Who is afraid of the Friedman rule?

Bhattacharya, Joydeep ; Haslag, Joseph ; Martin, Antoine 等

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] > 0 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 = H,L,

where 0 < [beta] < 1 is the agent's subjective rate of time preference; for a type-i agent, [c.sup.i] is the quantity of the consumption good, and [m.sup.i.sub.t] = [M.sup.i.sub.t]/[p.sub.t] denotes the quantity of real money balances carried over from period t to t + 1. We assume that [U.sup.i.sub.j] > 0 and [U.sup.i.sub.jj] < 0, i = L, H, j = m, c, where [U.sup.i.sub.j] = [partial derivative][U.sup.i]/[partial derivative]j and [U.sup.i.sub.jj] = [[partial derivative].sup.2][U.sup.i]/[partial derivative][j.sup.2]. Also, as is standard we posit that there exists a satiation level of real money balances such that [U.sup.L.sub.m]([c.sub.L], [m.sup.*L]) = [U.sup.H.sub.m]([c.sup.H],[m.sup.*H]) = 0 with [m.sup.*H] not less than [m.sup.*L]. Finally, we assume [U.sup.H.sub.m]([??],[??]) > [U.sup.L.sub.m]([??],[??]), [for all][??] [less than or equal to][m.sup.*H], for i = L, H. In words, for the same values of consumption and real balances, the type H derives greater marginal utility from the services associated with money than does a type-L agent.

Every period, an agent allocates its real balances from last period, current endowment, and transfers received from the government between current consumption and money balances to be carried over to the next period. Formally, the budget set of an agent i is defined by

(2) [bar.y] + [m.sup.i.sub.t-1]/(1 + [z.sub.t])+ [[tau].sub.t] [greater than or equal to] [c.sup.i.sub.t] + [m.sup.i.sub.t],

where 1 + [z.sub.t] = [P.sub.t] - l/[P.sub.t], [P.sub.t] is the price level in period t, and [tau] denotes transfers from the government. There are two maximization problems, one for each type of agent. The optimal choice for the type-i agents, i = L, H, is characterized by a sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that maximizes [W.sup.i] as given by Equation (1) subject to its sequence of budget constraints (Equation 2). It is easy to check that the relevant first-order condition is given by:

(3) [U.sup.i.sub.c]([c.sup.i.sub.t], [m.sup.i.sub.t]) = [U.sup.i.sub.m]([c.sup.i.sub.t],[m.sup.i.sub.t]) + [beta][U.sup.i.sub.c]([c.sup.i.sub.t+1],[m.sup.i.sub.t+1])/(1 +[z.sub.t]).

Equation (3) has a standard interpretation. At the margin, an agent is indifferent between consuming a unit this period versus carrying it over and consuming next period. The factor 1 + [z.sub.t] in the denominator of the second term captures the notion that carrying over a unit of nominal balance this period is worth 1/ (1 + [z.sub.t]) in the next.

The government runs a balanced budget period by period. At each date t [greater than or equal to] 0, the government finances a lump-sum tax or transfer, denoted [tau], by altering the money supply. Formally, the date-t government budget constraint is: [[tau].sub.t] = ([M.sub.t] - [M.sub.t-1])/Pt, where [M.sub.t] denotes the per capita quantity of nominal money at date t. We assume that the government follows a constant-money growth rule given by [M.sub.t] = (1 + z)[M.sub.t-1], where z > -1. The money supply expands if z > 0, so that [[tau].sub.t] > 0 is a transfer. Conversely, the money supply contracts if -1 < z < 0, so that [[tau].sub.t] < 0 is a tax.

B. Stationary Equilibrium

In a stationary environment, the price level increases at the same rate as the money supply. Hence, [p.sub.t] = (1 + z)[p.sub.t-1] is obtained. Thus, the money market-clearing condition can be represented as follows:

(4) [m.sub.t] = [mu][m.sup.H.sub.t] + (1 - [mu])[m.sup.L.sub.t],

where [m.sub.t] [equivalent to] [M.sub.t]/[p.sub.t] is the economywide stock of real balances. Further, in steady state, consumption and real money balances are constant over time so that [c.sup.i.sub.t] = [[bar.c].sup.i], [m.sup.i.sub.t] = [[bar.m].sup.i], and [m.sub.t] = [bar.m] for all t. Notice that [[tau].sub.t] = z[M.sub.t-1]/ [p.sub.t] = (z/(1 + z))[M.sub.t]/[p.sub.t] which in steady states reduces to [tau] = (z/(1 +z))[bar.m]. We assume that the amount of tax or transfer [tau] must be the same for both types of agents. This is the precise sense in which type-specific tax/transfer schemes are disallowed in our model. We justify this assumption by appealing to the implausibility of a tax/transfer scheme that attempts to identify people on the basis of their marginal preference for money, an object that is almost impossible for the government to observe.

Imposing steady state on Equation (3) yields

(5) [U.sup.i.sub.m]([[bar.c].sup.i],[[bar.m].sup.i])/ [U.sup.i.sub.c]([[bar.c].sup.i], [[bar.m].sup.i]) = 1 - [beta]/(1 +z) [equivalent to] [pi](z),

where [pi](z), by definition, is the opportunity cost of holding real balances. (4) For future reference, note that as 1 + z [right arrow] [beta], or [tau](z) [right arrow] 0, that is, when the money growth rate approaches the Friedman rule, the money holdings of each type reach their satiation levels. Finally, note that Equation (5) implies that, given z, a higher level of consumption is associated with a higher level of real money balances.

Using the agents' budget constraints (Equation 2), the government's budget constraint [tau] = (z/(l+z))[bar.m], and noting that Equation (4) in steady state implies [bar.m] = [mu][[bar.m].sup.H] + (1 - [mu])[[bar.m].sup.L], the agents' steady-state consumption is given by:

(6a) [[bar.c].sup.L] = [bar.y] + [mu](z/(1 + z))([[bar.m].sup.H] - [[bar.m].sup.L]),

(6b) [[bar.c].sup.H] = [bar.y] - (1 - [mu])(z/(1 + z)) ([[bar.m].sup.H] - [[bar.m].sup.L]).

Thus, [[bar.m].sup.L], [[bar.m].sup.H], [[bar.c].sup.L], and [[bar.c].sup.H] solve Equations (5)-(6b) simultaneously. Furthermore, it is easy to see that all the allocations can be implicitly represented as functions of z.

Notice from Equations (6a) and (6b) that heterogeneity in money balances affects consumption of each type. This is because an agent pays a type-specific seigniorage, (z/(1 +z))[[bar.m].sup.i], whereas the transfer, rebated by the government, (z/(1 +z))[bar.m], is type independent. Thus, (z/(1 +z))([bar.m] - [[bar.m].sup.i]), which is the second term in both equations, is the net transfer to an agent i. In the absence of any heterogeneity, this net transfer would be zero. Henceforth, we identify the second terms in Equations (6a) and (6b) as capturing the transfer effect. (5) Evidently, the transfer effect depends on the money growth rate and the difference between the real balances held by the two types.

Below, we will establish sufficient conditions under which the H types hold more money than the L types, that is, [[bar.m].sup.H] [greater than or equal to] [[bar.m].sup.L] will be obtained. We will further specify conditions under which both [[bar.m].sup.H] and [[bar.m].sup.L] monotonically decrease with z. The reason why we are unable to obtain condition-free results is the following. On the one hand, depending on whether the inflation tax rate is positive or negative, one or the other type is getting a net income transfer; the type that gets the transfer can afford to hold more money. However, the different marginal utilities from holding money also dictate whether they actually hold more money or not.

C. Money Growth Rate and Allocations

For analytical convenience, we assume a separable utility form given by:

[U.sup.i](c,m) = u(c) + [v.sup.i](m); i =- L,H,

where [v.sup.i](m) [equivalent to] [[lambda].sup.i][w(m) - mw' ([m.sup.*i])], and both u and w have constant elasticity of substitution (CES) forms, [c.sup.1-1/[sigma]]/(1 - 1/[sigma]. To conform to our assumptions made in the section "The Environment," we assume that [[lambda].sup.H] > [[lambda].sup.L] and [m.sup.*H] [greater than or equal to] [m.sup.*L] hold. Then, for any [??],

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We are then able to show the following.

