**I think this is right** (but I can't do the math to work out an example to be sure, though any competent grad student could).

**The idea is that you can get negative interest rates, despite productive investment and impatient consumers, because the prices of the goods you**It might be a theoretical curiosum, but it might also be empirically relevant, with falling prices of manufactured goods and falling share of expenditure on manufactured goods, relative to services. And all that.

*can*invest to produce more of will be falling over time relative to goods you*can't*invest to produce more of.[And of course, I figure out this model of negative real rates just as it looks like the Bank of Canada will be raising rates soon. Oh well. They are still not very high.]

Here's a toy model, just to illustrate the basic idea:

There are two goods.

**The first good is apples.** You don't need labour to produce apples. Apples just grow; they reproduce themselves. Apples are like Frank Knight's Crusonia plant, which has a stock K(t), and grows at a constant rate g, so produces an output gK(t) which can be consumed A(t) or re-invested I(t). A(t) + I(t) = gK(t) where I(t)=dK(t)/dt. **The Marginal Product of Capital, measured in units of apples, equals a constant g>0.**

**The second good is haircuts,** which have no investment technology, and cannot be stored. Haircuts are produced with labour only, and the technology never changes. The supply of labour is perfectly inelastic, so employment and **output and consumption of haircuts H is constant over time.**

There is an infinitely-lived representative agent with preferences U=U(A,H) and intertemporal discount factor 0<(1/(1+p))<1, where p is the pure rate of time-preference.

The economy is in competitive equilibrium.

**If we use apples as numeraire, the rate of interest must equal g.** If you borrow 1 apple this year, you promise to pay back 1+g apples next year. We know there will be positive investment in apples, and consumption of apples will be growing over time, if we assume g>p. We also know that the Marginal Utility of an extra apple dU/dA must be declining over time at a rate g-p as the consumption of apples rises.

**If instead we use haircuts as numeraire, the rate of interest will equal g-a, where a is the deflation rate of the price of apples relative to haircuts.** The price of apples will be falling over time, relative to haircuts, because consumption of apples is rising over time relative to haircuts (unless apples and haircuts are perfect substitutes so always have the same relative price). Arbitrage ensures that the haircut interest rate equals the apple interest rate minus the deflation rate of the price of apples in terms of haircuts.

**How quickly will apples fall in price, relative to haircuts?** The Marginal Utility of an apple, relative to the Marginal Utility of a haircut, must equal the price of an apple relative to haircuts. MU(A)/MU(H)=Pa/Ph. So to ask the same question a different way, how quickly will the Marginal Utility of apples fall relative to the Marginal Utility of haircuts?

**That depends on preferences U(A,H).** For example, if we assume separable log preferences U=log(A)+log(H), then MU(A) will be falling at rate g-p (and A will be growing at rate g-p), and MU(H) will be constant. So a=g-p. So the haircut rate of interest will be equal to p>0. But if we assume complementarity between apples and haircuts, so the Marginal Utility of haircuts rises as consumption of apples rises, the relative price of apples will fall faster than this. With the right preferences, we can make the relative price of apples fall as fast as you want. **With the right preferences, we can make the haircut interest rate negative (I think).**

**What is the interest rate if we use the CPI basket of apples and haircuts as numeraire?** (Which is how we normally talk about real interest rates, as nominal rate minus CPI inflation.) It's going to be a weighted average of the apple interest rate and the haircut interest rate, with the weights equal to the expenditure shares. **We have a positive apple interest rate, and we can have a negative haircut interest rate, so a weighted average on those two interest rates can also be negative, if the expenditure share of apples is small enough.**

Can anyone do the math to come up with a utility function to check I've got it right, so you can get a negative real rate?

Dunno if this is original. Probably someone said something similar somewhere sometime.

I hope someone verifies the math. The economic world needs more examples of negative rates being natural and normal.

Posted by: Benoit Essiambre | January 06, 2018 at 06:48 PM

Benoit: so do I. We know that in the separable log utility case we can get the haircut interest rate =0 if pure time preference p=0. So I think if we drop separability we *should* be able to get a case where it's negative. But I don't 100% trust my intuition.

Posted by: Nick Rowe | January 06, 2018 at 07:14 PM

Question. If I am infinitely lived and have access to a good that that grows at a constant rate with no other factor applied and of which I can consume any fraction of the total stock in a given time period, why wouldn't I (if I have sufficiently low TP) control my consumption in the early periods so that the stock of the good becomes so high that growth alone produces sufficient quantities of the good for the marginal utility of the nth unit to be 0 ?

Likewise, f my TP is too high won't I whittle away at the stock of the good until its all gone ?

Posted by: Market Fiscalist | January 07, 2018 at 10:14 AM

MF: let's simplify, by deleting haircuts from my model. So apples are the only good consumed.

