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The Big Squeeze on Government: Consequences of Baumol’s Cost Disease

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Government Share of GDP and Baumol's Disease, 1952 to 2012

A.  Introduction

A point on which all agree, whether conservative or liberal, Republican or Democrat, is that the cost of government keeps rising.  Whether it is the cost of building new roads or new military jet fighters, or the cost of schools or health services, the cost now is much more than in the past.  And this is not simply general inflation.  The cost of government services has risen at a significantly faster pace than general inflation.

This is true.  But what is not generally recognized if the fundamental cause, nor the implications as we as a nation have struggled to maintain government services.  The fundamental cause is not waste and corruption, nor lazy government workers.  Rather, it lies in the nature of the goods and services used for the public services the government provides.

This blog post will first review the facts on what has happened to expenditures on government goods and services (which for brevity, will hereafter often simply be referred to as government goods) over the past 60 years.  The 60 year period is taken so as to encompass most of the post-World War II period, but to begin once the numbers had stabilized from the very high levels during the war and the immediate post-war fluctuations.

The post will then review the fundamental cause, drawing on the work that has come to be called “Baumol’s Cost Disease”.  The post will discuss how this applies to the government sector, and the implications.

B.  The Share of Government Expenditures in GDP

The share of government spending in GDP has declined over the last 60 years, from almost 25% of GDP in 1952 to less than 20% in 2012, a fall of a fifth.  It is shown as the blue line in the graph at the top of this post.  [Note:  The definition of “government spending” used here is for government as it appears in the GDP demand accounts.  It includes all level of government – federal, state, and local – but only includes direct government spending on goods and services.  Hence it excludes government transfers payments, such as for Social Security or farm subsidies.  Transfer payments are spent by those receiving the funds.]

A fall of a fifth is a significant reduction in the government share.  But it does not show the true extent of the fall, as such GDP share calculations are based on the prices of each given year.  One also needs to know how much one received in real terms for what was spent, and this depends on how prices have changed.

The GDP accounts issued by the BEA do include estimates of the changes in the relative prices of the different components of the GDP accounts.  Over the 60 years from 1952 to 2012, the GDP deflator (the index of inflation for all the goods and services making up GDP) rose at an annual average rate of 3.3%.  Over this same period, the deflator for government spending rose at a somewhat higher rate of 4.1% a year.  This might seem to be only modestly higher, and for a short period it would be.  But compounded over 60 years, this difference in inflation rates cumulates to a difference of 58% in the prices of goods and services used for government vs. goods and services used in overall GDP.

With this relative price change, it now (in 2012) on average costs 58% more (compared to 1952) to produce goods to be used for government expenditures, than it does to produce goods for overall GDP.  Since GDP is also our income (i.e. what we as a nation receive for what we produce), it takes a higher share of our income today to buy the same real goods used for government expenditures as it would have at the relative prices of 60 years ago.

Put another way, to get the same real goods used for government expenditure in 2012 as one would have gotten at the relative prices of 60 years ago, one will now have to spend 58% more.  Or if one spends the same dollar amount adjusted for general (GDP) inflation, one will receive only 1/1.58 = 63% as much.

This impact is huge.  We are indeed receiving far less now in government services for a given dollar expenditure than we would have at the relative prices of 1952.

One way to view this is to ask how much would we have spent as a share of GDP in 1952, for the same real level of goods used for government in that year, if the prices then were instead the relative prices we had in 2012.  The result is the red line in the graph at the top of this blog.  The same goods used for government in 1952 would, at the prices of 2012, have been equivalent to expenditures of 39% of GDP in that year.

Over time, this red line then fell.  It fell in part because the share of GDP used for government (in the contemporaneous prices of each year) was reduced over time and by a fifth by 2012, but more importantly also because the relative prices of the goods used for government provided services rose by 58% over the period.  The government share fell until it reached just 19 1/2% of GDP in 2012.  That is, correcting for the fact that prices of goods used for government were rising (relative to other prices) over time, in addition to the cut-back in the share at contemporaneous prices by a fifth, real government expenditures in terms of GDP share were only half as much in 2012 (19 1/2% of GDP) as what they were in 1952 (39% of GDP).

