In my recent National Post column, I make reference to some back-of-envelope calculations to the effect that replacing the fiscal anchor of balanced budgets to one of a fixed debt-GDP ratio allows the federal government to increase spending by 1.2 percentage points of GDP, or by about $25 billion.

I'm going to work through the math here, and I'm going to take it very slowly - partly for the benefit of people who are seeing these sorts of manipulations for the first time, but mostly so I don't screw it up.

Some notation first:

*G*is government spending*T*is tax revenue*B*(for bonds) is government debt*i*is the interest rate

**Balanced budget case**

If the budget is balanced, then government spending must be equal to the revenues left over after serving the debt. Debt service payments are calculated by multiplying the interest rate *i* by the size of the debt *B*:

*G = T - iB*

Divide everything by GDP – denoted by *Y* – to express everything in ratios to GDP:

*G/Y = T/Y - iB/Y*

Note that *B/Y* is the debt-to-GDP ratio, denoted by *DR* (for debt ratio), so write this as

*G/Y = T/Y - iDR*

**Fixed debt ratio case**

We have to compare across years, so we’ll add subscripts to denote which year we’re talking about. If (say) debt in a given year is *B _{t}* then debt in the following year is

*B*. Next year’s debt

_{t+1}*B*is the previous year’s debt

_{t+1 }*B*plus the deficit accumulated from the previous year (

_{t}*G*). If there’s a surplus, then the ‘deficit’ is negative, and debt falls.

_{t}+ iB_{t}– T_{t}*B _{t+1} = B_{t} + iB_{t} + G_{t} - T_{t}*

Collecting terms:

*B _{t+1} = (1 + i)B_{t} + G_{t} - T_{t}*

Again, divide through by GDP to express everything in ratios to GDP. For reasons that will become apparent in a minute, let’s divide by *Y _{t+1}*, GDP in the next year:

*B _{t+1}/Y_{t+1} = (1 + i)B_{t}/Y_{t+1}+ G_{t}/Y_{t+1 }- T_{t}/Y_{t+1}*

Let’s denote the growth rate of GDP by *g*, so that *Y _{t+1}/Y_{t} = (1+g)*, or, equivalently,

*Y*. Substitute [

_{t+1}= (1+g)Y_{t}*(1+g)Y*] for

_{t}*Y*on the right-hand side, so that all terms are expressed as a ratio of the GDP for the same year. This shows how the debt ratios in years

_{t+1}*t*and

*t+1*are related:

*B _{t+1}/Y_{t+1} = [(1 + i)/(1+g)]B_{t}/Y_{t}+ [1/(1+g)]G_{t}/Y_{t }- [1/(1+g)] T_{t}/Y_{t}*

Recall that the ratio *B/Y* is the debt ratio (*DR*) for a given year. Let’s say that the government sets *B/Y* at a constant ratio in both years, so we can write them as *DR* without the year subscript:

*DR = [(1 + i)/(1+g)]DR+ [1/(1+g)]G _{t}/Y_{t }- [1/(1+g)] T_{t}/Y_{t}*

Move the right-hand side term with *DR* to the left-hand side and collect the *DR* terms:

*(1 - [(1 + i)/(1+g)])DR = [1/(1+g)]G _{t}/Y_{t }- [1/(1+g)] T_{t}/Y_{t}*

Common denominator for the term in brackets on the left-hand side:

*[(1+g) - (1 + i)]/(1+g)DR = [1/(1+g)]G _{t}/Y_{t }- [1/(1+g)] T_{t}/Y_{t}*

Multiply through by (1+*g*):

*[(1+g) - (1 + i)]DR = G _{t}/Y_{t }- T_{t}/Y_{t}*

The left-hand side simplifies a bit:

*(g - i)DR = G _{t}/Y_{t }- T_{t}/Y_{t}*

A little bit of re-arranging so that *G/Y* is on the left-hand side and to get an expression we can compare easily with the balanced budget case:

*G _{t}/Y_{t} = T_{t}/Y_{t} - iDR + gDR*

We don’t need to distinguish between years anymore, so write this as

* G/Y = T/Y - iDR + gDR*

**Comparing spending with the two anchors**

Let’s compare the two expressions for spending-to-GDP ratios. Here, once again, is spending under a balanced budget:

*G/Y = T/Y - iDR*

And here is that ratio when the government follows a fixed debt ratio rule:

*G/Y = T/Y - iDR + gDR*

You can see that the spending-to-GDP ratio is higher under the debt ratio rule, and the difference is equal to the GDP growth rate *g* times the targeted debt ratio. That term *gDR* is also the size of the deficit, expressed as a ratio to GDP.

It’s interesting that the interest rate *i* doesn’t show up here: the increase in spending comes entirely from the anticipated growth in GDP, not whether or not interest rates are low or high.

**Some back-of-envelope calculations**

According to the October 2017 Fiscal Update, nominal GDP is expected to grow by about 4% between 2017 and 2018, so let’s set *g* = 0.04. Multiply this by the current debt ratio of 30%, and you get 0.04 x 0.30 = 0.012. In other words, going from a balanced budget anchor to a debt ratio anchor increases government spending by about 1.2% of GDP.

GDP is projected to be around $2.1 trillion in 2017, so 1.2% of GDP is roughly $25 billion.

Don't really see the point of this exercise, given that neither a balanced budget nor a fixed debt:GDP ratio make much sense.

The optimum deficit, as Keynes explained, is whatever reduces unemployment as far as is consistent with not too much inflation. Or as he put it, "Look after unemployment and the budget will look after itself."

Posted by: Ralph Musgrave | October 30, 2017 at 10:58 AM

I apologize for wasting your time.

Posted by: Stephen Gordon | October 30, 2017 at 11:38 AM

Out of curiosity, has there been much work on exploring the feedback effects on government debt and interest rate? Your napkin-math assumes that the interest rate is known a priori, but at some level of government debt we'd expect to begin seeing crowding-out effects that would change the underlying interest rate even for a fixed central bank response function.

Posted by: Majromax | October 30, 2017 at 01:25 PM

Stephen: good. Glad your bottom line seems to match my intuition.

Another back-of-the-envelope calculation I like to make is: suppose we held the level of debt constant in real terms (so there is no deficit if we adjust everything for inflation). Multiplying the 30% debt ratio by the 2% inflation target (so the nominal debt rises at the same rate as the price level) gives us an extra 0.6% of GDP of government spending (or tax cuts). That's $12.5 billion, or half of your own calculation.

Ralph: I see the point of this exercise is to allow the government to run deficits in bad years and surpluses (or smaller deficits) in good years, while having some Long Run target for the debt ratio that makes that policy sustainable over time.

Posted by: Nick Rowe | October 30, 2017 at 01:51 PM

Will you do the math for the US?

Posted by: tom mullaney | October 30, 2017 at 02:47 PM

G/Y = T/Y - iDR + gDR

What happens when one year treasury rate matches growth? Here is the USA chart:

https://fred.stlouisfed.org/graph/?id=A191RL1Q225SBEA,#0

In the last eight years, we would have balanced the budget three times to keep DR constant.

Posted by: Matthew Young | October 30, 2017 at 06:45 PM

Matthew,

The average remaining maturity of U. S Treasury debt is almost 6 years (up from a low of 4 years in 2008).

https://www.yardeni.com/pub/usfeddebt.pdf

Page #15

The interest rate on 5 year Treasuries is about 2.06%.

https://fred.stlouisfed.org/series/DGS5

More importantly total interest paid / total debt outstanding = 2.43% and U. S. Federal debt held by the public has been growing at about roughly 5%.

https://fred.stlouisfed.org/series/A091RC1Q027SBEA

https://fred.stlouisfed.org/series/GFDEBTN

U. S. Nominal GDP growth has been about 4%.

https://fred.stlouisfed.org/series/GDP

Assuming that government receipts and government spending grow at the same rate as economic growth, interest expense is growing about 3.43% faster than receipts.

Right now interest expense consumes about 13.5% of all U. S. government receipts.

Assuming no changes (interest rate, economic growth, debt growth), interest expense will equal government receipts in:

I1 / T1 = 13.5%

I2 = I1 * exp (X * ln (1 + 5% + 2.43%))

T2 = T1 * exp (X * ln (1 + 4%))

I2 / T2 = 0.135 * exp (X * ln (1.0743/1.04))

Setting I2 / T2 = 100% (1.00) and solving for X

X = ln (1.00 / 0.135) / ln (1.0743 / 1.04) = 62 years.

Mind you, that is after all other government expenditures (defense, education, social benefits, etc.) have been cut to zero.

Posted by: Frank Restly | October 30, 2017 at 08:06 PM

Frank,

Let us make it even simpler. Congress pays the ten year rate (interest expenses/debt equals ten year rate) So lets plot NGDP vs the Ten year rate:

https://fred.stlouisfed.org/graph/?id=DGS10,#0

Using this scale, I see two points in the last eight years that would have balanced the budget, and a third point coming up in 2018.

The theory proposed has a reversion to zero in it, it is a predictive filter and will bounce around zero to keep DR constant. A good idea, actually, but it means balancing the budget every 3-6 years if you do it right.

Posted by: Matthew Young | November 01, 2017 at 12:53 PM

Matthew,

The only problem is that interest rates on federal debt are unlikely to go negative.

Who other than central banks would buy them?

Nominal GDP growth can go negative. See:

https://fred.stlouisfed.org/series/GDP

And so guaranteeing a stable debt ratio (DR) is dubious at best.

Posted by: Frank Restly | November 02, 2017 at 07:26 PM

I'll take your word for it, and you can count on my commitment to oppose this terrible idea.

Just one small question though... what "fiscal anchor" you are talking about? I haven't seen a balanced budget since 2007.

Posted by: Tel | November 03, 2017 at 05:00 AM

I would be very interested to lean your thoughts about a recent analysis of monetary and fiscal policy. http://necsi.edu/research/economics/econuniversal

Posted by: Yaneer Bar-Yam | November 12, 2017 at 11:34 AM

Why not have a debt service cost ratio target, rather than a debt ratio target ?

Posted by: Tony Holmes | November 13, 2017 at 07:04 AM

Tony,

"Why not have a debt service cost ratio target, rather than a debt ratio target ?"

That's not a terrible idea. Limit interest expense to say 15% of available tax revenue.

To do that, the federal government would need to do one of several things:

1. Manage interest rates directly (bye bye central bank).

2. Manage it's quantity of debt issuance with regard to interest rate changes instead of with regard to the business cycle (bye bye countercyclical debt financed fiscal policy).

3. Switch to all zero coupon bonds and manage it's debt duration with regard to interest rate changes (100 year, 1000 year, 10000 year bonds).

4. Sell equity claims on it's future revenue in lieu of or in conjunction with bonds.

With #4, central bank independence is preserved, countercyclical fiscal policy becomes possible without additional debt issuance, and duration of government securities (debt / equities) can be limited.

FYI, with traditional government bonds, coupon interest payments are made semi-annually and thus those coupon payments are regularly occurring government expenditures. With zero coupon bonds, interest and principle are returned to the investor when the bond matures - no coupon payments are made.

Posted by: Frank Restly | November 13, 2017 at 05:04 PM