Optimal State-dependent Monetary Policy Rules ∗ Christian Baker†

Richard W. Evans‡

April 2014 (version 14.04.a)

Abstract

This paper defines a monetary equilibrium and computes an optimal nonlinear, full-information, state-dependent monetary policy rule to which the monetary authority commits at the beginning of time. This type of optimal monetary policy represents a combination of the flexibility of discretion with the time consistency of commitment. The economic environment is a closed-economy general equilibrium model of incomplete markets with monopolistic competition, producer price stickiness, and a transaction cost motive for holding money. We prove existence and uniqueness of the competitive equilibrium given a monetary policy rule and prove existence of the optimal rule. We show that the optimal state-dependent monetary policy rule satisfies the standard results of the discretionary policy literature in that it keeps inflation and nominal interest rates low (Friedman rule) and reduces inefficient variance in prices. Lastly, we compare the optimal monetary policy rule to a limited-information Taylor rule. We find that the Taylor rule, based on observable macroeconomic variables, is able to closely approximate the economic outcomes of the model under the optimal full-information rule. keywords: Optimal monetary policy; Money supply rules; Time consistency; Nonlinear solution methods JEL classification: E52; E42; E31; C68. ∗

We thank Ian Fillmore for excellent research assistance and substantial contributions early in this project. We also thank Dave Spencer, Kerk Phillips, Jeff Humpherys, Scott Condie, Val Lambson, and Brennan Platt for helpful comments and suggestions. This paper benefitted from comments and suggestions from seminar participants at the BYU-USU Macro Workshop. This work was supported in part by the National Science Foundation, Grant No. DMS-0639328. † Brigham Young University, Department of Economics, B086A JSFB, Provo, UT 84602, [email protected]. ‡ Corresponding author: Brigham Young University, Department of Economics, 167 FOB, Provo, Utah 84602, (801) 422-8303, [email protected].

1

Introduction

Fischer (1990) claimed that the “rules versus discretion debate in monetary policy is at least 150 years old.” In their survey on how modern macroeconomic theory is shaping policy, Chari and Kehoe (2006) highlight how this debate is still central to questions of what optimal monetary policy should look like. The optimal optimal monetary policy rules arising from models in which the monetary authority is characterized as being committed to a rule generally end up being less inflationary on average and easier to compute than optimal rules arising from models in which the monetary authority is characterized as having discretion. However, Fischer (1990) also notes that every monetary system in the last 150 has operated with “substantial discretionary authority.” This paper attempts to bridge the gap between the literature on monetary rules under commitment and the discretionary monetary policy literature, building from the rules and commitment side. We define a monetary equilibrium and compute an optimal nonlinear state-dependent monetary policy rule to which the monetary authority commits at the beginning of time. The economic environment is a closedeconomy general equilibrium model of incomplete markets with monopolistic competition, producer price stickiness, and a transaction cost motive for holding money on the part of households. We show that this optimal state-dependent rule satisfies the standard results of the discretionary policy literature in that the optimal rule manages the two competing incentives of generating low inflation and nominal interest rates (Friedman rule) and reducing inefficient variance in prices.1 This optimal monetary policy rule assumes full information on the part of the monetary authority in being able to observe the underlying fundamental shocks to the economy. 1 The Friedman rule of low inflation and nominal interest rates comes from the transaction cost friction. Because households have to hold money to make transactions, the opportunity cost of holding money is the nominal interest rate. The social cost of producing money is near zero. So the policy maker equates the social cost and the private cost of money by creating negative inflation (often deflation), which can be interpreted as lowering nominal interest rates or raising the return on money holdings. The incentive to reduce price variability comes from the sticky price friction causing relative price inefficiencies when the economy is hit by a shock. Optimal monetary policy tries to formulate policy rules that reduce these relative price inefficiencies.

1

An additional theoretical contribution of this paper is that we prove existence and uniqueness of the functional competitive equilibrium in this model given a money growth rule. We then use the existence and uniqueness of the competitive equilibrium to prove the existence of the optimal monetary rule. This proof, adapted from the mathematical literature on Hammerstein integral equations,2 provides analytical and computational support for an important class of models in monetary economics. We then compare the optimal full-information monetary rule to an optimal Taylor rule that allows monetary policy to respond only to current observed values of macroeconomic variables without observing the underlying fundamental shocks. We find that the limited-information Taylor rule results in approximately the same average interest rate and a nearly identical firm pricing function in comparison to the optimal full-information monetary rule. However, we highlight that the slight loss in household utility from the limited-information Taylor rule comes from its inability to condition on the current productivity level in the economy, causing household consumption to respond with a slight lag to the underlying shocks to the economy. We interpret this finding as evidence that policy rules based on observable macroeconomic variables can closely approximate the economic outcomes of more flexible policy rules based on the underlying fundamentals of the economy, which are often unobservable. The primary difficulty of solving for general equilibrium monetary policy when the monetary authority has discretion arises from dynamic inconsistency of the model in which a plan that the monetary authority wants to commit to today becomes suboptimal in the future. The cannonical example is provided by Kydland and Prescott (1977). Barro and Gordon (1983) were the first to characterize the set of multiple reputational equilibria that can be supported in these models of monetary equilibrium and how these equilibria are supported by a grim trigger strategy mechanism. Marcet and Marimon (2011), Khan et al. (2003), and Klein et al. (2008) solve for a restricted set of discretionary monetary equilibria that exclude any reputational equilibria by 2

See Li et al. (2006).

