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Wednesday, April 8, 2015

The Budget Constraint Does Not Mean The Government Will Pay Off Its Debt

Professor Brad DeLong made some assertions about Modern Monetary Theory (MMT) in this article about bond bubbles, which drew responses on the Mike Norman Economics web site (here and here). Professor DeLong's points have a lot of embedded assumptions, and I cannot deal with all of them here. But I do discuss one assumption in my upcoming eReportUnderstanding Government Finance. This is the idea government 'has to pay back its debt'.

UPDATE: The eReport has been released.

DeLong's thesis is built around theories that 'bond vigilantes' exist and are powerful. This was a dominant theory when he was at the U.S. Treasury in the early 1990s. However, this was exactly the investment thesis that lead to the humbling of the JGB bears in the 'Widowmaker Trade'.

He writes:
Why? Suppose people start to fear that the government will not raise enough in taxes to pay off its debts. They will then try to dump government liabilities for real goods and services.
This is a throw-away comment in a blog post, so I do not want to stretch the textual analysis of this too far. I think he is referring to the concept of 'fiscal sustainability', which is standard in mainstream economics. Or he may be referring to some version of the Fiscal Theory of the Price Level. But 'sustainability' has nothing to do with a common sense interpretation of the phrase 'pay of its debts'.

A government 'paying of its debts' seems to imply that the debt-to-GDP ratio will go to zero at some point. In fact, the inter-temporal governmental budget constraint, which defines 'fiscal sustainability' for the mainstream, says almost nothing useful about what will happen to the debt-to-GDP ratio. Why DeLong uses such a misleading phrasing is unknown to me.

The rest of his article revolves around whether a default can be forced by the bond market. The MMT response is no, but it is a fairly complex topic, which I do not think I can cover completely even within my upcoming report.

The rest of this article is an unedited first draft of an excerpt from my upcoming report, which has the working title: Understanding Government Finance. The estimated publication date: Before the Fed hikes rates. I will attempt to reduce the complexity of this text, possibly by moving material to other sections. (Those sections are not yet complete, so I cannot judge where is the best place for material to be relocated.)

UPDATE: This section has been extensively edited during the publication process. I am leaving this original text, as the comments below refer to it.  The eReport product page is here.

Please note that this text refers to sections that lie outside of this excerpt.

The Governmental Budget Constraint

The modern mainstream approach to macroeconomics revolves around the use of Dynamic Stochastic General Equilibrium (DSGE) models. These models are the object of controversy, and they are particularly opposed by heterodox economists, such as post-Keynesians. I am not going to discuss this wider controversy, rather I wish to discuss the concepts of the inter-temporal governmental budget constraint, which I will also refer to here as the ‘governmental budget constraint’.[1] This is tied to the notion of fiscal sustainability, in that any fiscal policy ‘rule’ is allegedly sustainable if and only if it meets the inter-temporal governmental budget constraint.

In its simplest form, the inter-temporal governmental budget constraint can be written without mathematical notation as:

   (Market Value of Government Debt) = (Discounted sum of all future primary fiscal balances).

The primary fiscal balance is the fiscal balance excluding interest payments. This concept only makes sense if we assume that monetary policy (which determines interest payments) and fiscal policy can be decoupled, a stance that appears dubious.

Additionally, the formulation above ignores how ‘money’ affects government finance through ‘seigneurage revenue’. This is often ignored in DSGE models, as those models implicitly assume that nobody holds money (as that would be suboptimal behaviour). I will discuss this complication later. [Note: outside this excerpt.]

For simplicity, I will assume that the economy is in a steady state, in which nominal interest rates and nominal GDP growth rates are constant. Within a DSGE model, this assumption is too restrictive, but I will assert that one could replicate my analysis in a more general fashion using mathematics that would be understandable to an undergraduate mathematician or electrical engineer.[2]

For now, we will assume that the interest rate on debt is greater than the growth rate of nominal GDP. This is a fairly important assumption with regards to the governmental budget constraint. If the interest rate is lower than the growth rate of GDP, the picture is quite different, as will be discussed later[Note: outside this excerpt. I discussed this topic in another article: If r<g, DSGE Model Assumptions Break Down. That article is slightly more complex, and has some equations within it.]
Chart: Debt-to-GDP Ratio Scenarios

The chart above shows how the debt-to-GDP ratio evolves for a set of scenarios. In each scenario, nominal GDP grows at 4% per year, whereas the interest rate on debt is 6%.  I assume that money balances are zero. The initial value of government debt represents 60% of GDP. The topmost line shows what happens if the primary surplus is held at 0% of GDP at all times: the debt-to-GDP ratio continues to grow without bound.

