(D)omar Comin’!

In 1944, Evsey Domar wrote one of the most important papers in macroeconomics. He proposed a model of the economy in which the government could stabilise total output at a desired level, by way of bond-financed expenditure. The exposition was clear, the conclusions were undeniable and, most importantly, the maths were relegated to an appendix. Almost anyone could understand the paper, which might be why almost no-one talks about it.

“Oh indeed,” said Mr Little, approvingly.

But Domar’s main contribution wasn’t that the government could secure a level of output that was consistent with full employment through deficit spending. Crucially, he also showed that if bonds were issued to cover these deficits, the interest payments on the accumulating debt would be very unlikely to bankrupt the government in most growing economies.

The process

Domar takes what we’ve labelled a ‘horizontalist’ approach to government expenditure. The private sector begins with funds in its bank account (1). The government issues a liability, bonds, which it offers to the private sector as an asset, in exchange for some of their money (2). The government empties its account into private sector accounts when it spends (3). The government then taxes the private sector to obtain money for interest payments (4) and disburses interest (5).

GOVERNMENT SECTOR PRIVATE SECTOR
Assets Liabilities Assets Liabilities
1 Deposits $0 Deposits $100
2 Deposits $20 Bonds $20 Deposits $80
Bonds $20
3 Deposits $0 Bonds $20 Deposits $100
Bonds $20
4 Deposits $5 Bonds $20 Deposits $95
Bonds $20
5 Deposits $0 Bonds $20 Deposits $100
Bonds $20

The only difference between rows (1) and (5) in the table above is the creation of government liabilities/private sector assets. While this might seem innocuous, there are two possible hitches. First, if Domar’s government wants to spend more money, it must repeat the process, which will increase the quantity of bonds outstanding. Second, the outstanding bonds attract interest, which will increase the tax take (even though this is ultimately returned to the private sector).

Some qualifications

The Domar model has no ultimate source of funds; everything begins in medias res. We can assume there’s a central or private bank lurking somewhere offstage, but all money in the system simply appears in the hands of currency users.

Because Domar’s model permits a government solvency problem, it might not be best suited to modelling the economies of Australia, Canada, Japan, the United Kingdom or the United States, whose governments tax, spend and issue bonds in their own respective currencies and can never ‘run out of money’ in a meaningful sense. Japan’s government-debt-to-GDP ratio has passed 100 and now 200 per cent on some measures, but investors know that the Bank of Japan would convert their bonds to cash if they sold en masse. Eurozone governments don’t have the luxury of obtaining euros on demand in such a situation, so increasing debt-to-GDP ratios can present a problem for them.

Omar Little, Neo-chartalist

Nevertheless, the prospect of skyrocketing debt-to-GDP ratios still commands the attention of some governments (who are always able to repay their debts if the need arises). So until economists and politicians in countries with sovereign currencies are as enlightened as you, dear blog reader, here’s some ammunition for any fights they might want to stage on old, probably obsolete ground.

Evsey Domar presents: When Debt-to-GDP Ratios Explode!

The burden of the debt

Domar is interested in two related questions. Can the quantity of bonds (‘the government debt’) increase without limit? And will the interest payments, taken as tax, ever become unpayable?

Let’s say that the government wants to hit a target flow of income every year, Y. By itself, the private sector currently spends C per year in consumption and saves the rest. To increase the total flow of income to the target, the government adds its own expenditure to the total stream, G = Y - C.

But the government needs to have money in its account first, so it issues new bonds to the private sector. If the government wants to sustain income at its target rate Y by continuously spending, it must keep adding to the stock of outstanding bonds by (\Delta GD) each year, and it must also pay interest on the outstanding stock of bonds i \times GD.

Domar therefore splits the government’s debt into two components: the principal and the interest. The principal is the total stock of bonds on issue, GD, which is a debt owed by the government (and, often neglected, an asset owned by the private sector). The interest component, i \times GD, Domar labels the ‘burden of the debt’, because he proposed it be paid by taxing the private sector.

If you’re interested to see how this proposal leads to anything other than ruin, read on! Domar sets up two scenarios in his model in order to give his detractors the best chance of success. (Feel free to skim through the next two sections for the highlights, if maths isn’t your thing.)

Do fo’-fives still beat a full house?

