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Article Excerpt
I. INTRODUCTION
In almost every standard monetary economy populated by representative infinitely lived agents, the optimal long run monetary policy is one in which the nominal interest rate is zero, also known as the Friedman rule (Friedman 1969). Researchers have demonstrated that this result is robust to a wide variety of modifications. (1) Starting with the seminal work of Levine (1991), a new burgeoning literature has emerged that studies environments with heterogeneity in which the Friedman rule is not optimal (see, e.g., Albanesi 2003, Edmond 2002, Green and Zhou 2002, Ireland 2004, Paal and Smith 2000, among others). This paper adds to this literature by characterizing the set of optimal monetary policies that is favored by heterogeneous agent types in a standard monetary economy. The novel punch line is that it is possible for every agent type to dislike the Friedman rule.
A major part of our analysis is conducted in a fairly standard pure exchange money-in-the-utility function (MIUF) economy modified to include the presence of two types of agents, distinguished by their different marginal utilities from real money balances. The introduction of this heterogeneity produces a nondegenerate stationary distribution of money holdings. Put simply, in a steady-state equilibrium, one type holds more money balances than the other. In this setting, faster money growth affects the welfare of each type through two channels. First, there is the rate-of-return effect: both types reduce their money holdings in the face of a higher opportunity cost of holding money. Second, if the central bank is restricted to making (imposing) the same lump-sum transfer (tax) on both types, a (general equilibrium) transfer effect emerges that alters agents' budget sets, affects their demand for money, and creates a divergence in their consumptions. (2) Indeed, for positive money growth rates, the type that holds more money contributes more to seigniorage than the other type but receives the same transfer, in effect causing a redistribution of income from the former to the latter. For negative money growth rates, the direction of the redistribution is reversed: now, the type that holds more money pays a smaller tax, in effect engineering an income transfer from the type that holds less money to the type that holds more money.
It is possible for the redistributive effect of an increase in the money growth rate to dominate the rate-of-return effect for some types of agents. In that case, an increase in the money growth rate may even be welfare enhancing. We are able to show that at least one of the types always dislikes the Friedman rule (locally), that is, they are better off in a lifetime welfare sense if the money growth rate increases locally around the Friedman rule money growth rate. At the Friedman rule, all agents are satiated with real balances. If the money growth rate (opportunity cost of holding money) increases infinitesimally, the envelope theorem implies that the resulting change in money demand can have at most a second-order impact on utility. However, since the two agent types hold different levels of real balances, this change in the rate of money growth has first-order distributional effects. These distributional effects are necessarily zero-sum: one type of agent benefits at the expense of the other. If social welfare is a population-weighted sum of individual types' utilities, then it follows that social welfare may be maximized at a rate of money growth away from that prescribed by the Friedman rule. This result lies at the heart of our analysis and serves to underscore the deeper connection between many other papers in the literature that question the optimality of the Friedman rule in environments with heterogeneous agents.
We go on to show that in most settings, the type that holds less money dislikes the Friedman rule (locally) but in special circumstances, which we discuss below, even the type that holds more money balances may join the other type in their shared distaste of the Friedman rule. Furthermore, if the type that holds more money dislikes the Friedman rule locally, their welfare is never maximized globally at a nonnegative money growth rate. Interestingly, a parallel result for the type that holds less money is that even if they like the Friedman rule locally, they may be globally better off at (possibly) a positive money growth rate. Perhaps most surprisingly, welfare of each type may be maximized away from the Friedman rule. In other words, it is possible for everyone to prefer positive nominal interest rates over Friedman's zero-nominal-interest-rate prescription.
An intuitive explanation for these results is in order. Recall that the type that holds more money contributes more to seigniorage than the other type but receives the same transfer. As a result, she receives net transfers when the money growth rate (i.e., inflation tax rate) is negative. The net transfer is simply the product of the inflation tax rate and the difference in money holdings of the two types. As the money growth increases starting from the Friedman rule money growth rate, the inflation tax rate rises; this rate-of-return effect lowers the net transfer and, therefore, always hurts the type that holds more money. The effect coming from the changes in agents' money holdings is more complicated. Much depends on the rate at which each type adjusts their money balances in response to an increase in the money growth rate, that is, on the elasticity of money demand. If both types reduce their money balances at similar rates in response to an increase in the inflation tax rate, then the aforementioned rate-of-return effect dominates; in this case, the type that holds more money likes the Friedman rule. Precisely for the same reason, the type that holds less money will not like the Friedman rule.
