Sraffa and "Own-Rates"

I own the only salt mine in our area. Salt is a very widely used commodity, but it is not yet money, as it is not universally accepted in essentially all trades. (J.P. Koning might say it has a high degree of "moneyness.") As such, I have opened a side-business: I buy and lend-out all sorts of commodities by paying for them in salt, and when they are returned to me I trade them back in for salt.

It is February, and I am looking at two possible loans. One borrower wishes to borrow tomatoes for six months, and the other ice. When I consider the tomato deal, I realize that in August tomatoes are plentiful and will fetch much less salt. In fact, although a pound in February fetches two pounds of salt, I expect that in August a pound of tomatoes will trade for only a pound of salt. Meanwhile, ice is in the reverse circumstances: in the winter, I can get only a pound of salt for a pound of ice, but in August the same amount of ice will fetch two pounds of salt.

Since I do my accounting in terms of salt, I must account for at least the changing tomato price or I will make a serious loss on my loan. And the person borrowing the ice will make sure I account for the changes in the salt-price of ice or he will make a loss. So, simply to come out even, I will lend out a pound of tomatoes in February and demand two pounds back in August, while when I lend out two pounds of ice in the winter, I will only require a pound back in the summer.

But the above terms give me no reason to make the loan. I spend two pounds of salt on each loan in February, and wind up with two pounds in August. Why give up my ownership of the salt under these terms? In fact, I have a preference for present over future goods, and therefore I will want somewhat more salt back in August than I lent in February: let us say 5% more. So I demand 2.1 pounds of tomatoes back in August, but only 1.05 pounds of ice. After accounting for price changes, I am charging 10% interest per year on my loans.

Hopefully it should be clear that to generate the problem of multiple "own-rates" of interest on different commodities, Sraffa muddled two concepts that should be kept separate: on the one hand, the lender must account for changing prices. On the other hand, even after accounting for those, there is some rate of return he wants simply for giving up access to the goods lent for a period of time.

Conceptually, I contend the second item is interest proper (or "originary interest," as Mises would put it). And the choice of salt here doesn't matter: this "natural" rate is the same, whatever numéraire is chosen. It wouldn't matter if we took as our numéraire tomatoes or ice in the above example, once we abstracted away from the price changes and found the difference left between the amount lent and the amount repaid, we would see it is 5% in each case.

Of course, Lachmann noted all of the above a long time ago, but some bits of knowledge keep getting lost.


  1. Are there not more complexities?

    Your interest rates takes into account the future prices of ice and tomatoes but when you get back your 2.1 pounds of salt you may want to barter it for other goods. What happens if those goods have all doubled in price in salt terms in those six-months ? I guess you could then charge 105% interest to compensate for that. But what if some good had doubled and some only gone up by 50%? You could use some sort of indexing and still work out the rate you need to charge to maintain your 5% "natural rate".

    But if everyone did this everyone would use different indexing to match their specific spending patterns. So even if everyone had a "natural" rate of 5% in a world of dynamic price changes it is hard to see what this would mean in terms of the rate that would emerge on the loan market.

    So while I agree that some sort of time-preference driven natural rate underlies people's willingness to enter into loans I think there are considerable difficulties in seeing how this maps onto a "natural" rate that would emerge on the market.

    1. These complications are red herrings, rob. To whatever extent these natural rates diverge, they allow arbitrage oportunities: borrow in the market with the lower rate, lend in the market with the higher one, and make pure profit. People will keep doing that until they equalize the rates.

    2. I agree that arbitrage would lead to a single rate per commodity (if one ignores risk-premiums) emerging on the market.

      However this rate will have to factor in the complexities I mentioned so I do not see how that makes them "red-herrings".

      I am claiming that to say that interest rates consist of 2 parts (originary interest and expectations of change in relative prices) is incorrect. It is impossible to fully separate the one from the other. Interest rates will rather emerge from people's subjective valuation of the loans available to them and it will be impossible to say which part of these valuations are based on pure time preference and which on expectations of future relative price levels.

    3. "I agree that arbitrage would lead to a single rate per commodity..."

      No, across all commodities.

      "I am claiming that to say that interest rates consist of 2 parts (originary interest and expectations of change in relative prices) is incorrect."

      I know you are. You are incorrect.

      Of course we can't in practice perfectly differentiate these things. But in the case of a single lender, we can say, "Hey, Rob: how much of what you charged was due to how you expected relative prices to change?"

