The Difference between Quantitative Easing and changing the treasury maturity structure

Everyone says Quantitative Easing and changing the treasury maturity structure are equivalent (e.g. Paul Krugman, James Hamilton, Robert Waldmann, Robert Hall, Robert Barro).


However, if you look at each bond market through surplus analysis, you can see the difference between the two operations.


A. Changing the treasury maturity structure

A-1. Long-Term Bond Market



When the government reduces the issuance of long-term treasury bond, the supply curve shifts from "Supply Curve 1" to "Supply Curve 2". As a result, total surplus declines (shaded area). The government surplus changes from OBCD to OB'C'D'. Bond buyers' surplus changes from DCE to D'C'E.

(Here I assumed that the government doesn't change the amount of issuance according to bid-price, so that the supply curve is vertical. The government may have the reserve price under which it cancels the issuance, but here I assume that the reserve price is zero for simplicity [Although this is unrealistic assumption, it doesn't affect the main observation of this post]. In that case, the government surplus equals the amount of issuance.)

Of the bond buyers' surplus decrease, DFC'D' transfers to the government. The other part of the decrease, FCC', is the deadweight loss.


A-2. Short-Term Bond Market



The government wants to offset the above reduction in the revenue it raises from the long-term bond market by increasing the short-term bond issuance. So, the supply curve shifts from "Supply Curve 1" to "Supply Curve 2" in the short-term bond market.

On the other hand, as short-term interest rate has hit the zero bound, the demand curve is flat. That is, T-bill and the money is equivalent in this market (That is the point many commentators - such as Krugman - emphasize). So, the bond buyers' surplus doesn't change before and after the issuance increase.

As a result, only the government's surplus changes to compensate the reduction it suffered in the long-term bond market (shaded area).


B. Quantitative Easing

Long-Term Bond Market




In Quantitative easing, the Central Bank participates in the purchase of the long-term bond. So the demand curve shifts from "Demand Curve 1" to "Demand Curve 2". In that case, the bond buyers' surplus doesn't change; it just shift from DCE to D'C''E' (*1). On the other hand, the government's surplus increases by the shaded area DCC''D', which equals ECC''E'.

That government's surplus increase is the amount of "seigniorage" which the Central Bank monetized. It shows up explicitly in the change in bond price as shown in the above figure, so its effect on inflation expectation is more "direct" than in changing the treasury maturity structure case.

(When the Central Bank doesn't purchase the bond directly from the government, some of the surplus is diverted to the arbitrage profit of the bank. But if the competition among banks works well, that arbitrage profit approaches zero in the limit.)


(*1) Of the bond buyers' surplus after the shift D'C''E', C'C''E'E is the surplus of the Central Bank. So the private buyers' surplus is D'C'E, which equals to the bond buyers' surplus when the bond supply was decreased (A-1). However, there is no deadweight loss this time.



In sum, changing the treasury maturity structure ends up in the decrease of the long-term bond buyers' surplus with deadweight loss. In contrast, QE increases the government's surplus while keeping the total bond buyers' surplus unchanged. (Although the private bond buyers' surplus decreases in the latter case as much as the former case, there is no deadweight loss.)


Some may say that decrease in the issuance of long-term treasury bonds, along with decrease in the surplus of their buyers, is not such a big deal. But didn't Ricardo Caballero and Brad Delong emphasize about the current shortage in safe assets? Decreasing the issuance of long-term treasury bonds exacerbates that shortage.

While the Central Bank's purchase of the bonds seems to be equivalent to the decrease in the issuance of the bonds on the consolidated government basis, total outstanding of the bonds is different in each case nonetheless. Thinner total outstanding makes the market more vulnerable to the demand shock, which is likely to occur considering the craving for safe assets Caballero and Delong noted.

Besides, that difference in total outstanding involves deadweight loss as shown above.


　

A hyperinflation model: circumventing Buiter's critique by holding nominal money growth constant

In hyperinflation model, Cagan money demand function is commonly used (See David Romer, "Advanced Macroeconomics").

　　m = C・exp(-bπ)

where m is real money stock, π is the rate of inflation, b and C are constant parameters.