LEMMA 1. Suppose

[[lambda].sup.H] > [sup.[chi]][[lambda].sup.L], (A. 1)

where [chi] [equivalent to] [u.sub.c] ([bar.y] - (1 - [mu])(1 - [beta])[m.sup.*L])/([u.sub.c]([bar.y] + [mu](1 - [beta])[m.sup.*L])) > 1. Then, [[bar.m].sup.H] > [[bar].m].sup.L] for all [m.sup.*]H > [m.sup.*]L, that is, the H types hold more money than the L types. If [m.sup.*H] = [m.sup.*L], [[bar.m].sup.H] > [[bar.m].sup.L] [for all] > [beta] - 1 and [[bar.m].sup.H] = [[bar.m].sup.L] at z = [beta] - 1.

Note that the assumption [[lambda].sup.H] > [[lambda].sup.L] is sufficient for H types to hold more money than the L types for all z [greater than or equal to] 0; Equation (A. 1) is only required when z [less than or equal to] 0. Intuitively, if with z [greater than or equal to] 0, L types held higher real balances than H types, there would be a net income transfer away from the L types. A lower income in addition to a lower marginal utility from money would imply that they are holding lower real balances than the H types, thus contradicting our initial supposition. For z < 0, suppose contrary to Lemma 1 that L types hold more money and thus receive net income transfers. Now, the income effect and the relatively lower preference for real balances work in opposite directions. Nevertheless, H types will hold larger real balances than L types if Equation (A. 1) is satisfied, that is, the preference for real balances of the H types is sufficiently larger than that of L types.

An immediate implication of Lemma 1 is the following.

COROLLARY 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

The type that holds more money gets the higher consumption if and only if there is deflation.

Further, differentiating Equations (6a) and (6b) yields

(9) (d[[bar.c].sup.L]/dz)/[mu] = -(d[[bar.c].sup.H]/dz)/(1 - [mu])

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notice first that a change in z affects income transfers between the two types, and thus, changes in consumption have opposite signs (see Equation 9). Lemma 1 ensures that the second term in Equation (10) is positive. Thus, a higher z brings more (less) income transfers for the L (H) types. The first term, on the other hand, depends on the differential rate of change of real balances of the two types. In general, away from the Friedman rule, it turns out that the second term in Equation (10) dominates the first, and thus consumption of L (H) types increases (decreases) with z. However, near the Friedman rule, as both types adjust their real balances relatively sharply toward satiation, the direction of consumption changes may depend on their rates of real balance adjustment relative to each other. If these adjustment rates are similar, the second term in Equation (10) still dominates and consumption of L (H) types increases (decreases) with z. However, with a specific set of parameters, we find that the difference in money holdings responds nonmonotonically with the money growth rate; near the Friedman rule, it rises for a while and then starts to fall again. Then, the direction of the changes in consumption is reversed.

Thus, in order to further study changes in allocations with respect to z, we need to first understand how real balances of both types change with z.

LEMMA 2. At the Friedman rule, real money balances of both types are decreasing in the money growth rate. Furthermore, suppose

(A.2) [bar.y] > [[bar.y].sup.**],

where [[bar.y].sup.**] [equivalent to] ([phi] - (1 - [beta])/[[lambda].sup.H])[m.sup.*H], and

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then,

real money balances for both types are decreasing in the money growth rate for all z [greater than or equal to] 0.

From Equation (5), it follows that if consumption remained the same, real balances would simply decrease with z, a pure price effect. However, as is clear from Equations (6a) and (6b), consumption of both types changes with z. Moreover, Equations (6a) and (6b) imply that (1) if the difference [bar.m].sup.H] - [[bar.m].sup.L] remained same, a higher z will bring more (less) income for the L (H) types and (2) [[bar.m].sup.H] - [[bar.m].sup.L] changes with z, which also impacts their income. The two income effects of z may combine or oppose each other but, in general, the first component dominates. As a result, as z increases, the total income of H (L) types decreases (increases). Thus, for H types, a higher z not only increases the opportunity cost of money but also decreases their income. As a result, [[bar.m].sup.H] is decreasing in z. On the other hand, the income of L types is increasing in z. Assumption (A.2) ensures that the income effect is dominated by the price effect of a higher z. (6) Thus, [[bar.m].sup.L] is also decreasing in z. (7)

We reiterate that Assumptions (A.1) and (A.2) are sufficient but not necessary.

III. WHO DOES NOT LIKE THE FRIEDMAN RULE?

In this section, we first show that for a general class of MIUF models, it is never the case that the Friedman rule is optimal for both types of agents. To verify whether this result holds under model specifications in which monetary policy has an output effect, we then study a cash-in-advance economy with production.

A. One Type Always Dislikes the Friedman Rule

We start by proving that for all the utility functions that incorporate satiation, the Friedman rule is disliked by one type. The marginal rate of substitution between consumption and real balances is given by Equation (5), which is repeated below for convenience:

(5) [U.sup.i.sub.m]([[bar.c].sup.i], [[bar.m].sup.i])/[U.sup.i.sub.c]([[bar.c].sup.i], [[bar.m].sup.i]) = 1 - [beta]/(1 + z) [equivalent to] [pi](z). (5)

Note that by assumption, [U.sup.L.sub.m]([[bar.c].sup.L], [m.sup.*L]) = [U.sup.H.sub.m]([[bar.c].sup.H], [m.sup.*H]) = 0. Therefore, at the Friedman rule, [[bar.m].sup.i] = [m.sup.*i].

The analysis in Section II implies that the equilibrium steady-state utilities of agents can be expressed as function of the money growth rate z. Further, using Equation (5), it follows that

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notice that the first term within brackets represents the transfer effect of changes in z, while the second term denotes its rate-of-return effect. Since real balances are decreasing in z, the rate-of-return effect hurts both types when z is increased. Note from Equation (5) that at the Friedman rule, the second term in Equation (11) vanishes. Thus, at the Friedman rule, a change in utility takes place solely through a change in consumption. From Equation (9), we know that the change in consumption for the two types has opposite signs. Thus, using Equation (9), it follows that

(12) d[W.sup.L]/dz = - ([U.sup.L.sub.c]/[U.sup.H.sub.c]) x ([mu]/(1 - [mu]))d[W.sup.H]/dz.

Hence, increasing z at the Friedman rule is always a local improvement for one type of agents. With homogeneous agents, at the Friedman rule all agents are satiated with real balances; the envelope theorem implies that a small increase in money growth will have at most a second-order impact on utility through the familiar inflation tax channel. When the two agent types hold different levels of real balances, this change in the rate of money growth has first-order distributional effects. But these distributional effects are necessarily zero-sum: one type of agent benefits at the expense of the other. We summarize the above discussion in the following proposition.

PROPOSITION 1. Given our assumptions, the Friedman rule is always (locally) disliked by one type.

Notice that at the Friedman rule, both types are optimally satiated with real balances. Hence, a small change in z (engineered via changes in real balances) has no rate-of-return effect on their welfare. However, changes in real balances do affect net transfers between agents; indeed, Equation (11) makes clear that the direct rate-of-return effect of an increase in z is washed out, leaving only the indirect transfer effect. As Equation (12) highlights, the transfer effect hurts one and benefits the other; as such, it can never be that, locally near the Friedman rule, both types will want money growth rates unchanged. Recall from Equations (6a) and (6b) that the transfer effect depends on the gap between real balances held by the two types. If this gap shrinks as z increases, net transfer to (from) H (L) types decreases. In that case, L (H) types will be made better (worse) off by a local deviation in z. On the other hand, if the aforementioned gap widens, net transfers will depend on changes in the product (z/(1 + z))([[bar.m].sup.H] - [[bar.m].sup.L]), which in turn will depend on the preference specification. Nevertheless, the change will hurt one type at the cost of the other. (8)

The following Lemma 3 establishes necessary and sufficient conditions to identify the agent type that would benefit from a marginal increase in z at the Friedman rule.

LEMMA 3. Given agents' preferences, L (H) types will prefer an increase in z at the Friedman rule, if and only if

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [c.sup.*L] and [c.sup.*H] denote consumptions of L type and H type, respectively, at the Friedman rule.

We can explain the condition (13) as follows. Suppose z is increased infinitesimally at the Friedman rule. Then, there will be a change in the net transfer between the two types attributable to two effects: (a) a change in inflation tax rate z/(1 + z) and (b) a change in the difference between the real balances of the two types [[bar.m].sup.H] - [[bar.m].sup.L]. Increasing z reduces z/(1 + z) in absolute value and thus reduces (increases) transfers to the H (L) types. However, if the difference between real balances widens, that is, d[m.sup.H]/dz - d[m.sup.L]/dz > 0, then H (L) types are better (worse) off by a larger transfer. The right-hand side (RHS) of condition (13) in Lemma 3 represents the tax rate effect, while the left-hand side (LHS) represents the effect of changes in real balances. If the widening of real balances dominates the tax rate change, the H (L) types will (will not) prefer a deviation from the Friedman rule. The situation is reversed if the widening of real balances is smaller or if it shrinks instead, that is, d[m.sup.H] / dz - d[m.sup.L] / dz < 0.