The middle line is what happens if the primary surplus is 1.2% of GDP: the debt-to-GDP ratio remains constant at 60% of GDP. Please note that in this case there is a total fiscal deficit at all times, but it is only allows the debt stock to grow at 4% per year, which is below the rate of interest. This illustrates the important property that continuous deficits are needed to stabilise the debt-to-GDP ratio if the economy is growing in nominal terms. This has the implication that balanced budgets are associated with a debt-to-GDP ratio converging towards zero, which would be problematic for the operation of the financial system.

The bottom line is what happens if a surplus larger than the stabilising 1.2% of GDP level: the debt-to-GDP ratio continuously falls, and would eventually become negative.

The top trajectory and the bottom both represent ‘unsustainable’ debt trajectories. If the government tried to force its debt stock to be negative (somehow), the banking system would cease to function given the lack of position-making instruments. The usual worry, however, is a debt-to-GDP ratio that becoming arbitrarily large. If this is projected to happen, bondholders would presumably be nervous about owning bonds, as they will presumably become worthless at some point. The debate however, is whether such an outcome could be achieved, and the post-Keynesian position is very simple: such an out-of-control spiral would not happen in practice, and this tells us very little about fiscal policy. Mainstream models do not specify fiscal policy correctly, and the possibility of a ‘debt spiral’ is just a degenerate[3] outcome that is the result of model misspecification. I discuss this in a later section [Note: outside this excerpt.]. But it is true that the budget constraint relation holds (under the high interest rate assumption). The issue is how to interpret it.

Paying the Debt Back?

One popular interpretation of the budget constraint is that ‘the government will have to pay back its debt’. This is a formulation which is often seen in internet discussion, which seems to imply is that the government must drive the debt-to-GDP ratio to zero. This is not true. A fiscal trajectory meets the constraint so long as any upper limit to the debt-to-GDP ratio time series exists. The ratio could drift towards 10,000%, and the trajectory is 'sustainable' on this measure.

The reason why you need a positive primary fiscal balance (a primary surplus) is that you need to apply a brake to the debt dynamics in order prevent the debt-to-GDP ratio from going to infinity. If you set the primary surplus such that the debt-to-GDP ratio is constant (that is, debt grows at the same rate as GDP), the present value of the series will equal the market value of the debt. Doing this calculation requires using some manipulations of infinite series.[4]

More generally, you could run a smaller surplus for a period, and then enter a steady state primary balance in which the debt-to-GDP ratio remains constant at a higher level. The initially smaller surpluses will be balanced by higher surpluses at later dates, since the higher debt-to-GDP ratio requires a larger primary balance to reach the steady growth condition. Despite the extra dynamics, the constraint equation still holds. In this manner, we can steer the debt-to-GDP ratio to any positive level, and remain there, and still satisfy the constraint.
Chart: Debt-to-GDP Ratios For Scenarios

The above chart shows an example of how this works. Once again, there are three scenarios, with the same parameters for the growth rate (4%) and interest rate (6%). The difference is that this time, the primary balance is set to a particular percentage of GDP for the first 10 years (years 0-9 on the chart), and then the primary balance reverts to a level that is consistent with a constant debt-to-GDP ratio. The initial primary balances for the scenarios are -5%, 1.2%, and 5% of GDP.
Chart: Primary Balances For Scenarios

The chart above shows the path of primary balances. The trajectory that starts with a large primary deficit (-5% of GDP) switches over to having the largest primary balance in ‘year 10’, almost 2.5% of GDP. The scenario that starts with large surpluses (5% of GDP) drops off to the smallest primary balance, as the debt-to-GDP ratio has been crushed down and a smaller surplus is needed to stabilise the debt-to-GDP ratio.
Chart: Cumulative Discounted Primary Balances

The chart above shows the cumulative discounted values of the primary surpluses for each scenario. (For each year, we calculate the dollar value of the primary balance, and then discount it by the factor (1.06) raised to the power of the number of years in the future.) In all cases, the cumulative discounted surpluses converge towards the market value of the initial amount of debt outstanding, which is 60% of GDP in ‘year 0’. This convergence will hold for any trajectory that has the debt-to-GDP ratio stabilising at a limit which is greater than or equal to zero.

Therefore, we cannot say the ‘debt will be paid back’, other than the trivial observation that individual bond issues are paid off as they mature, while the stock of debt is steadily increasing.

