Wednesday, 3 September 2014

Own Rates of Interest and Real Rates of Interest

David Glasner had a couple of posts recently (here and here) considering real rates of interest in a barter economy.  I'm not sure I quite agree with him even in his last post on the topic, so I thought it was interesting to look at further.  I'm going to use the numbers from his example.

The issue he is looking at is the pricing of loans in a barter economy.  In this economy, loans have to be constructed with commodities.  So I might lend you 100 onions at 5% interest, which would mean that at the end of the loan you would pay me back 105 onions.  We are assuming this is a proper loan, like a loan of money, so onions are simply the denomination and the settlement medium.  In other words, you do not have to return to me the same onions I loaned you, plus five more.  You simply deliver to me 105 onions of the required type.

We can say that 5% is then the "own rate" for loans of onions.  We then consider what own rates might apply to loans of other commodities, like tomatoes or cucumbers.  For this, we assume that normally onions, tomatoes and cucumbers all trade at par with one another, but currently there is lower demand for tomatoes and higher demand for cucumbers.  So, at the current prices, 100 tomatoes exchange for 90 onions and 100 cucumbers exchange for 110 onions.

Simple arbitrage then dictates what the own rates of interest must be for tomatoes and cucumbers.  For example, someone might borrow 900 onions and exchange them for 1,000 tomatoes.  To repay their onion loan they need 945 onions (principal plus 5% interest), which they can get at the end of the loan by exchanging 945 tomatoes at the then par rate.  So a loan of 1,000 tomatoes will require repayment of 945 tomatoes, an interest rate of -5.5%.  Anything else will allow endless profits from borrowing in one commodity and lending in the other.  A similar argument shows that the own rate on loans in cucumbers must be 10.5%.

We need to ask how there can ever be a negative own rate of interest, such as we have here on tomatoes.  Does this not mean that someone can borrow 1,000 tomatoes, repay 945 and walk away with 55 tomatoes in profit?  The answer relates to the time aspect of the loan and the implications for storage.  If tomatoes are perishable, it may not be possible to store them from one period to the next at all, so this arbitrage may not be available.  Even if we take a non-perishable commodity, such as a precious metal, there may be storage costs such as security and these may eliminate the potential profits.

It is worth considering what would happen if there were no storage costs.   People would then want to borrow more tomatoes in order to simply hold them and take the profit.  This would tend to bid up the own rate on loans of tomatoes, which would prompt other people to exchange cucumbers and onions for tomatoes to be able to make more tomato loans.  This in turn would increase the current price of tomatoes in terms of these other commodities.  So both the own rate on tomato loans and the current price of tomatoes increases.  This continues until the arbitrage is eliminated.

So we have a different own rate for each commodity, but the relationship between all the rates is tied to the price structure.  With prices given, if one rate changes, they must all change.

If we introduce money to this economy, it must fit the same structure.  There will be a money price for each commodity.  We can work out the appropriate rate for money loans by reference to the current and future price of onions and the own rate on onions in order to meet the no arbitrage condition.  If we do the same exercise using tomatoes or cucumbers, we will get the same rate.

The question now is what the real rate of interest is here.  We have a different own rate of interest on each commodity.  Which if any is the actual real rate of interest?  In fact, we cannot say what the real rate of interest is without specifying the commodity in which we are expressing it.  There is no absolute real rate of interest that can be expressed purely in terms of time value.

That is not to say that time value does not matter.  If the time preference for consumption of vegetables changes, all of the own rates will change.  It is simply that we cannot point to a single own rate and say that this is the one that reflects pure time value, abstracted away from relative price movements.

In practice, we determine real rates of interest by reference to baskets of commodities.  We often calculate a real rate of interest using a consumer price index.  If we were to use a different index we would get a different result.

We can also use baskets of commodities in our imaginary economy.  For example, we could calculate the own rate on a basket of 10,000 onions, 20,000 tomatoes and 30,000 cucumbers.  In this case, the own rate works out at 6.75% (i.e. you get back 10,675 onions, 21,350 tomatoes and 32,025 cucumbers).  This means that we would be indifferent between lending the basket at 6.75%, or lending in any of the individual commodities alone at their respective own rates.

As with the individual commodities, the appropriate rate depends on the basket we choose - there is no true rate that is independent of that choice.  However, whatever basket we choose to reference, the arbitrage free rate required on money loans will always be the same.

1. It looks like overcomplicating the issue by conflating exchange rates with a seperate topic, loan interest.

The first thing to do to form a general description of the barter economy is to come up with an hypothetical unit of exchange based on the basket of commodities.

Once that is done the the loan interest rate follows.

1. I don't deny that this is somewhat complicated (which I confess is partly why it interested me). But I think that's a bit unavoidable as I'm not simply trying to come up with an interest rate, but rather to show that the rate you come up with depends on the arbitrary choice of reference basket (or hypothetical unit of exchange, if you prefer).

2. You can arrive at an average level of interest by three steps.

1. work out an hypothetical unit of exchange from the basket of currencies and price the commodities using it.
2. note the interest rate each commodity lender is charging priced in the hypothetical unit of exchange.
3. calculate the average rate of all the the individual rates. Maybe weighted by the amount of loans issued at each rate.

3. "basket of currencies" - do you mean basket of commodities? I wasn't looking at different currencies here.

It's fairly easy to calculate the appropriate rate for any given basket - the problem is deciding what basket to use. You could choose the basket that reflects actual sales in any one period, but this will not be the same period to period and interest rates necessarily straddle two periods.