It is instructive to work through a special case. To that end, start by assuming that [m.sup.*H] = [m.sup.*L] holds; then it is obvious that [c.sup.*L] = [c.sup.*H] holds. In this case, notice that condition (13) in Lemma 3 reduces to

(14) 1/([[lambda].sup.H][w.sub.mm]([m.sup.*H)) - 1/([[lambda].sup.L][w.sub.mm]([m.sup.*L])) < ( > )0.

Since [[lambda].sup.H] > [[lambda].sup.L] and [w.sub.mm] < 0 holds, Equation (14) implies that the L types like the Friedman rule, but the H types would prefer a higher money growth rate. (9) Thus, in this case, even the H types (who always hold higher real balances relative to L types and, with z < 0, are the net receivers of income) dislike the Friedman rule. This can happen because of the following reason. Notice that while the Friedman rule obtains the agents a satiation level of real balances, it does not maximize their income from net transfers. Now as z rises, faced with a positive opportunity cost, both types reduce their real balances. However, the decrease in L types' real balances is sharper relative to that of the H types. Thus, with a marginal increase in z, the H types can obtain bigger transfers (which to them have a positive worth in terms of marginal utility of consumption), whereas losing real balances at the margin is costless to them since they are already satiated with real balances.

The same logic implies that L types will not prefer a local deviation from the Friedman rule. Note, however, that it is not clear from the above condition if the Friedman rule is globally preferred by L types. Finally, suppose that the condition stated in Lemma 3 holds in a way such that L types prefer a higher money growth rate than the Friedman rule. Again, even though now the H types do not prefer a local increase in z, it is not clear if the Friedman rule maximizes their welfare.

The above discussion raises two key policy questions. First, what are the most preferred type-specific money growth rules? And, more importantly, what is the socially optimal money growth rate? While the answer to the first question is postponed until Section V, the socially optimal level of z is studied next in Section IV.

B. Models in Which Superneutrality Fails

Is Proposition 1 simply an artifact of the assumptions in the model that yields superneutrality? If changes in the money growth rate distort output, do our results disappear? Below, we first present a simple extension of our model that adds a labor-leisure choice and which reaffirms the results stated in Proposition 1. Next, we contrast our results with a cash-in-advance set-up where monetary growth additionally creates an intertemporal price distortion that depresses output. Both extensions prove that the presence of superneutrality is not needed for the flavor of Proposition 1 to survive.

MIUF with labor-leisure choice. Here, each agent has a unit of time that it can divide between labor and leisure. Let agents' momentary utility be given by [U.sup.i](c, l, m), and let each type have access to an identical production technology described by f(l), where f has the standard properties of a production function. It is straightforward to show that the marginal rate of substitution between consumption and labor is given by

(15) -[U.sup.i.sub.l]/[U.sup.i.sub.c] = [f.sub.i]([l.sup.i]).

Now that each agent's output is given by f([l.sup.i]), using Equations (6a) and (6b), their consumption is given by:

[[bar.c].sup.L] = f([[bar.l].sup.L]) + (z/(1 + z)) [mu]( [[bar.m].sup.H] - [[bar.m].sup.L]), [[bar.c].sup.H] = f([[bar.l].sup.H]) - (z/l + z)) [mu] ([[bar.m].sup.H] - [[bar.m].sup.L]).

As before, each agent's allocations and utility can be implicitly expressed as a function of z. Differentiating the L types' utility with respect to z yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which using Equation (15) reduces to:

d[U.sub.L]/dz = [U.sup.L.sub.c](d[[mu](z/(1 + z))([m.sup.H] - [[bar.m].sup.L])]/dz) + [U.sup.L.sub.m](d[[bar.m].sup.L]/dz).

Again, the second term in the above equation vanishes at the Friedman rule. Combining the above with a similar equation for the H types replicates Equation (12).

As discussed above, the envelope theorem applies when considering small departures from the Friedman rule. Whether this means that the marginal utility of real money balances equals zero or that the marginal rate of substitution (MRS) between consumption and leisure equals the marginal product of labor at the Friedman rule is not crucial; either way, the fact that these efficiency conditions hold implies that the allocative effects of a small departure from the Friedman rule will have at most a second-order impact on welfare.

A cash-in-advance economy. The agents' heterogeneity now stems from their differential abilities to produce and therefore accumulate unequal real balances from the sale of their produce. Here, both types of agents have identical preferences in consumption (c) and leisure (1 - l), represented by a standard utility function u(c, 1 - l). Agents produce consumption goods by using the following technology

(16) [y.sup.i] = [[alpha].sup.i] f ([l.sup.i]), [[alpha].sup.H] > [[alpha].sup.L], f' > 0.

As is standard in these models, we assume that a household consists of a shopper-seller pair, who separate at the beginning of each period and then reunite in the end. While the seller works at the mill and sells the output, the shopper goes to the mills (other than her own) with cash to purchase goods. Note that the money accumulated through sales can only be used for purchases during the next period. Thus, once the inflation is taken into account, a unit of labor that earns [[alpha].sup.i]f'([l.sup.i]) units of goods today is worth only [[alpha].sup.i]f'([l.sup.i])/(1 + z) units tomorrow. At the optimum, an agent is indifferent between enjoying a unit of leisure today or working in the market and consuming [[alpha].sup.i]f' ([l.sup.i])/(1 + z) units of goods tomorrow. Thus, a household's optimal labor-leisure choice is given by:

(17) -[u.sup.i.sub.l] = [beta][u.sup.i.sub.c][[alpha].sup.i]f'([l.sup.i])/(1 + z),

where [u.sup.i.sub.j] [equivalent for] [u.sup.j]([[bar.c].sup.i], [[bar.l].sup.i]) Alternatively, Equation (17) equates the marginal rate of substitution between consumption and leisure [u.sup.i.sub.l]/[u.sup.i.sub.c] to its marginal rate of transformation [[alpha].sub.i]f' ([l.sup.i]) discounted by the gross nominal interest rate (1 + z)[[beta].sup.-1]. Were the labor earnings consumed during the same period, the relative price of earnings to consumption would identically equal 1. Thus, the cash-in-advance constraint lowers the price of earnings relative to consumption by 1 - [beta]/(1 + z), which discourages work relative to the case in which earnings are consumed contemporaneously.

Further, in the steady state, agents' consumption is given by (see Equations 42a and 42b in Appendix E)

(18a) [[bar.c].sup.L] = [[alpha].sup.L]f([[bar.l].sup.L]) + (z/(1 + z))[mu]([[bar.m].sup.H] - [[bar.m].sup.L]),

(18b) [[bar.c].sup.H] = [[alpha].sup.H]f([[bar.l].sup.H]) -(z/(1 + z))(1 - [mu])([[bar.m].sup.H] - [[sub.m]L).

Observe that the terms in the above expressions are identical to those in Equations (6a) and (6b), except that agents' output now depends on their optimal choice of labor which in turn depends on the money growth rate z.

Once again, agents' steady-state utilities can be expressed as functions of z. Then,

d[u.sup.i] / dz = [u.sup.i.sub.c] (d[[bar.c].sup.i] / dz) + [u.sup.i.sub.l] (d[[bar.l].sup.i] / dz),

which, using Equations (17)-(18b), yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Notice that the first term on the RHS of Equation (19) captures the rate-of-return effect, while the second term represents the transfer effect. As discussed above, the rate-of-return effect now stems from the intertemporal price wedge introduced by the cash-in-advance constraint. Under some mild restrictions on preferences, it can be shown that a higher rate of inflation z discourages work. (10) Then, as in our MIUF version, the rate-of-return effect implies that both types are hurt by an increase in z, while the net transfer effect benefits one type at the cost of the other.

Notice also that the intertemporal price wedge 1 - [beta]/(1 + z), and thus the rate-of-return effect, vanishes at the Friedman rule. The change in welfare can be attributed solely to the transfers, and once again, the result is identical to Equation (12) obtained for the MIUF version, that is,

d[W.sup.L]/dz = -([u.sup.L.sub.c]/[u.sup.H.sub.c])([mu]/(1 - [mu]))d[W.sup.H]/dz.

Thus, as before, one type dislikes the Friedman rule.

