Phil Pilkington has responded to my post about the latest version of my paper (co-authored by Paul Zimmerman) on the Sraffa-Hayek debate about the natural rate of interest. For those of you who haven’t been following my posts on the subject, here’s a quick review. Almost three years ago I wrote a post refuting Sraffa’s argument that Hayek’s concept of the natural rate of interest is incoherent, there being a multiplicity of own rates of interest in a barter economy (Hayek’s benchmark for the rate of interest undisturbed by monetary influences), which makes it impossible to identify any particular own rate as *the* natural rate of interest.

Sraffa maintained that if there are many own rates of interest in a barter economy, none of them having a claim to priority over the others, then Hayek had no basis for singling out any particular one of them as the natural rate and holding it up as the benchmark rate to guide monetary policy. I pointed out that Ludwig Lachmann had answered Sraffa’s attack (about 20 years too late) by explaining that even though there could be many own rates for individual commodities, all own rates are related by the condition that the cost of borrowing in terms of all commodities would be equalized, differences in own rates reflecting merely differences in expected appreciation or depreciation of the different commodities. Different own rates are simply different nominal rates; there is a unique real own rate, a point demonstrated by Irving Fisher in 1896 in *Appreciation and Interest*.

Let me pause here for a moment to explain what is meant by an own rate of interest. It is simply the name for the rate of interest corresponding to a loan contracted in terms of a particular commodity, the borrower receiving the commodity now and repaying the lender with the same commodity when the term of the loan expires. Sraffa correctly noted that in equilibrium arbitrage would force the terms of such a loan (i.e., the own rate of interest) to equal the ratio of the current forward price of the commodity to its current spot price, buying spot and selling forward being essentially equivalent to borrowing and repaying.

Now what is tricky about Sraffa’s argument against Hayek is that he actually acknowledges at the beginning of his argument that in a stationary equilibrium, presumably meaning that prices remain at their current equilibrium levels over time, all own rates would be equal. In fact if prices remain (and are expected to remain) constant period after period, the ratio of forward to spot prices would equal unity for all commodities implying that the natural rate of interest would be zero. Sraffa did not make that point explicitly, but it seems to be a necessary implication of his analysis. (This implication seems to bear on an old controversy in the theory of capital and interest, which is whether the rate of interest would be positive in a stationary equilibrium with constant real income). Schumpeter argued that the equilibrium rate of interest would be zero, and von Mises argued that it would be positive, because time preference implying that the rate of interest is necessarily always positive is a kind of a priori praxeological law of nature, the sort of apodictic gibberish to which von Mises was regrettably predisposed. The own-rate analysis supports Schumpeter against Mises.

So to make the case against Hayek, Sraffa had to posit a change, a shift in demand from one product to another, that disrupts the pre-existing equilibrium. Here is the key passage from Sraffa:

Suppose there is a change in the distribution of demand between various commodities; immediately some will rise in price, and others will fall; the market will expect that, after a certain time, the supply of the former will increase, and the supply of the latter fall, and accordingly the forward price, for the date on which equilibrium is expected to be restored, will be below the spot price in the case of the former and above it in the case of the latter; in other words, the rate of interest on the former will be higher than on the latter. (p. 50)

This is a difficult passage, and in previous posts, and in my paper with Zimmerman, I did not try to parse this passage. But I am going to parse it now. Assume that demand shifts from tomatoes to cucumbers. In the original equilibrium, let the prices of both be $1 a pound. With a zero own rate of interest in terms of both tomatoes and cucumbers, you could borrow a pound of tomatoes today and discharge your debt by repaying the lender a pound of tomatoes at the expiration of the loan. However, after the demand shift, the price of tomatoes falls to, say, $0.90 a pound, and the price of cucumbers rises to, say, $1.10 a pound. Sraffa posits that the price changes are temporary, not because the demand shift is temporary, but because the supply curves of tomatoes and cucumbers are perfectly elastic at $1 a pound. However, supply does not adjust immediately, so Sraffa believes that there can be a temporary deviation from the long-run equilibrium prices of tomatoes and cucumbers.

The ratio of the forward prices to the spot prices tells you what the own rates are for tomatoes and cucumbers. For tomatoes, the ratio is 1/.9, implying an own rate of 11.1%. For cucumbers the ratio is 1/1.1, implying an own rate of -9.1%. Other prices have not changed, so all other own rates remain at 0. Having shown that own rates can diverge, Sraffa thinks that he has proven Hayek’s concept of a natural rate of interest to be a nonsense notion. He was mistaken.