2

characterizing the equilibrium as a game between successive policy authorities.3 Ad˜ao et al. (2003) study optimal discretionary monetary policy in a closed economy with cash-in-advance constraints and a simple price friction mechanism. But their environment contains a complete set of riskless state-contingent bonds, thereby inducing complete markets and a zero lower bound on interest rates. In this discretionary environment, they also demonstrate a continuum of equilibria. In contrast, we focus on an incomplete markets environment, which requires a different set of both analytical and computational techniques. Our goal is to be able to match the results of the incomplete markets discretionary monetary policy literature of the particular recursive contract style of Khan et al. (2003) with a rule under commitment that has enough flexibility to have the flavor of discretionary policy. Using a rules-based approach allows us to better compare the optimal policy to the large literature on Taylor rules that has been implemented to some degree in many central banks. Further, a Taylor-type monetary policy rule in our model implies a type of limited information that is analogous to some observations about real world monetary policy. Our model environment closely follows Ireland (1997). The main differences in our model are the presence of aggregate uninsurable shocks to productivity and the possibility for the monetary policy rule to be any function of past information rather than a constant money growth rate. With this more flexible policy rule in a stochastic environment, Ireland’s proof of existence and uniqueness of the competitive equilibrium no longer applies. One nice property of using stochastic productivity as the shock process in our model is that those shocks satisfy the policy invariance criterion of Chari et al. (2009), whereas many other types of demand shocks might not (e.g., government spendint). Having only one shock in our model reduces its quantitative usefulness as many other monetary models include multiple shocks in order to match quantitative moments of the data. We, therefore, see our model as a simple benchmark that provides qualitative results and an analytical and computational roadmap. 3

The optimal policy equilibrium concept of these papers follows the approach of the optimal penal codes work of Abreu (1988) and Abreu et al. (1990).

3

Further, our solution method allows for fully nonlinear, albeit continuous, functional solutions. Notwithstanding, our optimal policies and equilibrium pricing rules exhibit only small amounts of curvature in the state variables, lending some support to the validity of linearization methods.4 Our modeling approach includes a cash-in-advance constraint, which is an extreme form of transaction cost motive for holding money. A more general approach is used in Khan et al. (2003) in which a continuous cost parameter governs the relative ease with which less liquid assets can be used to purchase goods. In addition, we use Ireland’s more extreme price friction of all firms choosing their prices at the beginning of the period before the productivity level is seen. Styles of price friction that could be more closely calibrated to match that of the real world include the staggered pricing models of Taylor (1980) and Calvo (1983). Again, our simple transactions and pricing frictions mean that the quantitative results of this paper should not be taken too seriously. However, our qualitative results match the findings of the discretionary policy literature and they also shed light on how closely optimal limited-information Taylor rules can match the welfare and economic outcomes of optimal full-information monetary rules. The rest of the paper is structured as follows. Section 2 outlines the household problem, firm problem, and market clearing conditions of the model given a monetary policy rule. Section 3 describes the competitive equilibrium given a monetary policy rule and its existence and uniqueness, the optimal monetary equilibrium and its existence, and how to compute the optimal monetary equilibrium. Section 4 shows the resulting optimal monetary policy rule, equilibrium price functions, and interest rates. This section also shows the comparison between the optimal full-information rule and the limited-information Taylor rule. Section 5 summarizes the findings and concludes. 4

King and Wolman (2004, pp. 1,542-43) suggest that linearization techniques in New Keynesian monetary models with discretionary policy might mistakenly lead to unique equilibria.

4

2

Model

Our model is similar to that of Ireland (1997), with the added characteristics of a persistent aggregate uninsurable productivity process zt and a money growth rate rule that can be a nonlinear function of the state. The monetary authority commits to a state-dependent money growth rule at the beginning of time that maximizes a measure of social welfare. A continuum of infinitely lived, differentiated goods firms are characterized by a degree of market power. They must set their prices before the aggregate productivity level is realized, and keep those prices for one period. The economy is also inhabited by a unit measure of identical and infinitely lived households that choose how much to consume of each type of good, how much to work for each particular firm, and how much of their wealth to save in the form of money and bonds. Figure 1: Timing of the model

The timing within each period is shown in Figure 1. Households enter period t with cash balances mt and outstanding bond balances bt from the last period t − 1. The differentiated product firms enter each period having had to commit to a price pt (i) before the productivity level zt is realized. Because all individual prices pt (i) have been determined at the beginning of period t, the aggregate price pt is also determined. 5

At the beginning of the period, the aggregate firm productivity level zt is realized. Then households are paid their bond balances due bt , and the monetary authority increases the money supply by the gross money growth rate gt with a non-proportional transfer. Households then split into worker-shopper pairs as in Lucas (1980). The workers supply labor ht (i) to the differentiated product firms, and the firms produce output Yt (i). Shoppers choose consumption of each particular good ct (i) that they purchase with money. Households then choose how much of their wealth to save in terms of bonds that mature in the next period bt+1 . To this point, a cash-inadvance constraint holds such that all goods and bond transactions must be made with currency balances.5 After the goods market, households are paid for their labor at the competitive nominal wage wt . They also receive a dividend dt (i) from each firm for any profits earned. Firms must then choose and commit to a price for the next period pt+1 (i) based on maximizing expected profits in the next period. Household consumption ct (i), bond savings bt+1 , and labor ht (i) decisions, along with the nominal wage wt , determine the amount of wealth held in the form of currency for the next period mt+1 . The pricing decision of the firm is made ex ante and is based on the productivity level from the previous period. All of the household decisions are made ex post and are based on the current value of the productivity level.

2.1

Money

The monetary authority sets the money growth rate after the current period price level pt and the current period productivity level zt are realized. Because prices are endogenously determined ex ante, the relevant primitive in the monetary objective function in addition to the current period shock zt is the shock from the previous period zt−1 . The objective of the monetary authority is to choose a state-dependent gross money growth rate rule gt = g(zt−1 , zt ) at the beginning of time to maximize 5

The cash-in-advance constraint is an extreme form of transaction cost originally introduced by Clower (1967). It implies an infinite cost of using certain types of wealth for goods transactions. Khan et al. (2003) use a less extreme specification of transaction cost in which less liquid wealth can be used to purchase goods at an increased cost that is not infinite.

6

social welfare. The money growth rate rule is a map g : R2 → (0, g¯], where g¯ is some arbitrarily large upper bound. Let MtS be the aggregate money supply at the beginning of period t. The monetary authority makes non-proportional transfers of (gt − 1)MtS to each household, so aggregate supply of currency obeys the following law of motion.

S Mt+1 = gt MtS

∀t

(2.1)

The objective of the monetary authority is to choose the state-dependent money growth rate rule g(zt−1 , zt ) in order to maximize the unconditional expected net present value of the representative consumer’s lifetime utility. The expected value is unconditional in the sense that that monetary authority must commit to the money growth rate rule at the beginning of time. We specify the monetary authority’s objective function in section 3.

