Sunday, February 28, 2021

r < g

r<g is an essay on the question whether r<g means the government can borrow and not worry about repaying debts. No. 


A situation that the rate of return on government bonds r is less than the economy's growth rate g seems to promise that borrowing has no fiscal cost. r<g is irrelevant for the current US fiscal problems. r<g cannot begin to finance current and projected deficits. r<g does not resolve exponentially growing debt. r<g can finance small deficits, but large deficits still need to be repaid by subsequent surpluses. The appearance of explosive present values comes by using perfect-certainty discount formulas with returns drawn from an uncertain world. Present values can be well behaved despite r<g. The r<g opportunity is like the classic strategy of writing put options, which fails in the most painful state of the world.

The essay is based on comments I gave at the spring NBER EFG meeting on Ricardo Reis' "The constraint on public debt when r<g but g<m." My discussion starts here at 4:48,  Ricardo presents the paper (very good, worth listening to, many points I didn't get to) at 4:30 

pdf for now, as translating equations to blogger is taxing. 


  1. "large deficits still need to be repaid by subsequent surpluses."

    Ha. Ha. Ha. Not in my life time (which is short), my children's lifetime (medium), my grandchildren's (ages 1,2,4) lifetimes.

  2. John,

    From the paper:

    "r < g seems to offer a delicious opportunity. Briefly, suppose the government borrows a huge amount, and simply rolls over the debt, borrowing new money to pay principal and interest on the old."

    If the government is borrowing new money to pay both principal AND interest on existing bonds (Ponzi finance), then it doesn't matter what either r or g or any other economic variable is.

    "Therefore there is a maximum debt/GDP ratio out there somewhere. The fiscal ex- pansion cannot be unlimited or go on forever."

    No there is not. There is a maximum interest expense as a percentage of federal revenues - 100%. Once you reach that point, then bondholders are paying interest to themselves with their own money, and that is when the bond market blows up.

    1. The point is that Ponzi finance can work if there are more suckers born each minute than the speed of the Ponzi bubble.

      Interest can exceed 100% of federal revenue. The government can, and does, borrow new money to pay interest on the old.

    2. John,

      "The point is that Ponzi finance can work if there are more suckers born each minute than the speed of the Ponzi bubble."

      You really think that individuals working in finance and that buy and sell government bonds professionally are a bunch of "suckers" that have no idea what they are doing? That's your opinion?

      If you are counting on Ponzi finance to work then it doesn't matter what either r or g or debt/GDP is - THAT'S THE POINT.

      "Interest can exceed 100% of federal revenue."

      Please tell me when that has ever happened in the United State (interest expense has exceed 100% of federal revenue). The St. Louis Fred website maintains that information and the closest it has ever gotten is about 50% of tax revenue (about 25-26% of total federal revenue).

      Remember George Bush ("Read my lips, no new taxes"). Ever wonder why he had to raise taxes? Because the bonds sold in the late 1970's and early 1980's (with double digit interest rates) were coming due.

      "The government can, and does, borrow new money to pay interest on the old."

      The government uses money to pay interest on existing debts. But because money is fungible (you can't tell the difference between a dollar of tax revenue and a dollar of borrowed money), the only way you can see if Ponzi finance is going on is by looking at aggregate tax revenue and aggregate interest expenditures.

    3. John (aka PT Barnum),

      Will the preface of your new book on the fiscal theory of the price level include the following disclaimer - "For suckers only."?

  3. Typos; feel free to delete this comment.

    p.2: Write "Six trillion dollars ago..." Consider merging this paragraph with the subsequent one.

    p.4: "such and earthquake" should be "an." I suppose "doom-loop" should be hyphenated always (see previous paragraph) or never.

    p.5 : "the =r−gdebate" seems incorrect.

    p. 10 There is no I in "Whch r?"

    p. 11 "This is an r = -pi < g" is difficult to read as a sentence. Separately, could you introduce pi in the previous sentence "printing money (at rate pi) to satisfy..." (if that's what pi is.)

    p. 12 "alsodiscount" needs a space

    p. 14 you use "iid" here, but "i.i.d." below.

    p. 19 "Large deficits" should not be capitalized. Consider whether "grow out of debt strategy" needs hyphens.

