Posts Tagged 'Fisher equation'

Keynes on the Fisher Equation and Real Interest Rates

Almost two months ago, I wrote a post (“Who Sets the Real Rate of Interest?”) about the Fisher equation, questioning the idea that the Fed can, at will, reduce the real rate of interest by printing money, an idea espoused by a lot of people who also deny that the Fed has the power to reduce the rate of unemployment by printing money. A few weeks later, I wrote another post (“On a Difficult Passage in the General Theory“) in which I pointed out the inconsistency between Keynes’s attack on the Fisher equation in chapter 11 of the General Theory and his analysis in chapter 17 of the liquidity premium and the conditions for asset-market equilibrium, an analysis that led Keynes to write down what is actually a generalized version of the Fisher equation. In both of those posts I promised a future post about how to understand the dynamic implications of the Fisher equation and the relationship between Fisher equation and the Keynesian analysis. This post is an attempt to make good on those promises.

As I observed in my earlier post, the Fisher equation is best understood as a property of equilibrium. If the Fisher equation does not hold, then it is reasonable to attribute the failure to some sort of disequilibrium. The most obvious, but not the only, source of disequilibrium is incorrectly expected inflation. Other sources of disequilibrium could be a general economic disorder, the entire economic system being (seriously) out of equilibrium, implying that the real rate of interest is somehow different from the “equilibrium” rate, or, as Milton Friedman might put it, that the real rate is different from the rate that would be ground out by the system of Walrasian (or Casselian or Paretian or Fisherian) equations.

Still a third possibility is that there is more than one equilibrium (i.e., more than one solution to whichever system of equations we are trying to solve). If so, as an economy moves from one equilibrium path to another through time, the nominal (and hence the real) rate of that economy could be changing independently of changes in expected inflation, thereby nullifying the empirical relationship implied (under the assumption of a unique equilibrium) by the Fisher equation.

Now in the canonical Fisherian theory of interest, there is, at any moment of time, a unique equilibrium rate of interest (actually a unique structure of equilibrium rates for all possible combinations of time periods), increasing thrift tending to reduce rates and increasing productivity of capital tending to raise them. While uniqueness of the interest rate cannot easily be derived outside a one-commodity model, the assumption did not seem all that implausible in the context of the canonical Fisherian model with a given technology and given endowments of present and future resources. In the real world, however, the future is unknown, so the future exists now only in our imagination, which means that, fundamentally, the determination of real interest rates cannot be independent of our expectations of the future. There is no unique set of expectations that is consistent with “fundamentals.” Fundamentals and expectations interact to create the future; expectations can be self-fulfilling. One of the reasons why expectations can be self-fulfilling is that often it is the case that individual expectations can only be realized if they are congruent with the expectations of others; expectations are subject to network effects. That was the valid insight in Keynes’s “beauty contest” theory of the stock market in chapter 12 of the GT.

There simply is no reason why there would be only one possible equilibrium time path. Actually, the idea that there is just one possible equilibrium time path seems incredible to me. It seems infinitely more likely that there are many potential equilibrium time paths, each path conditional on a corresponding set of individual expectations. To be sure, not all expectations can be realized. Expectations that can’t be realized produce bubbles. But just because expectations are not realized doesn’t mean that the observed price paths were bubbles; as long as it was possible, under conditions that could possibly have obtained, that the expectations could have been realized, the observed price paths were not bubbles.

Keynes was not the first economist to attribute economic fluctuations to shifts in expectations; J. S. Mill, Stanley Jevons, and A. C. Pigou, among others, emphasized recurrent waves of optimism and pessimism as the key source of cyclical fluctuations. The concept of the marginal efficiency of capital was used by Keynes to show the dependence of the desired capital stock, and hence the amount of investment, on the state of entrepreneurial expectations, but Keynes, just before criticizing the Fisher equation, explicitly identified the MEC with the Fisherian concept of “the rate of return over cost.” At a formal level, at any rate, Keynes was not attacking the Fisherian theory of interest.

