Skip to main content# Chapter 2| Preliminaries

## 2.1 The neutral interest rate, $r^*$

## 2.2 Safe rates and risky rates, $r$ and $r+x$

## 2.3 The role of central banks. Trying to achieve $r=r^*$

## 2.4 Why is “$r<g$” so important?

## 2.5 Nominal and real rates, and the ELB

## 2.6 Conclusions

Low interest rates are central to the story. With this in mind, the chapter introduces five notions related to interest rates, which will be useful throughout the book.

Published onMar 30, 2022

Chapter 2| Preliminaries

Low interest rates are central to the story. With this in mind, the chapter introduces five notions related to interest rates, which will be useful throughout the book:

Section 1 defines the neutral interest rate, $r^*$. It can be defined in two equivalent ways. The first is that it is the safe real interest rate such that saving is equal to investment, assuming output is equal to potential output. The second is that it is the safe real interest rate such that aggregate demand is equal to potential output. The two definitions are indeed equivalent but suggest different ways of thinking about the factors which determine the neutral rate, ways which will turn out to be useful later.

Section 2 introduces the distinction between safe rates and risky rates such as the rate of return on stocks. It shows how an increase in perceived risk or in risk aversion leads to both a higher risky rate and a lower safe rate. When looking at data in the next chapter and thinking about the factors behind low safe rates, the distinction will turn out to be empirically important. Are current low safe rates due to shifts in saving or investment, or instead to higher risk or risk aversion?

Section 3 looks at the role of central banks. One can think of the effective mandate of central banks as setting the actual safe real interest rate, $r$, as close as they can to the neutral interest rate, $r^*$, and in so doing, keep output close to potential output. The important conclusion is that, while central banks are sometimes blamed for the current low rates, the rates set by central banks reflect mostly low neutral rates, which themselves reflect the factors behind the movements in $r^*$, saving, investment, risk, and risk aversion. In other words, central banks are not to blame for low rates: these just reflect underlying fundamental factors.

Section 4 discusses the importance of the inequality $(r-g)<0$, where $r$ is the real safe interest rate and $g$ is the real growth rate of the economy. When $r$ is less than $g$, debt, if not repaid, accumulates at rate $r$, while output grows at rate $g$. Thus, if no new debt is issued, the ratio of debt to output will decrease over time, making for more favorable debt dynamics. As $r$ is forecast to be less than $g$ with high probability for some time, this will play a major role in our discussion of fiscal policy later.

Section 5 discusses the nature and implications of the effective lower bound (ELB). Because people can hold cash, which has a nominal interest rate of zero, central banks cannot decrease the nominal policy rate much below zero. This implies that they cannot achieve real policy rates much lower than the negative of inflation; call that rate $r_{min}$. This potentially reduces their ability to decrease $r$ in line with $r^*$ when $r^*$ is very low, leading to situations where $r>r^*$. In other words, it potentially reduces or even eliminates the room for monetary policy to maintain output equal to potential output. This is the situation in which many central banks find themselves today, and again has major implications for our discussion of fiscal policy later.

Section 6 concludes. As the neutral rate $r^*$ has declined over time, it has crossed two important thresholds. First, $r^*$, and by implication, $r$ has become smaller than $g$, with important implications for debt dynamics and fiscal policy in general. Second, in some cases, $r^*$ has become lower than the ELB rate $r_{min}$, limiting the room for monetary policy to maintain output at potential, and by implication, increasing the need to use fiscal policy.

Start with a much simplified view of what determines interest rates. Assume that there is just one interest rate, a real interest rate (that is a nominal rate minus expected inflation) which is such that saving is equal to investment. Write the equilibrium condition that saving equals investment as:

$\label{SI}
S(Y,r,.) = I(Y,r,.)$(1)

where $S$ is saving, which is assumed to depend on income $Y$, on the real interest rate $r$, and on other factors, captured by the dot, which shift saving around; $I$ is investment, which also depends on output $Y$, on the real interest rate, $r$, and other factors captured by the dot, which shift investment around.