IV. SOCIAL WELFARE

The preceding analysis showed that precisely one type of agents will prefer a local deviation from the Friedman rule. That is, the type-specific welfare of one of the types is not maximized at the Friedman rule money growth rate. Is the Friedman rule "socially optimal" in this case? In order to answer this question, we first define social welfare W as a population-weighted sum of type-specific utilities. Formally:

W [equivalent to] (1 - [mu])[W.sup.L] + [mu][W.sup.H],

where [W.sup.H] and [W.sup.L] are as given by Equation (1). A benevolent central bank chooses z to maximize W (where [??] [equivalent to] arg [max.sup.z] W), that is, pick the z that solves d W/dz [greater than or equal to] 0. (11)

When is the Friedman Rule Socially Optimal?

Differentiating W with respect to z and using Equation (12) it can be shown that at the Friedman rule, that is, at [z.sup.FR] [equivalent to] [beta] - 1,

dW/dz|[sub.z]FR = [mu](d[u.sup.H]/dz) + (1 - [mu])(d[u.sup.L]/dz) = (1 - [mu])(1 - [u.sup.H.sub.c]/[u.sup.L.sub.c])(d[u.sup.L]/dz)

holds. Notice that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

If [m.sup.*H] = [m.sup.*L] holds, then [[bar.c].sup.H] = [[bar.c].sup.L] and [u.sup.L.sub.c] = [u.sup.H.sub.c] holds; here, the Friedman rule is also globally optimal as it allocates consumption efficiently while simultaneously allowing both types to hold their satiation level of real balances. On the other hand, if [m.sup.*H] > [m.sup.*L], at the Friedman rule, [[bar.c].sup.H] > [[bar.c].sup.L] and thus [u.sup.L.sub.c] > [u.sup.H.sub.c]. The following proposition is then immediate from an examination of Equation (20).

PROPOSITION 2. If [m.sup.*H] > [m.sup.*L], the Friedman rule is socially optimal only if the L types do not prefer a higher money growth rate.

Proposition 2 states that for the Friedman rule to be socially optimal, it is necessary that the L types locally like it. Conversely, it is implied that the Friedman rule cannot be socially optimal if increasing z yields a higher utility for the L types. At the Friedman rule, all agents are optimally satiated with real balances. Therefore, a marginal increase in z which cause real balance holdings to decline marginally is costless in terms of lost marginal utility. However, since [m.sup.*H] > [m.sup.*L], at the Friedman rule, [[bar.c].sup.H] > [[bar.c].sup.L] (see Equation 8), and therefore, the L types value a unit of consumption more than the H types do. So it is efficient to redistribute some income from the H to the L types in order to allocate consumption more efficiently. This would make the L types better off and render the Friedman rule socially suboptimal.

On the other hand, if L types prefer the Friedman rule to any marginal increase, then dW / dz|[sup.z]FR < 0. But, it does not ensure that the Friedman rule is also globally optimal. In Section V, we show that even when the L types prefer the Friedman rule locally, their type-specific optimal choice may turn out to be z > 0. Arguably, under such a scenario, a social planner may choose a [??] > [beta] - 1.

A. Can a Positive Money Growth Rate Ever Be Socially Optimal?

Clearly, if the L types do not like the Friedman rule, the planner's choice is [??] > [beta] - 1. Even otherwise, the planner may choose [??] > [beta] - 1. But can [??] ever be positive? The following proposition asserts that [??] must be negative.

PROPOSITION 3. The socially optimal money growth rate is negative, that is, [beta] - 1 [less than or equal to] [??] < 0.

The intuition behind Proposition 3 is quite straightforward. By choosing z > 0, the planner imposes a needless opportunity cost on all agents' stock of real balances; additionally, as argued above, by making [[bar.c].sup.H] < [[bar.c].sup.L], the planner engineers an inefficient income redistribution. If the money supply is constant, that is, z = 0, there is no income redistribution and [[bar.c].sup.H] = [[bar.c].sup.L]. The marginal social cost of reallocating consumption at z = 0 is essentially zero. Thus, both types can gain by holding marginally higher real balances; this can be achieved by marginally cutting z from z = 0.

Thus far, we have argued that the Friedman rule ceases to remain unambiguously optimal once the standard representative-agent paradigm is replaced with an environment with heterogeneous agents. Proposition 3, however, makes clear that the latter environment does not go so far as to justify an expansionary monetary policy.

V. TYPE-SPECIFIC OPTIMAL RULES

We go on to study the question: which money growth rate is globally liked by each type? In particular, is it possible that both types would like money growth rates that are higher than that implied by the Friedman rule? Can they each prefer positive money growth rates? Our analysis below shows that the type-specific welfare-maximizing values of z for both types, denoted as [[??].sup.L] and [[??].sup.H] crucially depend on their relative preference for real balances, particularly the money demand elasticities.

First, we specialize to a special functional form first popularized by Greenwood, Hercowitz, and Huffman (1988). Let utility be defined as follows:

(21) [U.sup.i]([c.sup.i], [m.sup.i]) = u[[c.sup.i] + [[lambda].sup.i](ln [m.sup.i] - [m.sup.i]/[m.sup.*i])]; i [equivalent to] H, L, [[lambda].sup.H] > [[lambda].sup.L]. (21)

We choose this form for two reasons. First, it enables us to make analytical progress and compute a closed-form solution for the optimal z that is liked by each type. Second, it differentiates between the rate-of-return and transfer effects with changes in z more sharply. Note that the basic dispute between the two types over the choice of z arises from the fact that their unequal real balances lead to unequal net transfers from the government, which in turn generates income effects for both the types. With a more general utility form, the income effect will affect agents' real balances as well as consumption. With Equation (21), real balances are insulated from the income effect and the changes in income are completely absorbed by the changes in consumption. As a result, the choice of real balances solely depends on the rate of money growth z.

Using Equation (5), the optimal demand for real balances is given by:

(22) [[bar.m].sup.i] = 1/([pi](z)/[[lambda].sup.i] + 1/[m.sup.*i]) = [m.sup.*i]/(1 + [pi](z)[m.sup.*i]/[[lambda].sup.i]),

where [pi](z) [equivalent] 1 - [beta]/(1 + z). It is clear from Equation (22) that both types are satiated with real balances at the Friedman rule. Further, real balances of both types decrease as the money growth rate is raised implying that the flavor of Lemma 1 continues to hold.

We maintain our assumption that [m.sup.*H] [greater than or equal to] [m.sup.*L] and [[lambda].sup.H] > [[lambda].sup.L] hold. In addition, if we further assume that

(A.3) [[lambda].sup.H]/[[lambda].sup.L] [greater than or equal to] [m.sup.*H]/[m.sup.*L]

holds, then as evident from Equation (22), a stronger version of the result in Lemma 2 also holds; indeed, under Equation (A.3), the H type's preference for real balances is uniformly stronger than the L type at all z. Both '=' and '>' in the above assumption are studied below.

B. Equal Elasticities of Money Demand

We further assume that the money demand elasticities of the two types with respect to z, denoted as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are equal. (12) First, note that

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [k.sup.i] [equivalent to] [m.sup.*i]/[[lambda].sup.i]. Then, for the money demand elasticities of the two types to be equal, it is required that

(24)[m.sup.*i] /[[lambda].sup.i] = [[kappa].sup.i] = [kappa], [for all]i,

holds. Further, notice that since [[lambda].sup.H] > [[lambda].sup.L] holds, it is implied that [m.sup.*H] > [m.sup.*L]. It directly follows from Equation (22) that

(25) [[bar.m].sup.H] - [[bar.m].sup.L] = ([m.sup.*H] - [m.sup.*L])/(1 + [kappa][pi](z)).

From Equation (25), it is obvious that [[bar.m].sup.H] - [[bar.m].sup.L] increases as money growth rate is lowered. In particular, this difference peaks at the Friedman rule.

Note from Equation (6b) that the net transfer to H types, which equals -(1 - [mu])(z/ (1 + z))([m.sup.*H] - [m.sup.*L])/(1 + [kappa][pi](z)), is positive when z < 0. A simple differentiation verifies that these transfers decrease as z increases. Clearly, at the Friedman rule, the H types enjoy the maximum consumption feasible at any z [greater than or equal to] [beta] - 1, in addition to satiating themselves with real balances. Thus, the Friedman rule is the best rule for the H types, that is, [[??].sup.H] = [beta] - 1.

