The Sustainable Bond-Finance Laffer Curve

Imagine an economy growing at rate g, with an interest rate on government bonds r, and a constant debt/GDP ratio D/Y. The government gains revenue from issuing new bonds gD each year, and loses revenue from paying interest rD each year. If r<g then the government earns positive net revenue from having and maintaining a positive Debt/GDP ratio, and can run a primary budget deficit forever with the debt/GDP ratio staying constant. But r is an increasing function of the debt/GDP ratio (the interest rate must rise to persuade people to hold a higher stock of government bonds, relative to their annual income), so past a certain point r will exceed g. So there is a Laffer Curve, like the one I've drawn above.

You can get a Laffer Curve like that in an OverLapping Generations model like Samuelson 1958, where there's a shortage of assets to save for your old age. But if you add a fixed stock of land to an OLG model, it eliminates any equilibrium where r<g. The rate of return on holding land (rents plus capital gains) would exceed the growth rate of the economy. But land is less liquid than government bonds. In my view, it is the liquidity of government bonds that makes it possible, in some countries, for r<g on government bonds. (Extremely liquid currency typically pays a negative real interest rate, equal to minus the inflation rate.)

But maybe if the stock of bonds is very small, they won't be very liquid, because there aren't enough buyers and sellers. So the Laffer Curve might not look exactly as I've drawn it. And it probably won't look like that for some countries -- it might always be in the negative quadrant where r>g. Depends.

In a first-best world, the optimal debt/GDP ratio is where r=g, for dynamic efficiency. (Roger Farmer's back-of-the-envelope estimate is that this means a debt/GDP ratio around 70%.) But in a second-best world, where the government needs tax revenue, and taxes are distorting, it would be better for the government to choose a lower debt/GDP ratio than this, so it gets some revenue from bond-finance, which reduces the amount of revenue it needs from distorting taxes.

That's what I was trying to say in my earlier post. But pictures are often clearer than words.

Comments

[Roger Sparks]

When we look at the government sector of an economy, we are looking at a peculiar entity. Government both has ownership of the entire economy and at the same time has no ownership at all. We can say that government has control.

Further developing the peculiarities, we see that government has income only from the private sector, collected either from taxes or continuously increasing debt.

Now you imagine that the private economy is growing at a rate g, which is measured as g*GDP. If the economy is balanced, both the private sector and government sector will be growing at the same rate of g.

Next you propose that government borrow to keep government debt at a constant ratio to GDP. Government can only borrow from the private sector which government already has control of. Hence, you are proposing that the economy borrow from itself. Of course, this is being done by every fiat currency nation in modern times, so this is not a unique proposal.

Unfortunately, from the perspective of the macro economy, this proposed policy of constantly increasing government debt must result in a private sector is repeatedly relinquishing resources to government (as an entity) control and receiving debt (promises to pay) in return. Who is it in the private sector that can afford to accept debt (promises to pay) rather than trades-of-services? Certainly is is not the poor in the economy who have no money to lend!

Now I would agree that government taxation distorts the economy, but repeatedly increasing debt also distorts the economy. I think you are trying to show that a balance between taxation and debt financing of government needs to be found.

[Ralph Musgrave]

The above seems to assume that maximising government revenue is some sort of fundamental economic objective. The basic economic objective is actually to maximise GDP (within environmental constraints, of course).

The role that government debt can play in that connection is as a private sector asset (which is what government debt is). I.e. the larger the private sector’s stock of financial assets, the more it will spend. Ergo the optimum size of the debt is whatever induces the private sector to spend at a rate that brings full employment.

As to interest, as MMTers keep pointing out, a government can pay any rate of interest it likes on its debt. So why pay anything at all? I don’t see a good reason for paying anything much above zero. Milton Friedman and Warren Mosler argued for zero interest on government debt, i.e. they argued that the only state liability should be zero interest yielding base money.

[Nick Rowe]

Roger: "...but repeatedly increasing debt also distorts the economy"

Why? Maybe people want to hold those bonds, to save for their retirement.

Ralph: think of all the combinations of {Debt/GDP ratio; interest rate r} compatible with full employment. Among those combinations, I'm assuming that D/Y and r are positively related.