All the governmental budget constraint says is that for every dollar in debt, the government will need to run a future primary surplus which has a discounted value (present value) of $1. From the point of view of the hypothetical infinitely long-lived representative household which inhabits a DSGE model, it assumes that for every dollar in per capita debt, it will get a future tax bill that is worth $1 now.[5] This property supposedly limits the effectiveness of fiscal policy, which is one reason why the mainstream consensus switched towards an emphasis upon monetary policy versus fiscal policy in the 1990s. This topic is rather complex, so I will not address it here. I will merely assert my view that the DSGE framework is ill-posed, and does not handle fiscal policy correctly. Fiscal policy is always effective, regardless of the level of interest rates.[6]


[1] This is technically too simplistic, as there are two parts to the governmental budget constraint. The first part is the non-controversial accounting identity which describes debt dynamics (the increase in government liabilities outstanding equals the budget deficit, after controlling for market value changes). The second part is the inter-temporal component, which I describe here. Since the first component is trivial, I ignore it.

[2] Instead of assuming that growth rates are constant, we can pin down the actual trajectory between two trajectories that have growth rates that are slightly above, and slight below the ‘long-term’ growth rate. As long as nominal GDP growth rates do not become unbounded (‘tend to infinity’) this can always be done. And if the growth rate does tend to infinity, we are in the realm of an ‘unsustainable trajectory’. A hyperinflation would be such a case, as the growth rates are hyper-exponential. An economy will collapse, breaking the model assumptions, before it reaches infinite size.

[3] I am using degenerate in a formal sense that is used by mathematicians; it does not imply anything about moral standing of a model. The meaning is that the model is technically correct, but the results make no sense. Assumptions embedded in the model do not correspond to real world behaviour.

[4] This is left as an exercise to the reader.

[5] This leads to the concept of Ricardian Equivalence.

[6] After the financial crisis, there has been a move to resurrect fiscal policy within DSGE models by arguing that fiscal policy is effective (only) at the zero lower bound on interest rates.
(c) Brian Romanchuk 2015


  1. Brian,

    The identity:

    (Market Value of Government Debt) = (Discounted sum of all future primary fiscal balances).

    is not true always.

    For example consider a non-growth model in which the output reaches a constant value. The identity is true for r ≠ 0 but doesn't work for r = 0.

    For non-growth models, first high growth rates, the public sector can even a primary deficit in the long run and the public debt converging to a constant value.

    The series summation for the primary surpluses is actually negative, while the public debt at present is not.

    So the identity

    (Market Value of Government Debt) = (Discounted sum of all future primary fiscal balances).

    needn't hold.

    1. "For non-growth models, first high growth rates, the public sector can even a primary deficit in the long run and the public debt converging to a constant value."

      should be "for growth models ..."

    2. Yes, the cases where the discount rate is greater than the growth rate is covered in the next section of the report (which I have not yet written). People might need to buy the eReport to get those juicy details...

  2. "For simplicity, I will assume that the economy is in a steady state, in which nominal interest rates and nominal GDP growth rates are constant."

    That is a steady 'growth' state to my way of thinking. A 'steady state' would be a no-growth condition where one year's GDP equaled the next year's GDP.

    Another topic, to my way of thinking, interest rates are like rent. They are just a cost of doing business. They have nothing to do with the money supply.

    1. I am using "steady state" as it used in the SFC model literature. The economy and all of its components all have the same growth rate, so the ratios of variables remains constant. A no-growth economy is just a special case. This may not match the way people would think about a physical system, but the tendency is for economies to have relatively steady real growth rates.

    2. This is a conventional answer -- perhaps that is why it bothers me.

      I consider that an economy can grow in three ways:

      (1)The velocity of money can increase which really means that goods are exchanged more quickly.

      (2) The money supply can increase in a steady fashion, taking all prices with it in an inflationary manner. An inflationary economy may be 'steady' in the sense that the amount of real goods and serves are constant -- only the prices change.

      (3), the amount of goods and services can increase in quantity, absorbing an increase in money supply without inflation.

      From this 'three option perspective', an economy modeled to have a "relatively steady real growth rate." would need to have THREE variables to allow tracking of each option, and would have preassigned growth attributes to each of the three variables. I dislike the idea of preassigned results.

      That said, I think the world economy has seen all three options of growth over the years of world trade expansion. We all, as participants in a world economy, simply make more trips to the store, pay increased prices for most everything, and take more variety home, than we did 50 years ago.