Scenario 1: A zero-growth economy

The first scenario assume that national income (or expenditure, or output, or GDP) is constant every year. The economy just doesn’t grow at all.

Y = Y_{0}

To ensure this constant flow of income is achieved, the government contributes a given proportion, G/Y, or \alpha for short. Government debt, GD, accumulates from its initial level, GD_{0}, by this constant fraction borrowed each year

GD = GD_{0} + Y_{0} \times \alpha \times t

The debt-to-GDP ratio is found by dividing the second equation by the first:

\frac{GD}{Y} = \frac{GD_{0}}{Y_{0}} + \alpha \times t

But as time marches on, with the government constantly taking a fixed slice of a fixed pie, its accumulation of pie slices will be truly gigantic. ‘In the limit’, or when time equals infinity, the government’s debt is infinite:

\lim \limits_{t \to \infty} \frac{GD}{Y} = \infty

Granted, we’re not in year infinity just yet, but won’t we be in trouble when we get there! Thankfully, we don’t have wait that long before we realise that, in reality, the government debt-to-GDP moves up, down and sideways from year to year. A constant stream of deficits isn’t necessarily paving the road to infinite debt.

Domar acknowledges the unlikelihood of expanding debt too, attributing this extreme case to the extreme setup of this scenario. In an economy where income has to be kept at the same level every year, the government would essentially be trying to find unproductive investments. (Government spending in this dreary scenario would not only have no multiplier effect, it would nullify the multiplier effect of any private spending.) As badly as some governments have performed, few have done that badly.

Scenario 2: A constant-percentage-growth economy

So Domar proposes a more realistic case. The GDP of most developed economies for the past 100 years has grown at about an exponential rate. This sounds pretty drastic, but it amounts to saying that income grows by around the same percentage, g, every year:

Y = Y_{0} \mathrm{e}^{g \times t}

In this scenario, the government still borrows a constant proportion \alpha of income, but because that income is now growing, so too is the amount borrowed. Accordingly, debt accumulates:

GD = GD_{0} + Y_{0} \times \alpha \int_0^{t} \mathrm{e}^{g \times t} dt

GD = GD_{0} + \frac{Y_{0} \times \alpha}{g} \times (\mathrm{e}^{g \times t} - 1)

The debt-to-GDP ratio is:

\frac{GD}{Y} = \frac{GD_{0}}{Y_{0}\mathrm{e}^{g \times t}} + \frac{\alpha}{g} \times (1 - \mathrm{e}^{-g \times t})

In the limit, government debt-to-GDP ratio in a growing economy reaches a constant value:

\lim \limits_{t \to \infty} \frac{GD}{Y} = \frac{\alpha}{g}

Moreover, “MONEY BE GREEN!”

The bottom line

Even though the government adds a constant stream of deficits into the total flow of expenditure in this second scenario, because income is growing, the debt-to-GDP ratio stabilises. And because the debt-to-GDP ratio stabilises, so too does the interest burden, levied as taxes on the private sector.

Deficits and debt might not be as disastrous as we’re told to think. Indeed, Domar paints a much brighter picture:

‘The problem of the debt burden is the problem of an expanding national income.’

In other words, there’s no compulsion to pursue a surplus to ‘pay off’ past deficits, if a growing GDP–to which productive government expenditure contributes–can do the trick. And even then, if you’re in a country whose government spends in its own currency, debt-to-GDP ratios might not be all that meaningful.

But Domar’s point remains sound. The goal of sustained GDP growth seems agreeable (sustainable growth we can leave for another blogpost). An expanding national income is enabled, in part, by government expenditures on the demand side and increased investment in productive capacity on the supply side. These policies can be achieved jointly; for example, raising expenditure on health and education boosts incomes and raises the productive capacity of workers.

And if leaf raking or ditch digging are the only options for employing idle capacity, and the idle minds of policymakers? In a depressed, low-inflation economy, the choice is easily stated. The output lost from a day of unemployment is gone forever. The extra debt incurred in employing productive workers to contribute to a growing economy eventually stabilises, and it provides an asset to the private sector in the process. The trade-off is difficult to spot.

References

Domar, E. (1944), ‘The “Burden of the Debt” and the National Income‘, The American Economic Review, 34(4), pp 798–827.
Lavoie, M. (2014), ‘Post Keynesian Economics: New Foundations‘, Edward Elgar: Cheltenham.

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