"The above seems to assume that maximising government revenue is some sort of fundamental economic objective."

No. I'm supposing the government wants to spend on public goods, poverty-reduction, whatever. And it wants to minimise the distortions needed to finance that spending.

[Frank Restly]

Nick,

"The government gains revenue from issuing new bonds gD each year, and loses revenue from paying interest rD each year."

Debt is cumulative. The government gains revenue from issuing new bonds g * ( D(1) - D(0) ) each year and loses revenue from paying interest r * D(1) each year.

We can have a situation where r < g, and yet r * D(1) > g * ( D(1) - D(0) ).

[Frank Restly]

Nevermind, I got it wrong. Just had to work through the math a little bit.

D(1) - D(0) = g * D(0) as long as the government maintains a constant debt / GDP ratio.

Proof
D/Y = D(0) / Y(0) = K : Constant Debt / GDP ratio.

dD / dt = K * dY / dt

[ D(1) - D(0) ] = K * [ Y(1) - Y(0) ] = D(0) * Y(1) / Y(0) - D(0)
D(1) = Y(1) * D(0) / Y(0)

Y(1) * D(0) / Y(0) - D(0) = g * D(0)
Y(1) / Y(0) - 1 = g

( Y(1) - Y(0) ) / Y(0) = g

[Ralph Musgrave]

Whence the assumption that the extent of distortion brought about by taxes is of significant proportions? A uniform sales tax on all goods and services would be totally non-distortionary. As for where sales taxes are NOT uniform (e.g. high taxes on motor vehicle fuel in the UK) that is likely to be for reasons generally regarded as beneficial (e.g. trying to cut down on the use of carbon based fuels). So the latter form of distortion is in a sense non-distortionary.

Re heavy taxes on high earners, that is also normally regarded as beneficial, and thus in a sense, not distortionary.

[Nick Rowe]

Ralph: "A uniform sales tax on all goods and services would be totally non-distortionary."

Leisure, home production, DIY, etc., are not taxed. Plus there's collection costs and compliance costs.

[Jussi]

"But r is an increasing function of the debt/GDP ratio"

Doesn't seem to hold true for Japan or for the US and Euro area after the Financial Crises.

Doesn't higher debt mean higher future taxes, which means lower growth and lower r - other things being equal?

"the interest rate must rise to persuade people to hold a higher stock of government bonds, relative to their annual income"

Expecting higher taxation in the future will increase propensity to save? So people will need to hold a higher stock of government bonds?

[Nick Rowe]

Jussi: think of a simple OLG model. People live 2 periods. The young buy bonds from the old. The bigger the debt, the lower is consumption when young, and the higher is consumption when old. So the higher the r needed to give people the incentive to postpone consumption.

[Jussi]

"The bigger the debt, the lower is consumption when young"

Why is that? Usually it is argued that more debt means more consumption?

[Nick Rowe]

Jussi: consider a young person: for a given level of income, if you spend part of that income buying government bonds from the old, you will be spending less on other things, like consumption and investment.

[Nick Edmonds]

Say you're at r=g and you decide that there is some benefit in a lower D/Y and reduced on-going taxes. To get there you need to temporarily raise taxes. How do you weigh up that cost / benefit?

[Jussi]

Nick: are you saying the aggregate saving doesn't change but r will be higher? Why more debt means the old are consuming more? Aren't they in turn spending part of their income buying government bonds from the government?

[Majromax]

> But in a second-best world, where the government needs tax revenue, and taxes are distorting

I think throughout this you're making the reasonable but hidden assumption that bond finance is less distorting than taxes, I'm worried about this hidden assumption because some level of distortion is built-in to interest rate elasticity in D/Y: the rate is elastic because other viable investments are crowded-out by the bond issue. All other things being equal, a higher D/Y ratio means that some factories won't be built and hence the GDP level should be lower over time than with a lower D/Y ratio.

If government taxes are highly-distorting, then maybe this is a fair assumption and the benefits of reducing taxes outweigh any negative to capital accumulation. On the other hand, if the bond income is used to reduce otherwise efficient lump-sum, land-value, or consumption taxes then the long-term best outcome might involve no debt at all.

