Sunday, April 14, 2013

Debt and growth in 10 minutes

This is a short video from last year. I only just found out it exists. It still seems pretty topical, and (for once) condensed because Lars Hansen really forced me to obey the 10 minute time limit!

There is a better link here from the BFI page here that covers the whole event, but I couldn't figure out how to embed those.


  1. John,

    A couple of things. First:

    %INT = Nominal Interest Rate
    MSD() = Maturity Structure of Government Debt
    DBT = Debt
    MAT = Average Remaining Maturity on Debt
    IE = Interest Expenditures

    IE = DBT * MSD ( %INT )

    For coupon securities (simple interest) the interest expense is:
    IE = DBT * %INT

    Long term coupon debt does not solve the problem even if market prices on the debt fall because the owner of the debt can simply hold it to maturity.

    For accrual securities (interest is compounded and repaid at maturity):

    IE = DBT * 1/MAT * ( exp ( 1 + %INT) ^ MAT - 1 )

    For the government to avoid default:

    GDP = Gross Domestic Product
    TR = Tax Revenue as a percentage of GDP

    TR * GDP > IE

    Allowing for different durations of debt:

    TP = Term Premium

    IE = DBT * 1/MAT * ( exp ( 1 + TP * %INT) ^ MAT - 1 )

    And so a government can set its annual interest expense (how much tax revenue goes towards paying bondholders) independently of the interest rate by adjusting either the amount of debt OR the average remaining maturity of its debt. This does not work for coupon securities for the reasons described above.


    The federal government need not fund deficits with debt. The interest payments on federal debt are guaranteed expenditures (courtesy of section 14 of the federal reserve act and the 14th amendment to the Constitution). What happens when the federal government sells non-guaranteed securities:

    GEQ = Government Equity
    %ROR = Rate of Return on non-guaranteed government securities (equity if you prefer)
    MSE() = Maturity Structure of Government Equity
    FR = Fail Rate of Government Equity
    MAT = Average Maturity on Equity
    RE = Return on Government Equity
    TP = Term Premium

    RE = GEQ * MSE ( %ROR )

    Again for compounding accrual securities the returns on government equity are as follows:

    RE = GEQ * 1/MAT * ( exp ( 1 + ( 1 - FR ) * TP * %ROR ) ^ MAT - 1 )

    Finally on Friedman:

    M * V = P * Y
    Subsitute Debt (D) for Money (M):

    D * V = P * Y = Income * ( 1 - Liquidity Preference ) + Change in Debt with Respect to Time

    Income is simply sale of goods + interest payments on debt

    D * V = ( D * V + INT * D ) * ( 1 - LP ) + dD/dt

    f(t) = Demand for new debt
    D = exp ( f(t) )
    dD/dt = f'(t) * exp ( f(t) )

    V * LP = INT * ( 1 - LP ) + f'(t)
    V = [ INT * ( 1 - LP ) + f'(t) ] / LP

    What about inflation and productivity?

    P * Y = Nominal GDP = Real GDP * ( 1 + Inflation Rate )
    P * Y = RGDP * ( 1 + IR )

    Productivity ( or economic efficiency if you prefer ) can be expressed as:

    Productivity = Real GDP / Debt = Velocity / ( 1 + Inflation Rate )
    PRO = RGDP / D = V / ( 1 + IR )

    PRO = [ INT * ( 1 - LP ) + f'(t) ] / [ LP * ( 1 + IR ) ]

    Solving for the inflation rate:

    IR = -1 + [ INT * ( 1 - LP ) + f'(t) ] / [ LP * PRO ]

    One of the first things you should notice is that interest rates and the inflation rate are positively correlated. Meaning that if productivity, liquidity preference, and credit demand are unchanged, a rise in interest rates will result in higher inflation.

    1. Back on the interest paid by the federal government for accrual securities:

      IE = DBT * 1/MAT * ( exp ( 1 + %INT) ^ MAT - 1 )

      If we differentiate this with respect to time

      dIE/dt = d/dt [ DBT * 1/MAT * ( exp ( 1 + %INT) ^ MAT - 1 ) ]

      And if we want to set dIE/dt = 0 so that the federal government pays the same amount of annual interest every year:

      d/dt [ DBT * 1/MAT * ( exp ( 1 + %INT) ^ MAT - 1 ) ] = 0

      Solving for dMAT/dt:

      dMAT/dt = [ dDBT/dt * 1/DBT * [ 1 - ( 1 + INT ) ^ -MAT ] + dINT/dt * MAT / ( 1 + INT ) ] / [ 1 - ( 1 + INT ) ^ -M - M^2 * ln ( 1 + INT ) ]

      For a given interest rate (INT), term premium (dINT/dt), debt level (DBT), average remaining maturity (MAT), and deficit (dDBT/dt) we can solve for the change in average maturity needed to keep annual interest expense paid out of taxes at the same amount.