2.2

Household problem

The economy is populated by a unit measure of identical households that choose how much to consume of each type of differentiated product ct (i), how much labor to supply to each differentiated product firm ht (i), and how much wealth to save for the next period in terms of bonds bt+1 that have a gross rate of return on bonds Rt and money balances at the end of the period mt+1 . Expected lifetime utility takes the following form. " U0 = E0

∞ X

#

(ct )1−γ − 1 where u (ct , ht ) = − χht 1−γ

t

β u (ct , ht )

t=0

γ ≥ 1, χ > 0 (2.2)

The term ct in (2.2) represents the constant elasticity of substitution (CES) aggregator proposed by Dixit and Stiglitz (1977) and takes the following form, Z ct ≡

1

ct (i)

θ−1 θ

θ  θ−1

∀t where θ > 1

di

0

7

(2.3)

where θ represents the elasticity of substitution among each type of differentiated good i. The term ht in (2.2) simply represents the sum of all the labor for each differentiated product firm. 1

Z

ht (i)di ∀t

ht ≡

(2.4)

0

The definition of the CES consumption aggregator in (2.3) gives rise to the demand equation for consumption of good i (2.5) as well as the expression for aggregate prices (2.6). If a household takes prices as given, the following expressions for individual consumption demand ct (i) and aggregate prices pt result if a household chooses ct (i) to minimize the expenditure on consumption given some aggregate consumption level C t ≤ ct . 6  ct (i) = Z pt =

pt (i) pt

−θ

1

pt (i)1−θ di

ct

(2.5)

1  1−θ

(2.6)

0

Households are subject to two constraints. A cash-in-advance constraint requires that consumption ct and bond savings bt+1 must be paid for with currency. A nominal version of the cash-in-advance constraint is the following, Z

1

Pt (i)ct (i)di + 0

Bt+1 ≤ Mt + (gt − 1)MtS + Bt Rt

(2.7)

where Pt (i) is the price of consuming good i, Bt+1 is the nominal amount of bond savings for next period, Rt is the gross nominal interest rate on bonds, Mt is nominal money balances held at the beginning of the period, gt is the gross money growth rate, MtS is the aggregate money supply, and Bt is amount of bonds held at the beginning of the period that will mature in period-t.7 6

A derivation of (2.5) and (2.6) is available in the Technical Appendix and is available upon request. 7 We note here that the gross nominal interest rate Rt is determined by a zero-net supply market clearing condition on these risky bonds, as described in (2.24) in Section 2.4. Because there is no riskless asset which can be demanded without limit, Rt will not be constrained by a zero-lower-bound Rt ≥ 1. Or rather, the gross nominal interest rate Rt can be strictly between 0 and 1 as well as Rt ≥ 1.

8

The nominal version of the household budget constraint simply requires that all expenditures are less than or equal to household income, 1

Z 0

Bt+1 Pt (i)ct (i)di + + Mt+1 ≤ Mt + (gt − 1)MtS + Bt + Rt

Z

1

Dt (i)di + Wt ht (2.8) 0

where Mt+1 is the nominal money balances held at the end of the period, Dt (i) is the dividends or profits paid by each differentiated products firm i to the representative household, and Wt is the nominal wage rate. Normalizing the cash-in-advance constraint (2.7) and the budget constraint (2.8) by the nominal money supply MtS renders the problem stationary and allows us to write the household’s problem in the following way,8

max ct (i),ht (i),bt+1 ,mt+1

Z s.t. and

E0

∞ X

β t u(ct , ht )

(2.2)

t=0

1

bt+1 gt ≤ mt + gt − 1 + bt (2.9) Rt 0 Z 1 Z 1 bt+1 gt pt (i)ct (i)di + + mt+1 gt ≤ mt + gt − 1 + bt + dt (i)di + wt ht Rt 0 0 pt (i)ct (i)di +

(2.10) where the lower case variables pt , wt , bt , and mt are simply their nominal counterparts divided by the aggregate money supply MtS , and where bt+1 and mt+1 are equal to their nominal counterparts divided by the aggregate money supply in the next period S Mt+1 . The Lagrangian for this problem is the following.

(∞ X

(ct )1−γ − 1 − χht + ... 1−γ t=0   Z 1 bt+1 gt λt mt + gt − 1 + bt − pt (i)ct (i)di − + ... Rt 0  ) Z 1 Z 1 bt+1 gt µt mt + gt − 1 + bt + dt (i)di + wt ht − pt (i)ct (i)di − − mt+1 gt Rt 0 0 L = E0

β

t



8

Although there is no growth in the real variables in this model, the equilibrium money growth rate can generate growth in the nominal variables. Dividing by MtS renders the nominal variables stationary.

9

where λt and µt are the multipliers on the cash-in-advance constraint and budget constraint, respectively. The first order conditions for the household maximization problem are the following, 1

ctθ

−γ

1

ct (i)− θ = (λt + µt )pt (i)

(2.11)

χ = µt w t

Z

gt (λt + µt ) = βEt (λt+1 + µt+1 ) Rt

(2.13)

µt gt = βEt (λt+1 + µt+1 )

(2.14)

1

pt (i)ct (i)di + 0

Z 0

1

(2.12)

bt+1 gt = mt + gt − 1 + bt Rt

bt+1 gt pt (i)ct (i)di + + mt+1 gt = mt + gt − 1 + bt + Rt

(2.15) Z

1

dt (i)di + wt ht

(2.16)

0

The budget constraint (2.16) holds with equality because of the first order condition for ht (2.12). We assume that the cash-in-advance constraint (2.15) holds with equaility, even when λt = 0. Written recursively, the household problem given the monetary policy rule g(z−1 , z) is characterized by the following Bellman equation in which the state in each period is the price level p and the current period productivity level z, and the monetary policy rule g(z−1 , z),     h 0 0 0 |z V V h p, z|g(·) = max u(c, h) + βE p , z |g(·) z 0 0 c,h,b ,m

(2.17)

where V h is the value to a household of entering a period with firm prices p, productivity level z, and monetary policy rule g(·). In equilibrium, the policy functions for consumption and labor supply will be functions of the state and the monetary policy     rule: c p, z|g(·) and h p, z|g(·) .