On the other hand, if the type that holds less money changes her money holdings at a faster rate than the other type, then the difference in money holdings grows as the money growth rate is raised. In such a setting, the type that holds more money would increase its net transfers and therefore dislike the Friedman rule; indeed, their welfare may be maximized at a much higher money growth rate. Under certain parameter sets, we find that the difference in money holdings responds nonmonotonically to the money growth rate; near the Friedman rule, it rises for a while and then starts to fall again. This makes the size of the redistribution respond nonmonotonically to the money growth rate. This explains why money growth rates higher than that implied by the Friedman rule, including positive money growth rates, may be welfare maximizing for one or both types. What is novel here is that while all agents may prefer some deviation from the Friedman rule, different types may want deviations of different sizes.
Thus far, we have deliberated on the effects of an increase in the money growth rate on type-specific welfare. What about societal welfare, a population-weighted aggregate welfare of both types? We are able to show that a sufficient (but not necessary) condition for societal welfare to not be maximized at the Friedman rule is that the type that holds less money locally dislikes the Friedman rule. This is because at the Friedman rule money growth rate, the rate-of-return distortion is absent and all agents are optimally satiated with real balances; however, the type that holds more money has the higher consumption but values it marginally less. As such, it may become efficient to redistribute some income away from these people, and this benefits the type that holds less money (hence, their "local dislike" of the Friedman rule). Somewhat interestingly, we can prove that the societal welfare--maximizing money growth rate is nonpositive. The intuition here is straightforward. Both types increase their money holdings as the money growth rate falls. Additionally, a zero money growth rate is preferred to a positive money growth rate because at the former, the transfer effect is absent and consumption is efficiently equalized across the types. At the other extreme of the Friedman rule money growth rate, as discussed above, it may become efficient to redistribute some income away from those who hold more money. This redistribution is achieved by choosing a money growth rate at which the transfer effect reallocates consumption such that the combined gain in utility from consumption dominates the combined loss of utility from the holding of smaller money balances. The novelty here is that the Friedman rule, contrary to received wisdom from many representative infinitely lived agent models, is not necessarily welfare maximizing. However, our analysis with heterogeneous agents does not go so far as to justify the use of an expansionary monetary policy.
A version of our result that the Friedman rule may not appeal to all types appears in Bhattacharya, Haslag, and Martin (2005). There, they show that it is quite possible (in a wide range of monetary environments) that one type may not like the Friedman rule. Unlike Bhattacharya, Haslag, and Martin (2005), we conduct our analysis in a standard representative infinitely lived agent model and go much further and characterize the set of monetary policies that each type likes. We show that it is possible that both types dislike the Friedman rule (something that is not possible in Levine 1991) and that the rule may not even maximize ex ante social welfare. Indeed, our analysis highlights several crucial components of the underlying political economy dimension of the larger question of the optimal monetary policy. It bears emphasis here that while the MIUF environment permits "closed-form" characterization of these results, many of the insights themselves are not specific to the chosen environment; indeed, they are applicable in standard cash-in-advance, turnpike, and shopping-time models of money.
The rest of the paper proceeds as follows. Section II presents the model economy, while Section III studies whether the Friedman rule is optimal for both types of agents. In Section IV, we study the optimal money growth rule that would be chosen by a social planner, while Section V studies the money growth rates that maximize type-specific welfare. Section VI concludes. Proofs of many of the results are relegated to the appendixes.
II. THE MODEL
In this section, we modify the standard representative-agent MIUF economy to include two types of agents distinguished by their preference for real money balances. The economy is populated by a continuum of unit mass of infinitely lived agents. Time is discrete and denoted by t = 0, 1, 2, ... , [infinity]. Let [micro] be the fraction of agents that place a relatively high value on the services from real money holdings, a notion that will be made precise below.
A. The Environment
There is a single consumption good which is perishable. Every period both types of households are endowed with constant [bar.y] > units of this good. (3) Money is the only asset in the economy. All agents maximize the discounted sum of momentary utilities over an infinite horizon. Agents who place a relatively high (low) value on the services of real money balances are referred to as type H (L). The preferences of the type i where i = H, L agents are represented by:
(1) [W.sup.i] [equivalent to][[infinity].summation over (t=0)][[beta].sup.t][U.sup.i] ([c.sup.i.sub.t],[m.sup.i.sub.t]), i...
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