The short answer to your question: diminishing Marginal Utility of Consumption. In a 2-period example, I maximise:

U(C(1)) + (1/(1+p))U(C(2)) subject to C(2) = (1+g)(K(0)-C(1))

The First Order Condition is MU(C(1))/MU(C(2)) = (1+p)/(1+g)

If MU(C) is very high when C is very low, and very low when C is very high, we get an interior solution where C is positive in both periods. And you would never have a solution where MU(C(2))=0 and MU(C(1)) > 0, because that would violate the first order condition. A simple example is U(C)=log(C), so MU(C)=1/C.

Posted by: Nick Rowe | January 07, 2018 at 11:27 AM

Maybe it's a glitch in the matrix, but if you have

A=g K-dK/dt

you have two steady state solutions: K=exp(g*t) and exp(-g*t). If you solve for steady state A, the first solution gives no consumption of apples and the second solution gives *declining* consumption over time. Is there a sign mistake somewhere?

I would like to solve the problem for CES functions (obviously log-linear - I.e. Cobb-Douglas - doesn't give you negative rates) but I'm tripped up on this hill. I went ahead and solved it for Cobb-Douglas

U=A^r H^(1-r)

And I got an interest rate of p+g*r but I'm pretty sure you're thinking this should be p-g*r so for low time preference and high production rate and rent you get a negative rate? But I'm not seeing it for the above constraint...

Posted by: C Trombley | January 08, 2018 at 08:00 PM

CT: I don't think there are any steady state solutions, except in the special case where we assume g=p so there is no investment, and K just stays at whatever level it is at initially.

But maybe I'm not sure what you mean by "steady state".

And when you say you solved for *the* interest rate, which good were you using as numeraire? Were you using the CPI basket? Or haircuts?

BTW, thanks for doing this.

Posted by: Nick Rowe | January 08, 2018 at 09:12 PM

I *think* I get the intuition here - if there is a good that can't be stored and whose supply is fixed then if demand (and price) in the future is expected to be higher than in the present then (depending upon time preference) it can have a negative own-rate of interest, and if this good (or similar goods ) make up a sufficient share of expenditure we may also get negative interest rates using a monetary unit that is based on a basket of all goods.

In your model this effect is driven by the fact that haircuts are complementary to apples (eating apples makes your hair grow faster?). Would it be possible to have the same effect in a model where supply of haircuts is expected to decline in the future (people get less good at haircuts as the get older) , or demand for haircuts increases through time for purely exogenous reasons (people just like shorter hair as they get older). In other words could you get negative interest rates purely through the demographics changing in the economy leading to changes in supply and demand that drive the same effect as in your simple model?

Posted by: Market Fiscalist | January 09, 2018 at 10:39 AM

MF: yes. Think of a model where haircuts are the only good, and can only be produced by the young, but the old want to get their hair cut too. Then it is very easy to get negative real interest rates, and a baby boom would reinforce that.

Posted by: Nick Rowe | January 09, 2018 at 11:51 AM

If utility is CES over apples and haircuts then then the relative price of apples is (a/h)^(s-1) where a is consumption of apples and h is consumption of haircuts, and s is the substitution coefficient. Hence the growth of the relative price is -(1-s)(g(a)-g(h))=-(1-s)g. So the deflation rate is (1-s)g. So g-a=g-g+sg=sg. If 1>s>0 then apples and haircuts are substitutes and the interest rate is positive. If s<0 then apples and haircuts are complements and the interest rate is negative.

Is this what you're talking about? Or is it too simple?

Posted by: notsneaky | January 13, 2018 at 04:54 PM

Second part. Let the cpi interest rate be r=wg+(1-w)(g-a)=g-(1-w)a, where w is the share of expenditure in apples. From before we have a=(1-s)g so r=g(s+w(1-s)). Note that if 1>s>0 then r>0. We need complements. We need to know what happens to w over time if s<0.

w=p(a)a/(p(a)a+p(h)h) and from U-max we have p(h)=p(a)*(a/h)^(1-s). Since h is constant anyway let's just set it to 1. So w=1/(1+a^(-s)). Hence the growth of w is gs(a^-s)/(1+(a^-s)). And if s<0 then growth of w is negative so the w(1-s) part in r gets smaller over time.

w is converging asymptotically to a constant (determined by parameter in the utility function which I ignored above)) from above. You can make that parameter as small as you want so yes, you can get negative interest rates.

The utility would be U=(m(a^s)+(1-m)(h^s))^(1/s)

For a concrete example take s=-2 and m=1/2 (or just drop those coefficients). Then w-->.5 and r=g*(-2+.5*3)=-.5g

Posted by: notsneaky | January 13, 2018 at 05:15 PM

(there might be a slight mistake in the utility function above, you might have to raise the m and the 1-m to s as well to get w=1/2 when m=1/2, i can't remember)

Posted by: notsneaky | January 13, 2018 at 05:24 PM

notsneaky: I *think* that sounds right. Thanks!

Posted by: Nick Rowe | January 13, 2018 at 05:30 PM