Rewriting this as an equation for inflation gives us

　　π = (-1/b)・ln(m/C)

So, government seigniorage d can be written as

　　d = Δm + πm
　　　 = Δm - (m/b)・ln(m/C)

Therefore,

　　Δm = d + (m/b)・ln(m/C)

Differentiating this with m gives us

　　∂Δm/∂m = (1/b)・{ln(m/C) + 1}

Which becomes zero when m = C・exp(-1).

Second derivative of Δm is

　　∂²Δm/∂m² = 1/bm

which is strictly positive. So

　　Δm = d - (C/b)・exp(-1)

is the minimum value of Δm.


Here, d = (C/b)・exp(-1) gives us the maximum value of d presuming Δm=0.

Proof)
　∂(πm)/∂π = C・exp(-bπ) - bCπ・exp(-bπ)
　　　　　　　　= (1-bπ)・C・exp(-bπ)
　Therefore, πm, or seigniorage when Δm=0,
　becomes maximum when π = 1/b.
　At that point,
　　m = C・exp(-1)
　and
　　d = (C/b)・exp(-1)

So, if we denote this maximum d under stationary condition as d^*, the above Δm minimum value becomes

　　Δm = d - d^*

Hyperinflation is thought to happen when d > d^*. However, in that case, Δm becomes positive. Which means, m increases, and π＝(-1/b)・ln(m/C) decreases. This situation should be called hyperdeflation, not hyperinflation. This is the paradox noted by Willem Buiter (for detail, see Alexandre Sokic, "Monetary hyperinflation, speculative hyperinflation and modelling the use of money").

And if we think about nominal money growth rate (let's denote this g_M) in this hyperdeflation case, it can be shown that

　　Δg_M / g_M = - (Δm / m)

This is because d is assumed to be constant. So, if Δm is positive, Δg_M is negative. In this paradox case, nominal money growth decreases, but as inflation decreases further, real money growth becomes positive.

If you describe people's behavior in this case, it's something like this:
They watch government move and say, "Oh dear, the government is trying to obtain seigniorage beyond possible point. Let's suppress our economic activity and bring inflation down, so that real money stock increases and the government can obtain and maintain that seigiorage. Then, the central bank doesn't need to print more money. It can even print less money."

Obviously, this just cannot happen -- at least, not in the secular world.

In the real world, g_M is the only political tool. So, if the government wants more seigniorage, it makes the central bank to increase g_M. At any rate, it never lets g_M decrease.


Now, let's see what happens if we hold g_M constant.

If g_M is constant, Δm becomes

　　Δm = g_Mm + (m/b)・ln(m/C)

Differentiating this with m gives us

　　∂Δm/∂m = g_M + (1/b)・{ln(m/C) + 1}

Which becomes zero when m = C・exp(-1-bg_M).

Second derivative of Δm is

　　∂²Δm/∂m² = 1/bm

which is strictly positive. So

　　Δm = -(C/b)・exp(-1-bg_M)

is minimum value of Δm. And this is strictly negative, which means hyperinflation can occur.

And, if g_M is constant, we can obtain general solution for π, which is
　　π = g_M + A・exp(t/b)
where A is constant and t denotes time.

Proof)
　Cagan money function
　　m = C・exp(-bπ)
　gives us
　　(dt/dm)/m = -b(dπ/dt)
　And seigniorage is
　　d = Δm + πm = g_Mm
　So, the following differential equation holds
　(Note that I use Δm and dm/dt interchangeably).
　　g_M = -b(dπ/dt) + π
　Solving this equation gives the above solution.

As noted above, the government would like to increase g_M. This constant-g_M-solution may be fit for some short period Δt, but it's unlikely to be adequate description for longer period.

However, maybe increasing g_M stepwise in this solution can serve as the ad-hoc but tolerable way to describe the hyperinflation. The following graphs are from an example of such a simulation (click to enlarge).

(d=left-hand-axis, g_M=right-hand-axis)

(d=left-hand-axis, π=right-hand-axis)

(m=left-hand-axis, Δm=right-hand-axis)

In this simulation, I set b=1/3, C=0.1, Δt=0.1, and g_M(initial value)=0.5. Hence, initial d=g_M・C・exp(-bg_M)=0.0423.
"Constant g_M" case is where g_M is always 0.5.
In not-constant-g_M-case, I increased g_M each period so that d becomes approximately constant.