I am sorry to have gone on a rather extended hiatus from posting, but I have been struggling to come up with a new draft of a working paper (“The Fisher Effect under Deflationary Expectations“) I wrote with the encouragement of Scott Sumner in 2010 and posted on SSRN in 2011 not too long before I started blogging. Aside from a generous mention of the paper by Scott on his blog, Paul Krugman picked up on it and wrote about it on his blog as well. Because the empirical work was too cursory, I have been trying to update the results and upgrade the techniques. In working on a new draft of my paper, I also hit upon a simple proof of a point that I believe I discovered several years ago: that in the *General Theory* Keynes criticized Fisher’s distinction between the real and nominal rates of interest even though he used exactly analogous reasoning in his famous theorem on covered interest parity in the forward exchange market and in his discussion of liquidity preference in chapter 17 of the General Theory. So I included a section making that point in the new draft of my paper, which I am reproducing here. Eventually, I hope to write a paper exploring more deeply Keynes’s apparently contradictory thinking on the Fisher equation. Herewith is an excerpt from my paper.

One of the puzzles of Keynes’s *General Theory* is his criticism of the Fisher equation.

This is the truth which lies behind Professor Irving Fisher’ss theory of what he originally called “Appreciation and Interest” – the 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. It is difficult to make sense of this theory as stated, because it is not clear whether the change in the value of money is or is not assumed to be foreseen. There is no escape from the dilemma that, if it is not foreseen, there will be no effect on current affairs; whilst, if it is foreseen, the prices of existing goods will be forthwith so adjusted that the advantages of holding money and of holding goods are again equalized, and it will be too late for holders of money to gain or to suffer a change in the rate of interest which will offset the prospective change during the period of the loan in the value of money lent. . . .

The mistake lies in supposing that it is the rate of interest on which prospective changes in the value of money will directly react, instead of the marginal efficiency of a given stock of capital. The prices of existing assets will always adjust themselves to changes in expectation concerning the prospective value of money. The significance of such changes in expectation lies in their effect on the readiness to produce

newassets through their reaction on the marginal efficiency of capital. The stimulating effect of the expectation of higher prices is due, not to its raising the rate of interest (that would be a paradoxical way of stimulating output – in so far as the rate of interest rises, the stimulating effect is to that extent offset), but to its raising the marginal efficiency of a given stock of capital. (pp. 142-43)

As if the problem of understanding that criticism were not enough, the problem is further compounded by the fact that one of Keynes’s most important pre-*General Theory* contributions, his theorem about covered interest parity in his *Tract on Monetary Reform* seems like a straightforward application of the Fisher equation. According to his covered-interest-parity theorem, in equilibrium, the difference between interest rates quoted in terms of two different currencies will be just enough to equalize borrowing costs in either currency given the anticipated change in the exchange rate between the two currencies over reflected in the market for forward exchange as far into the future as the duration of the loan.

The most fundamental cause is to be found in the interest rates obtainable on “short” money – that is to say, on money lent or deposited for short periods of time in the money markets of the two centres under comparison. If by lending dollars in New York for one month the lender could earn interest at the rate of 5-1/2 per cent per annum, whereas by lending sterling in London for one month he could only earn interest at the rate of 4 per cent, then the preference observed above for holding funds in New York rather than in London is wholly explained. That is to say, forward quotations for the purchase of the currency of the dearer money market tend to be cheaper than the spot quotations by a percentage per month equal to the excess of the interest which can be earned in a month in the dearer market over what can be earned in the cheaper. (p. 125)

And as if that self-contradiction not enough, Keynes’s own exposition of the idea of liquidity preference in chapter 17 of the *General Theory* extends the basic idea of the Fisher equation that expected rates of return from holding different assets must be accounted for in a way that equalizes the expected return from holding any asset. At least formally, it can be shown that the own-interest-rate analysis in chapter 17 of the *General Theory *explaining how the liquidity premium affects the relative yields of money and alternative assets can be translated into a form that is equivalent to the Fisher equation.

In explaining the factors affecting the expected yields from alternative assets now being held into the future, Keynes lists three classes of return from holding assets: (1) the expected physical real yield (*q*) (i.e., the ex ante real rate of interest or Fisher’s real rate) from holding an asset, including either or both a flow of physical services or real output or real appreciation; (2) the expected service flow from holding an easily marketable assets generates liquidity services or a liquidity premium (*l*); and (3) wastage in the asset or a carrying cost (*c*). Keynes specifies the following equilibrium condition for asset holding: if assets are held into the future, the expected overall return from holding every asset including all service flows, carrying costs, and expected appreciation or depreciation, must be equalized.

[T]he total return expected from the ownership of an asset over a period is equal to its yield

minusits carrying costplusits liquidity premium, i.e., toq–c+l. That is to say,q–c+lis the own rate of interest of any commodity, whereq,c, andlare measured in terms of itself as the standard. (Keynes 1936, p. 226)

Thus, every asset that is held, including money, must generate a return including the liquidity premium *l*, after subtracting of the carrying cost *c*. Thus, a standard real asset with zero carrying cost will be expected to generate a return equal to *q* (= *r*). For money to be held, at the margin, it must also generate a return equal to *q* net of its carrying cost, *c*. In other words, *q* = *l* – *c*.

But in equilibrium, the nominal rate of interest must equal the liquidity premium, because if the liquidity premium (at the margin) generated by money exceeds the nominal interest rate, holders of debt instruments returning the nominal rate will convert those instruments into cash, thereby deriving liquidity services in excess of the foregone interest from the debt instruments. Similarly, the carrying cost of holding money is the expected depreciation in the value of money incurred by holding money, which corresponds to expected inflation. Thus, substituting the nominal interest rate for the liquidity premium, and expected inflation for the carrying cost of money, we can rewrite the Keynes equilibrium condition for money to be held in equilibrium as *q* = *r* = *i* – *p ^{e}*. But this equation is identical to the Fisher equation:

*i*=

*r*+

*p*.

^{e}Keynes’s version of the Fisher equation makes it obvious that the disequilibrium dynamics that are associated with changes in expected inflation can be triggered not only by decreased inflation expectations but by an increase in the liquidity premium generated by money, and especially if expected inflation falls and the liquidity premium rises simultaneously, as was likely the case during the 2008 financial crisis.

I will not offer a detailed explanation here of the basis on which Keynes criticized the Fisher equation in the *General Theory* despite having applied the same idea in the *Tract on Monetary Reform* and restating the same underlying idea some 80 pages later in the *General Theory* itself. But the basic point is simply this: the seeming contradiction can be rationalized by distinguishing between the Fisher equation as a proposition about a static equilibrium relationship and the Fisher equation as a proposition about the actual adjustment process occasioned by a parametric expectational change. While Keynes clearly did accept the Fisher equation in an equilibrium setting, he did not believe the real interest rate to be uniquely determined by real forces and so he didn’t accept its the invariance of the real interest rate with respect to changes in expected inflation in the Fisher equation. Nevertheless it is stunning that Keynes could have committed such a blatant, if only superficial, self-contradiction without remarking upon it.