10

2.3

Firm problem

The economy is populated by a unit measure of firms that produce a differentiated good indexed by i. We assume that the capital stock for each firm is fixed. Each firm produces output Yt (i) according to the following linear production technology, Yt (i) = ezt Ht (i) ∀t

(2.18)

where Ht (i) is the firm-specific labor demand in time t and ezt is a persistent aggregate productivity level such that, zt = ρzt−1 + εt

where ρ ∈ (0, 1) and εt ∼ N (0, σε2 )

(2.19)

Let the support of the productivity level be some closed subset of the real line z ∈ Z ⊂ R.9 One of the key frictions in the model is in the firms’ pricing decision. Firms must set their price pt (i) before the productivity level is realized, as shown in Figure 1. After the productivity level is realized, firms then hire labor production at the normalized nominal wage wt in order to meet demand.10 So the firm’s decision at the end of each period is to choose a price for the next period pt+1 (i) in order to maximize expected profits dt (i) in the next period. max Et [dt+1 (i)]

pt+1 (i)

where dt+1 (i) = pt+1 (i)Yt+1 (i) − wt+1 Ht+1 (i)

(2.20)

Even though the firm’s pricing decision for pt+1 (i) occurs at the end of each period t, the decision is static because it does not depend on its period-t price level pt (i). In other words, because firms face no price adjustment costs from period to period, the objective of the firm pricing decision only incorporates the maximization of expected 9

The assumption of the support of z being closed and bounded is a minor deviation from the assumption of z being normally distributed. We need the support of z to be closed and bounded for the existence and uniqueness proof in Proposition 1, but our approximation of the normal distribution in the computation of the solution is essentially a truncated normal. 10 The lack of an i subscript on the wage wt reflects our implicit assumption that labor is homogeneous and mobile.

11

profits in the next period. So the policy function for firm prices in the next period given the monetary policy rule g(·) will simply be a function of the current-period shock. Written in recursive form, the pricing problem of the firm is the following,   V f z|g(·) = max Ez0 |z [p0 (i)Y 0 (i) − w0 H 0 (i)] 0

(2.21)

p (i)

  where V f z|g(·) is equilibrium expected firm dividends or rather the value to the firm of entering a period with a productivity level z and with monetary policy rule g(·). Firm output Y 0 (i), wages w0 , and labor demand H 0 (i) are also equilibrium objects that are each functions of the chosen pricing rule. The pricing policy function that solves the static firm maximization problem (2.21) can be written in recursive   form as p0 (i) = ψi z|g(·) .

2.4

Market clearing

Four markets must clear in this economy—the goods market, the labor market, the bond market, and the money market. Equations (2.22), (2.23), (2.24), and (2.25) characterize these market clearing conditions. Yt (i) = ct (i) ∀i, t

(2.22)

Ht (i) = ht (i) ∀i, t

(2.23)

bt = 0 ∀t

(2.24)

mt = 1 ∀t

(2.25)

Output supply must equal consumption demand, labor supply must equal labor demand, bonds are in zero net supply, and nominal money supply must equal nominal money demand.

12

3

Equilibrium

In addition to the household and firm optimization conditions and the market clearing conditions, a transversality condition must hold. This is analogous to a final condition on the problem.11 The transversality condition for this model is the following: i lim β E0 mt (λt + µt ) = 0 t

h

t→∞

(3.1)

Because all the household decisions are made after the productivity level is realized, both the household CIA constraint (2.15) and the household budget constraint (2.16) hold with equality and bonds are in zero net supply (2.24). The household decision problem becomes static in equilibrium because none of the choice variables show up in the continuation value term of the Bellman equation. All firm pricing decisions are also static because they face no price adjustment costs at the end of each period. So firm pricing decisions for the next period are simply a function of the   current period shock p0 (i) = ψi z|g(·) . Because equilibrium household policy functions are functions of the current price and the current productivity level (p, z) and equilibrium firm prices are a function of the previous productivity level p = ψ(z−1 ), the equilibrium state of this model is the most recent two-period history of productivity realizations (z−1 , z). We define a stationary distribution over the state space of two-period histories Γ (z−1 , z) as the unconditional expectation of the fraction of time the economy will spend in each possible two-period history over an infinite number of periods. The stationary competitive equilibrium given an arbitrary state-dependent monetary policy rule g = g(z−1 , z) is characterized by the following definition.

11

The transversality condition states that the present value of the state (mt in this case) in terms of marginal rates of substitution tends to zero as time goes to infinity: limt→∞ β t mt u0 (ct ) = 0. This is the infinite horizon analogy to the finite horizon result that you spend all your money in the last period of your life. The multipliers show up in (3.1) because the state mt only shows up in the Lagrangian in the multiplier terms. See Azariadis (1993, p. 211), Stokey and Lucas (1989, p. 98), and Ljungqvist and Sargent (2004, pp. 193).

13

Definition 1 (Stationary competitive equilibrium given g = g(z−1 , z)). Stationary competitive equilibrium given an arbitrary state-dependent monetary policy rule g = g(z−1 , z), where g : R2 → (0, g¯], is defined as a policy function for firm prices p(i), aggregate prices p as a function of firm prices according to (2.6), household consumption c(i) and labor h(i) allocations, a wage rate w, a gross return on bonds R such that: i. the household consumption c(i) and labor supply decisions h(i) satisfy the conditions for household optimization in (2.11) through (2.16), ii. firms choose their next period price level p(i) = ψi (z−1 ) in order to maximize expected profits in the next period according to (2.21), iii. the goods market, labor market, bond market, and money market clear according to (2.22), (2.23), (2.24), and (2.25), iv. the transversality condition (3.1) holds,     v. and all allocation functions c z−1 , z|g(·) and h z−1 , z|g(·) and price functions       p(i) = ψi z−1 |g(·) , w z−1 , z|g(·) and R z−1 , z|g(·) are stationary functions of the state.

