# OnInterest

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There are various theories of interest. Need to look at them properly. But to start with some crude discussion of my intuitions. (Well, I say intuitions, but more likely residual memories of the neoclassical economics I was taught.)

I'm pretty certain it must have something to do with time, and something to do with production.

Robinson Crusoe's savings

There is the one agent - Robinson Crusoe. There is one type of good on the island - coconuts, say. There is a compressed type of time with two periods - today and tomorrow.

Robinson Crusoe begins with an initial stock of ten coconuts. His choice is how many to consume today, and how many to consume tomorrow. He also has a production technology - a coconut machine. He feeds into it coconuts today, and out come coconuts tomorrow.

There are thus two major factors in his decision - his relative preference for coconuts today vs coconuts tomorrow; and how the machine works.

I phoned up an economist friend of mine and asked him 'where does interest come from?' He said - 'time preference.' On Robinson Crusoe's island time preference can be a factor in determining a primitive kind of interest rate, but it's not everything.

Assume to start with that RC is indifferent between coconuts today and coconuts tomorrow. This given, the determining factor can be the production technology.

The island is very hot. Without a machine, coconuts go rotten by day two. Production technology 1 - the machine is broken. RC will eat all his coconuts on day one.

Technology 2 - the machine simply preserves coconuts, one in one out. RC eats 5 on day one and 5 on day two.

Technology 3 - the machine increases the number of coconuts. Eg. 1.5 coconuts out for every one in. If RC is indifferent between consumption on diferent days, he puts them all in the machine and eats 15 on day two.

Perhaps more likely, his marginal utility of consumption on any given day is diminishing. (Note - still no time preference.) Depending on how strongly this is so he may, for example, eat six today, put four in the machine and eat six tomorrow.

Is there anything like an interest rate here? Call the coconuts he eats on day one his consumption in period one (C1). The coconuts he doesn't eat on day one are his savings (S1) = the capital he invests in the production process. The rate of return (r) on his savings = (coconuts on day two C2 - savings S1) / savings S1, is something like an interest rate.

Here the 'interest rate' r = the Rate of Transformation of the technology (need to check I've remembered the terminology right): i.e., the rate at which the machine transforms inputs into outputs.

Time preference can't change the machine's production function. It can effect the actual level of saving, and so of production output, but not the rate of return to saving. However if the machine has varying returns to scale, then the scale at which it is operated will effect the Marginal Rate of Transformation (MRT) - the rate at which the last coconut in is transformed into coconuts out, and indeed the average rate r. Thus in the (commonly assumed) case where there are diminishing returns to scale, time preference can be a determining factor for r.

Eg suppose Robinson Crusoe decides to stick four coconuts into the machine. With diminishing returns, the first coconut he puts in produces say two more coconuts and the last coconut only one more. Altogether he produces six for an input of four. Here MRT = 1, but r = 1.5.

(Now with added marginalism: generally, let f(s1) be the production function for the machine, and assume diminishing returns (f'>0, f"<0). Robinson Crusoe's preferences - for consumption within any one day, as well as between days - can be encoded in a utility function u(c1,c2). With the usual revealed preference theory, the utility function shows diminishing marginal returns to consumption on either day. We have that s1 = (y1 - c1), so can write the production function f(y1-c1). Robinson Crusoe's decision problem is then:

maximise U(c1,c2) subject to the constraint c2 = f (y1 - c1).

The first order conditions are u1 = l*f'(c1) and u2 = l where ui is partial derivative of U with respect to good i; l is the Lagrangean multiplier.

Then the (standard micro theory) result is: u1/u2 = f'(s1)

That is: the marginal rate of substitution between consumption on day one and on day two (MRS) = the marginal rate of transformation of the production function (MRT).

Would be easier to elucidate this with a picture. The important thing is: with diminishing returns, the slope of the production function f'(s1) changes whatever level of s1 (and so c1) is decided. The slope of the indifference curves that show Robinson Crusoe's attitudes between consumption today and tomorrow are also changing depending on what level of c1 we are at. The actual chosen level c1 depends on where these two coincide. Thus - both time preference and technology are determining factors in the level of savings chosen.