      In the case of the market as a whole, we just look at the spot and forward prices for each commodity. If the "own-rate" on salt is higher than the difference between the spot and forward prices plus originary interest, while that on tomatoes is lower than the difference between the spot and forward prices plus originary interest, we lend salt, buy it spot and sell it forward, while we borrow tomatoes, sell them spot and buy them forward, and earn pure profit. This will bring all the "own-rates" into line.

      Of course the economy is not always in equilibrium and the spot and forward prices might be wrong. That doesn't mean we can't analytically differentiate these items.

  2. I agree that in equilibrium the spot and forward prices and own-rates of interest would align just as you describe. I think this means that by comparing the difference in spot and future prices with the own-rate of interest one could find the natural rate of interest. So I guess I was wrong.

    Thanks for the clarification - I now see that your post is a very elegant restatement of Lachmann's refutation of Straffa.

    1. Rob, you need to be given an award and a large cash grant as "The only guy on the Internet who listens to others' arguments and therefore modifies his own view"!

    2. I will forgo my award for a post commenting on this paper...

    3. I will comment Rob: Bob makes the same mistake as Sraffa. He sets up a table (in response to Lachmann) and then calls account5ing for the difference between spot and future prices *interest*. Thus he can't resolve Sraffa's problem, and actually thinks Lachmann is a "regression." I am very surprised that Bob would think a brilliant economist like Lachmann would not grasp a simple table like the one he created. Lachmann clearly expects the reader to see that we have to *net out* the difference between spot and forward prices, and what is *left over* is the natural rate. In fact, in Bob's table, there is *no* originary interest: all he has are price changes, and the rate of originary interest is 0. (And the fact that Bob failed to include any payment for giving up control of the resource for a period of time shows that he failed to grasp what originary interest consists of.)

  3. Gene, this is a good explanation of what can be a tricky topic.

    1. This is the first time I have caught JP Koning in a demonstrable error.

    2. Bob, are you saying that the topic isn't tricky, or that you have a problem with Gene's post? I've just started reading up on this stuff now, so I'm still really green, but this seems like a good explanation to me.

    3. You need to read Bob's dissertation:

      Unanticipated Intertemporal Change
      in Theories of Interest [2003]

      My guess is once Bob gets the time he'll be cooking up a smackdown to post on his blog.

    4. JP, I was reading that 12 years ago as it was being written! And even back then, Kirzner and I were telling Bob, "You're just not grasping the concept of originary interest."

      The fact Bob has been silent on this for two days makes me think he is saying to himself, "Holy spit, how did I miss this 12 years ago!"

  4. I'm having trouble following how the rate stays the same with a different numeraire. Can you do a quick run through with tomatoes or ice as the numeraire?

    1. Sure, Jim.

      We use ice as the numeraire. I lend out two pounds of ice in Feb. and get back 1.05 in August.

      Hmm, looks like -45% interest. But we haven't abstracted away from the difference in spot and forward prices yet. When we look at that, we see that 1.05 lbs. of August ice is worth 2.1 lb.s of Feb. ice. So, in Feb ice, I get back my 2 lbs, plus 5%, or 10% / year interest. I.e., the same amount.

    2. Aaah, now I think see your error. I was having trouble pinning it down, but Bob is actually right on this one.

      You didn't really assume ice to be the numeraire.

      Assume ice is the numeraire:

      2 ice feb: 1.05 ice aug (-47.5% interest)

      All of the other exchange rates are consistent with this.

      Now, for salt:

      1 salt feb: 1.05 salt aug.

      But, per your reasoning, 1.05 salt in Aug. is worth .525 salt in Feb.!

      That's a -47.5% return, if we're ACTUALLY assuming ice as the numeraire. The same goes for tomatoes.

    3. Here's another demonstration of the arbitrariness of the 5%. I can literally take your comment and replace it with

      "Sure, Jim.

      We use salt as the numeraire. I lend out 1 pounds of salt in Feb. and get back 1.05 in August.

      Hmm, looks like 5% interest. But we haven't abstracted away from the difference in spot and forward prices yet. When we look at that, we see that 1.05 lbs. of August salt is worth .525 lbs of Feb. salt. So, in Feb. salt, I get -47.5%. I.e., the same amount as when ice was assumed to be the numeraire.

    4. No, Jim, no error here. I am merely differentiating the real interest rate from the nominal one, an absolutely standard move, which you and Bob are failing to do. Even with money, if we have 100% inflation and inteerst charges are nominally 110%, then the rate of originary interest is 10%, not 110%. One must account for inflation (or deflation) in the numeraire, or one will lose a heck of a lot of money!