So what I want to suggest is that, in attacking the Fisher equation, Keynes was really questioning the idea that a change in inflation expectations operates strictly on the nominal rate of interest without affecting the real rate. In a world in which there is a unique equilibrium real rate, and in which the world is moving along a time-path in the neighborhood of that equilibrium, a change in inflation expectations may operate strictly on the nominal rate and leave the real rate unchanged. In chapter 11, Keynes tried to argue the opposite: that the entire adjustment to a change in expected inflation is concentrated on real rate with the nominal rate unchanged. This idea seems completely unfounded. However, if the equilibrium real rate is not unique, why assume, as the standard renditions of the Fisher equation usually do, that a change in expected inflation affects only the nominal rate? Indeed, even if there is a unique real rate – remember that “unique real rate” in this context refers to a unique yield curve – the assumption that the real rate is invariant with respect to expected inflation may not be true in an appropriate comparative-statics exercise, such as the 1950s-1960s literature on inflation and growth, which recognized the possibility that inflation could induce a shift from holding cash to holding real assets, thereby increasing the rate of capital accumulation and growth, and, consequently, reducing the equilibrium real rate. That literature was flawed, or at least incomplete, in its analysis of inflation, but it was motivated by a valid insight.

In chapter 17, after deriving his generalized version of the Fisher equation, Keynes came back to this point when explaining why he had now abandoned the Wicksellian natural-rate analysis of the Treatise on Money. The natural-rate analysis, Keynes pointed out, presumes the existence of a unique natural rate of interest, but having come to believe that there could be an equilibrium associated with any level of employment, Keynes now concluded that there is actually a natural rate of interest corresponding to each level of employment. What Keynes failed to do in this discussion was to specify the relationship between natural rates of interest and levels of employment, leaving a major gap in his theoretical structure. Had he specified the relationship, we would have an explicit Keynesian IS curve, which might well differ from the downward-sloping Hicksian IS curve. As Earl Thompson, and perhaps others, pointed out about 40 years ago, the Hicksian IS curve is inconsistent with the standard neoclassical theory of production, which Keynes seems (provisionally at least) to have accepted when arguing that, with a given technology and capital stock, increased employment is possible only at a reduced real wage.

But if the Keynesian IS curve is upward-sloping, then Keynes’s criticism of the Fisher equation in chapter 11 is even harder to make sense of than it seems at first sight, because an increase in expected inflation would tend to raise, not (as Keynes implicitly assumed) reduce, the real rate of interest. In other words, for an economy operating at less than full employment, with all expectations except the rate of expected inflation held constant, an increase in the expected rate of inflation, by raising the marginal efficiency of capital, and thereby increasing the expected return on investment, ought to be associated with increased nominal and real rates of interest. If we further assume that entrepreneurial expectations are positively related to the state of the economy, then the positive correlation between inflation expectations and real interest rates would be enhanced. On this interpretation, Keynes’s criticism of the Fisher equation in chapter 11 seems indefensible.

That is one way of looking at the relationship between inflation expectations and the real rate of interest. But there is also another way.

The Fisher equation tells us that, in equilibrium, the nominal rate equals the sum of the prospective real rate and the expected rate of inflation. Usually that’s not a problem, because the prospective real rate tends to be positive, and inflation (at least since about 1938) is almost always positive. That’s the normal case. But there’s also an abnormal (even pathological) case, where the sum of expected inflation and the prospective real rate of interest is less than zero. We know right away that such a situation is abnormal, because it is incompatible with equilibrium. Who would lend money at a negative rate when it’s possible to hold the money and get a zero return? The nominal rate of interest can’t be negative. So if the sum of the prospective real rate (the expected yield on real capital) and the expected inflation rate (the negative of the expected yield on money with a zero nominal interest rate) is negative, then the return to holding money exceeds the yield on real capital, and the Fisher equation breaks down.

In other words, if r + dP/dt < 0, where r is the real rate of interest and dP/dt is the expected rate of inflation, then r < –dP/dt. But since i, the nominal rate of interest, cannot be less than zero, the Fisher equation does not hold, and must be replaced by the Fisher inequality

i > r + dP/dt.