Note: again r, the average rate of transformation, is determined by but not the same as the marginal rate of transformation f'(s1).

Two person economy

In the Robinson Crusoe example you can define a thing called 'rate of return on savings/capital/investment.' But it's not really an interest rate - if an interest rate is meant to be a price in a market. To get a price, we need to have someone else for Robinson Crusoe to trade with.

Now RC meets Friday. Friday also has 10 coconuts on day one. Friday doesn't have a machine. Maybe the climate has cooled a bit - coconuts do not go off. Before he met Robinson, Friday's plan would have been: eat 5 coconuts today, 5 tomorrow. Before RC met Friday, his plan would have been: consume 6 today, put 4 in the machine and get 6 tomorrow.

Robinson Crusoe's machine has an increasing production function (coconuts in greater than coconuts out), although with decreasing returns to scale (later coconuts in produce less coconuts out). 4 coconuts in then 6 out. 8 coconuts in then 11 out. After 8 coconuts in it stops working.

It wouldn't cost RC to let Friday use his machine, after he has already done so. Friday could then put 4 coconuts in and get 5 out tomorrow. RC would be doing just as well as before they met, and Friday better than before - he can have have 6 today and 5 tomorrow, or 5 today and 6 tomorrow if he keeps one aside, or 5.5 each day if you can split coconuts in half. They would get returns on their savings rc = 50% for Crusoe and rf = 25% for Friday.

But Crusoe could do better still - he could charge Friday to use the machine. For example, he could consume half of Friday's extra coconut. Suppose coconuts are only divisible in half. For the right shape of utility functions this in fact looks like the equilibrium bargain, the free market solution: Friday's choice is to invest 4 coconuts with Crusoe at 12.5% return or stay as before and eat 5 coconuts a day. Crusoe's choice is to take Friday's investment and make a profit of half a coconut or to stay as before and eat six coconuts a day. The bargain is the best choice for both of them.

In this mini-economy can we say: Robinson Crusoe's machine is the (only available) investment opportunity, and it offers a market interest rate of 12.5%?

I would still be inclined to say that Crusoe gets a 50% return on his savings. The extra half coconut he has bargained out of Friday does not represent a return on his investment - it is a pure profit from trade.

Lots of machines.

If the idea of Crusoe's profit obscures the point about interest, consider this scenario.

Now there are lots of agents and lots of machines. A perfectly competitive market for investments, leaving no room for profits. If someone tries to charge me half a coconut to use their machine, I just move on to the next machine owner. Now coconuts are infinitely divisible - machine owners undercut each other to a whisker over nothing. The market interest rate is the actual rate of return on an investment into the machine.

Alternatively, you could go the other way. Could think of technological set-ups where machine-owners need investors even more than investors need machines - for example if there are increasing then decreasing returns to scale, which models something like start-up costs to get the machine running efficiently. Then investors may be in a position to haggle the interest rate above the machine's rate of return.

In general, could construct simple models like this where bargaining power works to either party's advantage. In such cases there are three determinants of interest rate: time preference; technology; market power.

The perfect competition ideal abstracts away all considerations of market power - both investors and machine-owners undercut to cost. Then there are two determinants of the interest rate: time preference; technology.

Okay, now I'll go and read what real economists say about this, and see whether my 'intuitions' stand up.

My web starting point will be: http://cepa.newschool.edu/het/essays/capital/capital.htm

Interest and money.

Note in all the above haven't mentioned anywhere - labour; money.

In terms of money, from what I remember the standard line goes something like this:

'The' interest rate is the price - the opportunity cost - of holding money. This is because if you choose to hold money you are choosing not to put your savings into investments (eg. bonds). You are paying for the liquidity that money gives you by foregoing interest. So it is the interest rate that clears the money market, achieving equilibrium between money demand and money supply. But the interest rate is not itself determined in the money market. (Or something like that - perhaps more accurately, demand in the money market and in the bond market are two sides of the same investment decision, and if the price is set in one it is set in the other ...)

Anyway, what I'm trying to say is - if my intuitions as above are on the right lines, it could make perfect sense (in theory) to talk about interest even if there were no money.

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