Note that, for simplicity, there is no explicit treatment of the role of the government here: Think of saving as capturing both private and government saving, and investment as capturing both private and government investment. How fiscal policy affects saving and investment will be central and discussed at length later, but it is not essential to the point I want to make here. Note also that I ignore the fact that the economy may be open, so saving may not be equal to investment—although this has to be true for the world as a whole. I shall also return to the issue later when I discuss whether we should think of $r^*$ as being determined by country-specific factors or by global factors.

Assume that output is equal to potential output, denoted $Y^*$. Then, the equilibrium condition determines the value of the real rate such that saving is equal to investment; call it the *neutral rate* and denote it by $r^*$. Thus, $r^*$ satisfies

$\label{neutralrate1}
S(Y^*,r^*,.) = I(Y^*,r^*,.)$(2)

The equilibrium is represented in 2.1, with saving and investment on the horizontal axis, and the interest rate on the vertical axis. Both saving and investment are plotted against the interest rate, assuming $Y=Y^*$. Saving is increasing in the interest rate; investment is decreasing in the interest rate. The equilibrium is given by point $A$, with associated neutral rate $r^*$.

This gives us a first definition of the neutral rate: *The neutral rate, *$r^*$*, is the real interest rate at which, assuming output is equal to potential output, saving is equal to investment.*

A positive shift in saving shifts the saving relation to the right, shifting the equilibrium to point B, with higher saving and investment, together with a lower neutral rate. A negative shift in investment shifts the investment relation to the left, shifting the equilibrium to point C, with lower saving and investment, together with a lower neutral rate.

There is another, equivalent, way of looking at $r^*$, which will turn out to be useful as well. Define consumption $C = Y-S$, and rewrite the equation ([SI]) as

$\label{YCI}
Y = C(Y,r,.) + I(Y,r,.)$(3)

Output is equal to aggregate demand, the sum of consumption and investment (recall that consumption and investment include government consumption and government investment). Then, we can define $r^*$ as the interest rate such that aggregate demand is equal to potential output, so:

$\label{neutralrate2}
Y^* = C(Y^*,r^*,.) + I(Y^*,r^*,.)$(4)

The equilibrium is represented in Figure 2.2, with output on the horizontal axis and aggregate demand on the vertical axis, and will be familiar as a Keynesian cross diagram. Aggregate demand, $C+I$ is drawn as a function of $Y$ for a given value of the interest rate $r$. $C+I$ is increasing in $Y$ (and less than the 45-degree line, under the standard assumption that the sum of the marginal propensity to consume and the marginal propensity to invest is less than one); a higher interest rate decreases consumption and investment and shifts aggregate demand down; a lower interest rate shifts aggregate demand up. For a given value of $r$, the equilibrium value of output is given by the intersection of $C+I$ and the 45 degree line ($Y=Y$). The neutral rate, $r^*$ is such that the equilibrium level of output is equal to potential output $Y^*$. If $r$ is greater than $r^*$, then aggregate demand is lower, and equilibrium output is lower than potential output, $Y<Y^*$; if $r$ is lower than $r^*$, then aggregate demand is higher, and equilibrium output is higher than potential, $Y>Y^*$.

This gives us a second definition of the neutral rate: *The neutral rate, *$r^*$*, is the real interest rate such that aggregate demand generates output equal to potential output.*

Why two definitions, given that they deliver the same value of $r^*$? Because they naturally lead to a focus on different determinants of $r^*$:

The first one leads to a focus on low frequency determinants of saving and investment, such as demographics.

The second leads to a focus on short term determinants, such as the decrease in aggregate demand at the onset of the Global Financial Crisis, or, thinking about fiscal policy, the increase in demand triggered by the Biden stimulus of early 2021.

