## Old School Econ Versus Modern Financial Economics

OK kids, I am working on something that touches on the Sraffa-Hayek debate. (You know, the one you spent 3 classes on in grade school, right after the Boston Massacre.) Sraffa said matter-of-factly that in (long-run, steady-state) equilibrium, the spot and forward price is the same for all commodities, and that the “natural” or own-rate of interest on each commodity equals the nominal interest rate.

At first this struck me as wrong, because I vaguely remembered the arbitrage-free relation between spot and forward prices, and they sure as heck weren’t equal. So I went to my trusty research assistant to find:

For an asset that providesno income, the relationship between the current forward (F_{0}) and spot (S_{0}) prices is

F_{0}=S_{0}e^{rT}

wherer is the continuously compounded risk free rate of return, andT is the time to maturity. The intuition behind this result is that given you want to own the asset at time T, there should be no difference in a perfect capital market between buying the asset today and holding it and buying the forward contract and taking delivery. Thus, both approaches must cost the same in present value terms.

I’m pretty sure Sraffa is right, and that therefore the above block quotation is a little off. For one thing, how in the world is “buying the asset today and holding it” equivalent to “buying the forward contract and taking delivery”??

Instead, I think the argument should be that the above condition is an *upper bound* on the forward price, because if it were higher than it would be cheaper for the investor to simply pay the money upfront (i.e. with present dollars) and hold the asset. But in that case, the person gets strictly more out of it; he has the asset at least as long as in the original scenario (i.e. starting at time T), and also in the interim from now until then. So he can sell the asset if he changes his plans before then; he has strictly more options at his disposal, and for a lower price (in present value terms).

Now wait a minute, let’s give the Wikipedia article a chance. After all, Sraffa was a neo-Ricardian; maybe he screwed up here too. After giving the above “intuition” (which made no sense to me) behind the result, the article then tries to prove it like this:

IfS_{t}is the spot price of an asset at timet, andr is the continuously compounded rate, then the forward price at a future timeT must satisfyF_{t,T}=S_{t}e^{r(T − t)}.

To prove this, suppose not. Then we have two possible cases.

Case 1:Suppose thatF_{t,T}>S_{t}e^{r(T − t)}. Then an investor can execute the following trades at timet:

go to the bank and get a loan with amountS_{t}at the continuously compounded rate r…

It’s not important for us to worry about this case. We want to see what Sraffa did wrong. Remember, Sraffa is claiming that in a steady-state equilibrium, the spot and forward prices on commodities are the same, even though the nominal interest rate is positive. So that means we need to see why Case 2 leads to a contradiction, so let’s jump ahead in the Wikipedia article:

Case 2:Suppose thatF_{t,T}<S_{t}e^{r(T − t)}. Then an investor can do the reverse of what he has done above in case 1. But if you look at the convenience yield page, you will see that if there are finite stocks/inventory, the reverse cash and carry arbitrage is not always possible. It would depend on the elasticity of demand for forward contracts and such like.

Ohhhh, so you know that “rule” that we were told about at the beginning of the article? It turns out it only works if there are infinite stocks/inventory.

I just love this kind of thing. I bet you if a modern student of financial economics were reading the Sraffa-Hayek debate for some reason–maybe the poor guy lost a bet or was being hazed during frat initiation or something–he would read Sraffa’s claim that spot and forward prices were equal in equilibrium, and think, “What an idiot. We learned in class that the forward price has to rise with the risk-free interest rate. I’m so glad we are rigorous nowadays.” The guy probably wouldn’t even know that the proof relied on an ability to short an indefinite amount of the commodity.

**Disclaimer:** I am not betting my life that Sraffa is correct, I’m just saying I starting thinking it through and couldn’t spot any error in his claim. And then when I refreshed my memory by going to Wikipedia, I saw that the proof for the different relation relies on something that obviously isn’t true when talking about physical commodities. So my hunch is that Sraffa is right, especially since he wrote a book on this stuff.

Beyond me. I should probably go back and re-watch the zombie interview, or take some statistics courses.

Of course, technically, it’s possible to reconcile the two.