      An interesting economic world!

    3. Yes, you have separate nominal and real growth rates, and inflation is the residual between the two.

      But in a steady state, all nominal stock and flow variables will grow at the same rate as GDP. This means that money growth is equal to GDP growth, and velocity is constant.

      You can grow nominal GDP and keep money constant. But this means that the ratio of money holdings to household income falls. This behaviour is generally not seen. In my report, I have a chart of Canadian "currency" (dollar bills) as a % of GDP, and it is fairly steady over the decades.

    4. I don't have your confidence that the variables grow at the same rate. To back up this unease, I put together a post "Macroeconomic Stimulus Leaves a Remainder" that can be found at

      The chart showing a calculation for the stimulus received by the United States economy over the last 67 years shows large swings. This makes me believe that the variables such as velocity must also have wide swings. If any parameter was steady, I would be suspicious that the steadiness is an artifact of central bank management.

      No matter what, thanks for your post. I always appreciate the effort you put into your blog.

    5. I have the "steady state" assumption there solely to be able to calculate the sum of the discounted primary surpluses. Without that assumption, it is very hard to characterize the trajectory. The constraint equation will still hold in a more general case (under the rate and growth assumption), but the results would be harder to understand and replicate. I am trying to explain how the constraint works, by showing how different trajectories behave, and I am not saying that the real world is in a steady state.

  3. I think a very simple solution is - don't issue debt! There is no need to issue bonds.

    1. Within the context of how the Canadian monetary system currently operates, there is a need to issue bonds. (Why Canada? I am using the Canadian model as a baseline example in my articles. Other countries can then be examined as modifications of the Canadian system.) Operating procedures could be changed, but there would be costs associated with such a shift. I will eventually write a longer article about this; I have written some comments elsewhere.

  4. Very interesting post, I look forward to reading your book. I made a few attempts to put DSGE into an SFC but I didn't get to far.

    It seems like another fundamental flaw is that the economy has to be supply constrained and have loanable funds model deciding the interest rate. This is the only way for the interest rate to be fixed at a level which fulfills the budget constraint (inter temporal maximization right).

    If the economy is demand driven (not supply constrained) then their is no reason to believe that the interest rate used to discount current government debt will be the same one to discount future surpluses.

    is this more or less right?

    1. I am not sure I follow you on the supply or demand constrained point. The long-term discount rate in a DSGE model is the expected path of short-term rates, as this is determined by arbitrage. Until you add a term premium model, that seems to be the best you can do.

      The determination of the short term rate within a DSGE model to me looks like a portfolio mix decision, and it does not seem to me to be just loanable funds. Since everything is determined simultaneously, the supply and demand of funds are not fixed, which is how I would interpret loanable funds. (People may choose to interpret the equations as being the same as loanable funds, but that is just a verbal projection, which does not appear to capture what the equations are saying.)

      A SFC model does not attempt to characterize expectations out to infinity, but it seems it should at least try to capture expectations for the determination of bond yields.

    2. What do you mean by arbitrage?

      Looking at Cochrane's model in his fiscal theory of the price level paper, the real interest rate comes from the household utility function given they receive a fixed endowment ever period. Isn't this a supply constraint and essential to the discount rate used to price bonds?

    3. I was thinking mainly about nominal rate determination. They are equal to the expected path of short norminal rates, which is an arbitrage relationship.

      The Real Business Cycle (RBC) models have a flexible price level, and the price level adjusts so that the short-term real rate is always equal to the real discount rate in the utility function. Since there is no choice in the matter, I do not see this as a supply constraint issue. This description of the real world is trivially wrong, however.

      The New Keynesian models have price stickiness. This allows real rates to depart for awhile from the "natural rate", but there is a strong presumption that the short-term real rate will return to the natural rate, so presumably long-term real yields would not be volatile. But since this is reversion to a "natural" level, I do not know whether this is a supply constraint issue.

    4. It just shows how question begging these models are. If one assumes a perfect market, of course assets are going to be priced correctly.

    5. Since I am a fairly strong believer in the "rate expectations determine bond yields" view, I cannot complain about the DSGE models on that front.

      For an _economic_ model, it seems fair to assume that bond yields are priced in a consistent fashion with the rest of the model (some version of efficient markets). The only exception is if you want to study the effect of bond market inefficiency. But if you are an interest rates analyst looking to build models to make money, if you want to remain employed, you cannot just assume that the bond markets are always perfectly efficient. However, based on my experience, bond markets are hard to beat based solely upon trades predicated upon the direction of the rate of interest; most practitioner effort is spent looking for nooks where you can make money based on exploiting badly priced risk premia (credit, options structures, rates relative value). This behaviour is consistent with bond yields normally remaining near "fair value".