Substituting the market clearing conditions (2.24) and (2.25) into the cash-inadvance constraint (2.15) gives the following equilibrium condition for total expenditures on consumption in terms of the monetary policy rule. Z

1

pt (i)ct (i)di = gt

∀t

(3.2)

0

Combining (2.3), (2.6), (2.11), and (3.2) and summing over i gives the following expression. ct1−γ = λt + µt gt

∀t

(3.3)

Equations (2.5), (2.11), and (3.3) give the aggregate analogue to (3.2) that says total expenditures equal the money growth rate. ct =

gt pt

∀t

(3.4)

Plugging (3.4) into the expression for individual demand (2.5) that came from the cost minimization problem gives individual demand in terms of exogenous variables 14

at the time of the household’s decision.  ct (i) =

pt (i) pt

−θ

gt pt

∀i, t

(3.5)

The equilibrium expression for the normalized wage wt is found by combining the two household first order conditions (2.12) and (2.14) and substituting in (3.3) and (3.4) to obtain the following. "

χgt wt = β

Et

−γ gt+1 p1−γ t+1

#!−1 ∀t

(3.6)

Likewise, the equilibrium expression for the gross interest rate on bonds Rt is found by combining the household first order condition (2.13) with (3.3) and (3.4) to obtain the following. 1 Rt = β

 1−γ gt pt

" Et

−γ gt+1 1−γ pt+1

#!−1 ∀t

(3.7)

Equilibrium labor supply can be expressed in terms of exogenous variables by substituting the two market clearing conditions for the goods market (2.22) and labor market (2.23), as well as individual demand from (3.5), into the firm’s production function (2.18). ht (i) = e

−zt



pt (i) pt

−θ

gt pt

∀i, t

(3.8)

We then plug (3.5) and (3.8) into the firm’s profit function from (2.20) and set the derivative of expected profits equal to zero in order to obtain the following expression for the optimal price. pt (i) = pt =

  θ Et−1 e−zt wt θ−1

∀i, t

(3.9)

The symmetric equilibrium in the firm pricing decision where pt (i) = pt for all i and t is shown in (3.9) because all the terms on the right-hand-side are not functions of i. With pt (i) = pt from (3.9), the equilibrium expressions for ct (i), ct , ht (i), and ht

15

are simplified to the following. ct (i) = ct =

gt pt

ht (i) = ht = e−zt

∀i, t gt pt

(3.10)

∀i, t

(3.11)

Lastly, plugging (3.3) and (3.4) into the transversality condition (3.1) gives the following equilibrium version of the transversality condition. t

lim β E0

t→∞



 gt−γ =0 pt1−γ

(3.12)

And the first order difference equation in pt is found by substituting the equilibrium equation for wages (3.6) into (3.9).

pt =

χ β



θ θ−1





"

Et−1 gt e−zt

Et

−γ gt+1 1−γ pt+1

#!−1  

∀t

(3.13)

The stationary equilibrium given the monetary policy rule g = g(z−1 , z) as stated in Definition 1 is computed by first solving for the optimal firm equilibrium pricing rule as the fixed point price function of the difference equation in (3.13). Let the policy function for the firm pricing rule for prices today given productivity level in the previous period z−1 be p = ψ(z−1 ). Then the recursive form of (3.13) is as follows.

ψ(z−1 ) =

χ β



θ θ−1





"

Ez|z−1 g z−1 , z e−z


Ez0 |z


0 −γ

g z, z ψ(z)1−γ

#!−1  

(3.14)

Once the optimal pricing decision rule p = ψ(z−1 ) is computed from (3.14) given a state-dependent monetary policy rule g(z−1 , z), the expressions for equilibrium wage, gross interest rate, consumption, and labor supply can be computed from (3.6), (3.7), (3.10), and (3.11), respectively. From (3.14), it is clear that the policy rule for prices in the current period are a function of the shock from the previous period and the monetary policy rule   p z−1 |g(·) . Substituting the equilibrium firm pricing rule from (3.14) into (2.17) 16

as well as the market clearing conditions on bond holdings and currency holdings from (2.24) and (2.25), the equilibrium household Bellman equation can be rewritten in the following way,     V h z−1 , z|g(·) = max u(c, h) + βEz0 |z V h z, z 0 |g(·) c,h

(3.15)

where the equilibrium policy functions for consumption and labor supply can now be written as functions of the last two shocks z−1 and z and the monetary policy rule     g(z−1 , z): c z−1 , z|g(·) and h z−1 , z|g(·) .12 These conditions allow us to make the following proposition about the existence and uniqueness of the equilibrium defined in Definition 1 given an arbitrary money growth rate rule g(·).

Proposition 1 (Existence and uniqueness of stationary competitive equilibrium given g = g(z−1 , z)). Let z ∈ Z ⊂ R, where Z is closed and bounded. Let g(z−1 , z) be the set of money growth rate rules and let p(z−1 ) be the set of firm pricing rules for which g ∈ (0, g¯] and p ∈ (0, ∞) for all z−1 and z. Also let the coefficient of relative risk aversion in the period utility function (2.2) be restricted to γ ≥ 1. Let the equilibrium Bellman equation be (3.15), where the given state-dependent money
growth rate rule is g z−1 , z , the equilibrium values for maximized consumption c and labor supply h are given by (3.10) and (3.11), respectively, and equilibrium firm prices are a function of the previous period’s productivity level p(z−1 ) according to (3.14). Then the equilibrium from Definition 1 exists and is unique. Proof. See Appendix A-1 for proof.

With Definition 1 characterizing the equilibrium firm and household policy functions given an arbitrary monetary policy rule g(z−1 , z), we define a stationary optimal monetary equilibrium under commitment. Let the objective function of the monetary authority be the weighted sum of the present value of lifetime utility of the representative agent over all two-period histories of productivity realizations. Define the 12

The equilibrium policy functions for the wage and the gross interest as  rate can be written  functions of the last two shocks and the policy rule as well w z−1 , z|g(·) and R z−1 , z|g(·) .