      Notice that this does not work for coupon securities. Extending duration for coupon securities increases annual interest payments - always.

  2. Fertilizer, weeds, good stuff. I knew you had it in you. Philosophy really is the better way to go. Educated creative People understand philosophy much better than numbers. I hope to see more fertilizer etc. Perspective
    Numbers and graphs prove nothing to people who can see visions. The numbers and graphs are for those genetically inferior monkeys who won't understand even if they see them. They can't visualize because they have no imagination. First comes the personal vision. Then comes the proofs. Descartes meditations. General thesis. Numbers proove nothing to the visionary. Descartes triangle proof of GOD. Mathematics is purely a religious fanatics obsessive compulsive disorder/ Descartes meditations
    I am sure you will refrain from mathematical proofs as your main claim to legitimacy once you have a sufficient volume of philosophy under your belt. And you will feel much more confidence in your work with the perspective that comes only through philosophy. PhD doctor of philosophy. Really is something to be proud of. If it is lagitamit.

  3. And in other news:

  4. Not only should they drop math in public school, they need to drop public school completely. Public school has no place in a capitalist society. Teachers determining the value of their product. = Labor theory of value. Americans are so uneducated that they can't understand theirs is the most communist society on the earth. There is no such thing as capitalism in America

  5. Aristotle thought otherwise.

    A functioning market system cannot exist without a legal system to enforce contracts agreed to by two or more parties. A legal system cannot exist without a system of public education to inform the citizenry of its rights and obligations.

    1. The legal system predated public education in America (and most countries founded before the Prussian model spread). Massachusetts was early in REQUIRING youths to get an education, but that didn't mean they actually PROVIDED it publicly.

    2. Um, yes they did provide it publicly:

  6. Prof. Cochrane,

    I suggest you’ve missed the elephant in the room – put another way, there is a phenomenally easy way out of the “debt” problem (growth or no growth). It’s QE.

    Any country which issues its own currency can simply print money and buy back the debt. And the inflationary effect of that is minimal, if the record of QE over the last two years is any guide.

    But to the extent that QE is in fact inflationary, that is easily countered via increased tax (and/or reduced public spending). The latter cut in net government spending may pose POLITICAL problems, but there is absolutely no strictly ECONOMIC problem there.

    But all that assumes that the US/UK/Japan etc actually NEED to cut their debt. The truth is that as long as the yield on their debt is about equal to inflation (which it is), they are paying a zero real rate of interest. What’s the problem there?

    And if the real interest rate rises on “rolled over” debt, then just print money and pay back the debt plus raise taxes as explained above. Easy.

    P.S. You looked very cheerful rather than grumpy in the video.

    1. Mr. Musgrave,

      The truth is as long as the interest payments on the debt are less than the amount of tax revenue collected, there is no funding problem - inflation or no inflation, growth or no growth.

      And all this assumes that the federal deficit must be funded with debt. There are two deficit funding means - debt and equity.

    2. Your first paragraph is bizarre. Suppose interest amounted to 90% of tax collected. You seem to claim there is no problem there. I’d suggest there is one heck of a problem: there’d be next to nothing to pay for defence, education, law enforcement, and the hundred other things government does!!

      Re your second paragraph, I’m baffled as to what funding government via “equity” consists of. You mean one can buy shares in the US government? That’s news to me. But perhaps you can expand on that point.

    3. Mr. Musgrave,

      First the distinction between the amount of annual interest a goverment pays and the interest rate it pays on any one security.

      For annual coupon securities, the federal government pays annual interest equal to the interest rate times the principle value of the bond. For accrual securities, the federal government pays total accrued interest (usually compounded) on all securities that reach maturity in a given year.

      And so a government by selling accrual securities can limit the amount of annual interest it pays independently of the interest rate that it pays by simply extending the duration of the debt that it sells.

      "I’d suggest there is one heck of a problem: there’d be next to nothing to pay for defence, education, law enforcement, and the hundred other things government does!!"

      You are assuming that the government cannot access the capital markets to obtain funding? Can you honestly tell which dollar spent on education was borrowed and which one came from tax revenue?

      Equity (definition)

      "In accounting and finance, equity is the residual claim or interest of the most junior class of investors in assets, after all liabilities are paid."

      A government bond is a senior claim on a government's tax revenue. It pays a rate of return (interest rate) that is guaranteed and paid out of tax revenue. Equity as a junior class of investment offers a rate of return that is not guaranteed. And so it would simply be a junior claim on the government's tax revenue.

  7. Whilst I'm not an economist, (and barely a historian)so I will even attempt to draw conclusions, I wrote the following after being quite entertained by the views of National Debt by a Victorian, about 150 years ago and about 150 years after its instigation


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