Keynes and the Fisher Equation

The history of economics society is holding its annual meeting in Chicago from Friday June 15 to Sunday June 17. Bringing together material from a number of posts over the past five years or so about Keynes and the Fisher equation and Fisher effect, I will be presenting a new paper called “Keynes and the Fisher Equation.” Here is the abstract of my paper.

One of the most puzzling passages in the General Theory is the attack (GT p. 142) on Fisher’s distinction between the money rate of interest and the real rate of interest “where the latter is equal to the former after correction for changes in the value of money.” Keynes’s attack on the real/nominal distinction is puzzling on its own terms, inasmuch as the distinction is a straightforward and widely accepted distinction that was hardly unique to Fisher, and was advanced as a fairly obvious proposition by many earlier economists including Marshall. What makes Keynes’s criticism even more problematic is that Keynes’s own celebrated theorem in the Tract on Monetary Reform about covered interest arbitrage is merely an application of Fisher’s reasoning in Appreciation and Interest. Moreover, Keynes endorsed Fisher’s distinction in the Treatise on Money. But even more puzzling is that Keynes’s analysis in Chapter 17 demonstrates that in equilibrium the return on alternative assets must reflect their differences in their expected rates of appreciation. Thus Keynes, himself, in the General Theory endorsed the essential reasoning underlying the distinction between real and the money rates of interest. The solution to the puzzle lies in understanding the distinction between the relationships between the real and nominal rates of interest at a moment in time and the effects of a change in expected rates of appreciation that displaces an existing equilibrium and leads to a new equilibrium. Keynes’s criticism of the Fisher effect must be understood in the context of his criticism of the idea of a unique natural rate of interest implicitly identifying the Fisherian real rate with a unique natural rate.

And here is the concluding section of my paper.

Keynes’s criticisms of the Fisher effect, especially the facile assumption that changes in inflation expectations are reflected mostly, if not entirely, in nominal interest rates – an assumption for which neither Fisher himself nor subsequent researchers have found much empirical support – were grounded in well-founded skepticism that changes in expected inflation do not affect the real interest rate. A Fisherian analysis of an increase in expected deflation at the zero lower bound shows that the burden of the adjustment must be borne by an increase in the real interest rate. Of course, such a scenario might be dismissed as a special case, which it certainly is, but I very much doubt that it is the only assumptions leading to the conclusion that a change in expected inflation or deflation affects the real as well as the nominal interest rate.

Although Keynes’s criticism of the Fisher equation (or more precisely against the conventional simplistic interpretation) was not well argued, his intuition was sound. And in his contribution to the Fisher festschrift, Keynes (1937b) correctly identified the two key assumptions leading to the conclusion that changes in inflation expectations are reflected entirely in nominal interest rates: (1) a unique real equilibrium and (2) the neutrality (actually superneutrality) of money. Keynes’s intuition was confirmed by Hirshleifer (1970, 135-38) who derived the Fisher equation as a theorem by performing a comparative-statics exercise in a two-period general-equilibrium model with money balances, when the money stock in the second period was increased by an exogenous shift factor k. The price level in the second period increases by a factor of k and the nominal interest rate increases as well by a factor of k, with no change in the real interest rate.

But typical Keynesian and New Keynesian macromodels based on the assumption of no capital or a single capital good drastically oversimplify the analysis, because those highly aggregated models assume that the determination of the real interest rate takes place in a single market. The market-clearing assumption invites the conclusion that the rate of interest, like any other price, is determined by the equality of supply and demand – both of which are functions of that price — in  that market.

The equilibrium rate of interest, as C. J. Bliss (1975) explains in the context of an intertemporal general-equilibrium analysis, is not a price; it is an intertemporal rate of exchange characterizing the relationships between all equilibrium prices and expected equilibrium prices in the current and future time periods. To say that the interest rate is determined in any single market, e.g., a market for loanable funds or a market for cash balances, is, at best, a gross oversimplification, verging on fallaciousness. The interest rate or term structure of interest rates is a reflection of the entire intertemporal structure of prices, so a market for something like loanable funds cannot set the rate of interest at a level inconsistent with that intertemporal structure of prices without disrupting and misaligning that structure of intertemporal price relationships. The interest rates quoted in the market for loanable funds are determined and constrained by those intertemporal price relationships, not the other way around.

In the real world, in which current prices, future prices and expected future prices are not and almost certainly never are in an equilibrium relationship with each other, there is always some scope for second-order variations in the interest rates transacted in markets for loanable funds, but those variations are still tightly constrained by the existing intertemporal relationships between current, future and expected future prices. Because the conditions under which Hirshleifer derived his theorem demonstrating that changes in expected inflation are fully reflected in nominal interest rates are not satisfied, there is no basis for assuming that a change in expected inflation affect only nominal interest rates with no effect on real rates.

