### Why negative interest rates might not work

**Matthew Martin**9/04/2014 01:58:00 PM

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^{1}so that we can eliminate liquidity traps by implementing negative interest rates. Here's why I'm uncertain whether negative interest rate policy is expansionary.

To start, let's go through the standard logic: the fisher equation relates nominal interest, real interest, and inflation: [$$]r_t=\tilde{r}_t+\pi_t[$$] where [$]r_t[$] is the nominal interest controlled by the Fed, [$]\tilde{r}_t[$] is the real interest rate, and [$]\pi_t[$] is the inflation rate, in period [$]t[$]. The consumption/savings tradeoff depends on the returns to saving rather than consuming, so that consumption, and therefore output and employment, is decreasing in [$]\tilde{r_t}.[$] If prices are sticky, then [$]\pi[$] will respond to policy relatively slowly, so that in the short run reducing the nominal interest rate [$]r_t[$] leads to a reduction in [$]\tilde{r}_t[$] and thus an increase in consumption and reduction in saving. This is the liquidity effect, which is expansionary.

What this analysis has left out, however, are the feedback effects of inflation on output. This happens through two additional mechanisms: the New Keynesian Phillips Curve and the Euler consumption equation. We won't worry about exact analytical specifications for each, just let [$]f_t[$] describe how the household choice of [$]C_t[$] depends on the real interest rate and inflation (consumption Euler equation), and [$]g_t[$] describes the New Keynesian Phillips curve relationship between current and expected inflation, so that we have:

^{[update]}

\begin{align}

r_t&=\tilde{r}_t+\pi_t\\

C_t&=f_t\left(\tilde{r}_t,\pi_{t+1}\right) \\

\pi_t&=g_t\left(\pi_{t+1}\right)

\end{align}

where [$]g_t[$] is an increasing function and [$]f_t[$] is decreasing in [$]\tilde{r}_t[$] but increasing in [$]\pi_{t+1}[$].

^{2}It is now apparent that the liquidity analysis in the preceding paragraph was incomplete--inflation is actually a free variable! Conventional wisdom says that lowering the nominal interest rate causes inflation to increase, because we typically think of lowering the interest rate as being achieved by increasing the supply of money. But strictly from the mathematics, this is actually ambiguous--there are two paths to lower nominal interest rates: a monetary expansion that lowers real interest rates via sticky prices, or a monetary contraction that lowers expected inflation. Suppose we have monetary expansion, then expected inflation [$]\pi_{t+1}[$] rises via the money market (as I showed here), which induces more consumption via [$]f_t[$] and raises current inflation via the Phillips curve [$]g_t[$], which further reinforces the liquidity effect by lowering the real interest rate [$]\tilde{r}_t[$] for the given nominal rate target via the Fisher equation.

However, when we instead assume that lower nominal rates are achieved via monetary contraction, all of these reinforcing effects reverse signs: expected inflation falls, which reduces current inflation via [$]g_t[$], reduces current consumption via [$]f_t[$], and puts reverse pressure on the real interest rate in the fisher equation since, holding [$]r_t[$] at the target, a decrease in [$]\pi_t[$] implies higher, not lower, real interest [$]\tilde{r}_t[$]. So despite the simplistic logic of the first paragraph, lowering the Fed funds rate can potentially be contractionary if it is associated with a decrease in monetary aggregates.

For positive rates, the empirical evidence overwhelmingly suggests that lowering the Fed funds rate increases the money supply. This is not obvious from theory alone--lower nominal interest rates does reduce "money printing" in some respects, for example by reducing interest on reserves. But net effect is pretty unambiguous because the Fed typically engages in ample Open Market Operations that increase base money by far more than the reduction in interest payments. In this respect, what passes for conventional wisdom is quite wrong in saying that the Fed balance sheet doesn't matter--this neutrality is an illusion that arises from taking too many modelling shortcuts (like most New Keynesian papers, Christiano, Eichenbaum, and Rebelo (2011) for example, do not explicitly model the money market that drives their key assumptions, see footnote).

But under Miles Kimball's proposal, the Fed would lower interest rates to below zero by taxing away balances of e-currency. This is a reduction in monetary base, just like the case of IOR, and by itself would be contractionary, not expansionary. The expansionary effects of Kimball's policy depend on the assumption that households will increase consumption in response to the taxing of their cash savings, rather than letting their savings depreciate. That needn't be the case--it depends on the relative magnitudes of income and substitution effects for real money balances. The substitution effect is what Kimball has in mind--raising the price of real money balances will induce substitution out of money and into consumption. But there's also an income effect, whereby the loss of wealth induces less consumption and more savings. Thus, negative interest rate policy can be contractionary even though positive interest rate policy is expansionary.

Indeed, what Kimball has proposed amounts to a reverse Bernanke Helicopter--imagine a giant vacuum flying around the country sucking money out of people's pockets. Why would we assume that this would be inflationary?

[1] To be clear, this is something we should do regardless of whether we also enact Kimball's negative interest rate policy. Any business with an internet connection already has everything it needs to conduct payments electronically. Paper is costly and inefficient and should be killed.

[2] See Christiano, Eichenbaum, and Rebelo (2011). The consumption Euler is equation (11), while the New Keynesian Phillips Curve is equation (9). While Christiano et al do not explicitly model the money market, the NK model is equivalent to a money-in-utility model with nominal rigidities (as in this post but with monopolistic competition and Calvo pricing), where interest rate policy is enacted by targeting the money supply. This equivalence is invoked when Christiano et al assume the direction of causality from inflation to policy rate in equation (6).

*[update]What follows here is meant to provide intuition, not a formal proof. An earlier version omitted subscripts from [$]g_t, f_t[$] which was a bit misleading--these functions do have other time-dependent arguments that have been suppressed here for simplicity. For a proof that reversing the causal assumption embedded in the NK Taylor rule implies that lowering rates can be contractionary, see Schmitt-GrohÃ© and Uribe (2012), which has also been covered by David Andolfatto here.*