    1. All fixed. Thank you so much! Can I get you to read Fiscal Theory of the Price Level next? : )

  4. This comment has been removed by the author.

  5. OT, but maybe worth pondering.

    If you own the digital currency Bitcoin, the world is in a violent implosive deflation.

    So, a private-sector currency backed by nothing---nothing!---and which is worthless in paying taxes (albeit gaining some acceptance in commercial markets) is appreciating.

    This has been going on for 12 years and running.

    That is not a theory; that is what has happened in real markets. So what theory is validated by what has happened in the real world?

    Perhaps a currency is worth what people say it is, like gold.

    Also, I do not understand the premise that government must borrow to finance deficits. Michael Woodford says that central bank QE, concurrent to deficit spending, is a helicopter drop, or money-financed fiscal program.

    The US can finance federal spending by printing money. See Japan. Where is the inflation? I don't know.

    Maybe inflation is the wrong topic. If you look at the graphs, in the next downcycle we see deflation and negative interest rates, even in the US.

    1. Ben,

      "Also, I do not understand the premise that government must borrow to finance deficits."

      You are correct. The U. S. Constitution gives Congress the sole ability to borrow on the credit of the United States (as well as the ability to print / coin money).
      But noticeably lacking from the Constitution is the word deficits and here is why.

      Governments don't borrow to fund deficits, instead deficits place a limit on the amount of government bonds that are sold. Economists, politicians, financial pundits, and everyone else have this completely backwords.

      A government could if so inclined sell bonds on demand regardless of it's fiscal position (surplus / deficit). In that event what would happen is that banks with access to the Fed discount window would be hopping over themselves to borrow from the Fed at a short term interest rate and lend to the federal government at a longer term rate. Over a very short period of time (with on demand bond sales), the interest expense owed by the federal government on it's debt would reach 100% of available tax revenue.

      "The US can finance federal spending by printing money. See Japan. Where is the inflation? I don't know."

      The U. S. central bank (Fed) was created in 1913 and the FOMC was later created in 1935. The purpose behind it's creation was to act as a lender of last resort because of the perceived cronyism that was prevalent during the late 19th and early 20th century.

      The Fed's mandate for price stability didn't come later until the Federal Reserve Reform Act of 1977 and the Humphrey Hawkins Act of 1978.

      Instead of looking for inflation in Japan, perhaps you should instead be looking for cronyism.

      Imagine if the federal government awarded $5 trillion dollars to the inventor of the Covid 19 vaccine and paid that amount using printed money (or central bank QE). Would that result in higher inflation? Probably not because handing a large lump sum payment like that to one person is not going to have that big of an effect on consumer prices (how much bread and milk and gasoline does one person need anyway).

      Would that be a cronyistic approach, absolutely.

  6. For what it's worth, my non-expert view after reading some of the leading opinions on the topic (including yours) is as follows: (1) deficits "don't matter" in and of themselves, BUT (2) deficits are the best leading indicator of unwise government spending that we currently have, and (3) unwise government spending does matters a lot.

    Or said another way: we can run deficits forever if our technocrats are smart enough. If Biden's huge infrastructure bill turns out to be the next interstate highway system (and not a bunch of bridges to nowhere), it will not matter how we paid for it. If our military spending hits the mark with the modern equivalent of aircraft carriers (and avoids the modern equivalent of battleships), it won't matter how we paid for it. If our educational system is transformed to churn out brilliant young minds (and not just squandered on stuff that doesn't work), it won't matter how we paid for it. But the rub of it is that if your idea requires r<g to sell to voters or wrangle bipartisan support, then it's pretty likely you're on the wrong side of the examples above.

    1. "deficits are the best leading indicator of unwise government spending that we currently have"

      Or unwise taxing or both

    2. Anonymous,

      See Irving Fisher's separation theorem - the investment decision is separate and distinct from the financing decision.

      And both decisions are equally important.

      You are arguing from the premise that "the end justifies the means", ie, it doesn't matter how something is accomplished as long as that accomplishment is deemed worthy.

      Does the construction of any of these buildings justify the slave labor that was used to construct them?

      With separate financing and spending decisions a government should be able to find a way to satisfy both the wants and concerns of the recepient of the spending as well as the wants and concerns of the financier for that spending.

      See Dale Carnegie - In any negotiation find the resolution that satisfies both parties - seek win-win scenarios.