This fall by half is huge, and explains why we seem to get less and less from our government expenditures (whether on roads, or for military equipment, or in schooling), even though government spending as a share of GDP only fell by a fifth when measured in the current prices of each year.

Another way to look at this would be to ask what government spending would be now, if it had been allowed to grow over the 60 years at the same pace as GDP grew.  If there had been no relative price change, then at equal rates of growth the share of government in GDP would not have changed.  But with the relative price changes, a higher share of GDP would have been spent on government to provide such a bundle of goods for government. The fact that we spent less than that is a measure of how much government spending has been squeezed.
The result is the green curve in the graph at the top of this post.  It shows what would have been spent on goods used for government in each year, if government spending had grown at the same rate as GDP in that year, and valued in the prices of each year.  From 1952 to 2012, real GDP grew at an average rate of 3.1% a year (note this is total, not per capita, GDP).  Real government spending grew only at a rate of 1.9% a year over that period.  Cumulated over 60 years, the difference in growth rates meant that GDP grew by a total of twice as much as government did.  And the goods used for government provided services would have totaled 39% of GDP in 2012 at this constant growth share, or double the 19 1/2% of GDP actually spent in 2012.
[Note:  It is not a coincidence that the 39% of GDP in 2012 on the green line is the same as the 39% of GDP in 1952 on the red line.  The red line figure shows what the spending would have been at the government share in 1952 but at 2012 prices.  The green line figure projects forward this same 1952 share, leaving it unchanged relative to GDP in real terms, and in 2012 shows what this share would then have been at 2012 prices.  But the paths between 1952 and 2012 will differ, and are not mirror images.]
Whichever way one looks at it, this reduction by half in the government share is a huge squeeze on public services.  It goes a long way to explaining why our roads are so much more inadequate now compared to decades ago, with extreme congestion and poor repair.  It explains why our state universities are charging so much more in tuition, while state support has declined.  It explains why what we receive today in public services simply is not what it used to be.  But why have these costs of goods for public use risen so much faster than the cost of other goods?

 

C.  Baumol’s Cost Disease

What has come to be called Baumol’s Cost Disease (or sometimes simply Baumol’s Disease) was developed by William J. Baumol (then Professor of Economics at Princeton), together with William G. Bowen (then also Professor of Economics at Princeton, and later President of Princeton) in the mid-1960s.  They were engaged in a research project looking at why the cost of tickets to live performances of the fine arts had to rise continually, at rates above the general inflation rate, and yet still could not keep up with costs.  A recent re-statement of the issue (but with a particular focus on health care), is provided in the 2012 book by Baumol and others, titled “The Cost Disease:  Why Computers Get Cheaper and Health Care Doesn’t”.

In a nutshell, the fundamental cause of the cost problem is that labor productivity, while perhaps rising in all sectors, will not rise as fast in some sectors as in others.  The sectors where labor productivity rises relatively less fast will face increasing costs, as labor in such sectors will need to be paid more, due to competition for such labor from those sectors where productivity is rising faster.  Yet those sectors with the relatively slower productivity growth will not be able to offset that rising cost of labor with a rate of productivity growth that is as high as that enjoyed by the other sectors.  If we still want or need what the sectors with the relatively slower productivity growth produce, we will need to pay a higher relative price to cover those higher costs.

This is clear in the example of the performing arts.  A Mozart string quartet that required four performers 20 minutes to play in 1780, still required four performers 20 minutes to play in 1966, or in 2013.  Their productivity has not grown at all in over two centuries. Such performers could instead be employed in other sectors, and paid at increasing rates over time there, as labor productivity rose in those other sectors.  If they are going to be employed still to perform Mozart, they will need to be paid more, even though their productivity in playing a Mozart string quartet has not risen in centuries.

Baumol’s Cost Disease will arise whenever productivity growth in the sector being examined is less than productivity growth in the rest of the economy.  There has been a good deal of discussion recently of the implications of this in health (as for example in the Baumol “The Cost Disease” book cited above), as well as in education (explaining why university tuition has steadily risen at a pace greater than general inflation).  But it applies generally for sectors where the goods produced are labor intensive or hand crafted.