The net transfer to the L types, on the other hand, is negative as long as z is negative. However, they do enjoy the benefits of a lower inflation by holding a higher stock of real balances. The optimal z for them, thus, depends on the trade-off between these two effects. At the Friedman rule, the rate-of-return effect vanishes as discussed in Section III. However, both the seigniorage tax rate z/(1 + z) and the difference between the real balances of the two types [[bar.m].sup.H] - [[bar.m].sup.L] decrease in absolute value at the Friedman rule, as z is increased. Thus, L types would benefit from an increase in the money growth rate as the absolute value of net transfers from them decreases. Then, the question is what is the optimal money growth rate for the L types? In particular, is a positive z ever optimal for them? To compute [[??].sup.L], we first obtain the consumption of L types by substituting Equation (25) in Equation (6a):

(26) [[bar.c].sup.L] = [bar.y] + [mu](z/(1 + z))([m.sup.*H] - [m.sup.*L])/ (1 + [kappa][pi](z)). (26)

Thus, [[??].sup.L] is obtained by maximizing L types' utility, that is, as a solution to

d[u.sup.L]/dz = ([u.sup.L])'[d{[[bar.c].sup.L] + [[lambda].sup.L](ln [[bar.m].sup.L] - [[bar.m].sup.L] / [m.sup.*L])} / dz] = 0.

Substituting Equations (22) and (26) into the above equation implies that [[??].sup.L] solves

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [pi]'(z) = [beta][(1/(1 + z)).sup.2]. The above equation simply states that at the optimum, the marginal cost of raising z in terms of its rate-of-return effect equals the marginal benefit of a higher z in terms of its transfer effect. Some algebra yields

(27) [[??].sup.L] = [[beta].sup.2][m.sup.*L/([beta][m.sup.*L] - [mu]([m.sup.*H] - [m.sup.*L]) x ((1 - [beta]) + 1/[kappa])) - 1.

The following lemma establishes the necessary and sufficient conditions which determine when [[??].sup.L] is positive.

LEMMA 4. The L types prefer a positive money growth rate if and only if

[m.sup.*H]/[m.sup.*L] > 1 + [beta](1 + 1/((1 - [beta])[kappa]))/[mu].

The higher the ratio [m.sup.*H]/[m.sup.*L] and higher the fraction of H types in the population, [mu], the higher is the transfer to the L types under a positive money growth rate. Then, it may be optimal for the L types to sacrifice utility from real balances in favor of higher income transfers. As an example, for [beta] = 0.96, [mu] = 0.5, [[lambda].sup.h] = 1, [[lambda] = 0.1, [m.sup.*H] = 100, and [m.sup.*L] = 10, the above condition is satisfied. Substituting these values in Equation (27) yields an optimal value [[??].sup.L] = 0.2539.

It is not possible to make any analytical progress toward the issue of globally optimal z, even using a logarithmic functional form. Below, we will present the results of a numerical exercise that will shed light on the questions that motivated this section. For the example below, we set [bar.y] = 2.28, [beta] = 0.96, and [mu] = 0.5.

EXAMPLE 1 (Logarithmic utility). Suppose [u.sup.i]([[bar.c].sup.i], [[bar.m].sup.i]) = ln [[bar.c].sup.i] + [[lambda].sup.i](ln [[bar.m].sup.i] - [[bar.m].sup.i]/[m.sup.*i]) where [[lambda].sup.H] = 1 > [[lambda].sup.L] = 0.1. Assume [m.sup.*H] = 100 and [m.sup.*L] = 10. Then, as illustrated below in Figure 1, the L types like a positive value of z, while the H types like the Friedman rule.

The exact story as told by this example is fairly robust to numerous changes in the parametric specifications.

[FIGURE 1 OMITTED]

C. Unequal Elasticities

In this section, we show that it is possible that neither type likes the Friedman rule. For this purpose, we drop the Assumption (24) and allow the elasticities of money demand to be unequal across the two types. In particular, we assume that

(28) [m.sup.*H]/[[lambda].sup.H] = [[kappa].sup.H] < [m.sup.*L]/[[lambda].sup.L] = [[lambda].sup.L].

For simplicity, we assume that the satiation level of real balances is same for both the types, that is, [m.sup.*H] = [m.sup.*L] = [m.sup.*]. However, we maintain our earlier assumption that [[lambda].sup.H] > [[lambda].sup.L].(13) Thus, Equation (22) can be rewritten as:

(29) [[bar.m].sup.i] = [m.sup.*]/(1 + [[kappa].sup.i][pi](z)).

Thus, [[bar.m].sup.H] > [[bar.m].sup.L] for all z > [beta] - 1. Assumption (28) implies that close to the Friedman rule, the elasticity of money demand for the L types exceeds that of the H types. Indeed, note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, our assumptions on preferences essentially imply that although the H types always hold a higher stock of real balances relative to the L types, the closer is the z to the Friedman rule, the faster is the rate of adjustment of real balances (to changes in z) of the L types relative to the H types.

Next, using Equations (6a) and (6b) with Equation (29), the steady-state consumptions can be rewritten as:

(30) [[bar.c]sup.L] = [bar.y] + [mu][m.sup.*](z/(1 + z))[1/(1 + [[kappa].sup.H][pi](z)) - 1/(l + [[kappa].sup.L][pi](z))], (30)

(31) [[bar.c].sup.H] = [bar.y] - (1 - [mu])[m.sup.I](z/(1 + z)) x [1/(1 + [[kappa].sup.H][pi](z)) - 1/(1 + [[kappa].sup.L][pi](z))].

We know from Equation (12) that one of the types would benefit if the central bank deviates from the Friedman rule. The following lemma clarifies that it is now the H types that dislike the Friedman rule. Indeed, as we show below, it is even possible for both types to disfavor the Friedman rule.

LEMMA 5. The Friedman rule is disliked by the H types; they would prefer a positive nominal interest rate. However, [[??].sup.H] [member of] ([beta] - 1, 0).

The result that [[??].sup.H] > [beta] - 1 has the following intuition. Recall from Equation (6b) that the net transfer to the H types depends on the gap between real balances of the two types. Although this gap is always positive, it may shrink or widen as z is decreased depending on the relative elasticities of the two types at any given z. Since the L types have a relatively higher elasticity of money demand close to the Friedman rule, the gap shrinks as z gets closer to the Friedman rule. Thus, it turns out that the net transfer to the H types becomes smaller as z gets closer to the Friedman rule. As the rate-of-return effect vanishes at the Friedman rule, [[??].sup.H] > [beta] - 1. On the other hand, it is clear that [[??].sup.H] < 0. At such a money growth rate, the H types gain on both dimensions: they receive positive net transfers from the L types and also benefit from the rate-of-return effect.

Also from Equation (12), it is clear that the L types would dislike a small deviation from the Friedman rule; hence, the L types like the Friedman rule locally. It remains to be checked whether the Friedman rule is also their global optimum. Below, we show that under certain parameter restrictions, the L types will be better off at some z > [beta] - 1.

The following lemma asserts that [[??].sup.L] either equals [z.sup.FR] or is positive. In addition, it establishes sufficient conditions when [[??].sup.L] > 0.

LEMMA 6. z [member of] ([beta] - 1,0) can never be optimal for the L types. Furthermore, if [[lambda].sup.H] >> [[lambda].sup.L], that is, if the preference of H types for real balances is sufficiently stronger than for the L types, [[??].sup.L] > 0 holds.

The intuition behind this result is straightforward. At the Friedman rule, not only the L types consume their total endowment but also they satiate themselves with real balances. The only way they can be induced to like any other z is if there is a net income and consumption gain that compensates for them for their resultant loss of real balances. When [[lambda].sup.H] is sufficiently large, the H type will hold a sufficiently large amount of money even when z > 0. As a result, at some z > 0, L types receive a level of net transfers that gives them a higher welfare than that available at [z.sup.FR]. We collect the punch line of the above discussion in the next proposition.

PROPOSITION 4. If [[lambda].sup.H] >> [[lambda].sup.L], both types dislike the Friedman rule.

Following the derivation in Appendix G, assume [m.sup.*] = e, [[lambda].sup.L] = 0.1. Then, for any [[lambda].sup.H] > 0.7699 even though the L types dislike a local increase in z at the Friedman rule, their global optimum now is [[??].sup.L] = 0.25.

VI. CONCLUDING REMARKS

By construction, monetary policy cannot have redistributive effects in representative-agent models. Yet, these effects are known to be quantitatively significant and important (see, e.g., Erosa and Ventura 2002). The purpose of this paper is to examine whether optimal monetary policy is sensitive to heterogeneity. The punch line is that the Friedman rule ceases to remain unambiguously optimal once the standard representative-agent paradigm is replaced with an environment with heterogeneous agents.