      The problem with the New Keynesian models is that they have a circular dependence upon the natural rate of interest. (The RBC models are just silly.) Central banks have a lot more freedom of action than those models imply. I hope to write some articles about this "circular dependence" problem, but the summary is that that "natural interest rates" are being constructed in such a fashion that the whole methodology cannot be falsified. Which means that the idea has no theoretical or predictive content.

  5. We cannot assume that interest rates are set by private markets when we are discussing government budgets, and simultaneously assume that interest rates are set by the central bank when we are discussing monetary policy.

    I don't agree that the debt dynamics are trivial; in fact, they are the only interesting thing here.

    If we write the debt-gdp ratio as d, the primary balance as b, the nominal interest rate on government debt as i, the (real) growth rate as g, and the inflation rate as p, then we have:

    d_t+1 = d_t (1+i)/(1 + p + g) - b_t

    For small values of i, p and g this is equivalent to:

    Delta-d = (i - p - g)d - b

    If you want a constant debt-GDP ratio, you need to choose the policy variables i and b so that Delta-d = 0 for the prevailing p and and g. If you want a rising or a falling path, pick different variables. Phrases like "present value of future primary surpluses" do not have any positive content.

    I do not think the notion of a steady state growth path has any value for explaining the behavior of real economies. And certainly the claim that the current market value of government debt is equal to the present value of expected future surpluses is laughably wrong as a description of any actual debt market.

    I know that playing with these kinds of models can be fun but I'm afraid you will be wasting your considerable talents if you go further down this road.

  6. I should add that this:

    "All the governmental budget constraint says is that for every dollar in debt, the government will need to run a future primary surplus which has a discounted value (present value) of $1"

    is equivalent to the DeLong claim you begin by criticizing. If the market value of outstanding government debt is equal to expected future primary surpluses, then a decline in expected future surpluses must lead to a decline in the market value of debt, implying a rise in interest rates and/or higher inflation. If you don't want to end up where DeLong is, you will need to find a different way of thinking about this.

    1. Hello,

      When someone says they "paid off their mortgage", the usual interpretation is that they no longer have a mortgage, not that they replaced it with an even larger mortgage. Therefore, the naive interpretation of DeLong's statement is that the government debt would be driven to zero. This is not equivalent to the constraint, as I show. I assume that DeLong does not believe that, but why use that wording? It may have been a bad idea to drag his statement into this, but I was in the process of writing the chapter, and he used the phrasing that I take objections to. It gave me a topical intro section.

      I object to the decomposition into primary surplus plus interest expense, but if you do it, you do end up with the relationship holding if the interest rate is greater than the growth rate of the economy. In a later section, I explain problems with the interest rate assumption. It includes the some of the comments in the article I reference.

      The steady state assumption is just for the ease of presenting simulation results. I have reworded things, and I will try to downgrade the emphasis on "steady state". I am writing this report by almost completely avoiding equations. This is partially because ebooks have a hard time with equations (they have to be converted into image files), and partially as I am trying to aim this at a wider audience. (I might not succeed, but it is worth a shot.) I could add in the equations for a a steady state GDP ratio, but I was unsure whether they add much. I would probably just do a graph of the relationship with some parameters fixed.

      But as a final note, this section is the second half of a chapter that starts out with Functional Finance. My argument is this classical budget constraint is either trivial or wrong, and that Functional Finance arguments represent the true constraint on government finance. (Before anyone objects, I lump any notion of an external constraint in with Functional Finance principles.) This excerpt possibly is not the best picture of my views, but I ran it as it seemed topical and I want to keep the material for my blog flowing at a steady pace.

    This article seems on topic.
    The point being if the govt interest payments are spent, it generates tax revenues. If not then it is 'borrowed' back. If for some reason people don't want bonds the govt can 'print money.' That's just a QE/asset swap. The thing is they like interest payments so they don't.
    Bonds are not a claim on tax revenues, any more than cash or reserves. Cash is a zero interest bearer bond issued by the Central Bank. Reserves are variable rate bonds issued by the Central Bank.
    They are all money, it's just the amount of "welfare" paid with them that's the difference.
    There is little difference between a bond and a child from an MMT point of view, the child receives child benefit, if I die and gypsies look after the child, they get the benefit. The only difference is issuing criteria, etc :)


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