Much of the goods and services used by government for the services it provides are of this nature.  Roads, for example, are custom made for the specific site; military jet fighters (and most high tech military equipment generally) are made by highly skilled technicians in small batches of a perhaps a few dozen a year; elementary school teachers teach in classes that are similar in size now as they were 60 years ago; public health workers need to examine patients one on one; and public safety workers (police, firemen, prison guards, and other security workers) provide what they do by their direct presence; and so on.  Teachers, health care workers, and public safety workers, plus military personnel and postal workers facing similar issues, account for most public sector employees (keep in mind we are referring to all levels of government in this note).  And by its nature, the work of those in general public administration (“bureaucrats”) is also highly labor intensive.

It should be emphasized that productivity growth in the provision of government services has not been zero or negligible.  There have been efficiency gains in the government sector.  But the important point for Baumol’s Cost Disease is not that the productivity growth in that sector is zero, but rather that it is simply something less than the rate of productivity growth in other sectors.

And the nature of what government provides makes it impossible to match the productivity growth rates that one has seen most spectacularly in goods such as microchips and hence computers, but more generally in manufacturing and agriculture.  Government services, like many services, have had improvements in productivity, but at rates that simply cannot match the pace of productivity growth possible elsewhere.

Hence, because of Baumol’s Cost Disease the relative price of government services should be expected to go up over time.  This is precisely what has been observed.  There is no reason to attribute this rise in the relative price to allegations of corruption or lazy government workers.  It is of course possible that corruption and lazy workers exist, but for this to have caused the rise in the relative price over time one would need to make the case that corruption and lazy workers are not only worse now than before, but that they have become steadily worse over time.  There is no evidence that supports this.

It is also important to note that while the relative price of government services has risen over time in the past, with this also expected to continue going forward, this does not imply that we as a society will be unable to afford the government services at the higher relative price.  Labor productivity is growing, in the government sector as well as in the rest of the economy, and hence the cost in terms of labor time of the goods of government as well as this cost in the rest of the economy are both getting cheaper.  Hence we can afford to devote a higher share of GDP to government services over time, if we so choose, as the relative cost of government services rises.  And since what government provides, whether in education and health services, or infrastructure, or security and national defense, are all important, we should want to ensure they are adequately provided.

There is therefore nothing wrong for the share of government in GDP to rise over time, as Baumol’s Cost Disease will predict will happen if the services government provides are important.  They would need to be paid for, through higher taxes, but as the society grows richer from the productivity growth in both the government and non-government sectors, we can afford this.  The only problems that arise come from not recognizing this.

D.  Implications, and Conclusion

Since the implications of Baumol’s Cost Disease for government services has generally not been recognized, there are indeed problems.  There has been a tremendous squeeze on government, leading to government services that are an embarrassment for a rich country.  As the numbers above indicate, we are now spending only half as much on government in real terms as we would have had government been allowed to grow at the same pace as GDP since 1952.

Note that this is not an argument that government spending should have been twice as much in 2012 as it was.  This would have matched the real share that it was in 1952, and therefore is an indicator of how much government has been squeezed over this period, but the 1952 benchmark is arbitrary.  And with the 58% higher relative price for government goods over this period, it would be rational to try to scale back on the expenditures for the now more expensive goods.  But cutting back by half is extreme. Rather, the argument made here is that one should be making a well-considered decision at any point in time on whether particular government expenditures (whether for education, or for police, or for military jets) are worthwhile at the price of the time.  If so, one should do it.  But one should not subject total government expenditures to some arbitrary cap, and say that expenditures under that cap are fine while expenditures over that cap should not be allowed.  Since the higher prices over time (due to Baumol’s Cost Disease) reflect differential but still positive productivity growth rates, we can afford those higher government expenditures if we so choose.

Unfortunately, much of the budget discussion in recent years has focussed precisely on setting some fixed cap on government expenditures as a share of GDP.  There have been calls for such a cap directly, or indirectly by saying government revenues should be set at some cap as a share of GDP and that there should then also be a balanced budget (or a budget surplus).

