## 22 December 2015

### What's the Significance of Low Real Interest Rates?

[Note: I started writing this post a while ago, so it ostensibly has no connection with these two posts that Nick Rowe and Scott Sumner wrote recently. I just realized that this is somewhat relevant, so I decided to finish it]

For the last twenty years or so, real interest rates on government bonds have continued to fall from their high of about 9%. Determining the cause of such a fall is by no means an easy task; after all economic theory generally suggests that real interest rates on safe assets -- like government bonds -- should be relatively constant in the long run and reflect the rate at which consumers discount future spending relative to current spending. Economic theory tells us that low real interest rates mean that current consumption is high and future consumption is low relative to what it otherwise would have been. This certainly is a possibility; perhaps falling real interest rates are indicative of a shift in consumer spending patterns away from saving and into borrowing, although the causality seems to be backwards if that is truly the case, which leaves the question of what has caused this decline in real interest rates open once again.

Perhaps the basic models in which the government has no power over the real interest rate in the long run are incorrect; given the sharp increase in the real interest rate on government bonds during the 1980s, this certainly seems plausible. In this case, it may be useful to switch to looking at this problem through the lens of an OLG model instead of a basic representative agent RBC/Neo Classical one. Every period, a new young agent is born with the endowment $y$ which can be used to buy either consumption ($c^y_t$) or government bonds ($b_t$), or to invest in capital ($k_t$). The young agent faces the budget constraint
$$(1)\: y = c^y_t + b_t + k_t$$
In the next period, the young become old and use income from interest on government bonds, $R_t b_t$, and from income generated from capital, $f(k_t)$ to finance their consumption and the taxes levied by the government. Old agents face the budget constraint
$$(2)\: c^o_{t+1} = R_t b_t + f(k_t) - \tau_t$$
Agents are born wanting to maximize their consumption in both periods of their life, with consumption when old discounted at rate $\rho$. The agents' discount factor is $\beta = \frac{1}{1 + \rho}$. Utility it derived from the log of current young consumption and the log of future old consumption:
$$U = \log c^y_t + \beta \log c^o_{t+1}$$
Agents maximize their utility function subject to both of their budget constraints. Young agents choose their consumption so that
$$(3)\: \frac{1}{c^y_t} = \beta \frac{1}{c^o_{t+1}} R_t$$
That is, young agents take as given the interest rate the they can receive by saving now and consuming later or that they would pay if they consumed now and saved later and decide to save more if the interest rate is high -- since their lifetime income can be increased by their saving -- and save less if the interest rate is low. The government sets the number of bonds that it issues by discretion each period which, given the young agent's consumption decision, determines the level of capital investment.

Another first order condition of the model is the the real interest rate on government bonds is equal to the marginal productivity of capital. That is,
$$(4)\: R_t = f'(k_t)$$
Since the level of government bonds determines capital investment, it also determines the real interest rate on government bonds. More government debt means less capital which, per $4$, means a higher real interest rate (assuming that $f(k)=k^\alpha$ where $\alpha < 1$). This works because agents must be indifferent between holding more government bonds or more capital in equilibrium; otherwise they would end up demanding more or less capital than they wanted.

In this model, low real interest rates are a result of high capital expenditure and low government debt. The prescription for low interest rates, then, is to engage in a large fiscal expansion that would increase the amount of government bonds in the economy. Less capital demand would have to be justified by a higher real interest rate. Of course, this seems empirically slightly dubious. After all, the amount of government debt skyrocketed in 2008 and interest rates failed to rise. To understand why this wouldn't necessarily be consistent with higher real interest rates, it's important to think along the lines of a demand for government bonds.

Agents in this model are willing to demand more government bonds at higher interest rates, so if the government sets the supply of government bonds higher, then the demand must correspondingly rise through an increase in the real interest rate. The reason that massive increases in government debt in 2008 and 2009 are not consistent with higher real interest rates is that demand for government debt increased; perhaps even by more than the increase in supply. This was likely caused by the sudden illiquidity associated with other assets that were previously considered safe - e.g. mortgage backed securities or Greek government bonds. The resulting surge in demand for government bonds is known almost colloquially as a 'flight to quality.'

The ideal fiscal response to this is to satiate demand for government debt by running large deficits (note that this is the exact opposite of the policy actions taken by the majority of governments since 2008). In a way, this is a non-Keynesian reason for pursuing fiscal stimulus; more government debt would be useful for raising the real interest rate. Not only would this make the economy closer to a competitive equilibrium (one without government intervention), it would likely make monetary policy more effective. Narayana Kocherlakota, president of the Minneapolis Fed, made this point in a speech in July. The basic argument he presents is that the government can raise the long-run neutral real interest rate by increasing the amount of government debt. The higher neutral rate of interest (i.e. the real interest rate in this model, since there is no money) will make it so that the Fed will be less likely to hit the zero lower bound when trying to ensure that target is hit.

Effectively, fiscal policy should be used to remedy situations in which the demand for money is indeterminate and the central bank cannot adequately influence the real interest rate (see, e.g., here).

## 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.

## 15 December 2015

### Shut Up About Ricardian Equivalence

Economists that are both opposed to and in favor of fiscal stimulus frequently cite Ricardian equivalence as a reason that, in models with perfect credit markets, it doesn't matter whether stimulus is funded through increased taxes or through deficits. The problem with this analysis is that it assumes lump sum taxation. That is, taxes are not collected from things like consumption expenditures, which are effectively no different than deficits because 1) they don't discourage people from working, consuming, investing, etc and 2) they are expected to rise at some point in the future to retire the current debt, so the present value of taxes goes up with government spending. In reality, the argument is about distortionary taxes vs. deficits (= lump sum taxes, as per Ricardian Equivalence).

