DeLong and Summers and Self-Financing Fiscal Expansion (very wonkish)

Advance warning: This post will only be of interest to those who have read the DeLong and Summers paper.

As has been referenced on this blog a number of times, the recent paper by Brad DeLong and Larry Summers has had a significant impact on the debate over optimal fiscal adjustment in a depressed economy.    Although the authors themselves note that the balance of arguments may be quite different in a non-creditworthy economy, the paper is still important to our debate (see also here and here).   

The key innovation in the paper is to incorporate the fiscal implications of potential “hysteresis effects.”  Put simply, these effects refer to long-lasting effects on future output – and thus on the future fiscal position – of fiscal adjustment measures today that reduce today’s output.   For example, the loss of a job due to weaker growth this year could have long-lasting fiscal implications if it makes it harder for the person losing their job to gain employment in the future. 

A striking conclusion in the paper is that, under what appear to be a relatively undemanding set of conditions, an expansion of government spending (or alternatively not engaging in some planned expenditure cut) could be self-financing from a long-term fiscal perspective.    Put another way, fiscal expansion could bring about improvement rather than disimprovement in a country’s underlying creditworthiness.  Thus, fiscal adjustment could be self-defeating from a creditworthiness perspective.   Given the influence of their argument, it is important to look closely at the assumptions that underlie this result. 

A key equation in the paper is equation 2.7.  It gives the effect on the fiscal balance in any future year of a one unit increase in government spending today.   (See the paper for all notation and detailed explanations.)  The impact on the budget is:

-(r –g )(1 – μτ) + ημτ.

The first term gives the cost in any future period of today’s increase in government spending.   The assumption is that the government can allow the debt to GDP ratio to rise as a result of the increased spending, but then must pay of enough debt each year to keep the debt to GDP ratio constant at its higher level.   The second term captures the increase in government revenue in the future as a result of the permanently higher output due to hysteresis effects.  The condition for the overall impact to be positive – and thus for the increase in government spending to be self financing is:

r < g + ημτ/(1 – μτ).

For reasons that will become apparent below, it is useful to re-write this equation as:

g > r – ημτ/(1 – μτ).

There are two strong assumptions built into this result.   First, it is assumed that the optimal fiscal policy involves allowing the debt to GDP ratio moving to a permanently higher level.   There is a large literature on this question, which I cannot get into here.   Simon Wren-Lewis has some useful posts (see here and especially here).   I just note here that there is an active debate on whether there the debt to GDP ratio needs to return to some optimal level (and also on the optimal speed of that adjustment to that level).   The answer to that question could have a significant impact on the future fiscal costs of today’s expansion. 

I want to focus here on a second issue.   An implicit assumption is that the hysteresis effects are permanent.   There is the same adverse effect on future output one hundred years from now as there is next year.   This is not plausible.  It seems important to allow for some realistic “decay” in the hysteresis effect over time.   Today’s fiscal actions do lead to a “shadow” on future output; but that shadow is likely to diminish with the passage of time. 

Suppose that this decay rate is a constant δ.   That is, δ percent of the hysteresis effect disappears each year.   If the decay rate is, say, 7 percent, the half-life of the hysteresis effect would be 10 years – still quite long.  

To obtain the future fiscal effects of today’s one unit increase in government spending we have to identify the present value of both the costs and benefits.    Assuming a fixed real interest rate, r, the present discounted value is:

[-(r – g)( 1 – μτ)]/r   +  [ημτ][1/(r+δ)].

We can then rewrite the condition for the fiscal expansion to be self financing as:

g > r – [ημτ/(1 – μτ)][r/(r+δ)].

Note that we would obtain the DeLong and Summers condition if δ is zero. 

Allowance for this decay can have a substantial effect on the self-financing condition.  For example, with a real interest rate of 3.5 percent and a decay rate of 7 percent, the second term on the right-hand side of the condition above is just one-third its level with a decay rate of zero.   This would have a significant impact on any self-financing calculation. 

My purpose here is certainly not to detract from the importance of the DeLong and Summers paper.    The paper provides an extremely useful framework for thinking about future fiscal implications of today’s fiscal adjustment efforts.   But the results in the paper on the ease of passing the self-financing condition are stark (if comforting).   But they are based on important implicit assumptions.    This brief note has attempted to highlight some of those assumptions. 