For example, the Bowles-Simpson budget plan called for federal government revenues to be capped at 21% of GDP, with expenditures then set to match this.  The Paul Ryan budget plan called for federal revenues to be capped at 19% of GDP, with expenditures reduced to meet this and then to fall even further.  [Note that both of these figures are for total federal government expenditures, including transfers.  The figures in the graph at the top of this post are for government direct spending only, excluding transfers, but for federal, state, as well as local government.]

Understanding the underlying dynamics resulting from Baumol’s Cost Disease shows how misguided such constant share of GDP targets are.  They ignore that a growing economy, with a growing population, will need to be supported by growing government services.  Given the nature of government services, one cannot expect the rate of productivity growth  in government to match that enjoyed elsewhere in the economy.  There is nothing wrong with that, and does not necessarily reflect a lack of innovation or skill.  Some goods are simply more labor intensive than others, and productivity growth will generally be less for such goods.

By Baumol’s Cost Disease we can see that then the prices of the goods used for government will rise relative to others, and that if we still wish to obtain such goods, we will need to pay more.  The GDP share will rise, but we can afford it as productivity is rising in all the sectors.  They are simply rising at different rates.

The Recovery From the 2008 Collapse That Could Have Been: The Impact of the Fiscal Cuts

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GDP Recovery Path with Govt Growth at Historic Average, 2004Q4 to 2013Q2

A.  Introduction

Previous posts on this blog (including this older one from 2012) have discussed how the sluggish recovery from the 2008 economic collapse could have been avoided if one had allowed government spending to grow as it had during Reagan’s term.  This post will look in more detail at what the resulting path for GDP would have been if government spending had followed the path as it had under Reagan, or even had simply been allowed to grow at its normal historical rate.

The 2008 collapse was of course not caused by fiscal actions, but rather by the bursting of the housing bubble, and its consequent impact both in bankrupting a large share of an overly-leveraged financial system and in causing household consumption to fall as many homeowners struggled to repay mortgages that were now greater than the value of their homes.  Faced with the high unemployment resulting from this, an expansion of government spending would have supported the demand for output and hence for workers to produce that output.  And initially, fiscal spending did indeed grow.  This growth (along with aggressive action by the Fed) did succeed in turning around the steep slide of the economy that Obama faced as he took office.  But since 2009 government spending has been cut back, and as a result the recovery of GDP has been by far the slowest in any cyclical downturn of the last four decades.

This blog post will look at alternative scenarios of what the recovery path of GDP could have been, had government spending not been reduced.  Two primary alternatives will be examined.  In the first, government spending is allowed to grow from the point President Obama took office at a rate equal to its average rate of growth over the period 1981 to 2008.  In the second, government spending is allowed to grow from the onset of the recession (i.e. from the fourth quarter of 2007) at the same rate as it had during the Reagan years, following the downturn that began in the third quarter of 1981.

B.  Government Spending Growth at the Historic Average Rate

In the first scenario, real government spending on goods and services (as measured in the GDP accounts, and inclusive of state and local government as well as federal) is allowed to grow at a rate of 2.24% per year.  This is the average rate of growth for government spending over the 28 years from 1980 to 2008.  This was a modest growth rate, and spanned the presidencies of three Republicans (Reagan and the two Bushes) for 20 of the 28 years, and one Democrat (Clinton) for 8 of the 28 years.  The 2.24% growth rate was substantially below the growth rate of GDP of 3.04% over this same period (note this is for total GDP, not per capita).  As a result, real GDP grew by over 50% more over this period than government spending did.  But this modest pace of government spending growth was substantially more than the absolute fall in government spending during Obama’s term in office.

Note that this path for government spending is not some special rate faster than the historical average, as would normally be called for in a downturn when fiscal stimulus is needed because aggregate demand in the economy is less than what is needed for full employment.  Rather, it is just the historical average rate.  This should be seen as a neutral path, with government spending neither purposely stimulative, nor purposely contractionary.

This path for government spending is shown as the orange line in the following, where the path is superimposed on the graph presented in the earlier blog post of such paths of government spending in each of the downturns the US has faced since the 1970s:

Recessions - Govt Cons + Inv Expenditures Around Peak, 12Q before to 22Q after, with growth at avg historical rate

Maintaining government spending growth at the historical 2.24% rate would have led to government spending well below that seen during the Reagan years (in the recoveries from the July 1981 and January 1980 downturns), roughly where it was in the recoveries from the November 1973 and March 2001 downturns, and well above where it was in the recoveries from the July 1990 downturn (during the Clinton years) and of course the December 2007 downturn (under Obama).  Government spending in the current downturn (the brown curve in the graph) has fallen substantially during the period Obama has been in office.

The graph at the top of this post then shows what the GDP path would have been if government spending would have been allowed to grow at the historic average rate.  The impact will depend on the multiplier.  As was discussed in the earlier post on fiscal multipliers, the multiplier for the US in this period of high unemployment and short-term interest rates of close to zero will be relatively high.  But for the purposes here, we will run scenarios of multipliers of 1.5, of 2.0, and of 2.5.  This will span the range most economists would find reasonable for this period.

The results indicate that had one simply had government spending grow at its historic average rate, the economy would likely now be at or close to potential GDP, which is what GDP would be at full employment.  But because of the fall in government spending since 2009 rather than this increase, current actual GDP is over 6% below potential GDP, and unemployment is high.  (Potential GDP comes from the CBO estimates used in its May 2013 budget projections, but adjusted to reflect the methodological change made by the BEA in July 2013.  Due to these adjustments, including for the GDP deflators used, the potential GDP path is not as “smooth” as one would normally see.  But it will be close.)

Republicans have argued that we cannot, however, afford higher government spending, even if it would lead the economy back to full employment, as it would lead to an even higher public debt to GDP ratio.  But as was discussed in a recent post on this blog on the arithmetic of the debt to GDP ratio, it is not necessarily the case that higher government spending will lead to a higher ratio.  The Republican argument fails to recognize both that GDP will higher (due to the multiplier, and indeed a relatively high multiplier in the current conditions of high unemployment and close to zero short-term interest rates), and that a higher GDP will generate higher tax revenues due to that growth, which will off-set at least in part the impact on the deficit of the higher spending.

The resulting paths for the debt to GDP ratios by fiscal year, using a 30% marginal tax rate for the higher income, would be:

Public Debt to GDP with Govt Growth at Historic Average, FY2009 to FY2013

The impact of the higher government spending is to reduce the debt to GDP ratios over this period.  The higher government spending leads to a higher GDP, and this higher GDP along with the extra tax revenues generated at the higher output means the debt rises by proportionately less.  The ratios fall the most, as one would expect, the higher the multiplier.  If one is truly concerned about the burden of the debt, one should be supportive of fiscal spending in this environment to bring the economy quickly back to full employment.  The debt burden will then be less.

The debt to GDP ratios still rise over these years.  This serves to point out that the assertion made by the Republicans that the public debt to GDP ratio has risen so much during Obama’s term due to explosive spending under Obama is simply nonsense.  The debt to GDP ratios rose not due to higher government spending, but primarily due to the economic collapse and slow recovery, which has decimated tax revenues.  With higher government spending, the debt to GDP ratios would have been lower.

C.  Government Spending Growth at the Rate During the Reagan Years

The second set of scenarios examine what the path of GDP would have been had government spending been allowed to grow, following the onset of the downturn in December 2007, at the same pace as it had during the Reagan years following the onset of the July 1981 downturn.  The path followed is shown as the green line in the graph above on government spending around the business cycle peaks.

The resulting recovery in GDP during the current downturn would have been significantly faster:

GDP Recovery Path with Govt Growth at Reagan Rate, 2004Q2 to 2013Q2

If government spending had been allowed to grow under Obama as it had under Reagan, the economy likely would have reached full employment in 2011 (multipliers of 2.5 or 2.0), or at least by the summer of 2012 (multiplier of just 1.5).  That is, the economy would have been at full employment well before the election.

The deb to GDP ratios would also have been less than what they actually were:

Public Debt to GDP with Govt Growth at Reagan Rate, FY2008 to FY2013

Note that for these calculations I assumed that once the economy reached full employment   GDP (potential GDP), that government spending was then scaled back to what was then necessary to maintain full employment, and not over-shoot it.  Hence the curves for the 2.5 and 2.0 multipliers move parallel to each other (and are close to each other) once this ceiling has been reached.

D.  Conclusion

Fiscal spending was not the cause of the 2008 collapse.  Rather, the cause was the bursting of the housing bubble, and the resulting bankruptcy of a large share of the financial system, as well as the resulting reduction in household spending when many homeowners found that their homes were now worth less than their mortgages.

But following an initial increase in government spending, in particular as part of the fiscal stimulus package passed soon after Obama took office, government spending has been cut back.  The scenarios reviewed above indicate that had government spending merely been allowed to grow at its normal historical rate from when Obama took office (i.e. even without the special stimulus package), the US would by now be at or at least close to full employment.  And if government spending had grown as it had during the Reagan years, the economy would likely have reached full employment in 2011.

There is no need to introduce some special factor to explain why GDP is still so far below what it would be at full employment.  There is no need to assume that something such as “business uncertainty” due to Obama, or new and burdensome regulations, have for some reason led to this slow recovery in GDP.  Rather, the sluggish recovery of GDP and hence of employment can be explained fully by the policies that have kept government spending well below the historical norms.

An Increase in Government Spending Can Reduce the Debt to GDP Ratio: Econ 101

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Most people realize that it is not the absolute value of the government debt that matters, but rather the ratio of that debt to GDP.  A larger economy can support a larger debt.  But most people will also think instinctively that an increase in government spending will necessarily lead to an increase in the government debt to GDP ratio.  It is not surprising that they should think so.  But it is wrong.

Whether the government debt to GDP ratio will rise or will fall when government spending increases will depend on economic conditions and other structural factors.  In conditions of high unemployment and where the Central Bank has driven the interest rates it can control essentially to zero, such as exist now in the US and Europe, an increase in government spending will increase the demand for goods and services, and hence will increase the demand for labor to produce those goods and services.

Employment and output will then rise. How much they will rise will depend on the multiplier, but as was discussed in a previous Econ 101 post on this site, in conditions of high unemployment and close to zero Central Bank controlled interest rates such as currently exist, the multiplier will be relatively high.  The higher incomes that then follow from the higher employment and output will also then lead to higher tax revenues, as a share of the higher incomes will be paid in taxes.

Hence the addition to the deficit and thus the public debt will be less than simply the increase in government spending, due to the higher tax revenues.  With GDP higher due to the greater demand and with the debt also possibly higher but not by as much, the debt to GDP ratio could fall.  And indeed, under conditions such as currently exist in the US and Europe, the debt ratio will almost certainly fall.

To see this, one can start with a simple numerical example.  Suppose one starts with a GDP equal to 100 units (it could be $100 billion), a public debt of 50 (or 50% of GDP, roughly where it was in the US in 2009), a multiplier equal to 2.0 (a reasonable estimate for the US in recent years), and a marginal tax rate on additional income of 30% (also a reasonable estimate for what it is for US federal government level revenues; it would be higher if one included state and local government revenues).

In these conditions, suppose government spending rises by 1 unit.  With a multiplier of two, GDP will then rise by 2 units.  Tax revenues will then rise by 0.6 units, when the marginal tax rate is 30% on the additional 2 units of GDP.  The government deficit, and hence the public debt, will rise by 0.4 units, equal to the extra 1 unit of government spending less the 0.6 units of additional tax revenue.  The resulting public debt will be 50.4, while GDP will then be 102, and the ratio of 50.4/102 is equal to 0.494.  Hence the debt ratio fell from 50% to 49.4% when government spending rose by 1.  Higher government spending led to a reduction in the debt to GDP ratio.  While the total debt rose, GDP rose by proportionately more, leading to a fall in the debt to GDP ratio.

