## 23 March 2016

### 'Low Interest Rates Are Contractionary'

I wanted to delve a little bit deeper into Scott Sumner's claim that low interest rates are contractionary. Unfortunately, the model he uses to prove this is complete nonsense, since it requires the central bank to control the money supply and the nominal interest rate at the same time to achieve the result. Because of this, I think a dynamic analysis is pertinent.

Consider a dynamic and slightly altered version of Scott's money demand model:
$$(1)\: m_t - p_t = y_t - \alpha i_t$$
where $m_t$ is the (log) money supply, $p_t$ is the (log) price level, $y_t$ is the (log) of output, and $i_t$ is the nominal interest rate. Since the model is dynamic, I will also add the Fisher relation and the Euler equation:
$$(2)\: i_t = r_t + E_t\pi_{t+1}$$
$$(3)\: y_t = E_t y_{t+1} - \sigma r_t$$
where $r_t$ is the real interest rate and $\pi_t$ is the inflation rate ($\pi_t = p_t - p_{t-1}$).

Currently this model lacks an aggregate supply curve, so, for the time being, I'll go with a vertical AS curve for simplicity: $y_t = 0$. The central bank also sets the money supply in period $t+1$ such that $p_{t+1} = \bar p$. In this case, what happens when the central bank increases the money supply?

Well, if we simplify the model slightly, we can reduce it to
$$(4)\: p_t = \frac{m_t + \alpha\bar p}{1 + \alpha}$$
That is, an increase in the money supply causes the price level to rise and the nominal interest rate to fall ($\frac{\partial p_t}{\partial m_t} = \frac{1}{1 + \alpha} < 1$) -- interestingly the conventional result.

Of course Scott is now screaming at me for committing the same sin as Krugman by fixing the future price level. Don't worry, I'll get to a more complex model, I just wanted to show that Sumner's result doesn't make sense in this pseudo-dynamic context (that, I might add, is already much better than his lazy static model. The same goes for Krugman's 1998 model, which was also dynamic).

If we relax the assumption that the price level is fixed in the next period, the model instead simplifies to
$$(5)\: p_t = E_t\frac{1}{1 + \alpha}\sum^\infty_{j=t}\left(\frac{\alpha}{1+\alpha}\right)^{j-t}m_j$$
From this we know that the current price level is a function of the discounted sum of expected money supplies. This is basic market monetarist stuff: if an increase in the money supply is expected to be immediately reversed, the price level will not rise. Keep note of the fact that, in contrast with most market monetarist assertions, an expected reversal far in the future has a highly diminished effect on the price level. Expected monetary tightening ten years from now is as good as useless.

The point here, though is that a central bank can effect a reduction in the nominal interest rate without increasing the current money supply -- it simply has to reduce expected money supply growth. My point here is that Sumner never brought this up, and, since his reasoning is based entirely on the static version of the model, he has no right to say that his model implies contractionary low interest rates.

Changing aggregate supply only makes the model more difficult to understand, but, if the aggregate supply curve became
$$(6)\: y_t = f(p_t)$$
it would then be possible to argue that expected monetary tightening has an adverse effect on the economy. At this point, though, I don't think anyone should care. Sumner may have been claiming this, but his model certainly didn't justify the claim, so he should have been ignored.

1. "(1)mt−pt=yt−αit"

Am I correct here? If we change the equation back to pre-log form, we get

mt/pt = yt^-ait or mt = pt/yt^ait.

1. It should be $\frac{M_t}{P_t} = Y_t (1+i_t)^{-\alpha}$

2. Because exp(i) ≈ 1 + i for |i| << 1

3. Thanks John and Tom,

I was trying to understand the equations by considering how they would move, sliding one variable at a time.

I think we have "i" cast in a very disappearing role but term "a" becomes a driver.

If term "a" was a constant, we should be able to find a candidate value. On the other hand, if "a" is a variable, we are expecting that there are other driving factors at play.

4. In this case, $\alpha$ is a parameter, so it is constant.