11 replies on “DeLong and Summers and Self-Financing Fiscal Expansion (very wonkish)”

I think this was dealt as a topic in another op-ed a couple months back, do you have url. In general I think the theoretical assumptions are undermined relative to the severity of debt problems. In a stable system, the assumptions of the authors are probably perfectly valid, but in a surprisin and volatile system, with debt/gdp rations of (120 -168) for Ireland and Spain, and undeclared non stress tested bank losses, for example, in Spain, in fact, beyond the 100% debt/gdp, there are no theoretical models that can plan negotiable paths towards solvency, without default. But data models implying austerity as a route to market continue to be trundled out.

Foreward to the 2009 edition of “The Great Crash 1929″
” As in 1929,the archtects of disaster will form a rich rogue’s gallery to go shooting in. There will be Lawrence Summers, impassioned advocate in 1999 of the repeal of the Glass-Steagall Act, which since 1933 had provided a key protection against securities fraud by banks. Summers bounced from Harvard’s presidency to Obama’s White House, and is a man whose reputation remains to be rebuilt or buried by events.”

James K. Galbraith
Townsend, Vermont
August 17, 2009

If transitory expansionary fiscal policy is self-financing, then expansionary fiscal policy now moves the long-run debt-to-GDP ratio to a lower level if tax rates are kept constant.

The point that “hysteresis” effects are persistent rather than permanent is a good one–there are genuine “new deals” out there. But there are also genuine “new deals” that apply to public finances as well…

There is a third implicit assumption – that of jobs for life. It assumes that a worker today who is made unemployed would otherwise be first supported by direct or indirect government support for the industry they work in and second that the industry would return to ‘normality’.

Let’s look at Ireland of the bubble. I peaked at about 26% of GDP, with C something like 60% (I think) with retail sales growing at a 10% annual clip. The two sectors that have seen most unemployment are I (construction) and C (retail and leisure services).

This leaves us with two problems:
1. Should the government continue with the same levels of I? Not really sustainable I would argue. How about the same levels of C? Well, since much of C is imported, that would seem subject to leakages.
2. For construction, is the economy going to ‘recover’ to a state where all the former construction employees are gainfully employed without government support? Do retail employees ‘lose’ their skills through unemployment?

An additional wrinkle seems to be an assumption of a constant cost of debt – carrying a higher debt load into the future with the intention of reducing it through repayments requires:
1. Something that has never happened (governments don’t repay debt).
2. A surplus – something that rarely happens (governments are quite adept at spending money).
3. That any putative surplus is not eaten up by increased interest costs.
4. That the debt load can be reduced before the business cycle next turns.

Keynsianism with fancier sums is still just Keynsianism. It requires heroic believe in the prudence of future governments; a belief that the experience of past governments belies.


In your cash flow calculation, as I understand it the negative impact comes from the need to pay interest on the extra debt that results from the government spending. A dollar increase in spending raises the deficit (and debt) by 1 – μτ dollars. My interpretation of your calculation for this part is that you identify the additional cash outflow that would be required to keep the debt to GDP ratio at its new higher level, which is the interest rate minus the growth rate. But it seems to me that that the future cash flow cost is the interest rate times the increase in debt that has resulted from the fiscal expansion. Even if debt is reduced for other reasons — e.g. the hystersis effect — that does not change the fact that the debt (and thus interest costs) are diredtly higher than they would be if the fiscal expansion had not occured. Maybe I have misunderstood this part of your calculation.

Sorry, not sure what you mean by “new deals” in the second part of your comment.

We do identify the additional cash inflow that would be required, but unless you are on the knife-edge where it is exactly self-financing and no more, thereafter you can either (a) cut tax rates, (b) allow current tax rates to pay down the debt, or (c) some combination of the two.

For example…

Suppose you start out with a stable debt-to-GDP ratio of 1, supported by a nominal interest rate i on government debt of 6%/year, an inflation rate π=2%/year, a real interest rate on government debt r=4%/year, a real GDP growth rate of g=2%/year, and a primary budget surplus of 2% of GDP.