There are probably a huge range of possible scenarios of how changes in expected inflation could affect nominal and real interest rates. One should not disregard the Fisher equation as one possibility, it seems completely unwarranted to assume that it is the most plausible scenario in any actual situation. If we read Keynes at the end of his marvelous Chapter 17 in the General Theory in which he remarks that he has abandoned the belief he had once held in the existence of a unique natural rate of interest, and has come to believe that there are really different natural rates corresponding to different levels of unemployment, we see that he was indeed, notwithstanding his detour toward a pure liquidity preference theory of interest, groping his way toward a proper understanding of the Fisher equation.

In my Treatise on Money I defined what purported to be a unique rate of interest, which I called the natural rate of interest – namely, the rate of interest which, in the terminology of my Treatise, preserved equality between the rate of saving (as there defined) and the rate of investment. I believed this to be a development and clarification of of Wicksell’s “natural rate of interest,” which was, according to him, the rate which would preserve the stability of some, not quite clearly specified, price-level.

I had, however, overlooked the fact that in any given society there is, on this definition, a different natural rate for each hypothetical level of employment. And, similarly, for every rate of interest there is a level of employment for which that rate is the “natural” rate, in the sense that the system will be in equilibrium with that rate of interest and that level of employment. Thus, it was a mistake to speak of the natural rate of interest or to suggest that the above definition would yield a unique value for the rate of interest irrespective of the level of employment. . . .

If there is any such rate of interest, which is unique and significant, it must be the rate which we might term the neutral rate of interest, namely, the natural rate in the above sense which is consistent with full employment, given the other parameters of the system; though this rate might be better described, perhaps, as the optimum rate. (pp. 242-43)

Because Keynes believed that an increased in the expected future price level implies an increase in the marginal efficiency of capital, it follows that an increase in expected inflation under conditions of less than full employment would increase investment spending and employment, thereby raising the real rate of interest as well the nominal rate. Cottrell (1994) has attempted to make an argument along such lines within a traditional IS-LM framework. I believe that, in a Fisherian framework, my argument points in a similar direction.

 

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8 Responses to “Keynes and the Fisher Equation”


  1. 1 Miguel Navascués June 6, 2018 at 11:44 am

    Marvelous explanation of the problem.

  2. 2 George H. Blackford June 6, 2018 at 2:15 pm

    Before we can understand how the rate of in-terest is determined in Keynes’ general theory we have to begin by agreeing on whether we are talk-ing about the equilibrium rate of interest or the actually rate of interest that exists in the real world at any given point in time.

    It is fairly obvious (at least it was to Robert-son) that Keynes’ began by focusing on the deter-mination of the actual rate of interest that exists in the real world at any given point in time by way of a Marshallian partial-equilibrium analysis of this problem. In the process, he came to the conclusion that, given the fact that in a monetary economy money (or the creation of debt) is required as a medium of exchange, it is logically impossible to understand or explain the way in which decision-making units determined the rate of interest in terms of their choices with regard to the flows of saving and investment or of the supply and demand for loanable funds. The only way he could make sense out of the way in which decision-making units determined the rate of interest was in terms of their choices with regard to the supply and demand for the stock of money. I have explained the process by which Keynes came to this conclusion in detail here: http://www.rweconomics.com/htm/Pro.htm

    What I find interesting with regard to the way in which the real rate of interest is assumed to be determined in The General Theory is that Keynes did not actually examine the way in which this rate is determined. Instead, he took expectations to be exogenously determined and explained how the money rate of interest is determined given expectations. He then examined how changes in expectations affect the system. I believe it is worth noting that Fisher took the same approach in The Theory of Interest. (pp. 43-6)

  3. 3 Nick Edmonds June 7, 2018 at 1:08 am

    “In the real world, in which current prices, future prices and expected future prices are not and almost certainly never are in an equilibrium relationship with each other, there is always some scope for second-order variations in the interest rates transacted in markets for loanable funds..”

    I’m not sure I follow this. What sort of interest rates and what markets are you referring to?

    In the real world, nominal rates are transacted in financial markets, but I don’t think those are loanable funds markets are they? They simply involve swapping one form of financial asset (some form of money) for another, and do not involve any supply of or demand for actual savings.

    If there are any markets in the real world where both: a) transactions actually take place and b) loanable funds are exchanged (in the sense that one party saves and the other dis-saves), then these must actually be the markets like those for current commodities, where a current price is transacted, rather than an interest rate. Of course, given expectations of future prices, determination of the current price may imply some form of real interest rate. Is that what you meant?