We develop a model economy in which the equilibrium distribution of money holdings is nondegenerate. The analysis essentially plays off the two effects of an increase in the money growth rate. There is the rate-of-return effect which causes both types to reduce their money holdings in the face of a higher opportunity cost. In the absence of type-specific taxes and transfers, a transfer/redistributive effect emerges. For example, in the case of 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.

The possible benefits of a net transfer of income may easily overwhelm the negative rate-of-return effect for some types of agents. In that case, an increase in the money growth rate may even be welfare enhancing for some. Much depends on the rate at which each type adjusts their money balances in response to an increase in the money growth rate. We show that at least one of the types always dislikes the Friedman rule (locally). We go on to show that 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, it is possible for everyone to prefer positive nominal interest rates over Friedman's zero-nominal-interest-rate prescription. In terms of the question posed by the title of this paper, the answer may be that everyone is "afraid" of the Friedman rule.

We also show that societal welfare, defined as the population-weighted aggregate welfare of both types in our model, is almost never maximized at the Friedman rule. However, our environment with heterogeneous agents does not go so far as to justify an expansionary monetary policy. The upshot is that unlike in models with representative agents, here the prescription for "optimal" monetary policy depends on whether welfare of the individual or that of society is being maximized. In this context, our analysis highlights some crucial components of the inevitable political economy dimensions of the larger question of the optimal monetary policy.

ABBREVIATIONS

CES: Constant Elasticity of Substitution

LHS: Left-Hand Side

MIUF: Money-in-the-Utility Function

MRS: Marginal Rate of Substitution

RHS: Right-Hand Side

APPENDIX

A. Proof of Lemma 1: [[bar.m].sup.H] > [[bar.m].sup.L]

First, for z > 0, we prove that [[bar.m].sup.H] > [[bar.m].sup.L] by contradiction. Choose any z > 0. Suppose [[bar.m].sup.L] [greater than or equal to] [[bar.m].sup.H]. Then, it follows from Equation (7) that ([[lambda].sup.H]/[[lambda].sup.L]) (w'([[bar.m].sup.H]) - w' ([m.sup.*H]))/(w'([[bar.m].sup.L]) - w'([m.sup.*L])) > 1. Then, from Equation (5), [u.sub.c]( [[bar.c].sup.H])/[u.sub.c]( [[bar.c].sup.L])>l which in turn implies [[bar.c].sup.L] > [[bar.c].sup.H]. But, given Equations (6a) and (6b), this violates our assumption. Hence, [[bar.m].sup.H] > [[bar.m].sup.L] for all z > 0.

Now, choose any z < 0. A sufficient condition for [[bar.m].sup.L] < [[bar.m].sup.H] is that (w'([[bar.m].sup.L]) - w'([m.sup.*L]))/(w'([[bar.m].sup.H]) - w' ([m.sup.*H]))> 1. Since from Equation (5) ([[lambda].sup.L]/ [[lambda].sup.H])(w' ([[bar.m].sup.L]) - w' ([m.sup.*L]))/(w' ([[bar.m].sup.H]) - w' ([m.sup.*H])) = [u.sub.c] ([[bar.c].sup.L])/[u.sub.c]( [[bar.c].sup.H]), we need to show that ([[lambda].sup.H]/[[lambda].sup.L])[u.sub.c]([[bar.c].sup.L])/ [u.sub.c]([[bar.c].sup.H]) > 1 holds. Notice that for all z > [beta] - 1, an upper bound for the consumption of L types is [bar.y] + [mu](1 - [beta])[m.sup.*L]. Similarly, a lower bound for the consumption of the H types is [bar.y] - (1 - [mu])(1 - [beta])[m.sup.*L]. Then, a lower bound for ([[lambda].sup.H]/[[lambda].sup.L])[u.sub.c] ([[bar.c].sup.L])/[u.sub.c]([[bar.c].sup.H]) equals ([[lambda].sup.H]/[[lambda].sup.L])[u.sub.c]([bar.y] + [mu](1 - [beta]) [m.sup.*L])/[u.sub.c]([bar.y] - (1 - [mu])(1 - [beta])[m.sup.*L]). Thus, a sufficient condition for [[bar.m].sup.L] < [[bar.m].sup.H] is

[[lambda].sup.H] > [sup.[chi][[lambda].sup.L],

where [chi] [equivalent to] [u.sub.c] ([bar.y] - (1 - [mu])(1 - [beta]) [m.sup.*L])/[u.sub.c]([bar.y] + [mu](1 - [beta])[m.sup.*L]) > 1. Finally, note that when [m.sup.*H] = [m.sup.*L], [[bar.m].sup.H] = [[bar.m].sup.L] holds at the Friedman rule trivially.

B. Proof of Lemma 2: d[[bar.m].sup.L]/dz, d[[bar.m].sup.H]/dz [less than or equal to]

As we have assumed that the consumption utility has a CES form, let u(c) = ([c.sup.1-1/[sigma] - 1/(1 - 1)/[sigma]),where [sigma] is the intertemporal elasticity of substitution and [sigma] = 1 represents the logarithmic case. Note that Equations (5)-(6b) simultaneously determine consumption and real money balances in steady state for type-i agents. Totally differentiating them together yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where Z [equivalent to] 1/(1 + z). Note that [u.sup.i.sub.c] [equivalent to] [u.sub.c]([[bar.c].sup.i]) and [u.sup.i.sub.cc] [equivalent to] [u.sub.c]([[bar.c].sup.i]). (14) Using Kramer's rule and after some algebra, we obtain

(32a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(32b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [DELTA] [equivalent to] ([v.sup.H.sub.mm]/[u.sup.H.sub.c]) ([v.sup.L.sub.mm]/[u.sup.L.sub.c]) - [sigma][[mu]([v.sup.H.sub.mm]/[u.sup.H.sub.c])/[[bar.c].sup.L] + (1 - [mu]) ([v.sup.L.sub.mm]/[u.sup.L.sub.c])/[[bar.c].sup.H]]Z[pi](z). Below, we evaluate the above derivatives for z [greater than or equal to] 0, z = [beta] - 1 and z [member of] ([beta] - 1.0) in three steps:

step i: z [greater than or equal to] 0. Note first that d[m.sup.H]/dz < 0, since all terms on the numerators are negative, while all in the denominator are positive. However, a sufficient condition for d[m.sup.L]/dz < 0 is

1 - [mu]([sigma]/[beta])(([[bar.m].sup.H] - [[bar.m].sup.L])/[[bar.c].sup.L])[pi](z) > 0.

The above inequality holds if

[c.sup.L = [bar.y] + [mu]Z([[bar.m].sup.H] - [[bar.m].sup.L]) > [mu]([sigma]/[beta])([[bar.m].sup.H] - [[bar.m].sup.L])[pi](z), i.e.,

[bar.y] > [mu]([[bar.m].sup.H] - [[bar.m].sup.L]) (([sigma]/ [beta])[pi](z) - Z).

As d[m.sup.H]/dz < 0, an upper bound for the RHS equals [max.sub.z[greater than or equal to]0] {(([sigma]/[beta])[pi](z) - Z)} [mu][[bar.m].sup.H]|sub.z=0]. Thus, a sufficient condition for d[m.sup.L]/dz<0 is that [bar.y]>[phi][[bar.m].sup.H]|[sub.z=0], where [phi] = [mu]([sigma]/[beta])(1 - [beta]) if [sigma] < 1, [phi] = [mu] (([[sigma]/[beta]) - 1) if [sigma] > 1. Note that [[bar.m].sup.H]|[sub.z=0] = [[lambda].sup.H][bar.y] [m.sup.*H]/((1 - [beta])[m.sup.*H] + [[lambda].sup.H][bar.y]). Hence,

[bar.y] > [[bar.y].sup.**] [m.sup.*H] ([phi] - (1 - [beta]])/ [[lambda.sup.H]) [??] d[[bar.m].sup.L]/dz < 0,

which we have assumed in the main text.

Step II: z = [beta] - 1. Finally, at the Friedman rule z = [beta] - 1, [pi](z) = 0, and then

d[[bar.m].sup.L]/dz = [[beta].sup.-1][u.sup.L.sub.c]/[u.sup.L.sub.mm] < 0 and

d[[bar.m].sup.H]/dz = [[beta].sup.-1][u.sup.H.sub.c]/[u.sup.H.sub.mm] < 0.