[Nick Edmonds]

Also, I don't think you can take it as given that r is increasing in D/Y. In your 2 generation OLG model, it will be decreasing if the elasticity of substitution between C1 and C2 is low enough (for example if the preference is simply C1 = C2, then D/Y = 1/(2+r)).

[Roger Sparks]

"........... (the interest rate must rise to persuade people to hold a higher stock of government bonds, relative to their annual income), so past a certain point r will exceed g."

This problem (the need to raise r) is solved by allowing government to borrow from itself 'officially'.

Explanation:
Government represents all of the people in the economy. Therefore, when government borrows, it must borrow from the private sector. Then we see two ways of borrowing from the private sector.

First we have the normal way where government issues bonds to be sold to the private sector. This extracts funds from the private sector which government immediately spends, thereby reinserting funds back into the private sector.

Second, we have government borrowing from itself. This occurs when central banks (which are owned by government[which, in turn, is owned by the private sector]) makes a loan to government. Gov gives CB a bond; CB gives Gov funds. Government immediately spends funds into the private sector, thereby initially-inserting additional funds into the private sector.

You can see that, with either method, the private sector receives funds from government (which are payment for services and resources). The private sector can only provide a finite amount of services and resources each year so we can logically conclude that funds borrowed from either source can be used to claim a larger share of private output than would be claimed if government did not borrow from any source.

The growth rate of the economy has nothing to do with this process, nor does interest rates. Instead, government borrowing is a 'prime mover' that changes the distribution and motivation of the private sector.

[Nick Rowe]

Nick E: (Good to see you back on here!) "Say you're at r=g and you decide that there is some benefit in a lower D/Y and reduced on-going taxes. To get there you need to temporarily raise taxes. How do you weigh up that cost / benefit?"

I was hoping nobody would ask that question! It's too hard for my brain right now. But you would presumably want a slow transition towards the optimal steady state. There's something called a "Turnpike Theorem" that I think applies here. But I never did really understand that stuff.

With Leontieff preferences between C1 and C2, you get L-shaped intertemporal indifference curves (like right and left shoes). That would make r extremely sensitive to small changes in debt. It would go from minus to plus infinity if you added the marginal dollar. Doesn't seem to happen.

[Nick Rowe]

Majro: In a simple OLG model, the optimal level of bonds is where r=g, and as you move further away from that optimal quantity, the marginal cost will start at zero and be increasing. You can play with this model if you like (except that model has 0 growth and r=0 with no debt).

Jussi: you should have a look at that simple model too, to get the flavour.

[Scott Sumner]

Nick, You said:

"But in a second-best world, where the government needs tax revenue, and taxes are distorting, it would be better for the government to choose a lower debt/GDP ratio than this, so it gets some revenue from bond-finance, which reduces the amount of revenue it needs from distorting taxes."

Do you mean "higher" debt/GDP ratio, or am I confused?

[Nick Rowe]

Scott: *lower* debt/GDP ratio.

First, think of Friedman's Optimal Quantity of Money (applied to currency which pays 0% nominal). 1st best is to have a very large M/P, so the government earns zero seigniorage profits. 2nd best would be to have a smaller M/P, which means a higher inflation rate, which reduces the real interest rate paid on holding currency, so the government earns positive seigniorage profits.

It's the same with government bonds, which are not MoA or MoE like currency, but are liquid like currency, and have a downward sloping demand curve as a function of their opportunity cost (or upward-sloping demand curve as a function of their real rate of return).

[Scott Sumner]

Thanks, I stupidly compared it to the peak of the Laffer curve, not the optimal point.

[Frank Restly]

Nick,

Working the math a little heavier:

Again assuming a constant debt to GDP ratio
D = K * Y
D(0) = K * Y(0)
D(1) = K * Y(1)
%D = %Y = g

r * D(1) < D(1) - D(0) : Bonds are sold to pay off interest on existing bonds with no taxes and no other government expenditures

r * D(1) / D(0) < g
r * Y(1) / Y(0) < g
r * ( g + 1 ) < g

r - g < r * g : Not r < g

Even with a slightly positive r - g, this can still work. r = 3%, g = 2.95%, r * g = 0.0885%, r - g = .05%

Now include other government expenditures and a tax rate.