17

weighting function as the invariant distribution Γ(z−1 , z), which is the percent of time spent in each possible two-period history.13

Definition 2 (Stationary optimal monetary equilibrium under commitment). Stationary optimal monetary equilibrium under commitment is defined as a state-dependent monetary policy rule g(z−1 , z) that maximizes social welfare V (z0 ), which is defined as the unconditional expectation at the beginning of time of equilibrium discounted lifetime household utility from Definition 1 and equation (3.15) for a given monetary policy rule g(z−1 , z). h  i V (z0 ) = max E0 V h z−1 , z|g(·) (3.16) g(z−1 ,z)

The unconditional expectation in (3.16) is calculated using the ergodic distribution over all possible two-period histories Γ(z−1 , z). Then the objective function (3.16) can be rewritten in the following way. Z Z V (z0 ) = max

g(z−1 ,z)

Γ(z−1 , z)V z

h



 z−1 , z|g(·) dz−1 dz

(3.17)

z−1

This objective function is similar the Rawlsian veil of ignorance criterion used by Phelan (2006). As we mentioned at the end of section 2.1, the monetary authority’s ex ante decision of an optimal state-dependent monetary policy rule gˆ(z−1 , z) is dynamic only in the sense that the rule g(z−1 , z) influences both the household equilibrium period     utility through consumption c z−1 , z|g(·) , labor supply h z−1 , z|g(·) , and current     prices p z−1 |g(·) and also the optimal price level p0 in the next period V h z, z 0 |g(·) . Because the equilibrium interest rate sets bond holdings to their zero net supply value and because we assume that the cash-in-advance constraint binds, the monetary authority controls the household savings decision which is simply amount of currency held over to the next period.

13 This is equivalent to assuming that the initial state is not known at the time when the monetary authority commits to the policy rule. Another approach would be to choose an optimal money growth rule given a particular initial distribution. This is a Rawlsian objective criterion.

18

Proposition 2 (Existence of stationary optimal monetary equilibrium under commitment). Let the stationary optimal monetary equilibrium be characterized by Definition 2 and the stationary competitive equilibrium given g(z−1 , z) be characterized by Definition 1, and the continuous function g : R2 → [εg , g¯], where εg > 0 is a small positive value arbitrarily close to 0 and g¯ < ∞ is an arbitrarily large value. Then at least one stationary optimal monetary equilibrium exists. Proof. See Appendix A-1 for proof.

A description of how to compute the stationary optimal monetary equilibrium under commitment from Definition 2 is given in Appendix A-2. Because Proposition 2 only guarantees existence of the monetary equilibrium and non uniqueness, the computational approach must include multiple starting values to increase the confidence that any local optima are also a global optimum.

4

Results and Comparison to Taylor Rule

For the computation of the optimal monetary equilibrium in Definition 2, we calibrate the parameters of the model by using standard RBC parameter values. We treat one model period as one year. We calibrate the annual discount factor to β = 0.96, and the coefficient of relative risk aversion to γ = 3. We set the disutility of labor parameter arbitrarily to χ = 1, and we choose constant elesticity of substitution parameter to θ = 6 to reflect aggregate markups in the 20-percent range. We calibrate the parameters of the productivity process to be ρ = 0.8145 and σ = 0.013. We use an exponentiated fifth-order complete polynomial in (z−1 , z) to approximate g.14 Panel A of figure 2 shows the computed optimal money growth rate rule g(z−1 , z) from Definition 2. The optimal gross money growth rate is monotonically increasing in the previous period productivity level z−1 and monotonically decreasing in the current period productivity level z. The equilibrium price function p(z−1 ) induced by the optimal monetary rule is shown in Figure 3. The average interest rate in 14

Python and Matlab code for the computation are available upon request. Appendix A-2 gives a detailed description of how to compute the equilibrium.

19

¯ = 0.8342 and the ergodic mean of household utility is V (z0 ) = this equilibrium is R −25.0041.15 Figure 2: Comparison of optimal money growth rate rule g(z−1 , z) with Taylor rule

Panel A: Optimal money growth rule g(z(−1),z)

1.1

g(z(−1))

1 0.9 0.8 0.7

0.2 0.1 0 −0.1 −0.2

z(t−1)

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

z(t)

Panel B: Optimal Taylor−type money growth rule g(z(−1))

1 0.95

g(z(−1))

0.9 0.85 0.8 0.75 0.7 0.65 0.2 0.1 0 −0.1 z(t−1)

−0.2

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

z(t)

Recall the two competing incentives of the monetary authority in this environment of the Friedman rule of keeping inflation and nominal interest rates low and the 15

Again, because the bonds are determined by the zero-net-supply market clearing condition in (2.24) in Section 2.4 and because there are no riskless assets, the interest rate in every period Rt is not constrained by the zero lower bound. That is, Rt can be less than 1.

20

Figure 3: Comparison of equilibrium price function p(z−1 ) under optimal money growth rate rule g(z−1 , z) and under Taylor rule

1.05

1

0.95

p(z(−1))

0.9

0.85

0.8

0.75

0.7

0.65 Optimal Taylor 0.6

−0.1

−0.05

0 z(−1)

0.05

0.1

0.15

incentive of reducing inefficient variation in prices. It is clear from the computed example above that the Friedman rule is being followed in terms of generating low nominal interest rates. However, it is not clear from this computation alone how much price variance smoothing is being induced. To highlight these effects in the optimal policy g(z−1 , z), we compare this the equilibrium results from this rule to those of an optimized Taylor rule for the money growth rate. The Taylor (1993) rule is a simple rule to which a monetary authority could commit that was initially proposed in its simplest form as being a linear function of observed inflation and deviations of output from its long-run levels. The literature on this specific type of monetary rule arose from the finding in Kydland and Prescott (1977) that policy discretion can lead to welfare losses, and that these losses can be ameliorated by commitment to rules. McCallum (1988, 1990) and McCallum and Nelson (1999) suggested that monetary rules should be expressed in terms of monetary instruments that can be controlled on a high frequency basis and should be functions of information that could plausibly be possessed by a central banker at the time of

21

the policy action. The latter characteristic that a good monetary policy rule be a function of variables in the information set of the policy maker at the time of the policy action suggests that our full-information policy rule g(z−1 , z) might be too strong. That is, it is unlikely that a central banker can observe the actual underlying shocks of the economy. It is more likely that the central banker can observe the equilibrium aggregate economic outcomes that are functions of these shocks.16 Our Taylor-type rule is the money growth rate analogue of the form of Taylor-type interest rate rules used in Smets and Wouters (2007) and Christiano et al. (2011). Our Taylor rule for the gross money growth rate takes the following form, 

