# What Does It Mean for the Natural Rate of Interest to Be Negative?

Talk of the zero lower bound has permeated the debate about monetary policy in recent years. In particular, there is one consistent story across a variety of different thinkers involving the difference between the natural rate of interest and the market rate of interest. Specifically, the argument holds that if the market rate of interest is higher than the natural rate of interest then monetary policy is too tight. With regards to the current state of the world, this is potentially problematic is the market rate of interest is zero, but needs to be lower.

I find this way of thinking about monetary policy to be quite odd for several reasons. First, conceivably when one talks about the natural rate of interest, the reference is to a real interest rate. New Keynesians, for example, clearly see the natural rate of interest as a real rate of interest (at least in their models). Second, the market rate of interest is a nominal rate. Thus, it is odd to say that the market rate of interest is above the natural rate of interest when one is nominal and one is real. I suppose that what they mean is that given the nominal interest rate and given the expectations of inflation, the implied real market rate is too high. But this seems to be an odd way to describe what is going on.

Regardless of this confusion, what advocates of this approach appear to be saying is this: when the market rate of interest is at the zero lower bound and the natural rate of interest is negative, unless inflation expectations rise, there is no way to equate the real market rate of interest with the natural rate.

But this brings me to the most important question that I have about this entire argument: Why is the natural rate of interest negative?

It is easy to imagine a real market interest rate being negative. If inflation expectations are positive and policymakers drive a nominal interest rate low enough, then the implied real interest rate is negative. It is NOT, however, easy to imagine the natural rate of interest being negative.

To simplify matters, let’s consider a world with zero inflation. The central bank uses an interest rate rule to set monetary policy. The nominal market rate is therefore equal to the real market interest rate. Thus, assuming that the central bank is pursuing a policy to maintain zero inflation, they are effectively setting the real rate of interest. Thus, the optimal policy is to set the interest rate equal to the natural interest rate. Also, since everyone knows the central bank will never create inflation, this makes the zero lower bound impenetrable (i.e. you cannot even use inflation expectations to lower the real rate when the nominal rate hits zero). I have therefore created a world in which a central bank is incapable of setting the market rate of interest equal to the natural rate of interest if the natural rate is negative. My question is, why in the world would we ever reach this point?

So let’s consider the determination of the natural rate of interest. I will define the natural rate of interest as the real rate of interest that would result with perfect markets, perfect information, and perfectly flexible prices (the New Keynesian would be proud, I think). To determine the equilibrium real interest rate, we need to understand saving behavior and we need to understand investment behavior. The equilibrium interest rate is then determined by the market in our perfect benchmark world. So let’s set up a really simple model of saving and investment.

Time is continuous and infinite. A representative household receives an endowment of income, y, and can either consume the income or save it. If they save it, they earn a real interest rate, r. The household generates utility via consumption. The household utility function is given as

$\int_0^{\infty} e^{-\rho t} u[c(t)] dt$

where $\rho$ is the rate of time preference and $c$ is consumption. The household’s asset holdings evolve according to:

$\dot{a} = y - c + ra$

where $a$ are the asset holdings of the individual. In a steady state equilibrium, it is straightforward to show that

$r = \rho$

The real interest rate is equal to the rate of time preference.

Now let’s consider the firm. Firms face an investment decision. Let’s suppose for simplicity that the firm produces bacon. We can then think of the firm as facing a duration problem. They purchase a pig at birth and they raise the pig. The firm then has to decide how long to wait until they slaughter the pig to make the bacon. Suppose that the duration of investment is given as $\theta$. The production of bacon is given by the production function:

$b = f(\theta)$

where f’,-f”>0 and b is the quantity of bacon produced. The purchase of the pig requires some initial outlay of investment, $i$, which is assumed to be exogenously fixed in real terms and then it just grows until it is slaughtered. The value of the pig over the duration of the investment is given as

$p = \int_{-\theta}^0 e^{-rt} i dt$

Integration of this expression yields

${{p}\over{i}} = {{1}\over{r}}(e^{r\theta} - 1)$

Let’s normalize the amount of investment done to 1. Thus, we can write the firm’s profit equation as

$\textrm{Profit} = f(\theta) - e^{r\theta}$

The firm’s profit-maximizing decision is therefore given as

$f'(\theta) = re^{r\theta}$

Given that the firm makes zero economic profits, it is straightforward to show that

$r = {{f'(\theta)}\over{f(\theta)}}$

So let’s summarize what we have. We have an inverse supply of saving curve that is given as

$r = \rho$

Thus, the saving curve is a horizontal line at the rate of time preference.

The inverse investment demand curve is given as

$r = {{f'(\theta)}\over{f(\theta)}}$

The intersection of these two curves determine the equilibrium real interest rate and the equilibrium duration of investment. Since the supply curve is horizontal, the real interest rate is always equal to the rate of time preference. So this brings me back to my question: How can we explain why the natural rate of interest would be negative?

You might look at the equilibrium conditions and think “sure the natural rate of interest can be negative, we just have to assume that the rate of time preference is negative.” While, this might mathematically be true, it would seem to imply that people value the future more than the present. Does anybody believe that to be true? Are we really to believe that the the zero lower bound is a problem because the general public’s preferences change such that they suddenly value the future more than the present?

But suppose you are willing to believe this. Suppose you think it is perfectly reasonable to assume that people woke up sometime during the recession and their rate of time preference was negative. There are two sides to the market. So what would happen to the duration of investment if the real interest rate was negative? From our inverse investment demand curve, we see that the real interest rate is equal to the ratio of the marginal product of duration over total production. We have made the standard assumption that the marginal product is positive, so this would seem to rule out any equilibrium in which the real interest rate was negative. But suppose at a sufficiently long duration, the marginal product is negative. We could always write down a production function with this characteristic, but how generalizable would this production function be? And why would a firm choose this actually duration when they could have chosen a shorter duration and had the same level of production?

Thus, the only way that one can believe that the natural rate of interest is negative is if they believe that individuals suddenly value the future more than the present and that in a perfect, frictionless world firms would prefer to undertake dynamically inefficient investment projects. And not only that, advocates of this viewpoint also think that the problem with policy is that we cannot use our policy tools to get us to a point consistent with these conditions!

Finally, you might argue that I have simply cherry-picked a model that fits my conclusion. But the model I have presented here is just Hirshleifer’s attempt to model the theories of Bohm-Bawerk and Wicksell, the economists that came up with the idea of the natural rate of interest. So this would seem to be a good starting point for analysis.

P.S. If you are interested in evaluating monetary policy within a framework like this, you should check out one of my working papers, written with Alex Salter.

### 8 responses to “What Does It Mean for the Natural Rate of Interest to Be Negative?”

1. Bill Woolsey

No.

I don’t believe that the supply of saving is perfectly elastic. I think it is upward sloping

I think there is a good bit of empirical evidence for this.

So, what is up?

Why are you modeling a steady state? Do we really need to show that the natural interest rate might be negative for all time (literally, right, the past and the future.)

What happens if people expect lower income in the future? Couldn’t the saving supply curve shift to the right?

Pigs and bacon? OK.

What happens if people expect the demand for bacon to fall?

What happens to the demand for pigs today?

Doesn’t investment demand shift to the left?

Why cannot they intersect below the horizontal axis?

The natural interest rate is negative.

Now, I don’t have any interest in proving that expected future income could keep on decreasing. Of that if this had been expected since time began that there would have every been any wealth or capital.

I rather am interested in a scenario where we start with a capital stock and net worth. Suppose that the government plans impose draconian regulation to stop global warming in 5 years.

When that happens, income will fall. Petty much every investment project we can start today will stop being profitable in 5 years.

What does this do to the supply of saving and demand for investment? Why can’t they be equal at a negative real interest rate?

Now, once these lower real incomes materalize, dissaving will occur, net worth will fall. The capital stock will depreciate tremendosly. No doubt growth would resume from a new, lower path, and the natural interest rate would again be positive.

W

• Bill,

There is a lot to disentangle here, so I’ll try to touch on the big stuff.

1. You object to looking at the steady state. But I’m not talking about some constant steady state that always exists. Rather I’m pointing out that shocks to, say, the rate of time preference could push us to a new steady state. Thus, the business cycle would be the movement from one steady state to the next. If you prefer, we could think of this in terms of deviations from the steady state, but I don’t think the results would change.

2. The supply of savings curve is horizontal at the rate of time preference because duration is on the horizontal axis. This is a fairly standard equilibrium condition from Irving Fisher to Jack Hirshleifer.

If you prefer, I could add growth to the model and the natural rate of interest would be

r = rho + b*g

where b is a parameter that measures risk aversion and g is the equilibrium growth rate of the economy. If people are really risk averse and the growth rate falls, then the natural rate can become negative. But real GDP growth (or productivity growth) is stationary. If it is mean reverting, why are we worried about it?

3. Again, so what if the natural rate of interest ever becomes negative? Why would we want monetary policy to follow the natural rate down that mine shaft? The investment demand curve is just a locus of profit maximizing points. Thus, if the equilibrium natural interest rate is negative, then the marginal product of capital would be negative. Assuming this is possible, why should we believe that in a frictionless world (the world that determines the natural rate), firms would want to operate at this point? And even if we accept that firms want to operate at this point, why would we want policy to promote this?

2. Macrocompassion

If I found that my savings were giving me a negative rate of interest, I would take the money out and place it in a daily account and not a savings deposit one. This is the natural reaction, thus there is no such thing as a natural negative rate of interest. Equities where there are no dividends and falling share prices will have to bear the same kind of response from their holders.

3. I’m tempted to agree with you that, “the only way that one can believe that the natural rate of interest is negative is if they believe that individuals suddenly value the future more than the present and that in a perfect, frictionless world firms would prefer to undertake dynamically inefficient investment projects,” which seems absurd. BUT if you don’t accept the possibility of negative rates, I’m curious what your understanding of the Great Recession is.

I see only a limited set of possibilities:

1. The natural rate has been negative, and the Great Recession and weak recovery were caused by a natural rate below the ex ante real rate.
— The natural rate has for some time been below -2ish for some time, and, as the New Keynesians show, the resulting gap between the real and natural rates has created a sustained output gap.
— Can’t OLG models get us a negative natural rate, a la Eggertsson and Mehrotra without even the exogenous debt constraint? You use a representative agent model above.

2. The natural rate has always been positive, and the Great Recession and weak recovery were/are caused by a series of real shocks.
— I could be mistaken, but did not think you were a proponent of this theory.

3. The natural has always been positive, and the Great Recession and weak recovery were caused by what you might call “monetary policy news shocks”.
— The natural rate has always been positive and always will be in the future, but agents expect the real rate to be above the natural rate sometime in the future. In the NK model, the output gap today is equal to a discounted sum of all future expected deviations of the natural rate from the (ex ante) real rate, thus this would create a recession.
— I.e., perhaps the natural rate was never negative in the last seven years, but agents expect the Fed to keep the real rate at 3% when the natural rate is 2% several years down the line (say).
— (I don’t think this type of thing gets enough discussion.)

4. The Wicksellian/Woodfordian natural rate framework is not a useful one for thinking about monetary policy and my above comments are just confused
— Is that what you’re trying to get at in this post?
— It seems like this might be what you’re getting at in your third point to Bill (“Why would we want monetary policy to follow the natural rate down that mine shaft?”)

My own opinion oscillates between #1 and #4.

P.S. Have not read your paper with Salter yet, but it’s on my list!

• Basil,

I have written about Eggertson and Mehrotra here: https://everydayecon.wordpress.com/2014/05/28/on-secular-stagnation-and-money/

Basically, their argument only holds up in a world in which money does not exist. So, perhaps needless to say, I am unconvinced.

The purpose of this post was to get people thinking about what all of this actually means. It seems to me that the problem with the debate is that the very models that are used to discuss the ZLB often exclude capital. And herein lies a significant problem. But Wicksell’s definition of the natural rate of interest is “the marginal product of capital.” I think that this is a reasonable definition. But if I am correct, then typical assumptions would mean that the investment demand curve should asymptotically approach the horizontal axis. In other words, there can be no natural rate below zero.

But suppose that we drop the assumption that the first derivative of the production function with respect to capital is always positive. Maybe at some point there are decreasing returns to capital. Then it is possible for there to be an equilibrium natural rate below zero. But think about what this means. Under this scenario, the preferred outcome of those in the market is to invest in projects that are dynamically inefficient (i.e. they could invest much less capital to get the same level of output). If the natural rate of interest is below zero, then it must be the case that firms prefer to invest in dynamically inefficient projects. And advocates of the ZLB approach seem convinced that the problem with policy is that we cannot get the policy rate low enough for these firms to undertake these projects. To me, that seems like a ridiculous idea.

