If the rate of interest on government bonds is less than the growth rate of the economy (r<g), then the average cost of bond-finance is negative. A government that has issued bonds, and issues more bonds each year to keep the debt/GDP ratio (B/Y) constant (which is sustainable), can have permanently higher spending, or permanently lower taxes, than an otherwise identical government which has issued no bonds. Issuing bonds is a (permanent) source of government revenue. Bonds are not a "liability" in the normal sense of the word.

**But it is important to distinguish between average cost and marginal cost.** Unless the demand for government bonds is perfectly elastic, an increase in the debt/GDP ratio will increase the equilibrium rate of interest required for people to be willing to hold those government bonds. So the government is like a monopsonist facing an upward-sloping supply curve of finance, and the marginal cost of extra bond-finance will exceed the average cost of all bond-finance. **The marginal cost of increasing the debt/GDP ratio might be positive even if the average cost is negative. It all depends on the interest-elasticity of demand for government bonds.**

I will now try to do the math.

Start with the simplest case, where the economy is not growing (g=0) and inflation is zero. There is a stock of bonds B, that pay a rate of interest r, and GDP is constant at Y. With the stock of bonds constant over time, each year the government will need to pay rB interest, and rB/Y interest as a percentage of GDP. If r>0, the government will need to have spending on goods and services (excluding interest) lower than taxes to service the debt. (It will need to run a "primary" surplus. Debt service costs as a ratio of GDP will be:

s = r(B/Y)

Now lets keep inflation at 0, but assume GDP is growing at rate g. This means the government needs to issue gB new bonds each year just to keep the stock of bonds growing at the same rate as GDP, and so keep the debt/GDP ratio constant over time. Which means it can run a smaller primary surplus than if the economy were not growing. Debt service costs as a ratio of GDP are now:

s = (r-g)(B/Y)

If we add inflation to the model we have two choices: either we interpret r as the nominal interest rate and g as the growth rate of nominal GDP; or else we subtract inflation from both so r is the real interest rate and g the growth rate of real GDP. It makes no difference which way we do it, as long as we are consistent. Similarly, it makes no difference whether the numerator and denominator in the debt/GDP ratio B/Y are both nominal or both real, as long as we are consistent.

If r>g the average cost of bond-finance is positive (a government with a positive debt/GDP ratio will need to run a bigger primary surplus than a government with no debt). But if r<g the average cost of bond-finance is negative. A government with a positive debt/GDP ratio will need to run a primary deficit to keep the debt/GDP ratio constant. It borrows to pay the interest on the debt, and then borrows some more just to prevent the debt growing more slowly than GDP.

Yes this is a Ponzi scheme. But it is a stable Ponzi scheme. It is very similar to what a government does when it prints currency. Currency pays 0% nominal interest, and minus 2% real interest if there is 2% inflation. So r<g for currency. And each year the government prints more currency to keep the currency/GDP ratio constant. Which is a nice profitable little business for the government. We even have a special name "seigniorage" for the revenue the government earns from issuing currency. So you can think of the revenue the government earns from issuing bonds, if r<g, in the same way.

But there's a limit to the revenue, as a percentage of GDP, the government can earn from issuing currency. Because the demand to hold currency, as a ratio of GDP, is not perfectly elastic. There is a negative relationship between the inflation rate (the negative real interest rate earned by holding currency) and the currency/GDP ratio. Similarly there will be a positive relationship between the interest rate on government bonds and the debt/GDP ratio. The demand curve for government bonds is not perfectly elastic; people will hold a higher ratio of government bonds to GDP only if those bonds pay a higher interest rate. To say the same thing another way, if government bonds are in low supply, relative to income, the interest rate on those bonds will be low, other things equal. So the marginal cost of bond-finance is greater than the (r-g) on the marginal debt; it must include the marginal (r-g) on the initial debt. That's the second term in this equation, that we get from differentiating the above equation:

ds/d(B/Y) = (r-g) + (B/Y)d(r-g)/d(B/Y)

**Remember Intro Microeconomics, when the prof explained that the Marginal Revenue for a monopolist was less than Price (Average Revenue), because to sell an extra apple he needs to cut the price? And how it was possible to have P>0 and MR<0 if you are on an inelastic portion of the demand curve? That's what we are doing here. Except it's Marginal and Average Cost, and the Average Cost might be negative but Marginal Cost positive. And it's the elasticity of (B/Y) with respect to (r-g) that matters.**

Divide both sides by (r-g), rearrange terms, and we get:

[ds/d(B/Y)]/(r-g) = 1 + (B/Y)d(r-g)/d(B/Y)(r-g) = 1 + 1/e

where e is the demand elasticity of (B/Y) with respect to (r-g). It's the percentage (not percentage point) change in (B/Y) for a one percent (not percentage point) change in (r-g). **And if it's inelastic (e<1) then with r<g we have a negative average cost of debt-finance but a positive marginal cost of debt-finance.**

**I think that math is right. It feels right.**

Is it elastic or inelastic? I don't know. But notice how as r gets closer to g, elasticity approaches zero. Just like a linear demand curve for apples will get inelastic as the price approaches zero. So if r is only a little bit less than g, the average cost is negative but the marginal cost positive.