Distortionary taxes are different than their lump sum counterparts since they directly act to disincentivize working (in the specific case of income taxes, which I will limit my analysis to from now on) and can thus either partially or fully negate the effects of a fiscal stimulus. So, when John Cochrane says something like "'Ricardian Equivalence,' which is the theorem that stimulus does not work in a well-functioning economy," [1] he's clearly confusing the two types of taxation as well as ignoring the fact that neoclassical economics predicts a positive multiplier on government spending [2]. To illustrate this, I wrote down a standard Real Business Cycle model and ran two simulations: one in which a temporary fiscal expansion was financed entirely with an income tax and another in which the same fiscal stimulus was financed partially by deficits (see the appendix for a derivation of the model).
 Figure 1: Impulse Response Function of Output to the Stimulus
 Figure 2: Impulse Response Function of the Income Tax Rate to the Stimulus
 Figure 3: Government Spending in both simulations; Government Debt in the second simulation

The Ricardian Equivalence argument would be irrelevant if 1) the stimulus had a positive effect on output and 2) the tax funded stimulus was initially less effective than the partially deficit funded one. As you can see in figure 1, both of these are true; the stimulus positively impacted output in each simulation and the stimulus was initially more effective when taxes were not increased to fully finance the stimulus on impact. The effectiveness of the stimulus is slightly less sound of a result, though. The fiscal multiplier in neoclassical models is highly dependent on calibration (see, e.g., [2]) and can range anywhere from zero to one, without distortionary taxation, depending on the specific calibration used. Regardless, the most important part of this argument is sound; the Ricardian Equivalence argument against deficit funded stimulus is wrong and should be ignored completely as it applies to a form of taxation that doesn't actually exist.

References:

[1] John Cochrane, 2011. "Krugman on Stimulus" The Grumpy Economist.

[2] Woodford, Michael. 2011. "Simple Analytics of the Government Expenditure Multiplier." American Economic Journal: Macroeconomics, 3(1): 1-35.

Appendix:

The following is a derivation of model that I used to generate the impulse response functions in figures 1, 2, and 3.

Household:

There is a representative household who maximizes the utility function $U = E_0 \sum^\infty_{t=0} \beta^t\left(\frac{c_t^{1-\sigma}}{1-\sigma} - \frac{n_t^{1+\phi}}{1 + \phi}\right)$ where $E_t$ is the rational expectations operator given information available in period $t$, $c_t$ is the household's consumption, $n_t$ is the labor supply, and $\beta$ is the household's discount factor - the rate at which future utility is discounted relative to current utility. The household can use net-of-taxes income from labor ($(1-\tau^w_t)w_t n_t$, where $w_t$ is the real wage), government bonds carried from last period ($R_{t-1} B_{t-1}$, where $R_t$ is the interest rate that bonds maturing in period $t$ - $B_t$ - pay), and net-of-depreciation income capital, $(1 + r_{t-1} - \delta)k_{t-1}$ to purchase consumption, new government bonds, or new capital. The budget constraint can be written as

$$(1.1)\: (1-\tau^w_t)w_t n_t + R_{t-1} B_{t-1} + (1 + r_{t-1} - \delta)k_{t-1} = c_t + B_t + k_t$$

The household maximizes $U$ subject to $1.1$  in order to determine its behavior:

$$(1.2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta)$$
$$(1.3)\: R_t = 1 + r_t - \delta$$
$$(1.4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi$$

Additionally, it is useful to define investment, $i_t$ as the instrument of capital accumulation:

$$(1.5)\: k_t = (1-\delta)k_{t-1} + i_t$$

Firm:

The firm bundles capital carried from last period and labor using a Cobb-Douglas production function to form output, $y_t$

$$(2.1)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha}$$

The firm maximizes profits, $y_t - w_t n_t - r_{t-1}k_{t-1}$ subject to $2.1$ in order to determine labor and capital demand

$$(2.2)\: w_t = (1 - \alpha)\frac{y_t}{n_t}$$
$$(2.3)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t}$$

Government:

The government issues new government bonds and collects tax revenue to pay for both government spending and interest on government bonds carried from last period. The government budget constraint can be written as

$$(3.1)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1}$$

In the first simulation, it is assumed that the government ensures $B_t = 0\: \forall t$, so government spending is simply financed by taxes

$$(3.2)\: \tau^w_t w_t n_t = g_t$$

In the second simulation, the government sets the tax rate as a function of the tax rate consistent with the long run level of government spending, $\tau^w_{SS}$ and the level of government debt issued in the previous period, $B_t$. The rule for the tax rate in the second simulation is

$$(3.3)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1}$$

In both simulations, government spending follows an autoregressive process and returns to its long run trend trend at decay factor $\rho$. Government spending follows

$$(3.4)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t$$

Where $\eta_t$ also follows an autoregressive process with the same decay factor an is hit with with the shock $\epsilon^g_t$

$$(3.5)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t$$

Equilibrium:

Combining $1.1$, $1.5$, and $3.1$ yields the resource constraint for the economy

$$(1)\: y_t = c_t + i_t + g_t$$

Equations $1.2$-$3.5$ can be used to determine the equilibrium for the rest of the endogenous variables:

$$(2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta)$$
$$(3)\: R_t = 1 + r_t - \delta$$
$$(4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi$$
$$(5)\: k_t = (1-\delta)k_{t-1} + i_t$$
$$(6)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha}$$
$$(7)\: w_t = (1 - \alpha)\frac{y_t}{n_t}$$
$$(8)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t}$$
$$(9)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1}$$
$$(10a)\: \tau^w_t w_t n_t = g_t\: \mbox{in simulation 1}$$
$$(10b)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1}\: \mbox{in simulation 2}$$
$$(11)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t$$
$$(12)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t$$