If the Fisher equation can’t be satisfied, all hell breaks loose. Asset prices start crashing as asset owners try to unload their real assets for cash. (Note that I have not specified the time period over which the sum of expected inflation and the prospective yield on real capital are negative. Presumably the duration of that period is not indefinitely long. If it were, the system might implode.)

That’s what was happening in the autumn of 2008, when short-term inflation expectations turned negative in a contracting economy in which the short-term prospects for investment were really lousy and getting worse. The prices of real assets had to fall enough to raise the prospective yield on real assets above the expected yield from holding cash. However, falling asset prices don’t necessary restore equilibrium, because, once a panic starts it can become contagious, with falling asset prices reinforcing the expectation that asset prices will fall, depressing the prospective yield on real capital, so that, rather than bottoming out, the downward spiral feeds on itself.

Thus, for an economy at the zero lower bound, with the expected yield from holding money greater than the prospective yield on real capital, a crash in asset prices may not stabilize itself. If so, something else has to happen to stop the crash: the expected yield from holding money must be forced below the prospective yield on real capital. With the prospective yield on real capital already negative, forcing down the expected yield on money below the prospective yield on capital requires raising expected inflation above the absolute value of the prospective yield on real capital. Thus, if the prospective yield on real capital is -5%, then, to stop the crash, expected inflation would have to be raised to over 5%.

But there is a further practical problem. At the zero lower bound, not only is the prospective real rate not observable, it can’t even be inferred from the Fisher equation, the Fisher equation having become an inequality. All that can be said is that r < –dP/dt.

So, at the zero lower bound, achieving a recovery requires raising expected inflation. But how does raising expected inflation affect the nominal rate of interest? If r + dP/dt < 0, then increasing expected inflation will not increase the nominal rate of interest unless dP/dt increases enough to make r + dP/dt greater than zero. That’s what Keynes seemed to be saying in chapter 11, raising expected inflation won’t affect the nominal rate of interest, just the real rate. So Keynes’s criticism of the Fisher equation seems valid only in the pathological case when the Fisher equation is replaced by the Fisher inequality.

In my paper “The Fisher Effect Under Deflationary Expectations,” I found that a strongly positive correlation between inflation expectations (approximated by the breakeven TIPS spread on 10-year Treasuries) and asset prices (approximated by S&P 500) over the time period from spring 2008 through the end of 2010, while finding no such correlation over the period from 2003 to 2008. (Extending the data set through 2012 showed the relationship persisted through 2012 but may have broken down in 2013.) This empirical finding seems consistent with the notion that there has been something pathological about the period since 2008. Perhaps one way to think about the nature of the pathology is that the Fisher equation has been replaced by the Fisher inequality, a world in which changes in inflation expectations are reflected in changes in real interest rates instead of changes in nominal rates, the most peculiar kind of world described by Keynes in chapter 11 of the General Theory.

Advertisements

Who Sets the Real Rate of Interest?

Understanding economics requires, among other things, understanding the distinction between real and nominal variables. Confusion between real and nominal variables is pervasive, constantly presenting barriers to clear thinking, and snares and delusions for the mentally lazy. In this post, I want to talk about the distinction between the real rate of interest and the nominal rate of interest. That distinction has been recognized for at least a couple of centuries, Henry Thornton having mentioned it early in the nineteenth century. But the importance of the distinction wasn’t really fully understood until Irving Fisher made the distinction between the real and nominal rates of interest a key element of his theory of interest and his theory of money, expressing the relationship in algebraic form — what we now call the Fisher equation. Notation varies, but the Fisher equation can be written more or less as follows:

i = r + dP/dt,

where i is the nominal rate, r is the real rate, and dP/dt is the rate of inflation. It is important to bear in mind that the Fisher equation can be understood in two very different ways. It can either represent an ex ante relationship, with dP/dt referring to expected inflation, or it can represent an ex post relationship, with dP/dt referring to actual inflation.

