## Tuesday, November 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.

1. Evan: I disagree with your model (unless I misunderstand it).

Let me re-write your demand function as y = -ar + b(e-1). Then PPP can be interpreted as "the elasticity parameter b approaches infinity, because domestic and foreign goods become perfect substitutes."

1. Do you disagree with the assumption of PPP? If there's going to be a deviation from it, given flight-to-safety capital inflows into the US, it's going to be on the contractionary side.

2. I do disagree with PPP, but that's not the issue. Suppose I agreed with PPP. I would say PPP is a consequence of your demand function, in the limit as b approaches infinity. You cannot assert PPP as an independent equation, it is like having an overdetermined model.

Take the limit as b approaches infinity. A tiny real exchange rate depreciation would cause an infinitely big increase in output determined.

3. OK, I think I understand what went wrong; for PPP to hold, b must approach infinity, given the definitions. I'm not exactly sure if this can be fixed -- at this point, I'm in a little over my head -- but is the problem corrected by assuming e as the nominal exchange rate, rather than the real exchange rate? Or, is it just that PPP cannot hold in the context of the model? The reason I had invoked it was to connect exchange rates to the domestic price level, and then to monetary policy, which I saw as the underlying issue here. Thanks for your input.

4. The "e" in y=-ar+be should really be the real exchange rate (just as the 'r" should be the real interest rate). The equation is a simplified version of the IS curve in an open economy. Net exports depends on the relative price of foreign to domestic goods, which is the real exchange rate. For example, if the foreign price level doubled, and the exchange rate depreciated by half, net exports should stay the same, because the relative price of foreign and domestic goods would stay the same.

This is really just a specific example of a more general rule. Every equation in a model (except the central bank's reaction function/target/whatever) should be homogenous in nominal variables. If you double all the nominal variables, and leave the real variables unchanged, the equation should still be true. (Except for the central bank's reaction function/target/whatever, because otherwise the price level would be indeterminate). (I might be using the wrong math terminology when I say "homogenous. I got a 'D' in A-level math, a very long time ago.)

5. I think you need some sort of AS curve (or Phillips Curve) to pin down the price level (or inflation rate). Something like y=sP for an upward-sloping SRAS curve.

2. This comment has been removed by the author.

3. A few quick notes: this might be my mistake, but I don't quite follow how you've defined the real exchange rate. As I understand it, in your pseudo-IS function you explicitly label 'e' as the real exchange rate, but then go on to define the law of one price as being: P = P* x e, which doesn't quite make sense to me. The LOP is ordinarily written like that, but the 'e' in that equation corresponds to the 'nominal' exchange rate. That makes sense because, say, in a one good economy, if the consumption item is now more expensive (P is higher) and prices in the foreign country stay the same, the nominal exchange rate must appreciate in response so that the LOP holds.

It seems to me, then, that in case (1) you are simply keeping the real exchange rate constant, which Krugman effectively discusses in his proposed modelling. Price level targeting is, in this instance, a fixed nominal exchange rate regime.

As for (2), I don't quite understand what the monetary authority is supposed to be doing. Setting a target for nominal output is probably less informative than assuming the monetary authority simply sets a target of N for real output. In that event, the TR line in Krugman's model is a vertical line at the desired level of output, and the monetary authority will allow the interest rate to increase in the exact amount of the increase in the risk premium, which ought to have no effect on output (naturally). Setting a target in terms of nominal output, seems to me, would have, again, no effect on output: the increase in risk premium has a negative effect on investment which is perfectly offset by the expansionary effect of real depreciation.

Effectively, we need to think of Krugman's model in the following sense: the monetary authority has no control over actual interest rates, only nominal exchange rates. It uses the wedge between the long run exchange rate and the actual nominal exchange rate as the driving mechanism in the model: the CB sort of gains 'momentary' control of policy by increasing or decreasing that wedge. Thus, his model isn't actually a model of risk premia at all: note that an increase in the rate required by bond holders of the Bundesbank (the safe asset) has exactly the same implications as an increase in the risk premium, and the latter is therefore pretty irrelevant to his conclusions.

As he points out, the main result of the model is that an increase in interest (for whatever reason, I would add) implies currency depreciation via the wedge, and therefore always leads to an increase in output. The question is surely then whether this is the right way to model the relationship between the interest and nominal exchange rates.

1. Thanks -- I think I did mean the nominal exchange rate (I wrote this up in my notebook over the weekend). Your point on (1) is correct. On (2), yes regarding the real-stability-only central bank, but I think a nominal-output target isn't the same in my model. Any devaluation is going to boost the price level and real output; therefore, a return to the same Py will happen before the same y. I agree with your points about the original model, also.

4. Evan, I start from a different premise. The risk premium is currently too high and would be lowered given the appearance of "bond vigilantes". Rising yields would be a sign of recovery and return to normalcy.

1. I understand, but IIRC, didn't Krugman have a post in response to deLong re: a bond bubble, with (what I thought was) convincing evidence that bonds were appropriately priced given the mkt. forecast for future interest rates. You might respond that that path of interest rates is effectively a bet of too-low NGDP -- and I would agree -- but I don't subscribe to the reverse logic, i.e. that increasing interest rates will generate an increase in the NGDP path. Hopefully I'm not misreading you.

2. Evan,
I do not believe or advocate raising interest rates will generate an increase in NGDP path. http://macromarketmusings.blogspot.com/2012/02/can-raising-interest-rates-spark-robust.html

3. Evan,

I did a second post to clarify that I was staring from a different premise than most in this discussion.