r * D(1) + G < D(1) - D(0) + TR * Y(1)

G = Government expenditures (other than interest)
TR = Tax Rate

Again assuming a constant debt to GDP ratio
D = K * Y
D(0) = K * Y(0)
D(1) = K * Y(1)
%D = %Y = g

Divide all sides of equation by D(0)

r * D(1) / D(0) + G / D(0) < %D + TR * Y(1) / D(0)

Subbing in D(1) / D(0) = Y(1) / Y(0), D(0) = K * Y(0), and %D = g

r * Y(1) / Y(0) + G / (K * Y(0)) < g + TR * Y(1) / K * Y(0)

Subbing in Y(1) / Y(0) = %Y + 1 = g + 1

r * ( g + 1 ) + G / (K * Y(0)) < g + ( TR / K ) * ( g + 1 )

r * g + r - g < ( TR / K ) * ( g + 1 ) - G / ( K * Y(0) )

It would be interesting to look at how changes in both the tax rate (TR) and government expenditures (G) affect the equation.

[rsj]

Suppose, for example, that we are past the point of 'maximizing' government income, but a proposal for a new road is made. Say the rate of interest is g. That road will generate g + s benefit to the economy. Why wouldn't you sell debt to build the road? Why on earth would you raise taxes to pay for the road, when the road has a long life?

This is just very odd. The government is probably the only entity on earth that doesn't need money, since it makes money. What the government is trying to do is maintain the health of the private sector. That is what should be maximized. If the private sector functions better with more bonds or fewer bonds, then that's what you do. Similarly, the CB should be setting interest rates for the benefit of the private sector, not because the government wants to pay a certain rate on its bonds. Government spending and taxation affects the real economy, because it provides public goods. We are not living in a Monarchy where a royal class is managing things in order to extract the most from us. The government exists for the private sector, not the other way around.

[Jussi]

"Jussi: you should have a look at that simple model too, to get the flavour."

t=1: The government issues and spends $100. The old consumes $50 (50 beers?) and the Young the other $50. The Old, as owners, buy the debt. The government splits 50/50 the consumption out of the debt, thus the Young consume +$50 and the Old -$50 (+50 out of fiscal spending and -$100 as additional savings towards the debt).
t=2: Either the Old, now the Dead" sold the bonds ($100) to the Young, which then needed to save the whole $100, or the Old didn't consume more and left the bonds as a bequest.

I get it! And I get it that thinking this way gives a *perception* that the government debt, specifically, is "a burden on future generations".

But, that is just a perceptions. The driver here is whether the Old consumes more or not. So the story can be told without the government debt at all, E.g. with private debt:

t=1: The Old issues private debt, and the society spends $100 more than without it. Both cohorts drink the same amount of more beer (50/50). The Old saved and bought the debt, just like in the case above. Their consumption is down because of that (-$50 above).
t=2: Either the Old, now the Dead, sold the debt ($100) to the Young, which then needed to save the whole $100, or the Old didn't consume more and left the bonds as a bequest.

So, Nick, the part I do not get is why are you claiming that the government debt is somehow magically different than other types of debt in this regard?

[Nick Edmonds]

"With Leontieff preferences between C1 and C2, you get L-shaped intertemporal indifference curves (like right and left shoes). That would make r extremely sensitive to small changes in debt. It would go from minus to plus infinity if you added the marginal dollar."

I don't think that's right. There's no substitution effect, but there is an income effect. If interest rates are higher, people need to save a lower portion of income in order to achieve equal consumption. This changes the amount saved even though changes in taxation to compensate for the debt service cost ensure that total consumption is unchanged.

The exact amount saved does sepend on whether taxes are levied on old or young (not mentioned in my previous comment), but either way I think it gives a negative relationship between r and D/Y.

[Nick Edmonds]

On reflection, if taxes fall 100% on the old, then what you say is correct.

[Frank Restly]

RSJ,

"Say the rate of interest is g. That road will generate g + s benefit to the economy. Why wouldn't you sell debt to build the road?"

Because the road will eventually wear out and depreciate to $0 in benefit. Meanwhile the debt will still remain.

You need to compare the total cost of the road (principle + interest) to the total added benefit that the road generates.