What I want to discuss in this post is the tacit assumption that usually underlies our understanding, and our application, of the ex ante version of the Fisher equation. There are three distinct variables in the Fisher equation: the real and the nominal rates of interest and the rate of inflation. If we think of the Fisher equation as an ex post relationship, it holds identically, because the unobservable ex post real rate is defined as the difference between the nominal rate and the inflation rate. The ex post, or the realized, real rate has no independent existence; it is merely a semantic convention. But if we consider the more interesting interpretation of the Fisher equation as an ex ante relationship, the real interest rate, though still unobservable, is not just a semantic convention. It becomes the theoretically fundamental interest rate of capital theory — the market rate of intertemporal exchange, reflecting, as Fisher masterfully explained in his canonical renderings of the theory of capital and interest, the “fundamental” forces of time preference and the productivity of capital. Because it is determined by economic “fundamentals,” economists of a certain mindset naturally assume that the real interest rate is independent of monetary forces, except insofar as monetary factors are incorporated in inflation expectations. But if money is neutral, at least in the long run, then the real rate has to be independent of monetary factors, at least in the long run. So in most expositions of the Fisher equation, it is tacitly assumed that the real rate can be treated as a parameter determined, outside the model, by the “fundamentals.” With r determined exogenously, fluctuations in i are correlated with, and reflect, changes in expected inflation.

Now there’s an obvious problem with the Fisher equation, which is that in many, if not most, monetary models, going back to Thornton and Wicksell in the nineteenth century, and to Hawtrey and Keynes in the twentieth, and in today’s modern New Keynesian models, it is precisely by way of changes in its lending rate to the banking system that the central bank controls the rate of inflation. And in this framework, the nominal interest rate is negatively correlated with inflation, not positively correlated, as implied by the usual understanding of the Fisher equation. Raising the nominal interest rate reduces inflation, and reducing the nominal interest rate raises inflation. The conventional resolution of this anomaly is that the change in the nominal interest rate is just temporary, so that, after the economy adjusts to the policy of the central bank, the nominal interest rate also adjusts to a level consistent with the exogenous real rate and to the rate of inflation implied by the policy of the central bank. The Fisher equation is thus an equilibrium relationship, while central-bank policy operates by creating a short-term disequilibrium. But the short-term disequilibrium imposed by the central bank cannot be sustained, because the economy inevitably begins an adjustment process that restores the equilibrium real interest rate, a rate determined by fundamental forces that eventually override any nominal interest rate set by the central bank if that rate is inconsistent with the equilibrium real interest rate and the expected rate of inflation.

It was just this analogy between the powerlessness of the central bank to hold the nominal interest rate below the sum of the exogenously determined equilibrium real rate and the expected rate of inflation that led Milton Friedman to the idea of a “natural rate of unemployment” when he argued that monetary policy could not keep the unemployment rate below the “natural rate ground out by the Walrasian system of general equilibrium equations.” Having been used by Wicksell as a synonym for the Fisherian equilibrium real rate, the term “natural rate” was undoubtedly adopted by Friedman, because monetarily induced deviations between the actual rate of unemployment and the natural rate of unemployment set in motion an adjustment process that restores unemployment to its “natural” level, just as any deviation between the nominal interest rate and the sum of the equilibrium real rate and expected inflation triggers an adjustment process that restores equality between the nominal rate and the sum of the equilibrium real rate and expected inflation.

So, if the ability of the central bank to use its power over the nominal rate to control the real rate of interest is as limited as the conventional interpretation of the Fisher equation suggests, here’s my question: When critics of monetary stimulus accuse the Fed of rigging interest rates, using the Fed’s power to keep interest rates “artificially low,” taking bread out of the mouths of widows, orphans and millionaires, what exactly are they talking about? The Fed has no legal power to set interest rates; it can only announce what interest rate it will lend at, and it can buy and sell assets in the market. It has an advantage because it can create the money with which to buy assets. But if you believe that the Fed cannot reduce the rate of unemployment below the “natural rate of unemployment” by printing money, why would you believe that the Fed can reduce the real rate of interest below the “natural rate of interest” by printing money? Martin Feldstein and the Wall Street Journal believe that the Fed is unable to do one, but perfectly able to do the other. Sorry, but I just don’t get it.

