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Wednesday, July 2, 2014

If r < g, DSGE Model Assumptions Break Down

The relationship between interest rates and the growth rate of the economy is critical for government fiscal dynamics. In the literature for Dynamic Stochastic General Equilibrium (DSGE) models, the discussion of the governmental budget constraint appears to have an embedded assumption that the real interest rate on government debt is greater than the economic growth rate (“r>g”). However, there is no reason that this has to be true, and the mathematics of the budget constraint fails if the condition does not hold. This poses a problem for the constraint, as a true mathematical constraint is something that is always true. Once this constraint is dropped, a good portion of the recent academic literature discussing fiscal policy becomes irrelevant. (Despite my opportunistic use of “r” and “g” in the title of this article – in order to capitalise on the popularity of a recent book – it has nothing to do with inequality.)


This article discusses an “endowment” economy (economic output is fixed, without any labour input trade-offs) which was described in more detail in an earlier article. I will once again summarise the situation:
  1. The economy consists of a single “representative” household that receives a number of apples each year (without labour).
  2. The household starts off with an initial stock of government liabilities (money or 1-year Treasury Bills).
  3. The government taxes and spends via transfer payments, but does not purchase apples for government consumption.

In order to pin down the price level, I impose an assumption that velocity is constant (discussed further here, in my article on “monetary frictions”). For example, if velocity is fixed at 100, and the household produces 100 apples that year and it holds $1 in money, the size of the economy is $100 (implying the price of an apple is $1). I discuss the constant velocity assumption further in the Appendix, as it can be relaxed.

At the initial time point, the household calculates the economic trajectory that optimises its utility, which is based on consumption over time. The form of the utility function does not matter too much, other than the fact that future consumption is discounted at a certain rate.

Real Rates And Inflation

For simplicity, I will assume that prices are flexible (future prices can be set independently of the current price), but prices have to respect the velocity constraint. When prices are flexible, DSGE model assumptions force the real rate of return on Treasury Bills to equal the (real) discount rate in the utility function. This means that if the rate of nominal interest is set to any value, (expected) inflation will be equal to the nominal interest rate less the fixed real rate. For simplicity, I will assume that the authorities are targeting the price level, and so the expected inflation rate is zero. In this case, the expected nominal interest rate equals the real discount rate for all time.

We can see immediately that this particular model framework does not fit real world economic data. In recent years, central banks in a number of developed countries have kept real policy interest rates negative, which would be impossible in this framework (the real discount rate is assumed to be positive). The fact that DSGE models make no useful predictions about the real world should come as no surprise; most market practitioners figured that out years ago. I am only detailing these problems to explain why I will be ignoring the “state of the art” research that is being cranked out by academics in my future articles discussing fiscal policy.

Bringing In Economic Growth

I will now assume that the number of apples produced grows each year by some factor. Since I need to look at the household sector in aggregate, this growth rate includes the population growth rate (typically denoted n), as well as per-household productivity growth (typically denoted g).

It should be noted that the representative household framework breaks down in the case of a growing population. If every single household shrinks its holdings of government liabilities by 0.5% per year, but the number of households grows by 1% per year, the aggregate growth rates of government liabilities is positive, not negative. Very simply, the representative household assumption makes very little mathematical or economic sense.

A Counterexample

Take my model economy, and assume that we have price level stability. We then assume that the number of apples produces grows by 2% per year, but the real discount rate is only 1%.

The velocity relationship tells us that the money stock has to grow by 2% per year. However, the aggregate interest rates on government liabilities is less than or equal to 1%, since there is no interest paid on money. Even if interest were paid on “money” (it would have to be “reserves” at the central bank), the rate of growth of government liabilities due to interest compounding is 1%. The only way the desired economic trajectory could be achieved is that the government has to run a (growing) primary deficit every year, so that the money stock would grow in line with nominal GDP.

This appears unremarkable. But this solution violates the so-called governmental budget constraint, which states that the discounted trajectory of primary fiscal surpluses equals the stock of initial debt. Since the government never runs a primary surplus, this “constraint” is obviously not respected.

In particular, I will return to the Bond Valuation Formula (part of the Fiscal Theory of the Price Level) which I describe in this earlier post. The Bond Valuation Formula is a direct consequence of the governmental budget constraint; if you believe one holds, the other does as well. I look at this formula because it is one of the few references I have found where the mathematics of the budget constraint is actually fleshed out. Once again, formula is:
(The formula, when translated into English, says that the real value of government debt is the discounted value of the path of real primary surpluses.) The formula is obviously violated by my example economy. How is this possible, when it was supposedly derived using actual mathematics? We need to look at the proof in the Appendix of the Cochrane working paper to see where the divergence occurs. After some algebra, the single period accounting identity for government finance was rearranged to:
You will need to read the paper for the description of the notation, but the problem lies is in the second term. Translating the mathematics into English, the term is the discounted value of the real value of government debt outstanding as time goes to infinity. John Cochrane then states “I impose the usual condition that the last term is zero.” (In the literature, this condition is known as the “transversality condition”.) This “usual condition” is what is violated in my example; the real value of government debt is growing by 2% per year, but the discount rate is only 1%. Therefore, the second term does not converge (loosely, “it goes to infinity”). (UPDATE) Note that this example is a worst-case error; if r=g, then the second term may converge to a non-zero value, creating an "error" in the price level predicted by the Fiscal Theory of the Price Level. (Previous sentence added to make this more general; thanks to "srini" for pointing this out.)