    3. FRestly - thanks for your thoughtful reply!

      You are correct that my response violates Fisher's separation theorem... that's sort of the point. I've become convinced that for a fiat "reserve currency" government, taxing and printing money end up being the same thing in the end.

      And yes, I do think the "ends justify the means" so long as those "ends" are a large societal return on investment and the "means" are government debt. I appreciate your point that when the same logic is applied in other contexts the flaws become apparent, but for me the context changes the logic. Sometimes I think the ends justify the means (here, for example), while in other cases I don't (the criminal justice system, for example). Call me wishy-washy if you'd like, I'll plead to the charge ;)

      But either way, my overall point (which I may not have done the best job expressing) was supposed to be that when the "means" are bad, the "ends" are also more likely to be poor. So trying to stick with your specific example: the American South had a pretty weak economy relative to the North so, a few historic landmarks aside, the cruel means employed did not lead to better ends for ANYONE at a macro/societal level.

    4. Anonymous,

      "FRestly - thanks for your thoughtful reply!"

      You are welcome.

      "And yes, I do think the ends justify the means so long as those ends are a large societal return on investment and the means are government debt."

      You are presuming that a government must borrow / sell debt to begin with.
      Nothing in the US Constitution dictates that a government MUST borrow.
      What if the federal government sold equity instead?

      What I mean is, what if the US government sold a nontransferable zero coupon securities that could only be used to redeem future tax liabilities? Those securities would qualify as equity separate and distinct from either bonds or money itself.

      Unlike bonds where the interest payments are funded by all taxpayers as a group, the realized returns on equity would be specific to the individual buyer / taxpayer.

    5. Anonymous,

      "But either way, my overall point (which I may not have done the best job expressing) was supposed to be that when the means are bad, the ends are also more likely to be poor."

      Why do you believe that debt is the correct means for funding what should be in society's best interest to begin with?

      What I mean is why do you believe that financiers for government expenditures need a high level of protection implied with debt when the supposed benefits are shared by all (including the financiers)?

  7. Despite using math, or financial math, PV, etc. a bid of common sense would suggest it is very hard if ever possible to grow in to a bad balance sheet. Ask any over levered company if they are able to grow in to bad B/S and the answer will be uniformly no. And that does not take in to account any compounding, just simple interest on debt.

    As an aside what is the target audience of the Book? Academics, laymen?

    1. This is intuitive, but incorrect. Indeed a (dramatically oversimplified for effect) history of the origins of private equity funds might be the story of a few clever financiers figuring out it's exceedingly possible to grow into a bad balance sheet :)

    2. And how businesses with bad balance sheets have you restructured or companies that were over levered you provided advise to? Your balance sheet is not bad until markets say so which is the case with most sovereign debt.

  8. Closing the Social Security deficit would eliminate most of the projected rise in debt relative to GDP. If the FED were to issue a CBDC, they would effectively monetize most of the Federal Government's outstanding debts. And I would also suggest we issue GDP linked bonds to remove the uncertainty of the relationship between R and G.

    1. And closing the SS/Medicaid deficit, which I presume would be a component of or consistent with an optimal spending/taxing rule, could be accomplished with some reduction in deadweight loss by shifting financing from a wage tax to a VAT.

    2. Thaomas,

      The income / wage tax was instituted by the 16th Amendment to the U. S. Constitution. Please show me the Constitutional Amendment that allows for a VAT (Value added tax).

      Also, Japan has a VAT. Do you see their debt or deficit falling?

      Finally, what SS/Medicaid deficit?


      "And I would also suggest we issue GDP linked bonds to remove the uncertainty of the relationship between R and G."

      With GDP linked bonds, why shouldn't I (and basically every other American) take a market short position on GDP linked bonds and quit my job? I think we have been over this once before.

  9. "But this analysis suggests two ridiculous conclusions. First, it seems there are no fiscal limits at all. As above, the government can borrow, send us “stimulus” checks, and nobody has to work again. Well, obviously not."

    This is obviously impossible because if no one works g << r. This feels like a straw man right off the bat, unfortunately.

  10. Why focus on constraints to borrowing rather than expenditure and taxing rules that lead to optimal borrowing?

    1. Thaomas,

      Because the spending decision and financing decision are equally important.