[Nick Rowe]

rsj: "Suppose, for example, that we are past the point of 'maximizing' government income, but a proposal for a new road is made. Say the rate of interest is g. That road will generate g + s benefit to the economy. Why wouldn't you sell debt to build the road? Why on earth would you raise taxes to pay for the road, when the road has a long life?"

Borrow to build the road. But this is *very* long run analysis. Assume (to keep it simple) the road lasts forever. Would it be best to *slowly* reduce the debt/GDP ratio? My "model" here says yes (but it doesn't say how quickly).

[Nick Rowe]

Jussi: I can bequeath assets to my kids (they will accept them) but I cannot bequeath debts to my kids (they will reject them). But the government can bequeath debts to our kids, by making them pay future taxes. (Though the kids might collectively revolt, of course.) That's the difference between private and government debt.

[Roger Sparks]

Take the 'government builds a road' example.

Assume that the CB (which is owned by government) issues the money needed to build the road.

The money used remains to circulate in the economy. Any interest paid on the debt is paid to the CB (which is owned by government)

My observation: 'Road building' in the past is the source of the money we use today.

[Jussi]

Nick:

I think I now (finally) understand where you come from. Let me know if my interpretation is unfair? It looks to me that you are saying "they (young) will reject them (the debt liabilities)", and therefore the old cohort cannot sell the assets to the young cohort. So the old cohort's (debt) assets are worthless? No selling, no "extra" consumption.

Is this your argument? Why do you assume that the private debt is worthless? Who would buy it in the first place if it doesn't offer any value, real goods (beers), later on? You cannot eat the cake and have it, either your debt in the model has value or not. The private debt needs to be somehow backed / collaterilized by the assets. So the example model is now:

t=1: The Old issues private debt, and the society spends $100 more than without it. The debt is collateralized, maybe the issuer owns a beer distillery. Both cohorts drink the same amount of more beer (50/50). The old cohort saved (built a distillery?), just like in the case above. Their consumption is down (50 beers) because of that.
t=2: Either the Old, now the Dead, sold the asset ($100 = the distillery) to the Young, which then needed to save the whole $100, or the Old didn't consume more and left the bonds (distillery) as a bequest.

[Matthew Young]

I think Nick wants to incorporating the tax distortion into the bond and growth mix,get a three factor equation, as did Samuelson in his three generation overlapping. So, at each time period, three account have to be balanced, growth, interest and distortion, each a planar boundary and the objective has to be 'indiffent' planes at the boundaries. So we solve a recursive equation, and like Samuelson we get three zeros, a cubic equation to solve. This is triple entry accounting, which Samuelson pointed out.

[rsj]

Nick,

"Would it be best to *slowly* reduce the debt/GDP ratio? "

I don't see why. Seignorage income is a form of tax. Why do we want to maximize government revenue, that is taxation of the private sector? Instead, we want to minimize taxation of the private sector while maximizing all public provision, and the only question is whether the stream of benefit to the public exceeds the interest rate paid. No other question matters for deciding whether to fund the project or not, and we should use this decision on every single project, letting the debt ratios be whatever they happen to be.

[Monerty1]

higger inflation due to higher M that can lead to higher intrest rates therefore limitting the revenue an intrest baring currency or money '' Govt would need to raise r (increase the opportunity cost of holding non-interest paying currency which reduces demand for currency) to prevent inflation.''https://twitter.com/MacRoweNick/status/949279648441040896 , https://twitter.com/MacRoweNick/status/949260901068300288 can generate and putting it into a laffer curve therefore limitting the revenue bonds can generate https://twitter.com/MacRoweNick/status/949258699633283072 since the central banks cant anymore buy its own newly issued tresury bonds at a lower rate putting bonds into a laffer curve could only hold in either in a steady state or during a situation wher the goverment spends above the capacity utilisation limit and creates an inflationary gap which are the only cases equation of exchange MV=PT holds relative purchasing parity S1 / S0 = (1 + Iy) ÷ (1 + Ix) holds unless you unironically are a monetarist or believe in the neo classical theory of inflation which is empirically wrong however this might be an intresting concept as a rule of thumb that holds only in cases of high inflation