This is a follow up to my last post; here I am going to go more in depth into why fiscal stimulus never really impacts aggregate demand in any DSGE.

Start with the simple consumption euler equation that everyone should be familiar with:

$$(1)\:c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} \left(\frac{1 + i_t}{1 + E_t \pi_{t+1}}\right)$$

where $c_t$ is consumption, $\beta$ is the representative agent's discount factor, $i_t$ is the nominal interest rate, and $\pi_t$ is the rate of inflation. Assuming there is no capital accumulation, this equation can be simply edited to include real GDP and government spending:

$$(2)\:(y_t - g_t)^{-\sigma} = \beta E_t (y_{t+1} - g_{t+1})^{-\sigma} \left(\frac{1 + i_t}{1 + E_t \pi_{t+1}}\right)$$

This can now be written in log-linear form, so that all variables are expressed as a percentage deviation from their steady states (except for the various interest rates):

$$(3)\:\hat{y}_t = E_t \hat{y}_{t+1} + \hat{g}_t - E_t\hat{g}_{t+1} - \frac{1}{\sigma}(i_t - E_t \pi_{t+1} - \rho)$$

where $\rho$ is the discount rate, which is equal to $\frac{1-\beta}{\beta}$. Keep in mind that $\hat{g}_t$ is actually the deviation of government spending as a percentage of GDP from trend, but this shouldn't have much of an impact on the analysis.

If we define the 'natural rate of interest' as the real rate of interest at which the output gap remains constant, only a small amount of algebra is required to solve for it:

$$(4)\:r^n_t = \rho - \sigma(E_t\hat{g}_{t+1} - \hat{g}_t)$$

It is clear from this that fiscal stimulus, that is increases in $\hat{g}_t$ absent changes in $E_t\hat{g}_{t+1}$, causes the natural rate of interest to increase. Why is this important? In most cases, it isn't; fiscal stimulus will just have the effect it normally does in a frictionless model (that is, generally speaking, have a multiplier on output somewhere between zero and one depending on the calibration of that model) and the central bank will simply raise the real interest rate so that the output gap remains equal to zero.

This is exactly what all the market monetarists are talking about when they mention monetary offset; any demand-side effect that fiscal stimulus might have is simply the result of central bank inaction in the face of a higher natural rate. Say, for instance, that the central bank set the nominal interest rate according to the following rule:

$$(5)\:i_t = max(0,r^n_t + E_t \pi_{t+1})$$

Now, suppose the fiscal authority does a fiscal stimulus, which raises $r^n_t$. Assuming the zero lower bound is not binding, the central bank will simply raise the nominal interest rate one for one with the increase in the natural rate caused by the fiscal stimulus.

The only reason a New Keynesian model ever exhibits a fiscal multiplier greater than one is that the zero lower bound is at some point binding. In this case, the fiscal authority can lower the value of $r_t - r^n_t$ without central bank intervention ($r_t$ is the real rate of interest). Basically, fiscal stimulus appears effective because of central bank inaction.

In this sense, the actual act of fiscal stimulus

*never*has demand-side effects; it can only influence the natural rate of interest, which is only helpful to the extend that the zero lower bound is binding. Because of this, we should really stop thinking of fiscal stimulus as a way of manipulating aggregate demand; it's really only helpful to the extent that is has real effects -- if anyone says otherwise, then they are unduly influenced by IS-LM.