Step III: z < 0. Then, as Z < 0. the second term in the denominators of both Equations (32b) and (32a) is negative. However.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if

min{[v.sup.H.sub.mm]/[u.sup.H.sub.c], [v.sup.L.sub.mm]/ [u.sup.L.sub.c]}[max{[v.sup.H.sub.mm]/[u.sup.H.sub.c], [v.sup.L.sub.mm]/[u.sup.L.sub.c]} - [sigma][bar.y][pi](z)/([c.sup.L][c.sup.H])] > 0.

Thus. to ensure that the denominator is positive, one needs to restricts parameters such that

max{[v.sup.H.sub.mm]/[u.sup.H.sub.c], [v.sup.L.sub.mm]/ [u.sup.L.sub.c]} < - [sigma][bar.y][pi](z)/([c.sup.L][c.sup.H])] > 0.

With [U.sup.i](c, m) = ln c + [[lambda].sup.i](ln m - m ln [m.sup.*i]), for example. the above reduces to

max{[[lambda].sup.H][c.sup.H]/[([[bar.m].sup.H]).sup.2]} > [bar.y]/ [c.sup.L][c.sup.H]))[absolute value of min(Z[pi](z))].

Note that min(Z[pi](z)) = [-(1 - [beta]).sup.2]/4[beta]. For [beta] = 0.96. thus [absolute value of min(Z[pi](z))] [??] 0.0004, which implies that the RHS is relatively small in magnitude, and this condition can be easily made to hold. However, this is an implicit restriction on parameters, and the condition can at best be verified numerically after assuming plausible parameter values. If it is made to hold, the denominator is positive.

Similarly, as Z[pi](z) is sufficiently small, the term [sigma][bar.y]Z[pi](z)/([c.sup.L][c.sup.H]) can be ignored in comparison with other terms in the numerator of Equation (32b); the rest of the terms are negative. The sum of the first and the last term in the numerator of Equation (32a) is negative if 1 - [mu]([sigma]/[beta])(([[bar.m].sup.H] - [[bar.m].sup.L])/[[bar.c].sup.L]) [pi](z) > 0; a sufficient condition is then [bar.y] > [max.sub.z[member or]([beta]-1,0])] {(([sigma]/[beta])[pi] (z) - Z)}[mu][[bar.m].sup.*H], that is, [bar.y] > ([mu]/[beta])(1 - [beta])[m.sup.*H] if: [sigma] < 1 and [bar.y] > ([mu][sigma]/[beta])(1 - [beta])[m.sup.*H] if [sigma] < 1. Then, by making [bar.y] sufficiently large, the term [sigma][bar.y]Z[pi](z)/([c.sup.L][c.sup.H]) in the numerator of Equation (32a) can be ignored.

C. Proof of Lemma 3

At the Friedman rule, [pi](z) = 0. Then, using Equations (32a) and (32b), we obtain

d[[bar.m].sup.H]/dz - d[[bar.m].sup.L]/dz|[sub.[pi](z)= 0] = [[beta].sup.-1]([u.sup.H.sub.c]/[v.sup.H.sub.mm] - [u.sup.L.sub.c]/[v.sup.L.sub.mm]).

Thus, following Equations (11) and (9), a type-L (H) agent's utility will increase (decrease) with z at the Friedman rule if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where, from Equations (6a) and (6b), [c.sup.*L] = [bar.y] + [mu]((1 - [beta])/[beta])([m.sup.*H] - [m.sup.*L]) and [c.sup.*H] = [bar.y] - (1 - [mu])((1 - [beta])/[beta]) ([m.sup.*H] - [m.sup.*.sub.L]).

D. Proof of Proposition 3

Differentiating the social welfare function yields

(33) dW/dz = [mu](d[W.sup.H]/dz) + (1 - [mu])(d[W.sup.L]/dz).

Using Equations (6a), (6b), and (9) in Equation (11) and simplifying yield

(34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From Corollary 1, we know that for any z [greater than or equal to] 0, [[bar.c].sup.L] [greater than or equal to] [[bar.c].sup.H], [u.sup.L.sub.c] = [u.sup.H.sub.c]. Further, from Lemma 2, d[[bar.m].sup.H]/dz < 0 and d[[bar.m].sup.L]/dz < 0, and ([[bar.m].sup.H] - [[bar.m].sup.L]) > 0 by Lemma 1. Hence, all the terms on the RHS of Equation (34) are nonpositive. Thus, for all z [greater than or equal to] 0, dW/dz< 0 holds. Hence, z [greater than or equal to] 0 cannot be socially optimal.

E. A Cash-in-Advance Economy in Which Money Growth Affects Output

Let the agents be endowed with a unit of labor. Their period utility functions are identical

(35) [u.sup.i](c, l) [equivalent to] u(c, 1 - l), i [equivalent to] L, H,

where 1 - l is the amount of leisure they enjoy, and the function u(*,*) has the standard properties. The agents are differentially endowed with technologies

(36) [y.sup.i] = [[alpha].sup.i]f([l.sup.i]), [[alpha].sup.H] > [[alpha].sup.L],

and the f(*) has the standard properties of a production function. Each household consists of a shopper-seller pair. While the shopper goes to the market with cash to buy consumption good, seller works and sells output to the buyers who arrive at the factory outlet. Thus, at the end of period t, the seller accumulates the following money balances:

(37) [M.sup.i.sub.t] = [p.sub.t][y.sup.i.sub.t] = [p.sub.t] [[alpha].sup.i]f([l.sup.i.sub.t]).

In steady state, (Equation 37) can be rewritten as:

(38) [m.sup.i.sub.t] = [[y.sup.i.sub.t] = [[alpha].sup.i]f ([l.sup.i.sub.t]). (38)

The shopper, on the other hand, inherits nominal balances from the previous period, receives transfers from the government, and then goes out to shop. Thus,

(39) [p.sub.t][c.sup.i.sub.t] [less than or equal to] [M.sup.i.sub.t-1] + [T.sub.t].

Note that

[T.sub.t] = [M.sub.t] - [M.sub.t-1], or

[[tau].sub.t] = ([M.sub.t] - [M.sub.t-1])/[p.sub.t] = z[M.sub.t-1]/ [p.sub.t] = (z/(1 + z))[m.sub.t-1].

The cash-in-advance constraint (Equation 39) can be rewritten as:

(40) [c.sup.i.sub.t] = [m.sup.i.sub.t-1] + [t.sup.t].

Clearly, Equation (40) binds with equality in the steady state. Otherwise, exchanging excess real balances with consumption will be a strict improvement. The optimization problem maximizes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to constraints Equation (37) and Equation (40). The optimum is characterized by the following first-order condition:

(41) -[u.sub.l]([c.sup.i.sub.t], [l.sup.i.sub.t]) = [[alpha].sup.i]f' ([l.sup.i.sub.t])([beta]/(1 + z))[u.sub.c]([c.sup.i.sub.t+1],[l.sup.i.sub.t+1]); for i = L, H.

Steady State. In steady state, Equation (41) yields

-[u.sup.i.sub.l] = [[alpha].sup.i]f'([[bar.l].sup.i])([beta]/(1 + z)) [u.sup.i.sub.c],

which is Equation (17) in the main text. Further, Equation (40) can be rewritten as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the last step makes use of Equation (38). Since [bar.m] = [mu][[bar.m].sup.H] + (1 - [mu])[[bar.m].sup.L], the steady-state consumption is given by:

(42a) [[bar.c].sup.L] = [[alpha].sup.L]f([l.sup.L]) + (z/(1 + z)) [mu]([[bar.m].sup.H] - [[bar.m].sup.L]),

(42b) [[bar.c].sup.H] = [[alpha].sup.H]f([l.sup.H]) - (z/(1 + z)) (1 - [mu])([[bar.m].sup.H] - [[bar.m].sup.L]),

which are presented as Equations (18a) and (18b) in the main text.

F. Proof of Lemma 5

To check if the H types would like to deviate, we differentiate H type's utility aggregate with respect to z and use Equation (29) with Equation (31) to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Clearly, at the Friedman rule, [pi](z) = 0, the first and the last term vanish, while the second term is positive. Now, unlike the previous case, the H types would prefer z > [beta] - 1. But can it be that [[??].sup.H] > 0? The answer is negative, as seen from the derivative above. At z = 0, that is, [pi](z) = 1 - [beta], the second term vanishes and the first and the third term are negative. Thus, there lies a maximum for [[??].sup.H] [member of] ([beta] - 1,0). It is easy to see that this must be the H types' global maximum, as for any z > 0, they incur both a loss of consumption as well as a loss of real balances.