Further numerical examples will help give a feel to what is going on:

Impact on Debt/GDP Ratio from a One Unit Increase in Government Spending
        Scenario: (a) (b) (c) (d) (e)
GDP: Y 100 100 100 100 100
Public Debt: D 50 70 30 50 50
multiplier: m 2.0 2.0 2.0 0.5 3.5
marginal tax rate: t 0.3 0.3 0.3 0.3 0.3
pre-change D/Y 0.500 0.700 0.300 0.500 0.500
Change in G 1 1 1 1 1
Change in Y 2 2 2 0.5 3.5
Change in D 0.4 0.4 0.4 0.85 -0.05
Resulting D/Y 0.494 0.690 0.298 0.506 0.483

Scenario (a) is the case just discussed.  With an initial public debt ratio of 50%, a multiplier of 2, and a marginal tax rate of 30%, a unit increase in government spending will lead the debt to GDP ratio to fall to 49.4%.  This is robust to different initial debt to GDP ratios:  The debt to GDP ratio will fall with higher government spending with an initial debt ratio of 70% (scenario (b), with the debt ratio where it was in FY2012) or at 30% (scenario (c), almost what the debt ratio had fallen to at the end of the Clinton administration, before the Bush tax cuts).

Under conditions where the economy is close to full employment, so that the multiplier will be relatively small, the debt ratio could rise with the higher government spending.  GDP will not rise by much, if at all, if the economy is already producing at or close to full employment levels.  The denominator in the ratio hence will not rise by much, if at all, while the numerator (the level of debt) will rise by the level of extra government spending, with only limited or no extra tax revenues to offset this since GDP has not increased by much.   Scenario (d) provides an example, with a multiplier of 0.5.  The debt ratio will rise from 50% to 50.6% in this example, when government spending rises by 1.

At the other extreme, a very high multiplier may lead to such a large increase in GDP that the extra tax revenues thus generated are greater than the increase in government spending, leading to an actual decrease in the deficit and hence the debt.  Scenario (e) presents an example, with a multiplier of 3.5.  Debt actually falls from 50 units to 49.95 units, despite the increase in government spending by 1 unit, and the debt to GDP ratio falls from 50% to 48.3%.

One will also get this result if the extra tax revenues generated for a given increase in GDP is sufficiently high.  The above examples assume a marginal tax rate of 30%.  More generally, if the marginal tax rate times the multiplier is greater than one (e.g. 30% times 3.5 = 1.05 in the example above), then the absolute value of the debt will fall with the higher government spending.

It may well be unlikely, however, that the multiplier will be as high as 3.5, even with the current high unemployment in the US and Europe.  Thus it is unlikely that the absolute value of the debt will fall with higher government spending, even in conditions of high unemployment.  But as was discussed above, with a reasonable estimate of the multiplier at around 2, one will see the debt to GDP ratio fall, under conditions such as now exist in the US and Europe.

For those with some mathematical expertise, it is straightforward to derive the specific conditions which will determine whether the debt to GDP ratio will rise or fall with an increase in government spending.  This requires some elementary differential calculus, and I will not go through the derivation here.  But the final result is that the debt to GDP ratio will fall if:

(t + D/Y) – (1/m) > 0

and the debt ratio will fall if the sum on the left is less than zero.  That is, the debt to GDP ratio will fall if the marginal tax rate (t), plus the initial debt to GDP ratio (D/Y), minus the inverse of the multiplier (m), is greater than zero (and will rise if the sum is less than zero). Thus if t=30%, D/Y=50%, and m=2 (so 1/m=0.5), with a sum then of 0.3 + 0.5 – 0.5 = 0.3, which is greater than zero, the debt to GDP ratio will fall.

The material above is straightforward.  There is nothing deep or complex.  It also just examines the immediate impact on the public debt to GDP ratio from an increase in government spending.  For a more elaborate look at the long-term impact, see the paper of Brad DeLong and Larry Summers published in 2012.  They show there that higher government spending will not only spur GDP in the short run under conditions such as exist now, but also that such spending will likely pay for itself in the long run through its long term positive impact on growth.

But this post simply focuses on the short term, and shows that counter to what many people might at first believe, higher government spending can lead to a fall in the public debt to GDP ratio.  All this result requires is the recognition that under conditions such as exist now, when unemployment is high and Central Bank controlled interest rates are close to zero, there will be a significant multiplier effect from an increase in government spending.  The resulting increase in GDP along with the extra tax revenues thus generated could very well then lead to a fall in the debt to GDP ratio.  Indeed, with the conditions and parameters such as now exist in the US and Europe, one should expect this result.