Both sets of factors are obviously important, and we shall explore them, and in particular the role of fiscal policy, later.1

There are a lot of different interest rates and rates of return out there, safe/risky, short/long, rates on corporate/government bonds, rates of return on equities, housing, commodities, even bitcoin, etc. It will be important later to introduce a distinction at least between safe and risky rates. So assume there are two rates, safe and risky. Assume saving depends on the safe rate—think of it as the interest rate on government bonds—and investment depends on the risky rate—think of it as the expected rate of return on equity. Denote the risky rate by $r+x$ where $x$ represents the risk premium over the safe rate.2

In this case, the equilibrium condition takes the form:

$\label{SI2}
S(Y,r,.) = I(Y,r+x,.)$(5)

And the neutral safe rate is given by:

$\label{neutralrate3}
S(Y^*,r^*,.) = I(Y^*,r^*+x,.)$(6)

The equilibrium is represented in Figure 2.3, with the safe rate on the vertical axis, and saving and investment on the horizontal axis. Given potential output, saving is an increasing function of the safe rate. Given potential output, investment is a decreasing function of the risky rate, thus, for a given risk premium, a decreasing function of the safe rate. The equilibrium is at point A; the safe rate is given by $r^*$, and by implication the risky rate is given by $r^*+x$.

An increase in the risk premium, call it $\Delta x >0$, be it because of higher risk or higher risk aversion, shifts the investment relation down by $\Delta x$, and leads to a decrease in the safe rate of $\Delta r^*$; because the decrease in $r^*$ is smaller than $\Delta x$, it leads to an increase in the risky rate.

So, this gives us another potential reason why the safe interest rate may be low (in addition to shifts in saving and investment we saw earlier), namely an increase in the risk premium, due either to an increase in risk aversion or an increase in risk itself. A very similar argument holds if we think not of risk, but of liquidity—and not of a risk premium on risky assets, but a liquidity discount on safe assets. If we think of the asset paying the safe rate, for example a Treasury bill, as more liquid, and the other asset as being less so, an increased demand for liquidity will lead to a lower safe (liquid) rate, and a higher rate on the less liquid one.

This gives us potentially four sets of factors to explain movements in the neutral rate: shifts in saving, shifts in investment, shifts in the risk premium, and shifts in the liquidity discount. I shall discuss what we know about their relative contributions in the next chapter.

The role of central banks is to avoid overheating, which might lead to increasing inflation, as well as underheating, which leads to excessive unemployment.3 Given the definition of the neutral rate as the rate that keeps output at potential, we can think of central banks as trying to set $r=r^*$. If, for example, aggregate demand falls, implying a decrease in $r^*$, the central bank will typically try to decrease $r$ in line with $r^*$ in order to avoid or at least to limit the drop in output.

This has two important implications:

The first is that, while central banks cannot achieve $r=r^*$ all the time, they try to come close (subject to the effective lower bound constraint discussed below).4 So, most of the time,(so long as the effective lower bound is not binding) we can look at $r$ as being a good proxy for $r^*$.

The second is that, while central banks are often blamed (or congratulated) for the current low rates, the blame is misplaced. The low $r$ reflects primarily a low $r^*$, and thus the factors behind it, high saving, low investment, high risk, high risk aversion, and higher demand for liquidity.

That the neutral rate $r^*$, and by the implication the actual rate $r$, could be very low was first discussed by Alvin Hansen, 1939 [1] who worried at the time that fewer and fewer investment opportunities would lead to low investment and weak private demand. He also believed that the interest elasticity of private demand was low (in terms of the IS-LM model, that the IS curve was very steep), implying that a low or even negative neutral rate might be needed to generate enough demand to maintain output at potential. He called such an outcome “secular stagnation.”5 In any event, his worries were not confirmed at the time, and private demand remained strong. Given recent evolutions however, the same worries have reappeared, and in 2013, Lawrence Summers [2] argued that we might indeed have entered a period of “secular stagnation”, and that $r$ was going to remain low for a long time. I am not sure that the Hansen-Summers terminology is best, but it has become standard (I would prefer “structurally weak private demand” but it probably sounds too technical).

