Negative Average Cost but Positive Marginal Cost of Debt-Finance

December 4, 2017
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If the rate of interest on government bonds is less than the growth rate of the economy (r

But it is important to distinguish between average cost and marginal cost. Unless the demand for government bonds is perfectly elastic, an increase in the debt/GDP ratio will increase the equilibrium rate of interest required for people to be willing to hold those government bonds. So the government is like a monopsonist facing an upward-sloping supply curve of finance, and the marginal cost of extra bond-finance will exceed the average cost of all bond-finance. The marginal cost of increasing the debt/GDP ratio might be positive even if the average cost is negative. It all depends on the interest-elasticity of demand for government bonds.

I will now try to do the math.

Start with the simplest case, where the economy is not growing (g=0) and inflation is zero. There is a stock of bonds B, that pay a rate of interest r, and GDP is constant at Y. With the stock of bonds constant over time, each year the government will need to pay rB interest, and rB/Y interest as a percentage of GDP. If r>0, the government will need to have spending on goods and services (excluding interest) lower than taxes to service the debt. (It will need to run a “primary” surplus. Debt service costs as a ratio of GDP will be:

s = r(B/Y)

Now lets keep inflation at 0, but assume GDP is growing at rate g. This means the government needs to issue gB new bonds each year just to keep the stock of bonds growing at the same rate as GDP, and so keep the debt/GDP ratio constant over time. Which means it can run a smaller primary surplus than if the economy were not growing. Debt service costs as a ratio of GDP are now:

s = (r-g)(B/Y)

If we add inflation to the model we have two choices: either we interpret r as the nominal interest rate and g as the growth rate of nominal GDP; or else we subtract inflation from both so r is the real interest rate and g the growth rate of real GDP. It makes no difference which way we do it, as long as we are consistent. Similarly, it makes no difference whether the numerator and denominator in the debt/GDP ratio B/Y are both nominal or both real, as long as we are consistent.

If r>g the average cost of bond-finance is positive (a government with a positive debt/GDP ratio will need to run a bigger primary surplus than a government with no debt). But if r

Yes this is a Ponzi scheme. But it is a stable Ponzi scheme. It is very similar to what a government does when it prints currency. Currency pays 0% nominal interest, and minus 2% real interest if there is 2% inflation. So r

But there’s a limit to the revenue, as a percentage of GDP, the government can earn from issuing currency. Because the demand to hold currency, as a ratio of GDP, is not perfectly elastic. There is a negative relationship between the inflation rate (the negative real interest rate earned by holding currency) and the currency/GDP ratio. Similarly there will be a positive relationship between the interest rate on government bonds and the debt/GDP ratio. The demand curve for government bonds is not perfectly elastic; people will hold a higher ratio of government bonds to GDP only if those bonds pay a higher interest rate. To say the same thing another way, if government bonds are in low supply, relative to income, the interest rate on those bonds will be low, other things equal. So the marginal cost of bond-finance is greater than the (r-g) on the marginal debt; it must include the marginal (r-g) on the initial debt. That’s the second term in this equation, that we get from differentiating the above equation:

ds/d(B/Y) = (r-g) + (B/Y)d(r-g)/d(B/Y)

Remember Intro Microeconomics, when the prof explained that the Marginal Revenue for a monopolist was less than Price (Average Revenue), because to sell an extra apple he needs to cut the price? And how it was possible to have P>0 and MR<0 if you are on an inelastic portion of the demand curve? That’s what we are doing here. Except it’s Marginal and Average Cost, and the Average Cost might be negative but Marginal Cost positive. And it’s the elasticity of (B/Y) with respect to (r-g) that matters.

Divide both sides by (r-g), rearrange terms, and we get:

[ds/d(B/Y)]/(r-g) = 1 + (B/Y)d(r-g)/d(B/Y)(r-g) = 1 + 1/e

where e is the demand elasticity of (B/Y) with respect to (r-g). It’s the percentage (not percentage point) change in (B/Y) for a one percent (not percentage point) change in (r-g). And if it’s inelastic (e<1) then with r

I think that math is right. It feels right.

Is it elastic or inelastic? I don’t know. But notice how as r gets closer to g, elasticity approaches zero. Just like a linear demand curve for apples will get inelastic as the price approaches zero. So if r is only a little bit less than g, the average cost is negative but the marginal cost positive.

[There ought to be a simpler way to do this, where we could talk more intuitively about the demand-elasticity with respect to r, not with respect to (r-g), and compare that elasticity to (r-g)? Someone else can try to do the math, because I’m off to Tim’s.]

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