Y g(p, Y ) = g¯ Y∗

 φ1 

p p∗

 φ2 (4.1)

where g¯, Y ∗ , and p∗ are the long-run averages of the money growth rate, output level, and price level, respectively, from the optimal monetary equilibrium model.17 The parameters φ1 and φ2 will be optimized so that the Taylor-type money growth rate rule maximizes the ergodic mean of household lifetime expected utility. Even though the central banker committed to using the Taylor rule in (4.1) is using current information p and Y to formulate policy, the Taylor rule in this model implies a specific and intuitively appealing form of limited-information rule. In particular, it implies that the monetary policy rule is only determined based on the previous period productivity level z−1 , ignoring the value of the current period productivity z. This limited information occurs despite the equilibrium result that price is a function of the previous period shock p(z−1 ) and output equals the money growth rate divided 16

Alan Greenspan was famously quoted as saying at the American Iron and Steel Institute’s annual meeting in 1997, “Every day, I still look for the price of No. 1 heavy melt steel scrap” (Lahart, Dec. 4, 2003). This was an example of the Federal Reserve Chairman looking for observable indicators as close as possible to the basic market fundamentals. 17 Our Taylor rule for the money growth rate is a function of price deviations rather than the more traditional inflation deviations approach. We do not use inflation deviations because inflation devations change the equilibrium state, and therefore all of the equilibrium policy rules, to being dependent on the entire history of productivity realizations. We demonstrate the math behind this in the Technical Appendix, which is available upon request. The Taylor-type money growth rate rule is much better behaved if the observable is prices rather than inflation.

22

by the price level Yt = ct = gt /pt combining (3.10) and (2.22). In fact, it is precisely the equilibrium expression for output Yt that delivers this limited-information result. Substituting the equlibrium expression for Y = g/p into the Taylor rule gives the following result. 

Y g = g¯ Y∗ g 1−φ1

φ1 

p p∗

φ2

g¯p(z−1 )φ2 −φ1 = (Y ∗ )φ1 (p∗ )φ2



g/p = g¯ Y∗ ⇒

φ1 

p p∗

φ2

  1
g¯p(z−1 )φ2 −φ1 1−φ1 g p(z−1 ) = (Y ∗ )φ1 (p∗ )φ2

(4.2)

The final expression of (4.2) shows that the Taylor money growth rate rule is really only conditioned on observable prices p(z−1 ) which are only a function of the previous period productivity level z−1 as shown in (3.14). This limited information implied by this model is intuitively appealing, because it implies that the current period variables in the information set of the central bank only incorporate the fundamental economic shocks from the previous period and ignore any current period shocks. The optimal Taylor rule is computed in a similar way to the optimal full-information money growth rate rule. We first guess at the values of φ1 and φ2 . Given the guess for the Taylor rule, we can solve for the equilibrium price function p, consumption c, and labor supply h. These allow us to compute the household value function V h and the ergodic mean of lifetime household expected utility V (z0 ). We keep updating the guesses until we converge on a maximum ergodic mean of lifetime utility. Using our same calibrated model parameters from before, the optimal Taylor money growth rate rule has φ1 = 2.1742 and φ2 = 1.1484. Panel B of Figure 2 shows the computed optimal Taylor-type money growth rate rule as a function of the previous period’s productivity level, which underlies the price level on which the rule is based. The optimal limited-information Taylor rule has a similar shape in the z−1 dimension as the full-information optimal money growth rate rule in Panel A. However, the Taylor rule does not vary in the current shock z dimension, reflecting the limited-information condition of the rule. Figure 3 shows the equilibrium price function p(z−1 ) under the limited-information

23

Taylor rule as well as the equilibrium price function under the full-information optimal money growth rule. These two pricing rules are nearly identical. The reason for this can be seen from the two optimal money growth rate rules in Figure 2. Both optimal monetary rules are similarly increasing in the dimension of the previous period productivity level. This is the dimension on which prices are conditioned, which intuitively supports the ability of both rules to induce similar pricing behavior on the part of firms. The average interest rate in this equilibrium with the limited¯ = 0.8339, which is nearly equal to that of the fullinformation Taylor rule is R information optimal money growth rate rule. The ergodic mean of household utility under the limited-information Taylor rule is V (z0 ) = −25.0048, which is only slightly lower than that of the full-information optimal money growth rate rule. Where the Taylor rule differs most from the optimal full-information rule is in the its ability to manipulate the interest rate with the correct timing in order to reduce the transactions cost friction of needing to hold currency. This difference can be clearly seen by looking at a simulation of equilibrium household consumption under each rule. Figure 4 shows a simulation of consumption under both the optimal money growth rate rule and the Taylor rule for 50 periods. Although both time series have nearly equivalent means and variances, the equilibrium consumption under the Taylor-rule responds more slowly than consumption under the optimal rule. This is a result of the monetary policy under the Taylor rule not responding to current period shocks. The comparison above between the full-information optimal monetary policy rule and the limited-information Taylor rule clearly highlight the effects of limited information on policy effectiveness. However, we also find in this model with extreme forms of transaction and price frictions that the limited-information rule, based on observable macroeconomic variables, closely approximates the economic outcomes under the full-information rule. We interpret this as evidence that implementing Taylor-type monetary policy rules may be an effective way to both reduce the time-inconsistency problem of discretionary policy as well as come close to inducing optimal economic outcomes in terms of societal welfare. 24

Figure 4: Comparison of simulated path of equilibrium consumption c(z−1 , z) under optimal money growth rate rule g(z−1 , z) and under Taylor rule

1.025 c optimal c Taylor 1.02

1.015

c(z−1,z)

1.01

1.005

1

0.995

0.99

0.985

5

0

5

10

15

20

25 period

30

35

40

45

50

Conclusion

In this paper, define and compute a full-information optimal monetary policy rule to which a monetary authority commits at the beginning of time. The economic environment in which this type of rule is studied is one with incomplete markets, monopolistic competition, price frictions on the part of firms, and a transaction cost friction on the part of households. We prove the existence and uniqueness of the competitive equilibrium given an arbitrary monetary policy rule and prove the existence of an optimal state-dependent rule. We find that the optimal full-information monetary rule follows the prediction of the Friedman rule that prices and nominal interest rates should be low in order to mitigate the transactions cost friction in the model. We compare this rule to an optimal limited-information Taylor rule. The comparison to the Taylor rule is important because it implies a specific form of limited information on the part of the monetary authority that is intuitively appealing. 25

