## 03 May 2015

### Monetary Offset Is A Thing in New Keynesian Models (Sort of)

I was playing around with fiscal stimulus in a New Keynesian model (with capital) that I had written down in Dynare and I was surprised to see that increases in government spending (funded by lump sum transfers) caused deflation rather than inflation. Puzzled, I decided to remove the part of the Taylor Rule that reacts to the output gap and, as I had initially predicted, fiscal stimulus became inflationary and output increased more than in my first test.

The moral of the story is that the idea of monetary offset that Scott Sumner brought up (I think) is partially right: the effects of fiscal stimulus will be (partially) counteracted by the central bank. In my model, this happens not because the central bank is targeting inflation, but because it tightens monetary policy in response to increases in output above potential. Of course, in order to have fiscal stimulus completely counteracted by the central bank, I needed to put the output gap coefficient on the Taylor Rule upwards of 5 (rather than the normal 0.5), so complete monetary offset with Taylor Rules doesn't seem to work.

In a way, this partially affirms the pro-fiscal-stimulus crowd and the pro-monetary-stimulus crowd (I belong more to the latter). Fiscal stimulus does appear to work in the short term, but the central bank really has power over aggregate demand.

If anyone wants to check my analysis, here are the relevant model equations:

$$y_t = e^{z_t} k_t^\alpha n_t^{1-\alpha}$$
$$z_t = \rho z_{t-1} + \epsilon_t^z$$
$$k_{t+1} = (1-\delta) k_t + x_t$$
$$w_t = c_t^\sigma n_t^\phi$$
$$c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} \left(\frac{1 + i_t}{E_t \pi_{t+1}}\right)$$
$$y_t = c_t + i_t + g_t$$
$$g_t = \rho g_{t-1} + \epsilon_t^g$$
$$w_t = mc_t (1-\alpha) e^{z_t} k_t^\alpha n_t^{-\alpha}$$
$$r_t + \delta = mc_t \alpha e^{z_t} k_t^{\alpha -1} n_t^{1 - \alpha}$$
$$\log\pi_t = \beta \log E_t \pi_{t+1} + \frac {(1-\theta)(1-\beta \theta)}{\theta} \left( \log mc_t - \log \left(\frac {\epsilon - 1}{\epsilon}\right)\right)$$
$$\frac{1 + i_t}{E_t\pi_{t+1}} = 1 + r_t$$
$$i_t = \beta^{-1} - 1 + \phi_\pi (\pi_t -1) + \phi_y (\log y_t - y^n)$$

Where $y_t$ is real GDP, $z_t$ is the TFP, $k_t$ is the capital stock, $w_t$ is the real wage, $c_t$ is consumption, $x_t$ is investment, $g_t$ is government spending, $mc_t$ is the marginal cost, $r_t$ is the real interest rate, $\pi_t$ is the gross rate of inflation, and $i_t$ is the nominal interest rate.

Here is the response to a fiscal shock with a normal Taylor Rule:
Normal Taylor Rule Fiscal Shock
And here is the response to a fiscal shock where the Taylor Rule ignores the output gap:
 Modified Taylor Rule

1. Hi John,

Your nominal interest rate appears to go negative in the first case -- I believe the circumstances that allow that (program buying by big mutual funds or retirement funds or e.g. limits on ATM withdrawals or costs of holding cash) aren't in the model per se. Maybe I am misreading your graphs?

1. The nominal interest rate is in log deviation from steady state, but you're right there isn't a zero lower bound in the model. The program I'm using to simulate the models doesn't seem to work well with things like the zero lower bound and variables that don't revert to the same steady state after a shock (like the price level). In a cashless economy, I'm pretty sure that if the zero lower bound is a binding constraint, monetary offset stops happening, but I'm not quite sure how this would work out with something like a nominal income target or a money supply rule since there are no constraints on monetary policy.

2. I get it.

One way you could create a pseudo zero lower bound is to add a small amount of noise ... effectively creating a "noise floor" for the interest rate -- that would approximate what the data looks like.

http://research.stlouisfed.org/fred2/graph/?g=1bqC

http://en.wikipedia.org/wiki/Noise_floor

3. I can add a shock every period, but as long as monetary policy follows a rule (and I'm doing a stochastic/dynamic rather than deterministic simulation), a negative rate is always possible. Consider the policy rule in the model: $i_t = \beta^{-1} + \phi_\pi (\pi_t - 1) + \phi_y (\log y_t - y^n)$. If $\pi_t - 1$ or $\log y_t - y^n$ are sufficiently negative, the nominal interest rate will be lower than zero.

4. This comment has been removed by the author.

5. edit: $i_t = \beta^{-1} -1 + \phi_\pi (\pi_t - 1) + \phi_y (\log y_t - y^n)$