As $r^*$ has decreased over the last 30 years (much more on this in the next chapter), it has crossed two important thresholds. First, it became less than the growth rate: $r^*$ and by implication $r$, became lower than $g$. The second is that it became so low that, at times, monetary policy could not decrease $r$ to match the decrease in $r^*$, a constraint known as the effective lower bound.

This section focuses on the implications of what happens when the safe real interest rate becomes less than the growth rate. For lack of a better label, I shall call it the $(r-g)<0$ condition.6 The inequality clearly holds today, and, as I shall argue in the next chapter, is likely to hold for some time to come. (Here is a good place to clarify the relation between the statement that $r<g$, and Piketty’s main point that $r>g$ [3]. There is no contradiction as we are referring to different rates. Mine is the safe interest rate, which is indeed less than the growth rate; Piketty’s is the risky rate, the average rate of return on wealth, which is indeed greater than the growth rate).

The sign of $(r-g)$ indeed has strong implications for debt dynamics as well as for the welfare implications of debt.

Take debt dynamics first. For any borrower, a low interest rate is good news. But, if you or I borrow, we still must pay back the loan before we die. Governments do not need to do that. In effect, because they live forever, when debt becomes due they can issue new debt, i.e. “rollover debt”. All governments do that.

To see what this implies, start from a situation where the government has no initial debt and taxes are equal to spending, so the budget is balanced. Now suppose that the government increases spending for just one year but does not raise taxes, and thus issues debt to finance the deficit. From the second year on, with spending back to normal, taxes again cover non-interest spending, but not interest payments. Debt thus increases at rate $r$. Output, on the other hand, grows at rate $g$.7

If $r>g$, which has long been taken as the standard case, the ratio of debt to output, often called the “debt ratio” for short, will increase exponentially (at rate $(r-g))$, and sooner or later, if the government does not want the debt ratio to explode, it will have to increase taxes (or decrease spending, or both). But if $r<g$, the situation we are in today and expect to be in for some time to come, then the debt ratio will decrease over time. Indeed, if $r<g$ forever, then the government can spend more for a while, issue debt and never raise taxes... Thus, the standard notion that higher debt implies the need for higher taxes in the future seems no longer to hold.

Turn to welfare implications. A low $r$ is actually a signal that something is wrong with the economy: In effect, if we think of the safe rate as the risk-adjusted rate of return on capital, the low safe rate is sending the signal that, risk adjusted, the return on capital is low. Put another way, it is sending the signal that capital is in fact not very productive at the margin. If so, to the extent that we think of debt as crowding capital and thus decreasing capital accumulation, debt may not be costly. It may even be beneficial if there truly was too much capital to start with.

These are fairly dramatic results. If $r<g$ were really to hold forever, it would suggest a very relaxed attitude towards debt. As we shall see, it comes however with two strong caveats. First, debt and deficits increase aggregate demand, and thus increase $r^*$; if central banks set $r=r^*$, then $r$ increases, possibly changing the sign of the inequality and returning debt dynamics to the standard case. Second, we cannot be sure that $r<g$ will last forever (or at least for a very long time). And, if it does not, we have to think about what adjustment might be needed. Thus, a central question will be: What do we expect $r$ and $g$ to do in the future? This requires looking at history, at likely underlying factors, and their likely future evolutions. This will be the topic of the next chapter. The implications for debt dynamics, for welfare, and for fiscal policy in general will be the subject of the following chapters.

I have assumed so far that central banks were able to set $r$ equal to, or at least close to, $r^*$. So, in response to a sharp decrease in demand, leading to a sharp decrease in $r^*$, I assumed they were able to decrease $r$ to match the decrease in $r^*$. When $r^*$ is very low however, this may not be the case.