G. Proof of Lemma 6

Using Equation (30) in Equation (21), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Equation (29), the above condition can be rewritten as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that the RHS is always positive. But for any z [less than or equal to] 0, the LHS is nonpositive. Hence [U.sup.L] [[absolute value of [sub.z]FR > [U.sup.L]].sub.z] for all z [less than or equal to] 0.

For the second part, suppose [[kappa].sup.H] = 0. Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if

(43) ([[kappa].sup.L][pi](z)/(1 + [[kappa].sup.L][pi](z)))(1 + [mu]Z[[kappa].sup.L]) > ln(1 + [[kappa].sup.L.][pi](z)).

Fix any z = [??] > 0. It is easily shown that the above condition holds for all [[kappa].sup.L] [[??].sup.L] where [[??].sup.L] is obtained as an implicit solution of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By continuity, Equation (43) should hold for [[kappa].sup.H] > 0, provided [[kappa].sup.L] is then sufficiently larger than [[kappa].sup.H].

REFERENCES

Albanesi, S. "Optimal and Time Consistent Monetary and Fiscal Policy with Heterogeneous Agents." mimeo, Duke University, 2003.

Bhattacharya, J., J. H. Haslag, and A. Martin. "Heterogeneity, Redistribution, and the Friedman Rule." International Economic Review, 46(2), 2005, 437-54.

Chari, V. V., L. J. Christiano, and P. J. Kehoe. "Optimality of the Friedman Rule in Economies with Distorting Taxes." Journal of Monetary Economics, 37, 1996, 203-23.

Correia, I., and P. Teles. "Is the Friedman Rule Optimal When Money Is an Intermediate Good?" Journal of Monetary Economics, 38, 1996, 223-44.

da Costa, C., and I. Werning. "On the Optimality of the Friedman Rule with Heterogeneous Agents and Non-Linear Income Taxation." manuscript, MIT, 2003.

Edmond, C. "Self-Insurance, Social Insurance, and the Optimum Quantity of Money." American Economic Review Papers and Proceedings, 92, 2002, 141-47.

Erosa, A., and G. Ventura. "On Inflation as a Regressive Consumption Tax." Journal of Monetary Economics, 49, 2002, 761-95.

Friedman, M. "The Optimum Quantity of Money," in The Optimum Quantity of Money and Other Essays, edited by M. Friedman. Chicago, IL: Aldine Publishing Company, 1969, 1-51.

Gahvari, F. "Lump-Sum Taxation and the Superneutrality and the Optimum Quantity of Money in Life Cycle Growth Models." Journal of Public Economics, 36, 1988, 339-67.

Green, E. J., and R. Zhou. "Money as a Mechanism in a Bewley Economy." Federal Reserve Bank of Chicago Working Paper No. WP 2002-15, 2002.

Greenwood, J., Z. Hercowitz, and G. Huffman. "Investment, Capacity Utilization, and the Real Business Cycle." American Economic Review, 78, 1988, 402-17.

Ireland, P. "The Liquidity Trap, the Real Balance Effect, and the Friedman Rule." mimeo, Boston College, 2004.

Levine, D. "Asset Trading Mechanisms and Expansionary Policy." Journal of Economic Theory 54, 1991, 148-64.

Ljungqvist, L., and T. Sargent. Recursive Macroeconomic Theory. Cambridge, MA: MIT Press, 2000.

Paal, B., and B. D. Smith. "The Sub-Optimality of the Friedman Rule and the Optimum Quantity of Money," manuscript, UT-Austin, 2000.

JOYDEEP BHATFACHARYA, JOSEPH HASLAG, ANTOINE MARTIN and RAJESH SINGH *

* We thank an anonymous referee, Lutz Hendricks, Narayana Kocherlakota, Andy Levin, Randy Wright, Chris Waller, as well as seminar participants at University of California at Santa Barbara, the University of Alberta, the University of Kentucky, and the 2004 Missouri Economic Conference for useful comments. The views expressed here are those of the authors and not necessarily those of the Federal Reserve Bank of New York or the Federal Reserve System.

Bhattacharya: Associate Professor, Department of Economics, Iowa State University, 260 Heady Hall, Ames, IA 50011-1070. Phone 515-294-5886; Fax 515-2940221; Email [email protected]

Haslag: Professor, Department of Economics, University of Missouri--Columbia, 118 Professional Building, Columbia, MO 65211. Phone 573-882-3483; Fax 573-882-2697; Email [email protected]

Martin: Senior Economist, Payments Studies Function, Federal Reserve Bank of New York, 33 Liberty Street, New York, NY 10045. Phone 212-720-6943; Fax 212720-8363; Email [email protected]

Singh: Assistant Professor, Department of Economics, Iowa State University, 260 Heady Hall, Ames, IA 50011-1070. Phone 515-294-5213; Fax 515- 294-0221; Email [email protected]

(1.) See, for instance, Ljungqvist and Sargent (2000). Chari, Christiano, and Kehoe (1996) and Correia and Teles (1996) extended this to the case in which other distortionary taxes are present. More recently, da Costa and Werning (2003) examined a model with hidden actions, finding that the optimal policy is one with zero nominal interest rates.

(2.) Following Pigou and Patinkin, Ireland (2004) calls it the "real balance effect." If the government is allowed to make type-specific transfers, then the Friedman rule will again be optimal, as shown by Gahvari (1988).

(3.) The assumption of an endowment economy is harmless. It will be easy to see, in what follows, that introducing capital and endowing households with a production technology will yield a steady-state capital stock that is independent of monetary policy.

(4.) Note that the gross nominal interest rate 1 + i = [[beta].sup.-1](1 + z). Thus, [pi] = i/(1 + i).

(5.) An alternative explanation of the transfer effect is the following. Suppose there is no heterogeneity, and all agents were identically L types. As all seigniorage is rebated back to the agents, the net transfer will trivially be zero. Suppose instead that a fraction B of agents hold "excess real balances," [[bar.m].sup.H] - [[bar.m].sup.L] [greater than or equal to] 0. As the excess seigniorage (z/ ( 1 + z) )([[bar.m].sup.H] - [bar.m].sup.L]) raised from them is equally redistributed to all, it transpires that each agents (of both types) receive [mu](z/(1 + z))([[bar.m].sup.H] - [[bar.m].sup.L]) as "excess rebate," which equals the net transfer to an L type as in Equation (6a). On the other hand, each H type pays (z/(1 +z)) ([[bar.m].sup.H] - [[bar.m].sup.L]) but receives only [micro](z/(1 +z)) ([[bar.m].sup.H] - [[bar.m].sup.L]). As a result, each H type's loss of income equals (1 - [mu])(z/(1 +z))([[bar.m].sup.H] - [[bar.m].sup.L]). The above interpretation assumed z > 0. It is easy to argue that z < 0 simply reverses the direction of income redistribution.

(6.) Alternatively, (A.2) can be reformulated in terms of restrictions on the preference parameter [[lambda].sup.H]. We prefer to state it in terms of endowment because of its simple intuitive interpretation.

(7.) Lemma 2 asserts that real money balances for both types are decreasing in the money growth rate both locally near the Friedman rule and globally for all nonnegative money growth rates. While Lemma 2 does not claim a similar behavior for the allowable range of negative money growth rates, such behavior is in fact true. Numerical examples confirm it. We provide a sketch of the sufficient conditions in Appendix B. It bears emphasis that all our main results (see Propositions 1-3 below) only require that money balances be decreasing for all positive money growth rates and also at the Friedman rule, as predicated by Lemma 2.

(8.) Notice that the assumption of separability is not required for the result stated in Proposition 1.

(9.) By continuity, the same holds true even for cases where [m.sup.*H] > [m.sup.*L] , but [[lambda].sup.H] is sufficiently larger than [[lambda].sup.L].

(10.) For example, preferences of the form u(c, l) = u[c - [l.sup.v]/v], v > 1 will readily generate this result.

(11.) The inequality accounts for the case in which the Friedman rule money growth rate happens to be a corner solution.

(12.) The optimal value of z for the H type is critically affected by this assumption. In the next subsection, we allow the types to have different elasticities of money demand.

(13.) The equality of satiation levels is not necessary, and our results hold even if we allow [m.sup.*H] > [m.sup.*L]. If so, a [[lambda].sup.H.] sufficiently larger than [[lambda].sup.L] will generate the results that follow. See the discussion that follows Lemma 3.

(14.) Recall from our assumption in the section "Money Growth Rate and Allocations" that the functional form of the consumption utility is identical for both types.