[There ought to be a simpler way to do this, where we could talk more intuitively about the demand-elasticity with respect to r, not with respect to (r-g), and compare that elasticity to (r-g)? Someone else can try to do the math, because I'm off to Tim's.]

**Update: see Majromax's comment immediately below. The revenue-maximising debt/GDP ratio is where the local demand elasticity with respect to r (the interest elasticity of demand for bonds) is equal to r/(g-r). So if it's less elastic than that, the marginal cost of debt-finance is positive.**

For a fixed GDP ratio debt target, in each period the government receives gB/Y 'revenue' through issue of new bonds and it pays rB/Y in interest, for 'profit' of gB/Y - rB/Y (which is the negative of your definition of debt service costs)

Differentiating this with respect to B under the assumption that Y and g are independent of B gives:

dP/dB = 1/Y(g - r - B dr/dB)

Divide through by r (assuming r > 0) and multiply by Y to give:

dP/dB Y/r = g/r - 1 - (B/r) (dr/dB) = g/r - 1 - 1/e

Setting this to zero to find the maximum of P tells us that the optimal debt ratio is at at which the local elasticity is equal to r/(g-r).

This is equivalent to your math under the assumption that g is short-run independent of B. If it's not (that is, if we're incorporating a supply-side story about crowding out investment), then things get much more complicated.

Posted by: Majromax | December 04, 2017 at 02:32 PM

Majro: Thanks! (I wouldn't call that the "optimal" debt ratio though; I would call it the revenue-maximising debt ratio.) Gonna update the post.

Posted by: Nick Rowe | December 04, 2017 at 04:44 PM

Nick,

"Yes this is a Ponzi scheme. But it is a stable Ponzi scheme."

You and I must have different definitions for a Ponzi scheme.

In my mind a Ponzi scheme exists where the returns on investment are paid solely by additional investors and not by a third party.

In the case of government debt, the returns on investment are the interest payments and the third party are tax payers.

If r < g and B / Y is constant, then the only thing to calculate is the tax rate sufficient to avoid the Ponzi condition.

Revenue (R) = Tax Rate (TR) * Gross Domestic Product (GDP)

Interest Expense (s) = Value of Bonds Outstanding (B) * Interest Rate (r)

For a growing / contracting economy:

R = TR * [ t * dY/dt + Y(0) ]

s = r * [ t * dB/dt + B(0) ]

For non Ponzi condition R >= s.

TR * [ t * dY/dt + Y(0) ] >= r * [ t * dB/dt + B(0) ]

TR >= r * [ ( t * dB/dt + B(0) ) / ( t * dY/dt + Y(0) ) ]

If you want to keep a constant debt to GDP ratio ( dB/dt = B(0)/Y(0) * dY/dt ), that's fine and the equation simplifies to:

TR >= r * B(0) / Y(0)

But you can still oscillate back and forth across the Ponzi boundary depending on your tax rate (TR) and your interest rate (r).

Posted by: Frank Restly | December 04, 2017 at 08:37 PM

Frank: you are not getting it. Stop now.

Posted by: Nick Rowe | December 04, 2017 at 08:47 PM

"Debt service costs as a ratio of GDP are now:

s = (r-g)(B/Y)"

This quote seems incorrect. Term 'r' is the interest rate on bonds and term 'g' is the growth rate on the combined private sector and government sector. The two terms are mismatched so make as much sense as subtracting carrots from oranges.

I think that when the economy is growing, debt service costs as a ratio of GDP are still just

s = rB/Y

measured on an annual basis.

Posted by: Roger Sparks | December 05, 2017 at 10:13 AM

d (s/Y) = ds/s - dY/Y = ds/s - g

ds/s > g if dr/r > g

So, service of debt s rises when marginal r is higher than g, in spite of r < g

Posted by: Miguel Navascues | December 05, 2017 at 12:54 PM

@Roger Sparks:

> I think that when the economy is growing, debt service costs as a ratio of GDP are still just

The gross debt service cost is still rB/Y, but if the government maintains its a priori constant debt-to-GDP ratio then it will have to issue gB/Y bonds. The debt service cost net of the bond issue is then (r-g)B/Y.

It's the issue of new bonds that makes this a nontrivial optimization problem, and that continual issuing of new bonds is what makes this a sustainable Ponzi scheme.