Look at the accompanying chart. It tracks the three variables in the Fisher equation (the nominal interest rate, the real interest rate, and expected inflation) from October 1, 2007 to July 2, 2013. To measure the nominal interest rate, I use the yield on 10-year Treasury bonds; to measure the real interest rate, I use the yield on 10-year TIPS; to measure expected inflation, I use the 10-year breakeven TIPS spread. The yield on the 10-year TIPS is an imperfect measure of the real rate, and the 10-year TIPS spread is an imperfect measure of inflation expectations, especially during financial crises, when the rates on TIPS are distorted by illiquidity in the TIPS market. Those aren’t the only problems with identifying the TIPS yield with the real rate and the TIPS spread with inflation expectations, but those variables usually do provide a decent approximation of what is happening to real rates and to inflation expectations over time.

real_and_nominal_interest_rates

Before getting to the main point, I want to make a couple of preliminary observations about the behavior of the real rate over time. First, notice that the real rate declined steadily, with a few small blips, from October 2007 to March 2008, when the Fed was reducing the Fed Funds target rate from 4.75 to 3% as the economy was sliding into a recession that officially began in December 2007. The Fed reduced the Fed Funds target to 2% at the end of April, but real interest rates had already started climbing in early March, so the failure of the FOMC to reduce the Fed Funds target again till October 2008, three weeks after the onset of the financial crisis, clearly meant that there was at least a passive tightening of monetary policy throughout the second and third quarters, helping create the conditions that precipitated the crisis in September. The rapid reduction in the Fed Funds target from 2% in October to 0.25% in December 2008 brought real interest rates down, but, despite the low Fed Funds rate, a lack of liquidity caused a severe tightening of monetary conditions in early 2009, forcing real interest rates to rise sharply until the Fed announced its first QE program in March 2009.

I won’t go into more detail about ups and downs in the real rate since March 2009. Let’s just focus on the overall trend. From that time forward, what we see is a steady decline in real interest rates from over 2% at the start of the initial QE program till real rates bottomed out in early 2012 at just over -1%. So, over a period of three years, there was a steady 3% decline in real interest rates. This was no temporary phenomenon; it was a sustained trend. I have yet to hear anyone explain how the Fed could have single-handedly produced a steady downward trend in real interest rates by way of monetary expansion over a period of three years. To claim that decline in real interest rates was caused by monetary expansion on the part of the Fed flatly contradicts everything that we think we know about the determination of real interest rates. Maybe what we think we know is all wrong. But if it is, people who blame the Fed for a three-year decline in real interest rates that few reputable economists – and certainly no economists that Fed critics pay any attention to — ever thought was achievable by monetary policy ought to provide an explanation for how the Fed suddenly got new and unimagined powers to determine real interest rates. Until they come forward with such an explanation, Fed critics have a major credibility problem.

So please – pleaseWall Street Journal editorial page, Martin Feldstein, John Taylor, et al., enlighten us. We’re waiting.

PS Of course, there is a perfectly obvious explanation for the three-year long decline in real interest rates, but not one very attractive to critics of QE. Either the equilibrium real interest rate has been falling since 2009, or the equilibrium real interest rate fell before 2009, but nominal rates adjusted slowly to the reduced real rate. The real interest rate might have adjusted more rapidly to the reduced equilibrium rate, but that would have required expected inflation to have risen. What that means is that sometimes it is the real interest rate, not, as is usually assumed, the nominal rate, that adjusts to the expected rate of inflation. My next post will discuss that alternative understanding of the implicit dynamics of the Fisher equation.


About Me

David Glasner
Washington, DC

I am an economist in the Washington DC area. My research and writing has been mostly on monetary economics and policy and the history of economics. In my book Free Banking and Monetary Reform, I argued for a non-Monetarist non-Keynesian approach to monetary policy, based on a theory of a competitive supply of money. Over the years, I have become increasingly impressed by the similarities between my approach and that of R. G. Hawtrey and hope to bring Hawtrey's unduly neglected contributions to the attention of a wider audience.

Archives

Enter your email address to follow this blog and receive notifications of new posts by email.

Join 2,198 other followers

Follow Uneasy Money on WordPress.com
Advertisements