Evan Soltas
Nov 13, 2012

# Fearing the Bond Viligantes

Paul Krugman posted on his blog an interesting write-up of a model which proposes that an "attack" by what he has called the "invisible bond vigilantes" is expansionary. In other words, an increase in the lending premium on sovereign debts should increase aggregate demand and thereby real GDP.

Krugman's mechanism makes a great deal of sense: (1) the increase in the risk premium generates depreciation of the currency for all domestic interest rates; (2) the lower exchange rate of the dollar causes an increase in net exports; and (3) thus we have a shifting to the left of the IS curve in the IS-LM model for all interest rates.

I agree with Krugman on several points, but in this post I want to propose a simple model which might make the best form of the counter-argument -- i.e. that such an increase in the risk premium would be contractionary. I also note a few areas in which his model and also my own make some assumptions which I, well, wouldn't bet very much on at all.

(Nick Rowe also has some brief comments on Krugman's model here, and so do Tyler Cowen, Brad deLong, David Beckworth and Scott Sumner.)

Let me first note the general areas of agreement. It's pretty obvious that a country with a floating exchange rate and an independent currency (like the U.S. or U.K.) is in a totally different situation than a country without either (like Greece or Spain) insofar as debt can create default risk. I also agree with some key components of the model -- you'll see me re-use much of his framework in a moment -- and with the insight that an increase in the risk premium should generate higher net exports and thereby, cet. par., higher real GDP in the short run.

But here's how I thought about this. In terms of intuition, an increase in the risk premium is a supply shock. A central bank which follows some sort of rule regarding price stability should be somewhat constrained as to how much devaluation they can allow. From this, one can reason that the sign on the effect to real GDP should always be negative, given that the central bank will split the impact between a decline in real GDP and an increase in the price level according to its preferences.

And that's what one sees come through in this little model.

First, start with the linearized demand function y = -ar + be, where r is the real interest rate, e is the nominal exchange rate in terms of the price of foreign currency, and a and b are constants. Note the signs of the terms mean that an increase in the real interest rate decreases real GDP and an increase in exchange rate increases real GDP -- i.e. when foreign currency is more valuable, our net exports increase.

Second, assume a real domestic interest rate determined by r = r* + p, where r* is the real risk-free rate of return and p is the risk premium. An increase in the risk premium increases the interest rate.

Third, assume that purchasing power parity (PPP) holds to some reasonable extent in some reasonably short time frame. That is to say, an identical tradable goods and services should cost roughly the same whether I'm buying them in dollars, yen, etc. We can write this out as P = ePf, where P is the domestic price level, e is (again) the exchange rate, and Pf is the foreign price level.

Fourth, assume some monetary policy rule. Here are three which I think characterize the broad swath of options, which I proceed to consider individually.

(1) price-level targeting: P = P*
(2) NGDP targeting: Py = N
(3) a Taylor-type rule: P + Ty = k, T > 0

To consider (1) with the assumptions explained above:

e = P* / Pf

y = -a(r* + p) + b(P* / Pf)

∂y/∂p = -a.

This means that real GDP will fall in response to an increase in the risk premium, due to its effect on real domestic investment, when the central bank is targeting the price level. (This should also be the response of an inflation-targeting central bank.) The strength of this effect will depend on the "a" parameter, which is the responsiveness of domestic investment to changes in the real interest rate.

To consider (2):

$e = \frac{N}{yP_f} \\ y = -a ( r^* + \rho ) + b\frac{N}{y P_f }\\ y^2 + ay ( r^* + \rho ) + b\frac{N}{y P_f } = 0 \\ \frac{\partial y}{\partial \rho} = \frac{-ya}{2y + a ( r^* + \rho)} \\ \frac{\partial y}{\partial \rho} \approx \frac{-a}{2}$

Again, this means that real GDP will fall, though less than under the previous monetary policy specification, due to the supply shock's effect on domestic investment.

To consider (3),

$e = \frac{k-Ty}{P_f} \\ y = -a ( r^* + \rho ) + b\frac{k-Ty}{P_f} \\ y = \frac{-a ( r^* + \rho ) + \frac{bk}{P_f}}{1 + \frac{bT}{P_F}}\\ \frac{\partial y}{\partial \rho} = \frac{-a}{1 + \frac{bT}{P_F}}\\ 0 < \frac{\partial y}{\partial \rho} < -a$

This finding should make a lot of sense. If T = 0, then there is no weight on output stabilization, and the Taylor-type rule becomes a price-level rule. As T rises, the effect on output is dampened -- though, naturally, by progressively larger increases in the price level.

Making some small and I believe acceptable assumptions -- the aggregate demand function, the determination of an interest rate, a soft PPP, and a monetary policy rule -- we see that an increase in the risk premium is likely contractionary. The intensity of the contraction, furthermore, depends on the willingness of the central bank to accept large increases in the price level to cushion output in the short run.

The major differences with Krugman's model in terms of structure are as follows. First, I assume that domestic investment will feel the sovereign's risk premium. My understanding is that this is consistent with the actual operation of debt markets; the debt of large companies will be knocked down in terms of credit rating when the sovereign's credit rating falls. Second, there are some important real-nominal distinctions in my model versus his; it's not clear why an independent central bank would tolerate the inflation Krugman's model builds in but never directly addresses. I think these explain the opposite findings.

Three further notes. First, I think the assumption of a fixed risk-free rate i* in the context of a run on US sovereign debt is highly strained, for the same reason that the small open economy model is not the same as the large open economy model. Second, when the risk premium rises, the increase in the real interest rate is likely not to capture the full effect on domestic investment -- there are other mechanisms, most importantly tighter lending standards, which will cause an even larger decrease in investment. Third, in the context of an increase in the risk premium on U.S. debt, the U.S. dollar is -- by our experience in the last recession -- likely to appreciate, unless global debt markets are sufficiently strong to withstand a global risk-off. Heavy capital flows into U.S. Treasuries will prevent devaluation for the time period Krugman's model expects an expansionary effect. It is worth noting that all three reasons suggest that an increase in the risk premium is likely to result in a decrease of real GDP in excess of what is predicted by my model.

Correction: I originally had "e" marked as the real exchange rate. It should have been the nominal exchange rate.