## Real Interest rates.

Monetary PolicyPosted by Huw Dixon Mon, February 27, 2017 22:06:36Can there be an equilibrium real rate of interest that is negative? In general, the answer is no. If we think of a steady-state where consumption is constant, then because the household discounts the future, the real rate of interest has to be strictly positive. The real interest rate corresponds to the marginal product of capital. In a representative agent economy, a negative real interest rate is possible as a transitory phenomenon, and would correspond to a decumulation of capital indicating that the capital stock was too large and hence the household would seek to reduce it by maintaining a high level of consumption with possible dis-saving (consumption in excess of income). A negative real interest rate would be a temporary phenomenon on the path to steady-state: along the path, as the capital stock is reduced, the real interest would get back into the positive territory. In the Ramsey model, a very large initial capital stock yielding a negative marginal product would result in a high level of consumption which fell over time, with dis-investment.

Matters are different in OLG models, which are not in
general dynamically efficient. In a simple exchange economy without production,
it is possible to get a negative real interest rate in equilibrium. If current consumption is cheaper than future
consumption, you need to give up more now to get less in the future. The key assumption needed is that there is no
storage or capital: one generation trades with another. Eggertson et al (2017)
have a model where people live for three periods: they have endowments middle
age and when old: they borrow when young.
Assuming that the endowment is largest in middle-age, consumption smoothing
indicates that they will borrow when young, and save when middle aged to
augment their retirement consumption (the old consume everything they
have). At any time, there are all three
generations living together. The middle aged at time t can only save for when
they are old in t+1 by lending to the young at time t, who will repay the old
at t+1. Here, the young demand loans
(consumption) from the middle aged; the middle aged lend to them so that next
period they get paid back and their old aged consumption is increased. The real interest rate here can be positive
or negative, depending on the balance between the supply and demand for loans.

*a unique, locally determinate secular stagnation equilibrium”*

*(*

*Proposition 1, page 21. Figures 4 just above the proposition make the essential role of deflation clear).*However, the secular stagnation equilibrium must have deflation: negative inflation. If inflation is positive, then there will be full employment. This is because the mechanism reducing output is the increase in real wages. So, in a secular stagnation equilibrium, the nominal interest rate is at the ZLB (zero lower bound), output is below full employment, inflation is negative and real wages above their full employment level (due to downward rigidity). The actual real interest rate is positive (equal to minus the deflation rate): it is a hypothetical real rate that is negative (the real rate that would restore full employment). The ZLB does not lead to an equilibrium negative real interest rate: it prevents the real interest rate from becoming negative when inflation turns negative.

Have we observed negative inflation? In the UK and the US just the occasional month in 2016 and (depending on whether you use CPI or RPI) perhaps for a month or two at the height of the crisis. Japan has had more disinflation since the late 90s (disinflation “peaked” at just over -2% in 2009). The Eurozone is a mixed bag: the aggregate inflation rate has mainly been strictly positive with a few exceptions as in the UK and US. For individual countries the story is more heterogeneous. So, if we look at the major economies, there is no evidence of sustained disinflation that might give rise to the high real rates required for Eggertsson Stagnation. In fact we find the exact opposite. The ZLB is combined not with negative inflation, but positive inflation. Rather than positive real rates, we find real rates are negative.

This brings us to
the most important an obscure part of the paper: section 8, the model with over
100 equations. Here there are lots of
generations and capital is introduced. The key equations are buried in the
appendix: A81 and A82. The marginal productivity for capital A81 is the usual:
the marginal product of capital will be strictly positive. Then there is
A82. This is a little different: there
is a price of capital goods term. Greg Thwaites has developed a model of
falling real interest rates driven (in part) by the falling price of investment
goods. There has been a downward trend in the prices of investment goods
(relative to consumption goods), which means that savings leads to more
investment (but possibly lower investment expenditure). This can drive down the
marginal product of capital. However, in Thwaites model the real interest rate
may be low, but is always positive. So what is it in the Eggesrtsson model that
can give you their figure 7: secular stagnation with strictly positive
inflation (recall, this was impossible in the world of proposition1). I must
admit, that I have read the paper and am none the wiser about how this might be
possible. The paper just presents a
calibration and reports that this is what happens. Unlike the world of
proposition 1 there is no clear story or intuition.

I do not doubt that with sufficient inventiveness a model with equilibrium negative real interest rates can be constructed. But it would not be a basis for monetary policy. Monetary policy needs to be based on robust models that have passed the test of time, not on exotica. I will continue to believe that real interest rates should be strictly positive in equilibrium.

## Posted by Huw Dixon Mon, February 27, 2017 22:08:37

The two papers mentioned are:

A Model of Secular Stagnation: Theory and Quantitative Evlauation,Gauti B. Eggertsson, Neil R. Mehrotra, Jacob A. Robbins. NBER Working Paper 23093 (2017).Why are real interest rates so low? Secular stagnation and the relative price of investment goods, Greg Thwaites. Bank of England working paper 564 (2015).