Posted by: Majromax | December 05, 2017 at 01:23 PM

@Majromax

Hmmm. Here is the way I would do the math:

We want next year's bond total to be some constant ratio of GDP. We would write this as

B1 = K*GDP1

We have a second equation relating last year's bonds to next year's bond total, which we write as

B1 = B0 + dB

Combine the two equations to write

B0 + dB = K*GDP1

I prefer to consider debt service as a raw number, so I will depart from Nick. Use capital S for raw debt service. Then next year's debt service will be r*(B0 + dB) so we write

S1 = r*(B0 + dB) or S1 = r*K*GDP1

Now we could express GDP1 in terms of GDP0 by writing

GDP1 = (1+g)*GDP0

Then we could combine the last two equations to write

S1 = r*K*(1+g)*GDP0

Finally, we can return to Nick's terminology to express S as a ratio of GDP, s = S/GDP. We write

s1 = r*K*(1+g)*GDP0/GDP1

I am not finding any (r-g) term so far.

How does this look to you?

Posted by: Roger Sparks | December 05, 2017 at 04:59 PM

Roger: I thought that Majro's answer was good. I'm gonna leave this to him.

Miguel: I'm afraid I'm not following that. If the percentage increase in r (for a 1% rise in B/Y??) is bigger than g, the marginal debt service cost is positive?

Posted by: Nick Rowe | December 05, 2017 at 06:13 PM

debt/GDP ratio and this r/(g-r)

Nice work actually, I reread this a few times.

From the result it is easy to see:

g, after debt service, goes up and GDP goes up. r, debt service, goes up mean you carry more debt.

Make them both distributions, or better, sequences of values. And suppose we have a machine that can generate typical sequences of r or g. Then, can we do the algebra above with our machine, rather than the sequences themselves?

Yea, I think so. We have sequence generators that divide and subtract. We can take a subset of all possible sequences, g and r, each subset being a bundle of equally probable typical sequences, which have a generator. The generators will follow the rules, of this ratio, not the other equations.

We get measurable error, and can make that as mall as the original data is accurate. The system needs the typical sequence of tiny g things that make up a GDP. There is like a Euler condition for this to work. Everyone tries to be Euler, but are equally bad at it. That assumption is good enough.

Posted by: Matthew Young | December 06, 2017 at 03:05 AM

We should note these equation are from an exponential system, obtained usually by taking the log of multiplicative exponential functions. They are log linear which is fine. So, DSGE economists alway want process with exponential growth and decay, which is fine.

The limited version of euler conditions, is finite log with bound error. Math guys can do this, I think. There is a complete set of ratios and adds and subtracts, and derivatives should be available. While DSGE has the whole number line, we use a subset of the integers. The The operations are done on generators, which are like tensors. It is an algebra built around optimum queued spanning trees. Te have consistent distance measures, and we can do ratios on generator size. I bet the math folks back me up.

Posted by: Matthew Young | December 06, 2017 at 03:27 AM

Hello Nick,

I have a somehat tangential response to this. My argument is that fiscal policy settings can entirely determine the inflation rate and real rate, and hence the worries about the marginal rate are moot.

The argument is as follows. For simplicity, assume:

- no growth (fixed population, no productivity)

- only money is government issued, no Treasury bills (0% nominal rate),

- fixed government consumption,

- progressive tax system, with a “high” max marginal rate, and tax brackets credibly fixed,

- welfare payments/Job guarantee providing a fixed reservation wage for employees.

In any economic model with the above parameters and where there is an upper limit of money holdings to nominal income, the steady state has to have an inflation rate that equals the indexation rate for the tax system. By implication, fiscal policy can plant the real rate wherever it wants.

Since the government can set the real rate with this set of policy settings, it is clear that steady state real interest rate is a function of fiscal policy more generally.

(I have a longer form argument on my website.)

Posted by: Brian Romanchuk | December 06, 2017 at 08:58 AM

@Roger Sparks:

> How does this look to you?

You account for the interest paid on the marginal ΔB bond issue, but you're not accounting for the revenue raised by the bond issue itself. Your calculations seem to assume that the bonds are issued gratis, rather than sold.

@Brian Romanchuk:

You've reinvented modern monetary theory (neo-Chartalism), hitting on all the main points: strict government issue of money, government expenditure as the money issuing instrument, taxes as a money sink, and a job guarantee. Note that your model is even over-constrained, since you can't guarantee both a fixed government consumption and an unlimited job guarantee/welfare.

Posted by: Majromax | December 06, 2017 at 09:47 AM

Hi Brian: "- only money is government issued, no [Typo?] Treasury bills (0% nominal rate),"

I think that's the only assumption you really need to get your conclusion (if we are talking about the real rate of interest on those T-bills, and not the rate of interest on other assets).