There are at least two mistakes. First, the negative own rate on cucumbers simply means that no one will lend in terms of cucumbers for negative interest when other commodities allow lending at zero interest. It also means that no one will hold cucumbers in this period to sell at a lower price in the next period than the cucumbers would fetch in the current period. Cucumbers are a bad investment, promising a negative return; any lending and investing will be conducted in terms of some other commodity. The negative own rate on cucumbers signifies a kind of corner solution, reflecting the impossibility of transporting next period’s cucumbers into the present. If that were possible cucumber prices would be equal in the present and the future, and the cucumber own rate would be equal to all other own rates at zero. But the point is that if any lending takes place, it will be at a zero own rate.

Second, the positive own rate on tomatoes means that there is an incentive to lend in terms of tomatoes rather than lend in terms of other commodities. But as long as it is possible to borrow in terms of other commodities at a zero own rate, no one borrows in terms of tomatoes. Thus, if anyone wanted to lend in terms of tomatoes, he would have to reduce the rate on tomatoes to make borrowers indifferent between borrowing in terms of tomatoes and borrowing in terms of some other commodity. However, if tomatoes today can be held at zero cost to be sold at the higher price prevailing next period, currently produced tomatoes would be sold in the next period rather than sold today. So if there were no costs of holding tomatoes until the next period, the price of tomatoes in the next period would be no higher than the price in the current period. In other words, the forward price of tomatoes cannot exceed the current spot price by more than the cost of holding tomatoes until the next period. If the difference between the spot and the forward price reflects no more than the cost of holding tomatoes till the next period, then, as Keynes showed in chapter 17 of the *General Theory*, the own rates are indeed effectively equalized after appropriate adjustment for storage costs and expected appreciation.

Thus, it was Keynes, who having selected Sraffa to review Hayek’s *Prices and Production* in the *Economic Journal*, of which Keynes was then the editor, adapted Sraffa’s own rate analysis in the *General Theory*, but did so in a fashion that, at least partially, rehabilitated the very natural-rate analysis that had been the object of Sraffa’s scorn in his review of *Prices and Production*. Keynes also rejected the natural-rate analysis, but he did so not because it is nonsensical, but because the natural rate is not independent of the level of employment. Keynes’s argument that the natural rate depends on the level of employment seems to me to be inconsistent with the idea that the IS curve is downward sloping. But I will have to think about that a bit and reread the relevant passage in the *General Theory* and perhaps revisit the point in a future post.

UPDATE (07/28/14 13:02 EDT): Thanks to my commenters for pointing out that my own thinking about the own rate of interest was not quite right. I should have defined the own rate in terms of a real numeraire instead of $, which was a bit of awkwardness that I should have fixed before posting. I will try to publish a corrected version of this post later today or tomorrow. Sorry for posting without sufficient review and revision.

UPDATE (08/04/14 11:38 EDT): I hope to post the long-delayed sequel to this post later today. A number of personal issues took precedence over posting, but I also found it difficult to get clear on several minor points, which I hope that I have now resolved adequately, for example I found that defining the own rate in terms of a real numeraire was not really the source of my problem with this post, though it was a useful exercise to work through. Anyway, stay tuned.

David,

I think there is still some confusion here. Sraffa was not trying to prove that the natural rate as such was nonsense. But merely Hayek’s exposition of it. Hayek was trying to do an analysis in a pure barter economy and did not want to introduce money. Sraffa argued that in order to do this you have to take price-levels — as Wicksell had done — and this led to the selection of an arbitrary numeraire which was effectively the same as introducing a money standard. Sraffa says this quite explicitly:

“This, however, though it meets, I think, Dr. Hayek’s criticism, is not in itself a criticism of Wicksell. For there is a ” natural ” rate of interest which, if adopted as bank-rate, will stabilise a price-level (i.e. the price of a composite commodity): it is an average of the “natural ” rates of the commodities entering into the price-level, weighted in the same way as they are in the price-level itself.” (p51)

Hayek was trying to avoid introducing a monetary standard in this debate because he didn’t like the implications of this. Specifically he knew that it would lead to Keynes’ analysis in the Treatise on Money which would later develop into the liquidity preference theory of the General Theory. Basically Hayek could see which way the wind was blowing and tried to resist it by banishing money. Sraffa called him on this and he couldn’t answer.

Here is a full overview:

http://fixingtheeconomists.wordpress.com/2014/07/22/can-lachmanns-arbitrage-save-the-austrian-theory-of-the-interest-rate/

Aren’t your own rates the other way round?

If the current price of cucumbers is $1.1 and its forward price is $1, I would say there is an equilibrium lending rate of 10% in cucumbers, i.e. repayment of 110 cucumbers for every 100 cucumbers loaned.