Thus, the really important point that I am making has to do with capital. Yet, the discussion of those who think that ZLB is entirely limited to the supply of saving. There is no discussion of capital investment and, in fact, capital is often missing from their model completely.

Regarding my own views, I still think that one can tell a story about a shortage of safe/transaction assets. But this comment is already long-winded as it is. Perhaps this is the subject for a future post.

4. Bill Woolsey

I do not favor having any government intervention on interest rates.

I am not worried about a negative natural interest rate. I oppose monetary institutions that interfere with the market interest rate adjusting to the natural interest rate at all times, including when it is negative.

I favor full privatization of currency. At very low interest rates, none will be issued and all money will be of a deposit form. If banks earn low yields on earning assets, the derived demand for deposits will be low and so will the equilibrium interest rate on money.

Nothing keeps nominal interest rates above zero in this institutional framework.

I favor a 3% nominal GDP growth path and anticipate that this will keep the price level stable on average. And so, real and nominal interest rates will usually be the same.

If real income is expected to fall, then the price level will be expected to rise, resulting in a lower real interest rate for any given nominal interest rate.

If the increase in saving due to people wanting to shift current consumption to the future when real income is expected to be lower, and the decrease in investment demand due to expectations of lower sales in the future when income is expected to be lower requires a negative real interest rate now to keep nominal GDP on target, then I see no value in having the government intervene to push nominal interest rates high enough to keep real interest rates positive while forcing nominal GDP below target.

Suppose the government issues short and safe bonds–like T-bills. Now, I don’t really believe that the real market clearing yield on those bonds should be described as “the” natural interest rate. But I don’t favor government efforts to keep the real yield on those bonds positive.

For example, if there is a financial panic and people sell off stocks and seek to park their money in T-bills, then having the nominal and real yield on those T-bills go negative if necessary to clear the market is the least bad result.

Understand, that any argument that people can instead hold zero-interest hand-to-hand currency, which is perhaps slightly shorter and safer than T-bills, which puts on floor on the yield on the T-bills is an argument that the market rate on those bonds will be kept above their natural rate. Any spill over of bond demand to the demand for currency is disruptive monetary disequilibrium.

I think this is a bad thing.

Even with a deposit only monetary system we could require that all banks redeem their money with central bank deposits and then have the central bank pay a sufficiently high nominal yield to keep real interest rates on short and safe assets positive at all times.

Why would that be desirable?

I think the interest rates earned on any type of clearing balances should float with other market interest rates.

You seem to be arguing that keeping nominal and real interest rates high is protecting society from businesses who buy too many pigs and producing too much bacon. Right? Inefficient investment?

I don’t see this as a problem.

• Bill,

1. The issues surrounding nominal GDP targeting are distracting because you are assuming that the interest rate mechanism works to correct it. One of the points of my post is to highlight the limits of using the interest rate to discuss/manage policy.

2. I am not talking about the supply and demand for T-bills. If T-bill rates are negative, that doesn’t mean that the natural rate is negative. The supply of T-bills is exogenously determined. It might just mean that there is a shortage of safe assets, so the price of T-bills rises until the market clears.

3. The argument that I am making is as follows. Wicksell defines the natural rate of interest as the marginal product of capital. This is the first paragraph of Chapter 8 in interest and prices. Woodford defines the natural rate of interest as the interest rate that would exist in a world of perfectly flexible prices. This is RBC land. In RBC land, the natural rate of interest is equal to the marginal product of capital. So whether you’re a Wicksellian or a New Keynesian, this is your definition. That is the definition I am using.

So let’s start there. If the natural rate is negative, this must mean that the marginal product of capital is negative. In other words, given the natural rate, firms want to invest in projects that are dynamically inefficient in the sense that they could invest less capital and get the same level of output. Their desire is to waste capital by misallocating it.

To argue that the ZLB is some binding constraint is to argue that the problem with policy is that we cannot get the real market interest rate sufficiently low such that firms can invest in these inefficient investment projects.

I think this whole notion is silly.

4. I never said we need high interest rates. In fact, I never said we need higher interest rates either. I simply said that it is questionable whether the real natural rate ever goes below zero and that, even if it does, it might not be a good idea to chase it below zero.