I would re-state your point slightly differently. I would assume the only asset the govt issues is currency that pays 0% nominal interest, and that therefore pays a real interest rate of minus the inflation rate. The deficit then determines the money growth rate, which determines the inflation rate, and hence the real rate earned by holding that currency. The real demand for that currency (M/P) is a negative function of the inflation rate. This is monetarism, in a world with only currency and no bonds.

Posted by: Nick Rowe | December 06, 2017 at 09:57 AM

On re-reading, your Typo maybe wasn't a typo. I think you are making the same assumption as me. You are talking about a government that issues currency, and does not issue Treasuries (bonds)?

Posted by: Nick Rowe | December 06, 2017 at 09:59 AM

@Majromax

"You account for the interest paid on the marginal ΔB bond issue, but you're not accounting for the revenue raised by the bond issue itself."

Yes, you are correct.

Nick did not specify what would be done with the money received from a bond sale. Government has the management choice of spending that money immediately or retiring that money by holding it (or destroying it if we follow MMT).

Posted by: Roger Sparks | December 06, 2017 at 12:36 PM

@Majromax - yup, I’m a MMTer; just putting into a mathematical formalism (in the article on my site).

Nick - yes money only, no bonds. (Possibly with a legally enforced ZIRP: you can buy all the TBills you want -at 0%.)

The arguments in the mathematical version of my article (which was not completely formalised, nor am I sure whether the arguments are correct!) abstract away from any relationship between money and inflation, I just show in steady state the growth rates have to be zero as a result of “plausibility constraints”. I only covered the 0% inflation (price level stability) case mathematically, but it seems that indexing the tax brackets would give similar nominal income growth.

Posted by: Brian Romanchuk | December 06, 2017 at 03:17 PM

Brian OK. If you hold the price level constant then real and nominal interest rates are the same. And in a standard ISLM with a constant price level, the government could increase the money supply to whatever level was needed to push the interest rate down to 0%. But assuming a constant price level is a problem, because prices won't be sticky forever, and you can't have an unlimited increase in real output.

Posted by: Nick Rowe | December 06, 2017 at 05:56 PM

Nick - my assumption is a fixed consumption function that only uses labour (matching simpler econ models).

If there’s a steady state, argument is that the steady state has to feature a wage inflation rate of 0% (if the brackets do not move), as otherwise the tax take gets too high, or everyone quits their job and takes a fixed welfare payment. (I waved my hands about the price of goods relative to wages.) I don’t care how we get to the steady state.

Even if output oscillates insome fashion, we should be able to pin down its average.

Forcing a non-zero inflation rate with a fixed indexation factor seems straightforward, but I did not try the math. The steady state definiiton is more complex.

Posted by: Brian Romanchuk | December 07, 2017 at 06:26 AM

What on Earth is the point of borrowing money (and having to pay interest on it) when you can print the stuff or grab near limitless amounts of the stuff from taxpayers?

The only valid reason for borrowing is when you’re short of cash, and governments clearly are not short of cash since they can print or impose taxes. If a taxi driver needs a new taxi and happens to have enough cash to hand, then he won’t borrow to buy the new taxi. Which proves taxi drivers know more about economics than governments or the economists who advise governments.

Posted by: Ralph Musgrave | December 07, 2017 at 10:14 AM

Nick, before I delve too deeply into this, can you please clarify two things?

(1) When you say "negative cost," does that mean the same thing as "positive cost that is less than the benefit"?

(2) Is there something special about government bonds, or would your analysis work for corporate bonds too?

Posted by: Bob Murphy | December 09, 2017 at 10:15 PM

Bob: good to see you on here.

1. Maybe. I don't see the difference. You can maybe think of rB as a cost to the government, and gB as a benefit, so the net cost is minus (r-g) i.e. a net benefit of g-r.

2. If I've got my head straight on this, I think it could work in principle with corporate bonds. Imagine a corporation that issued bonds that were very safe and very liquid, so would be held at a low rate of interest. And the corporation grows at the same rate as the economy. Which makes them very similar to commercial banks, some of whose very short term "bonds" are used as medium of exchange (demand deposits). But competition between corporations (banks), with free entry, would likely drive up r until the profits went to zero. You need some sort of de jure or de facto monopoly power or barrier to entry to make it work. But that monopoly power might simply be some Mengerian process where whoever gets in the market first wins, because its bonds are seen as more liquid, and so get traded more often, and so are more liquid.

You need some sort of liquidity story. Otherwise land would dominate in an r < g world, and destroy that r < g equilibrium. But land is rather illiquid.

Posted by: Nick Rowe | December 10, 2017 at 08:18 AM