If I was the lender, I would pay $110 for 100 cucumbers which I would lend to you. At maturity, you would then deliver me 110 cucumbers (principal plus 10% interest), which I would sell for $110. Alternatively, I could buy 100 tomatoes for $90, lend them to you, get back 90 tomatoes (principal minus 10% interest) and sell them for $90. So I’m indifferent between lending cucumbers, tomatoes or dollars.

Maybe, I’ve misunderstood how you are defining the own rate of interest.

I reckon Nick is correct.

He also raises another point: even for the Wicksell natural rate to work we have to abstract from risk let alone uncertainty. I wrote a paper on this some time ago. The natural rate implicitly rests on the EMH.

Overview:

http://fixingtheeconomists.wordpress.com/2014/05/17/the-natural-rate-of-interest-does-not-exist/

I enjoyed your paper and agree with its main point, but I’d like to offer few comments on this post.

1) I also think that your definition of own interest rate is inverted. If tomatoes cost $0.9 today, and forward price is 1$ (i.e. I pay one dollar today for delivery of one tomato tomorrow – this is not what finance people usually mean by forward price, maybe that’s the source of confusion?), I can transform one current tomato into 0.9 future tomatos, so own rate of interest is -10%.

2) Even in a stationary equilibrium, interest rate doesn’t have to be zero. For example, in a simple Ramsey/Kass/Koopmans growth model with zero growth in steady state, quantities and prices (relative to the numeraire consumption good) are constant, but the interest rate is still positive, determined by rate of time preference. The “current forward price” would be equal to discounted future spot price, so it would be not the same as spot price itself.

3) But even if rate of time preference is zero, I’m not sure about the example with cucumbers and tomatoes. The fact that one has positive and the other negative own interest rate doesn’t necessarily mean there is a corner solution. The differences in own rates are caused precisely by expected appreciation/depreciation of spot prices, so a good with higher own rate will also experience decline in its “purchasing power”, and from an investor’s point of view, the two will cancel out.

David,

In the example that you gave, I think you are missing two items – time and quantity of goods.

You gave the example of cucumbers versus tomatoes. Suppose the price of cucumbers is $1.00 per pound and the price of tomatoes is $1.00 per pound. Also, suppose that it takes 2 months for a pound of cucumbers to mature and be sold but 4 months for a pound of tomatoes to mature and be sold. Meaning that in a four month time period, half as many pounds of tomatoes are grown and sold.

The own rate of interest is calculated as follows:

%INT = Own Rate of Interest

%RINT = Real Own Rate of Interest

Qt = Pounds of tomatoes that can be grown and sold in a given time period

Qc = Pounds of cucumbers that can be grown and sold in a given time period

Pt = Price per pound of tomatoes

Pc = Price per pound of cucumbers

Tt = Time period for tomato growth and sale

Tc = Time period for cucumber growth and sale

T = Time period for loan

%INT = [ Qt * Tt * Pt ] / [ Qc * Tc * Pc ]

If both the tomato and cucumber markets always clear then:

Qt * Tt * Pt = Qc * Tc * Pc

The relative price of a pound of tomatoes and a pound of cucumbers can change without affecting the own rate of interest between the two as long as either the production time or the quantity produced and sold per time period also changes.

The own rate of interest is a nominal variable. The real own rate would be:

%RINT = [ Qt * Tt * Pt ] / [ Qc * Tc * Pc ] – [ Pt / Pc ]

Here even if the relative prices are unchanged ( Pt / Pc = 1), a productivity improvement in the growth of tomatoes versus cucumbers (or vice versa) combined with a change in the quantity of pounds of tomatoes versus cucumbers will change the real own rate.

Fisher is wrong about a unique real own rate because he does not allow for improvements in productivity that are not homogenous nor does he allow for the outmoding of goods.

Phillip,

You say “Basically Hayek could see which way the wind was blowing and tried to resist it by banishing money”.

Perhaps I am taking this quote out of context but the whole point of Hayek’s theory was that in a monetary economy , if the money rate differed from the natural rates then distortions could take place.

Hayek didn’t like Wicksell’s “basket of goods” approach because he saw that there is no fixed set of goods that reflects total spending through time – but I’m not sure he ever came up with a good alternative explanation of how the money rate in a monetary economy could fully encapsulate all the various own-rates in a barter economy. And I agree that once you factor in the effect that risk and uncertainty have on interest rates then his theory starts to feel unworkable.

Let me try it like this Mr. Glasner: What’s the world’s real rate of interest right now? You’re saying there’s one objectively right answer to that question?