John Cochrane argues that this condition must be imposed in order to meet the conditions of household optimality, which is a standard argument. In my view, this argument is incorrect. I have a discussion in another article – A Contradiction At The Heart Of DSGE Models – which (partially) explains my logic. (I will add more comments in a later article.) Instead, I will finish this post discussing the implications of this disagreement.

What If The Transversality Condition Holds?

Even if the reader is unconvinced with my critique of the Transversality condition, the situation for DSGE modelling is still not very satisfying. If we impose the Transversality Condition on my model economy, we get the situation that is impossible (for some reason) for the government to target price level stability. It has to force the ratio of the money stock to GDP to shrink by (slightly more than) 1% per year in order for the term to converge to zero. This would imply a policy of deliberate deflation of (at least) 1% per year.

In the real world, we do not see any tendency for the ratio of government liabilities to GDP to go to zero, which is what the transversality condition says must happen.

What If r>g?

If the government does not pay interest on money, we can find a solution in which there are no surpluses, even if the real rate of interest is greater than the real growth rate. This is because the growth rate of government liabilities due to interest will be lower than the rate of interest if money holdings are non-zero. For example, if the nominal rate of interest on Treasury Bills is 4%, and half of government liabilities are in the form of money, the nominal stock of government liabilities is only growing at 2% per year.

By increasing the weighting of money in household portfolios, it is always possible to find a solution to the problem in which no primary fiscal surpluses are ever run. (If real economic output is shrinking, it is necessary to have an inflation rate that keeps nominal GDP growth positive.) Once again, this will violate the governmental budget constraint (and the Bond Valuation Formula).

The Case Of Pure Price Flexibility

If we do not impose monetary frictions of some sort, my logic breaks down. In this case, there is nothing to pin down the initial price level. One could use the Bond Valuation Formula to determine the initial price level (which implies that the governmental budget constraint holds). But since the price level is essentially arbitrary in this case, it could be set to anything and we can still find a solution to the household optimisation problem. Those solutions could violate the governmental budget constraint (and hence the bond valuation formula). There is no reason to prefer one solution over the other, other than ideological prejudices.

This is what happens if you work with mathematical models which are heavily under-determined: their solutions do not make a lot of sense.

Concluding Remarks

If we can find an example model economy where the “governmental budget constraint” is violated, it is not in fact a constraint. It is then a relationship which may or may not hold, which is an entirely useless piece of trivia. Dropping the government budget constraint from the DSGE framework is not easily done, as it is tied to a constraint for household consumption over time. The trajectory of the economy is supposed to be the optimal solution for all time going forward; but it we lack a constraint on future behaviour, we cannot solve backwards to get the solution in the present.

As seen above, the crux of my argument involves the transversality condition. I will return to transversality in a later article, adding to my earlier critique.

(UPDATE) The paper "Interest Rates and Fiscal Sustainability" by Scott Fullwiler covers a lot of this ground, but in more historical depth. The Fullwiler paper notes that this growth rate condition goes back to the Domar 1944 paper "The 'Burden of the Debt' and the National Income". The reason why that earlier work is ignored in the modern DSGE literature appears to be the fact that the transversality condition is assumed to hold, and so what happens to the debt/GDP ratio is completely ignored. The earlier literature on sustainability was based on how the debt/GDP ratio evolved. (h/t Michael Sankowski at Mike Norman Economics.)

Appendix: The Constant Velocity Assumption

In my previous post, I discuss broader monetary frictions than just the constant velocity constraint than I use within this article. To be clear, I do not think velocity is constant in the real world. But some form of monetary friction has to be imposed in order for DSGE models to make even a little bit of sense.

If one uses a more complex monetary friction, such as velocity that varies based on other variables, or money in the utility function, you could derive similar results to my example. The only requirement is that velocity has an upper bound as well as a lower bound above zero. Such a constraint seems reasonable.

  • If velocity fell arbitrarily close to zero, the implication is that people would have money holdings that are arbitrarily large relative to their nominal incomes. We see very few people who earn $50,000 per year walking around with billions of dollars in their pockets.
  • If velocity became arbitrarily large, the economy would have to somehow function even though people would have almost no money in notes and coins or bank accounts (deposits at the central bank – reserves – would be an arbitrarily small portion of their balance sheets). Although this situation appears feasible, but I argue it would be untenable in practice, as the private sector would lack liquid government liabilities to back up private sector short-term liabilities. The relative lack of government liabilities would make paying taxes extremely difficult (since gross taxes paid would presumably rise in line with GDP).

