## 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).