  11. deficit and debt is desirable because that could cause excess demand. And if r is bigger than g, what of it? That doesn’t mean the debt cannot rise for a while to deal with temporary emergenices like Covid. So what’s the relevance of the size of g relative to r? No relevance at all.

    So . . . . the crucial question is: what’s the optimum size for the debt and what rate of interest should be paid on it. Enter MMT with the killer answer: the debt and deficit should be whatever minimises unemployment as far as that is consistent with hitting the inflation target.

    As to the rate of interest, that ideally needs to be near zero (as advocated by Milton Friedman). Reason is that there is no good reason to reward anyone simply for hoarding money.

    MMT rocks.

    1. Ralph,

      Interest is not strictly speaking the cost of borrowed money.
      It is the cost of time.

      "As to the rate of interest, that ideally needs to be near zero."

      Is your time so worthless to you?
      Would you lend money or anything else at a near zero interest rate?

      "Reason is that there is no good reason to reward anyone simply for hoarding money."

      What are we in the 18th century now? Money / debt are two sides of the same operation. When someone borrows from a bank both money (borrower asset, bank liability) and debt (borrower liability, bank asset) are created. Both money and debt are extinguished when the debt is repaid. It's called an elastic currency.

    2. I see the first line of my above comment has been left out. Not sure whether that's my fault or the fault of the Cochrane blog. Anyway, the first line was, "If g is bitter than r, what of it? That does not necessarily mean more...".

  12. FRestly, Re your question as to whether anyone would lend money at a near zero rate of interest, my answer is that tens of billions if not hundreds of billions of dollars are being loaned at a near zero rate right now. The return on the bonds issued by numerous countries is currently negative in real terms (i.e. after taking inflation into account) and as to NOMINAL terms, the positive rate is in many cases less than 1%.

    Re the second part of your comment, you seem to suggest that all money is created as a result of a loan. Certainly that’s how commercial bank money is created. But in the case of CENTRAL bank created money (so called “base money”) there is no loan at all: the CB simply presses buttons on a keyboard, and hey presto, money is created.

    1. Ralph,

      I am not asking why anyone would lend money at a near zero rate of interest, I am asking if you would. Is your time that worthless?

  13. In section 2, the integral equation that immediately follows equation (1) provides no information -- the right-hand side of the integral equation is b(t)/y(t) as may conveniently be found by substituting for s(t+τ)/y(t+τ) of the integrand its equivalent {b(t+τ)/y(t+τ) - d[b(t+τ)/y(t+τ)]} from equation (1), and then integrating by parts. The result is b(t)/y(t) = b(t)/y(t). Whether rg, it matters not all. Whatever we do on the right-hand side of the equals sign, the result will always come out to be equal to b(t)/y(t).

    "First, it seems there are no fiscal limits at all." This would be so in the case of present value equation--there are multiple paths for s(t)/y(t) to take and they all lead to the same result. The answer to the question, "Which road leads to Rome?" is "All roads lead to Rome."

    "Second, it seems that a theoretical wall separates r > g from r < g." Actually, no--whether r > g or r < g, or r = g, the present value relation will always return b(t)/y(t) as the present value. It is not path dependent. The present value of the terminal state, s(t+T)/y(t+T) and the value of the integral between t and t + T will exactly offset one another to yield b(t)/y(t), whatever choices are made for s(t)/y(t) between the starting time, t, and the termination time, t + T.

    "... there is a maximum debt/GDP ratio out there somewhere. The fiscal expansion cannot be unlimited or go on forever." Perhaps, but one would never notice it without a measure to discriminate as between admissible paths of the steering function's s(t)/y(t), such as by imposing an objective function to optimize or 'satisfice'. Suppose the planner strives to maximize social welfare while at the same time minimizing child poverty, and he is externally constrained to limit the loss of the currency's foreign exchange value to some set rate of depreciation against a basket of world currencies. The planning horizon is T time units in length and the present time is equal to t. Or, suppose that the planner's goal is to maximize economic growth, g, while simultaneously minimizing the rate of interest, r. The rate of interest is set exogenously, for example, by the terms of trade in external markets where the country's export products are bought and sold, and where its imports are procured.

    Each planner's set of objectives will give rise to a set of 'optimal' paths from which he will choose one to follow. Only then can we say whether the chosen value of the steering variable, s(t)/y(t), is appropriate or not. To get ahead, we need more information than is provided in section 2.


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