Did Sraffa anywhere suggest that the long-run equilibrium rate of interest is zero? I know that there are some economists who have suggested such things, and wouldn’t be shocked if Sraffa were one of them, though I don’t actually know. If we assume that they’re right, it’s pretty obvious that the standard modern equation gives Sraffa’s result.

Anyway, I’m not sure about Sraffa’s argument. I’m not entirely convinced that forward contracts would exist in a long-run, steady state equilibrium, for reasons similar to Mises’s argument for no money in the ERE. The point of a forward contract is to decrease risk and uncertainty, if there is no risk or uncertainty to start with, then forward contracts seem to serve little purpose (unless I’m missing something).

Setting that aside, intuitively, I’m inclined to favor the modern view. Think of it this way: suppose I want to buy a rock to be used in 3 months (with no uncertainty – so I have no potential use for the rock until that point), and the world follows Sraffa’s pricing rule. In that case, I have the option of buying the rock for $10 now, or entering a forward contract to purchase it for $10 in the future. Assuming a positive rate of interest, it’s obvious that the best option is to enter the forward contract. Then, I can put my $10 out at interest for 3 months, and, 3 months from now, end up with both a rock and 3 months of interest. Everyone else is in the same boat, so there’s pressure that would drive the forward price price up – up toward the point where the forward price is equal to the spot price plus the interest that would be earned. I can’t see the argument for Sraffa’s F = S. Now, I agree that the modern equality is asking too much when we have a finite inventory, but Sraffa’s argument doesn’t seem to fly for me, as anyone considering both options wouldn’t buy spot under this condition – unless r = 0.

Thank you Lucas for making this my favorite blog on the Internet. Where else could one find such discussion? I rest my case.

But I think you are being too quick to reject Sraffa. At first I thought what you did, but then I hesitated.

All you have demonstrated is that people have no incentive to buy a rock now and carry it forward. But that’s consistent with the assumption that we are in a steady state; nobody is carrying goods forward, because then next period would be different from this period.

You’re right, if the nominal interest rate is zero, everything (trivially) checks out. But I think Sraffa might even be entertaining a case where the production of every commodity grows by x percent per year, and the spot prices remain the same but the nominal interest rate is x percent too.

So for example suppose the two goods in the economy are apples and oranges, and the physical output of both grows at a rate of ten percent per year. The spot price is $1 per pound for both fruits, and the nominal interest rate is 10 percent. The forward price for both is also $1 per pound.

If I sell a pound of apples today, I can raise $1. Then I lend it out at interest, and at the same time go long on 1.1 forward contracts for apples. Next period, I use my $1.10 to buy 1.1 pounds of apples (exercising my forward contract). But of course, the use of the forward contract was superfluous; I could have just used the spot market.

On the other hand, I could have taken my apple in period 1 and planted it, so that it would yield 1.1 apples in period 2.

So what is wrong with this vision? I don’t see any problems.

Ah!

I think I was ignoring the fact that the asset is “income producing.” That is, if I buy an apple now, I end up with 1.1 apples next period. So, if I buy the rock now, I end up with rocks equivalent to that rock + 3 months interest.

In that case, I think Sraffa’s right – but that you’re looking at the wrong modern model. You want the equation for forward pricing when there’s a known income – which is:

F0 = S0e^[(r − q)T]

r is the interest rate, q is the rate of return on the asset. Since the two are equal in long-run equilibrium, F0 = S0.

So, there’s really no contradiction between Sraffa’s claim and the modern view – as long as we look at the right modern view.

Very interesting, I was picturing it as the apple gaining in price, but you’re right, it would be equivalent to a stock paying dividends and having a constant price.

Verrrry interesting.

Lucas let me make sure I’m being clear: In long-run static equilibrium, spot prices are the same period after period. So in that setting, why in the world would anybody buy a forward contract that had a higher forward price than spot? It would be smarter to just buy in the spot market in the future. Thus this would put downward pressure on the forward price until it equaled the spot price, right?

I’m only vaguely familiar with the ERE and steady-state EQ, but does opportunity cost exist in those instances? Then it would make sense to pay a premium and not have to worry about storing things for extended periods of time.

You might be right in general, but to reiterate, nobody is storing anything in Sraffa’s scenario. Otherwise you would start next period in a different position from how you started this period.