflation target could be hit with a rate r that didn't exceed g, in the long run. What debt ratio this implies could vary greatly from country to country and time to time.

I agree that at r = g, the government gets no revenue benefit. I'm still wondering about whether or not I think this matters.

[JKH]

Nick,

A crude math example in an attempt to illustrate the point I made above (October 29, 2017 at 04:31 PM).

Case 1

Prior period:
NGDP = 400
T = 100
G = 100
Primary deficit = 0
Debt = 200
Debt/GDP = 50 per cent

g = 2 per cent
r = 2 per cent

This period:
NGDP = 408
T = 102
G = 102
Primary deficit = 0
Interest cost = 4
Deficit = 4
End of period Debt = 204
Debt/GDP = 50 per cent

Constant and sustainable

Case 2

Prior period:
NGDP = 400
T = 100
G = 100
Primary deficit = 0
Debt = 200
Debt/GDP = 50 per cent

g = 2 per cent
r = 1.75 per cent

This period:
NGDP = 408
T = 102
G = 102
Primary deficit = 0
Interest cost = 3.5
Deficit = 3.5
End of period Debt = 203.5
Debt/GDP = 49.9 per cent

Declining and “better than” sustainable

Case 3

Prior period:
NGDP = 400
T = 100
G = 100
Primary deficit = 0
Debt = 200
Debt/GDP = 50 per cent

g = 2 per cent
r = 1.75 per cent

This period

Issue additional bonds of .5
Spend additional G of .5

NGDP = 408.5
T = 102
G = 102.5
Primary deficit = .5
Interest cost = 3.5 (roughly)
Deficit = 4.0
End of period Debt = 204
Debt/NGDP = 50 per cent (roughly)

Constant (roughly) and sustainable

Case 4

Same as Case 3
Except:
Instead of issuing new bonds
Issue new money of .5
Instead of paying 1.75 per cent on bonds, pay 2 per cent
Pay 0 per cent on money

The amount of new money issued in case 4 is the same as the amount of new debt issued in case 3. So the capacity to issue new debt at an interest rate of r < g is equivalent to a capacity to issue the same amount of new money instead (at an interest rate of 0), while paying an interest rate of r = g on debt.

Conclusion:

Paying r < g on debt is beneficial relative to the case of paying r = g. The benefit is the capacity to issue new debt, incremental to an otherwise assumed debt growth rate of r, while still sustaining a constant debt/NGDP ratio.

This benefit is equivalent to a theoretical counterfactual in which the interest rate paid on debt is r = g while new money (instead of incremental debt) is issued at an interest rate of 0.

This expresses the benefit of r < g in terms of an equivalent benefit of earning seigniorage on new money that is issued instead of new debt (incremental to the usual growth in debt according to r), while paying r = g on debt instead of r < g. The debt/NGDP level is kept constant and sustainable in either the actual case or the equivalent theoretical counterfactual case.

The purpose of this is to show that the debt issuance benefit can be expressed according to an equivalent case seigniorage benefit, which in turn can be expressed in an accounting sense either as an ongoing interest margin benefit or an upfront capitalized value benefit (sometimes loosely described as “revenue”).

This is the point I attempted to make in my comment above (October 29, 2017 at 04:31 PM.)

It seems to me to be easier to make the case that currency seigniorage can be interpreted as either a beneficial interest margin (normal central banking accounting) or upfront “revenue” (economists’ optional accounting), than it is to suggest that r < g generates “revenue” in a more general case sense.

[Nick Rowe]

Nick E: Ah! OK, we are on the same page. (Funny thing is, when I first read your comment, that is what I interpreted you to be saying! It was only on my second reading I interpreted you to be saying something else!)

JKH: OK, I *think* I am following you now.

[john]

You said:

"john: "If r is less than g then interest paid is always less than marginal tax receipts."

I don't think that's right. We can imagine a government that collects very low taxes and spending, and a second government with very large taxes and spending, but both have the same debt/GDP ratio and interest rates."

I'm afraid I don't get the difference.

But try this. If a government is paying the interest out of tax receipts, and r is less than g, tax receipts (tracking GDP growth) will be get bigger faster than interest expense.

Now, I suppose you could say a government could be financing its interest expense, where taxes are less than interest. But even then, eventually taxes will be bigger than interest.

[Nick Rowe]

john: take the simplest example, where taxes are zero. Let r=2% and g=4%, and the debt/GDP ratio be 100%. Every year the government borrows an additional 4% of GDP (so the debt grows at 4% and the debt/GDP ratio stays constant), uses half if it to pay interest, and spends the other half.

[john]

Nick, you seem to be focused on debt/GDP, for what reason who knows. I'm just focused on whether a government can finance itself, and it certainly can, as your simplest example also shows. In my world debt/GDP is a mere curiosity.