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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.

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.


"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).

Frank: you are not getting it. Stop now.

"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.

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

@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.


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?

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?

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.

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.

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.)

@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.

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.

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)?


"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).

@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.

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.

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.

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.

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?

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.

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