Then you are running a nominal budget deficit of 4%/year–the 2%/year primary surplus minus the 6%/year of debt interest–so that the debt is growing at the same 4%/year that nominal GDP is growing at, and the debt-to-GDP ratio is 1.

Now give me a multiplier μ=1.5, a tax share τ=.33, and a hysteresis parameter η=0.1. Our self-financing condition:

(r-g)(1-μτ) < μητ

is clearly satisfied inasmuch as:

(.04-.02)(1-(1.5)(.33)) .01 < .05

So what happens if we spend 3.33% of a year’s GDP for three years on fiscal stimulus, and thereafter do not change tax rates?

1. During those three years output rises by 5%–from 1 to 1.05, and then falls back not to 1 but to 1.015 due to the hysteresis effects.
2. During those three years debt rises by 1.67%/year–from 1 to 1.0167 to 1.033 to 1.05
3. The debt-to-GDP ratio thus grows from 1 to 1.05 during the three years that the stimulus program is in effect.
4. Thereafter, the hysteresis benefits mean that:
* The primary surplus is not .02 but 0.025 because of higher tax revenues
* The primary surplus required to keep the debt-to-GDP ratio constant at its new higher year-four value of 1.035 is not 0.02 but rather 0.0207
* 0.0207 < 0.025
* The debt-to-GDP ratio thus falls by 0.0043 from 1.035 to 1.0307 in year four
* And the debt-to-GDP ratio keeps falling, and asymptotes to its new steady-state value corresponding to a primary surplus of 0.025–which is a debt-to-GDP ratio of 0.8
* So the long-run impact of this self-financing fiscal stimulus program is to cut the long-run steady-state debt-to-GDP ratio from 1.0 to 0.8. No fiscal drag; rather a fiscal dividend…

I’d suggest that DeLong and Summers’ model is likely to hold well in a relatively closed economy, but that in a very open economy another form of hysteresis is likely to also be significant.

In a very open economy, activity in internationally traded sectors and their supply chain will be driven primarily by exports, and will be largely unaffected by domestic stimulus spending. That large part of domestically traded sector activity that arises indirectly or is induced by internationally traded sector activity will also be largely unaffected by stimulus spending. The capability to conduct stimulus spending will also be constrained by the fact that it only exercises a multiplier on part of the economy.

It is therefore relevant to look at issues that may cause hysteresis specifically in internationally traded sectors. One would expect to see hysteresis in the volume of activity, whether measured by output, exports, employment, share of global markets or whatever else. All other things being equal, a loss of 10% in activity will mean 10% less accumulation of knowledge, 10% less investment and 10% less presence in foreign markets, with a 10% gain in activity having a symmetrical upward effect.

Ireland has, famously, one of the most open economies in the world. Ireland’s internationally traded sectors suffered a severe loss of competitiveness after 2000, and are now only slowly clawing back sufficient competitiveness to allow them to grow – as clear a case of hysteresis as one would hope not to see. Domestic stimulus spending will do almost nothing positive to rectify this, but by propping up prices in the economy will actually delay the improvements in price competitiveness required to restart growth in internationally traded sectors.

In Ireland’s case, therefore, hysteresis effects point in two opposite policy directions – the effect that DeLong at Summers identify, which favours stimulus spending, is valid, although only present for part of the economy. The internationally traded sector effect, by contrast, favours rapid fiscal adjustment, and an end to countercyclical spending.

I haven’t really studied this, so I’ll stay out of it. But this looks dubious: “It seems important to allow for some realistic ‘decay’ in the hysteresis effect over time.”

In an open economy especially, any prolonged slump is likely to lead to emigration. Throw in Krugman-style scale economies and you’re looking at the loss of entire industry clusters.

Thanks Brad,

Thinking in terms of the standard equation for the change in the debt to GDP ratio is helpful. But I think you underestimate the dramatic implications of your example. With a constant primary surplus as a share of GDP and a positive interest rate–growth rate differential, there is a unique unstable equilibrium for the debt to GDP ratio: ps/(r-g). Once the debt to GDP ratio is falling, it continues to fall. In your example, the unstable equilibrium debt to GDP ratio is 0.025/.02 = 1.25. Allowing for the decay rate would lead to less dramatic but possibly more plausible conclusions.

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