I believe that a constant velocity makes a lot of sense in a steady state growth model. The key is not adopting the logical fallacy that Monetarists fell for. If the economy is assumed to be growing at a steady rate, it makes sense that the central bank needs to grow the monetary base in line with that growth rate in order to avoid distortions. But the opposite direction of causality – attempting to drive steady GDP growth by growing the monetary base at a fixed rate – makes little sense. If we move away from the “steady state” condition, the balance of forces that led to a particular observed value of velocity would likely change.

See Also:
(c) Brian Romanchuk 2014


  1. Brian,

    You have not discussed another condition--where the second term converges but not to zero. So, there is always some stock of government debt outstanding--that is, debt is never fully repaid. The strong conclusions of DSGE literature come from making the transversality assumption and the further assumption that the net present value of debt is zero.

    1. Thanks. I will modify my text to show that I am giving a worst-case error, but any non-zero value creates an error.

  2. Here you go:

    Scott F's paper on this is great too.

  3. HI Brian

    Nice job. FYI, Charles Goodhart wrote a number of good papers in the mid-to-late 2000s critiquing the transversality condition. He referred to it as the most dangerous idea in macro, or the worst idea, or something like that. I think several of the papers can still be found online.

    Jamie Galbraith did a quick piece on r vs. g, too at Levy.

    Scott Fullwiler

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  5. Assume r > g what does it mean? It means that holders of goverment debt are getting bigger and bigger share of all the real wealth generated by the economy? That process has an obvious limit, the amount of real wealth. And the amount of real wealth has to be in some relation to growth variable g. Therefore I do not see how r > g can hold given infinite horizon?

    Btw. I think your coding project is really interesting! Thanks for all the effort!

    1. There is a big Piketty r > g debate, which I cheesily worked into the title.

      I am looking at this from the macro perspective, and all we can talk about is the household sector as a whole. How wealth is distributed is buried within the aggregation, and I don't have an opinion on that debate.

      But in order for the debt/GDP ratio to remain finite, the government needs to run *primary* fiscal surpluses to keep debt levels from growing faster than GDP. That's a straight application of limit theorems. However, the budget constraint summation used by the mainstream does not converge in that case.

      In other words, the only "discount rate" that can be used in these summations that makes any sense is the growth rate of GDP itself (or something slightly larger), not the interest rate.

      Furthermore, the overall budget would still be in deficit, with interest costs greater than the primary surplus.

  6. I liked a lot your post and there is nothing I would disagree with. I only thought that the whole r > g discussion looks moot to me because r ~ g needs to be true (w.r.t. “the only "discount rate" that can be used in these summations that makes any sense is the growth rate of GDP itself (or something slightly larger), not the interest rate.”).

    I brought in the wealth distribution because I think through that it can be seen that debt as an asset class cannot grow without limit (limit theorem) and also what are the possible limiting mechanisms. And also shows that the political choice is always between surpluses vs. a form of financial repression. All below assumes closed economy.

    The government debt is just an indirect way for a group of people (not holding government papers) to be indebted towards another group of people (holding papers). I think this tells us how government debt being unsustainable actually means that wealth inequality is too high or unsustainable in the sense that non-holders of government papers are not able to pay papers back in real terms, meaning goods and services. This can be defused a) by wealth taxation (holders “pay it back to themselves”), b) by some form of financial repression ( “wealth tax”) and c) by defaulting (“wealth tax”). I think it is just that simple. Usually some form of financial repression is preferred (e.g. current real yield vs GDP growth).

    So I would rather say that: “But in order for the debt/GDP ratio to remain finite, the asset prices will adjust [in a way that (expected) *primary* fiscal surpluses are enough] to keep debt levels from growing faster than GDP”. Or maybe that “asset prices will be made to adjust” by the central bank / government programs.

    The main mechanism is usually the inflation/financial repression, which will reduce the value of government debt or overall indebtness, which always makes government debt sustainable. I think Cullen Roche ( is right that hyperinflation (unsustainability?) is “more than just a monetary phenomenon”. So inflation is an important adjustment mechanism while hyperinflation (unsustainability) ensues only if there are other factors in play.

    So all in all I find it peculiar that in DSGE primary surpluses as flows/quantities are assumed to have the burden of adjustment in order to get things in equilibrium while usually prices, and esp. asset prices, are assumed to have that role (w.r.t, “DSGE model assumptions force the real rate of return on Treasury Bills to equal the (real) discount rate in the utility function”).

    1. My latest article gets back to this for the DSGE model case:

      From the SFC perspective, the way that fiscal policy is specified solves the problem. If the debt-GDP ratio gets "high," activity rises (since consumption is explicitly a function of wealth), and the higher activity drives up the tax take automatically. The real-world is far more complex than the model I use in that article, but the outcome is directionally correct.

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