Zero Nominal Interest Rates: Why They’re Good and How to Get Them

This study shows that in a standard one-sector neoclassical growth model, in which money is introduced with a cash-in-advance constraint, zero nominal interest rates are optimal. Milton Friedman argued in 1969 that zero nominal rates are necessary for efficient resource allocation. This study shows that they are not only necessary but sufficient. The study also characterizes the monetary policies that will implement zero rates. The set of such policies is quite large. The only restriction these policies must satisfy is that asymptotically money shrinks at a rate no greater than the rate of discount. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. In a classic essay, Milton Friedman (1969, p. 34) states that only monetary policies that generate a zero nominal interest rate will lead to optimal resource allocations. He argues that “it costs . . . no physical resources to add to real cash balances,” and hence it follows that “the optimum quantity of money . . . will be attained by a rate of price deflation that makes the nominal rate of interest equal to zero” (italics in original). This prescription of zero nominal interest rates has come to be known as the Friedman rule. Friedman’sargumentconvincinglyshowsthatzeronominal interest rates are necessary for efficient resource allocation. However, Friedman leaves three key questions unanswered. First, are zero nominal interest rates not only necessary, but sufficient to ensure an optimal allocation of resources? For example, suppose there is a severe price deflation at the same time that nominal interest rates are zero. Individuals might (inefficiently) lower their capital holdings to take advantage of the high real rate of return offered by money. Second, what kinds of monetary policies implement zero nominal interest rates, in the sense that the policies are consistent with the existence of an equilibrium with zero nominal interest rates? If money growth and inflation rates are equal in equilibrium, then one way to implement zero nominal interest rates would seem to be to shrink the money supply at the efficient rate of return on capital (net of depreciation). Is this true? And, if so, is it the only possible monetary policy that produces zero nominal interest rates? Finally, we must confront the question of unique implementation. For a particular specification of monetary policy, while there may be one equilibrium in which nominal rates are always zero, there may also be one or more equilibria in which they are not. A central bank cannot force individuals to coordinate on its desired equilibrium if other, less desirable equilibria are possible. Hence, we would like to know, What are the characteristics of monetary policies which only implement zero nominal interest rates? In this article, we use a simple economic model to address these questions of optimality, implementation, and unique implementation of monetary policy. The model is a standard one-sector neoclassical growth model that has one main friction: a cash-in-advance constraint that requires households to use cash balances accumulated before each period to buy consumption goods in that period. The cash-in-advance constraint is a simple way to motivate a transactions demand for money: when interest rates are positive, households do not hold money as a store of value, but rather only because they need money to purchase consumption goods. Similarly, the cash-in-advance constraint is generally viewed as a clean way to incorporate the quantity theory of money into a decision-theoretic framework. In particular, if nominal rates are positive, then in each period, households hold only enough money to fund their purchases of consumption goods in the next period. This implies that (consumption) velocity is constant at one, so the inflation rate in any period is equal to the difference between the rates of money growth and consumption growth (which is the essence of the quantity theory of money). We first use the model to assess the characteristics of interest rates when monetary policy is optimal. The cashin-advance constraint implies that households have to wait until next period to use their current wage earnings to buy goods. Consequently, households equate their marginal rate of substitution between consumption and leisure not to their marginal product of labor, but rather to their marginal product of labor discounted by the time value of money. We show that this wedge can be eliminated if and only if the time value of money—that is, the nominal interest rate—is zero in every period. Next, we completely characterize the set of monetary policy rules that implement zero nominal interest rates. Interestingly, the set is defined only by the long-run behavior of monetary policy; even extreme contractions and expansions of the money supply are consistent with zero nominal interest rates as long as such movements do not last for an infinite amount of time. Correspondingly, in these equilibria, real balances may vary considerably and, in fact, can grow exponentially. Finally, we show that, at least when households have utility functions that are logarithmic in consumption and additively separable in consumption and leisure, there is a large set of policies that uniquely implement zero nominal interest rates. An example of such a policy is one that leads money to shrink for a finite number of periods at a rate no slower than households’ psychic discount rate and to shrink thereafter exactly at the psychic discount rate. The intuitive explanation for this example is simple: if the nominal interest rate is positive in any period in this kind of economy, households hold only enough money to buy their desired level of consumption goods. Hence, if the nominal interest rate is to be positive, then the rate of price deflation has to equal the rate of money shrinkage; but this in turn implies a nonpositive nominal interest rate. Our results have a key theoretical implication. Most economists’ intuition about the (long-run) effects of changes in the supply of money is shaped by Friedman’s (1963 [1968, p. 39]) famous dictum that “inflation is always and everywhere a monetary phenomenon.” Our main message is that while inflation is a monetary phenomenon for any suboptimal monetary policy, inflation is entirely a real phenomenon for any optimal monetary policy (because the rate of deflation equals the real rate of interest). Our results also have a striking policy implication. Zero nominal interest rates are consistent with a large set of monetary policies. This means that the optimality of monetary policy can be verified only by looking at interest rates, not by looking at the growth rates of the money supply. The Environment Without Money . . . In this section, we set out the physical environment in which agents interact, and we characterize efficient allocations in that environment. We consider an infinite-horizon environment with a continuum of identical households. Each household has a unit of time in every period; this time can be split between leisure lt and work nt. There is a single consumption good. In period t, the typical household ranks streams of consumption and leisure (ct+s,lt+s) ∞ s=0 according to the utility function (1) ∞ s=0 βu(ct+s,lt+s). The utility function u is strictly concave and continuously differentiable and satisfies the conditions that uc(0,l ) = ∞ for all l and ul(c,0) = ∞ for all c. At the beginning of period 1, there are k0 > 0 units of capital. (All quantities are written in per capita terms.) In period t, capital and labor can be used to produce output according to the production function (2) yt = f (kt−1,nt). The production function f is continuously differentiable, homogeneous of degree one, and concave. Output yt can be split between consumption ct and investment xt: