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

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.

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 ?

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.

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...

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.

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?

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.

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?

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

(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)

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

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