## 16 December 2015

### A Novel New Keynesian View of Fiscal Policy

Usually when I read New Keynesian economists on fiscal policy, they tend to focus more on fiscal multipliers or the effectiveness of tax cuts at the zero lower bound. But what about fiscal policy in general? I have come across some literature on this, but it usually limits itself to comparing the relative roles of monetary and fiscal policy - e.g. what is the optimal coefficient for the output gap in the fiscal policy rule? Here, I'd like to present a somewhat novel approach to New Keynesian fiscal policy (at least I've never seen or read this anywhere else).

Consider first the basic Consumption Euler equation that determines how household's allocate consumption between the present and the future given an interest rate.

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

In simple New Keynesian models, real GDP is composed of just government spending and consumption since there is no capital accumulation, so $1$ can be rewritten as a function of output, $y_t$, and government spending, $g_t$.

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

It is useful to linearize $2$ to make it a bit easier to work with, but first, to make the math a little easier, it is helpful to notice that $y_t - g_t$ is the same as $y_t(1 - \frac{g_t}{y_t})$. Given this, defining $\theta$ as $\frac{1}{\sigma}$, and defining $\beta$ as the inverse of the gross time preference rate, $\rho$, it is possible to write $2$ in log-linear form - i.e. all equations are written as percentage gaps from their long run level.

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

Keep note that, in this case, $\hat{g}_t$ is the gap of the government spending to GDP ratio from trend rather than simply government spending from trend. For my purposes, this is basically irrelevant.

If the goal of fiscal policy is to ensure that the output gap is zero at all times - not too weak of an assumption in my opinion - then it's pretty simple to solve for optimal policy given $3$:

$$(4)\: E_t \Delta \hat{g}_{t+1} = -\theta(i_t - E_t\pi_{t+1} - \rho)$$

In English, equation $4$ tells us that the role of fiscal policy is simply to offset any failure of the monetary authority to set the right real interest rate ($i_t - E_t \pi_{t+1}$). If, for example, the monetary authority has set a real interest rate that is too high, then government spending should be expected to shrink relative to trend in the next period. This can be accomplished either through stimulus - raising current government spending now and reducing it in the future - or through causing expected temporary austerity - decreasing next period's government spending then allowing government spending to return to trend.

The first option is preferable for a couple of reasons. For one thing, government spending also has real effects (see my previous blog post), so austerity might have unintended supply side consequences. Also, the austerity must be reversed at some point for the policy to work - since $\hat{g}_{t+1}$ would fall to zero if the austerity were permanent - so future fiscal policy may be impaired if the central bank continues to set an interest rate that's too high.

Another way of approaching this is to rewrite $3$ to incorporate a 'natural real rate of interest.' In this case, $3$ can be rewritten as

$$(5)\: \hat{y}_t = E_t\hat{y}_{t+1} - \theta (i_t - E_t\pi_{t+1} - \rho + \frac{E_t\Delta\hat{g}_{t+1}}{\theta})$$

Defining the natural real rate of interest at which the output gap is zero, it is clear that $\rho - \frac{E_t\Delta\hat{g}_{t+1}}{\theta}$ is equal to the natural rate, $r^n_t$.

Assuming the central bank tries to set the real interest rate equal to the natural rate unless the zero lower bound is binding, i.e. $i_t = \max\left(0,\: E_t \pi_{t+1} + r^n_t\right)$, the job of the government can be seen as preventing the zero lower bound from ever binding, or, in other words, setting $E_t\pi_{t+1} + r^n_t > 0\: \forall t$.

$$(6)\: 0 < E_t\pi_{t+1} + \rho - \frac{E_t\Delta\hat{g}_{t+1}}{\theta}$$

or

$$(7)\: E_t\Delta\hat{g}_{t+1} < \theta(E_t\pi_{t+1} + \rho)$$

From $7$, it is clear that expected growth in government spending relative to trend should always be less than a function of the expected inflation rate. That is, the lower the expected inflation rate, the bigger the stimulus that should be undertaken. Effectively the goal of fiscal policy is to offset failures in monetary policy and to make sure that the zero lower bound never binds in the first place.