    5. Wrong-o, Jim. I am adjusting for the difference between spot and forward prices for a commodity in its own terms. Feb salt spot versus August salt futures were assumed to be 1-1 here.

    6. And, Jim, let me note you have utterly failed to deal with my arbitrage example.

    7. Gene, I'm pulling my hair out right now!!

      Let me lay everything out in a different way. I'm honestly struggling to see it your way, so if you can straighten things out for me, I'll be happy to admit defeat.

      First, just to be clear; the real interest rate, r, is given by:

      r = (1+n) /(1+i) - 1

      where n = nominal rate and i=inflation (in terms of the numeraire).

      Now, first suppose that salt is the numeraire.

      For tomatoes we have: 1 Feb. tomato ==> 2.1 Aug. tomatoes

      n = 1.1

      In Feb. it costs 0.5 tomatoes to get a lb. of salt and in Aug. it costs 1 tomato to get a lb. of salt. Hence:

      i = 1

      r = 2.1/2 - 1 = .05

      as in your example.

      Now suppose ice is the numeraire.

      We still have:


      For inflation, in Feb. it costs 0.5 tomatoes to get a lb. of ice and in Aug. it costs 2 tomatoes to get a lb. of ice. Hence:

      i = 3 (or 300%)


      r = 2.1/4 - 1 = -47.5% !!

      This is the result using the standard method for calculating the real interest rate. Where have I gone wrong?

    8. Here is the correct way to take ice as the numeraire:

      In Feb, I use 2 lbs. of ice to buy 1 lb. of tomatoes and lend them out. In August I will demand back 2.1 lbs of tomatoes. In August that gets me back 1.05 pounds of ice. But the price of ice in its own market has doubled, so that is worth 2.1 pounds of Feb. ice, and so...

      Surprise! I actually got 5%, or 10% / year, interest! Cause you know, if I didn't, there would be costless arbitrage across markets. As in the example you keep failing to address.

    9. And notice how messed up your calculations are: Ice is much more valuable in August then in Feb. so if it is the numeraire, that is obviously DEFLATION. And yet you calculate an INFLATION rate of 300%!

    10. Thanks, Gene. I'm on board with the calculation. But I must say, this is not the standard way to calculate interest. You're calculating interest by comparing the real rate to the "break-even" rate. You're not *just* netting out inflation.

      My calculation of -47.5% (in terms of ice) was correct. But the break even return is -50% (in terms of ice). So relative to the break-even return, you net 5%, as you've been harping.

      This has really brought my attention to the difference between your example and Bob's. You said above that Bob's table doesn't contain originary interest. That's actually not true. It's just that Bob never defines or discusses the exchange rate at which a person "breaks-even". His table is simply a system of exchange ratios. To figure out originary interest you have to make the additional assumption of a "break-even" rate for a given good.

      For example, Bob could have said that he doesn't want to "just" break even in terms of steel (as you said with salt). As a result, the table would then imply that he earns originary interest of 300%. And as you've demonstrated, that would hold true for all other goods.

      And that's now where my problem lies. In order to say that originary interest is invariant to the numeraire, you have to be able to say that the break-even rates are not independent of one another.

      Put it this way. Suppose instead you now own the only tomato farm in your area. *All* reasoning is in terms of tomatoes (not salt) - that's all you care about. What would it mean to come out even in terms of tomatoes? The only way your system of prices would still imply an originary interest of 5% is if "coming out even" in terms of tomatoes means an exchange ratio of 1 Feb. tomato for 2 Aug. tomato. But that ratio is simply an artifact of your assumption that 1:1 in salt is "coming out even".

      I could assume, for example, that I "come out even" if I get 2.05 Aug. tomatoes for 1 Feb. tomato. This would imply originary interest of (2.1-2.05)/2.05 = 2.44%. And that would hold true for all other goods.

      Apologies for the long post.

    11. "But that ratio is simply an artifact of your assumption that 1:1 in salt is "coming out even"."

      No it's not: a Feb. spot tomato trades for two Aug. future tomatoes.

      Look at the spot and future markets.

    12. And look at my arbitrage example.

  5. I filed that very helpful post under my Sraffa - Hayek dispute folder. Thanks!

    Gene, the avatar picture you use is huge when I copy the comments into a word file! ;)

    1. Nothing like seeing me up close, is there?

    2. It's ok. You look like a scholar who might teach philosophy and maybe geography.


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