Historically, over the nine US recessions since 1960, the decrease in the (nominal) policy rate during recessions ranged 2% to 8.8%, with an average decrease of 5%.8 This was possible because inflation was on average much higher than today, and so were average nominal interest rates.9

With low inflation and low neutral real rates today, central banks have lost much of the room they had to decrease $r$ in response to $r^*$. Central banks control directly the nominal rate, not the real rate. To a first approximation, nominal interest rates cannot go negative, a constraint known as the *zero lower bound, or ZLB*.10 This is because if they did, holding cash—which yields a nominal rate of zero—would become a superior alternative to holding bonds, and people would want to shift to cash. This has a simple but important implication:

Write down the relation between the real rate, $r$, the nominal rate $i$, and the expected rate of inflation $\pi^e$:

$r = i – \pi^e$(7)

If the nominal interest rate cannot be negative, then the lowest the real rate can be is the negative of expected inflation, $-\pi^e$.11 Depending on the country, today’s forecasts of inflation by firms and investors over, say, the next five years are around 2-3%, thus implying, if the nominal rate cannot be negative, a lower limit for the 5-year real rate of minus 2-3%.12

Central banks have learned however that they can actually set slightly negative nominal rates without triggering a major shift to cash. This is because holding very large quantities of cash is inconvenient, potentially dangerous, and, in some cases simply infeasible, for example for banks which, if they sold their bond holdings and replaced them with cash, would have to hold gigantic amounts of cash, at some security risk.13 Thus, as was shown in Table 1 in the introduction, a number of central banks have been able to set *negative* nominal rates, for example, at the date of this writing, -0.75% for the Swiss policy rate. To reflect this, the constraint on monetary policy is no longer called the zero lower bound but, rather, the *effective lower bound, or ELB*. The additional room from negative nominal rates is small however relative to the required size of decreases in $r$ in the past.14

The ELB has two clear implications for fiscal policy.

If, as is the case at the time of writing, the ELB constraint is strictly binding in many advanced economies, and the output gap is still negative, there is no room for monetary policy to increase demand and output; put another way, if $r^*$ is less than $r_{min}$, central banks cannot get $r$ down to $r^*$, and the burden of increasing demand to return output to potential must fall fully on fiscal policy.

If, as is likely to happen at some point, and will probably have happened when this book comes out, aggregate demand strengthens, leading to an increase in $r^*$ above the effective lower bound, the ELB may no longer strictly bind, and most nominal interest rates may become positive again. But they are still unlikely to be high enough to allow central banks to react to large adverse shocks to demand. In that case, fiscal policy must be the main policy instrument to respond to the decrease in demand.

Before moving on to conclusions, note an important conceptual difference between secular stagnation and the ELB constraint. Secular stagnation, i.e. a very low $r^*$, leading to $(r-g)<0$, reflects fundamental forces in the economy, which may not be easy to undo. The ELB constraint is instead more of a self-inflicted wound, or to use a soccer expression, an own goal. The low inflation rate of the last three decades reflects largely the target inflation rate chosen by central banks, typically around 2%.15 Had central banks chosen a higher target, inflation would likely have been higher, and so would have expected inflation; nominal rates would have been higher, and there would have been more room to decrease them, were it to be needed. The discussion of the right inflation target is a long one and it is far from settled; but it often ignores the fact that, if the target is chosen too low and the ELB is often strictly or potentially binding, more of the macro stabilization must be achieved through fiscal policy, which may have its own costs.16

We now have the tools we need to think about fiscal policy. And you can already see the basic implications of both $(r-g)<0$ and the ELB constraint:

To the extent that $r-g$ remains negative for a long time in the future, debt may not be very costly, either in fiscal terms or in welfare terms.

To the extent that the ELB constraint continues to bind, either strictly or potentially, fiscal policy, i.e. higher deficits, may be needed to maintain output at potential.

This suggests an economic environment with lower costs of debt, and higher (output) benefits of deficits (debt).

But what will happen to the neutral rate in the future? This is the topic of the next chapter.