  4. 4 Frank Restly June 7, 2018 at 3:28 pm

    David,

    Did Keynes or Fisher consider the effect of debt quantities on economic variables such as inflation?

    IntN% = dN / N = Nominal Interest Rate
    IntR% = dQ / Q = Real Interest Rate
    Inf% = dP / P = Inflation Rate
    D% = dD / D = Debt Growth Rate

    Fisher Equation (Interest rate only)
    IntN% = IntR% + Inf%

    This seems incomplete without a debt growth rate term.

    dN/N + dD/D = dQ/Q + dP/P
    IntN% + D% = IntR% + Inf%

    Do increases in the nominal interest rate drive the inflation rate higher (as implied by the Fisher equation) or is it some combination of both debt growth and the nominal interest rate?

  5. 5 Frank Restly June 7, 2018 at 4:16 pm

    Taking things a bit further and adding a productivity term:

    PY = Q / D = Productivity

    D * PY = Q

    dD/D + dPY/PY = dQ/Q
    dD/D = dQ/Q – dPY/PY

    Previous equation:
    dN/N + dD/D = dQ/Q + dP/P

    Substituting our equation for credit growth rate (dD/D):
    dN/N + dQ/Q – dPY/PY = dQ/Q + dP/P

    Cancelling out dQ/Q on both sides
    dN/N – dPY/PY = dP/P
    dN/N = dP/P + dPY/PY

    The real interest rate and the credit growth rate disappear from the equation and are replaced with the growth rate in productivity (dPY/PY).

    Nominal Interest Rate = Inflation Rate + Productivity Growth Rate

    This is the part that I believe Keynes and Fisher both miss on.

  6. 6 Frank Restly June 9, 2018 at 8:36 pm

    David,

    “One of the most puzzling passages in the General Theory is the attack (GT p. 142) on Fisher’s distinction between the money rate of interest and the real rate of interest where the latter is equal to the former after correction for changes in the value of money.”

    Hypothetical:
    dN/N = 5% : Nominal Interest Rate
    dD/D = -7% : Credit Growth Rate
    dPY/PY = 1% : Productivity Growth Rate

    New equations – Inflation Rate and Real Interest Rate:
    dP/P (4%) = dN/N (5%) – dPY/PY (1%)
    dQ/Q (-6%) = dD/D (-7%) + dPY/PY (1%)

    Notice that despite a positive nominal interest rate of 5% greater than a positive inflation rate of 4%, the real interest rate is negative 6%, not positive 1% as implied by the Fisher equation.

    What Fisher (and Keynes) fail to acknowledge is that both inflation AND credit / monetary contractions reduce the total value of money in circulation.

  7. 7 Frank Restly June 9, 2018 at 11:03 pm

    Or if you don’t like productivity being defined that way, consider defining changes is the total value of the money stock this way:

    PP = Purchasing power of total money stock

    PP = Debt (D) / Price Level (P)
    D = PP * P
    dD/D = dPP/PP + dP/P

    Again:
    dN/N + dD/D = dQ/Q + dP/P

    Substituting in for dD/D:
    dN/N + dPP/PP + dP/P = dQ/Q + dP/P

    Canceling out the inflation rate terms:

    dN/N = dQ/Q – dPP/PP

    Nominal Interest Rate = Real Interest Rate – % Change in purchasing Power

    Hypothetical:
    dN/N = 5% : Nominal Interest Rate
    dD/D = -7% : Credit Growth Rate
    dP/P = 4% : Inflation Rate

    dPP/PP (-11%) = dD/D (-7%) – dP/P (4%)

    dQ/Q (-6%) = dN/N (5%) + dPP/PP (-11%)

    Same result – real interest rate is negative 6% (not positive 1% as implied by the Fisher equation). That is because the positive inflation rate (4%) is coupled with a decline (7%) in the total amount of debt / money in circulation resulting in a negative 11% decrease in the purchasing power of the money stock.

    Obviously, I am equating debt with money via 100% reserve banking (a dollar can be only lent once). Things are complicated by fractional reserve banking (more debt exists than the total money in circulation) and by central bank open market purchases. But that is a subject for another day.

  8. 8 JKH June 11, 2018 at 5:13 am

    Chapter 17 is brilliant.

    Chapter 11 section 3 is a muddle.

    It does seem plausible that a change in inflation expectations might affect the MEC and the interest rate in ways that are not exactly the same, but that’s as far as I can get with it.


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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.

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