Specifically, the Taylor rule in this model implies that the monetary rule can only be conditioned on past fundamental economic shocks and is unable to condition policy on the current period shock. Similarly, full information on the part of a monetary authority is a strong assumption as it is not likely that central bankers are able to observe the fundamental shocks to the economy, neither present nor past. The characteristic that the Taylor-type monetary policy rule is conditioned on observable macroeconomic variables makes it a realistic alternative to the full-information optimal rule. We find that the equilibrium economic outcomes under the optimal limited-information Taylor rule closely approximate those under the optimal full-information monetary rule, despite the inability of the Taylor rule to condition on current period shocks. We interpret this finding as evidence that policy rules based on observable macroeconomic variables can closely approximate the economic outcomes of more flexible policy rules based on the underlying fundamentals of the economy, which are often unobservable.

26

APPENDIX A-1

Proofs

Proof of Proposition 1. This proof follows closely the proof of existence and uniqueness in Li et al. (2006) of a functional solution to a class of Hammerstein equations. In preparation for the proof, we introduce some notation. Let z ∈ Z ⊂ R. Let G be a bounded closed subset of RN .18 Let g(G) be a map g : G → (0, g¯], where g¯ is some arbitrarily large finite number. Let C(G) denote the usual real Banach space with the norm ||p||C = maxx∈G |p(x)| for all p ∈ C(G). Let L2 (G) denote the usual real reflexive Banach space with the norm
1/2 R ||p|| = G |p(x)|2Rdx for all p ∈ L2 (G) and the real Hilbert space with the inner product hp, qi = G p(x)q(x)dx for all p, q ∈ L2 (G).
θ . We can then write the optimal pricing decision rule p = Now let A = χβ θ−1 ψ(z−1 ) from (2.39) as follows, Z g(z, z−1 )ez p(z)1−γ R p(z−1 ) = A F (z|z−1 )dz (A.1.1) g(z 0 , z)−γ F (z 0 |z)dz0 Where F (z|z−1 ) is the conditional probability distribution function of z. We now define the function k : G → R1 as k(z, z−1 ) = R

g(z, z−1 )ez . g(z 0 , z)−γ F (z 0 |z)dz0

It can be shown that k is symmetric, nonnegative, and continuous, and that k 6= 0 on G. We also define the function f : G → R1 as f (z, p(z)) = p(z)1−γ F (z|z−1 ). Then we can rewrite (A.1.1) in the form of the Hammerstein equation as Z
p(z−1 ) = A k(z, z−1 )f z, p(z) dG

(A.1.2)

We also introduce several function operators for ease of notation. Define operator K : C(G) → C(G) by, Z Ku(x) = A k(x, y)u(y)dy, x ∈ G, ∀u ∈ C(G) (A.1.3) G

and define operator f : C(G) → C(G) by, x ∈ G, ∀u ∈ C(G).

f u(x) = f (x, u(x)),

(A.1.4)

We can now write the optimal pricing decision rule (A.1.2) as p = Kf p 18

In our example with g(z−1 , z), G ⊂ R2 .

27

(A.1.5)

Lemma 1 (Strongly monotone operator principle). (Deimling, 1985, Theorem 11.2, p. 100) Let (X, ||·||) be a real reflexive Banach space. Also let H : X → X ∗ be a continuous operator where there exists c > 0 such that hHp − Hq, p − qi ≥ c ||p − q||2 ,

p, q ∈ X,

so H is strongly monotone operator. Then H : X → X ∗ is a homeomorphism between X and X∗. Lemma 2. (Li et al., 2006, Remark 2.2) K : L2 (G) → L2 (G) is positive bounded linear and symmetric. Hence the square root operator of K, K 1/2 : L2 (G) → L 2 (G) exists and is unique, and is also bounded linear and symmetric with K 1/2 = ||K||1/2 . Moreover, K 1/2 : L2 (G) → C(G) is completely continuous. Lemma 3. (Li et al., 2006, Lemma 2.3) K 1/2 : L2 (G) → C(G) is a linear completely continuous operator. Then K 1/2 : L2 (G) → L2 (G) is also linear completely continuous. Lemma 4. (Li et al., 2006, Remark 2.3) K 1/2 p1 6= K 1/2 p2 for all p1 , p2 ∈ L2 (G) with p1 6= p2 . Lemma 5. (Li et al., 2006, Lemma 3.1) (i) The equation p = Kf p

(A.1.6)

has a solution in C(G) if and only if the equation q = K 1/2 f K 1/2 q

(A.1.7)

has a solution in L2 (G). (ii) The uniqueness of solutions for equations A.1.6 and A.1.7 is equivalent. (iii) If (A.1.7) has a nonzero solution in L2 (G), then (A.1.6) has a nonzero solution in C(G). If (A.1.7) has infinitely many solutions in L2 (G), then (A.1.6) has infinitely many solutions in C(G). We take the money growth rate g = g(z, z−1 ), g ∈ (0, ∞) as given. We also assume that γ ≥ 1. We first show that f (z, p) is a nonincreasing function in p. Let p1 , p2 ∈ R1 with p1 ≤ p2 . Then f (z, p1 ) = (p1 )1−γ F (z|z−1 ) and f (z, p2 ) = (p2 )1−γ F (z|z−1 ). Since γ ≥ 1, we have that (p1 )1−γ ≥ (p2 )1−γ . And since F (z|z−1 ) ≥ 0, it follows that f (z, p1 ) ≥ f (z, p2 ). We now show that equation A.1.6 has a unique solution. It follows from Lemma 5 that the operator equation A.1.6 has a unique solution in C(G) if and only if equation A.1.7 has a unique solution in L2 (G). Hence it is also equivalent to say that the operator equation Hq = 0 has a unique solution in L2 (G), where H = I − K 1/2 f K 1/2 . From Lemma 3 we know that K 1/2 : L2 (G) → C(G) ,→ L2 (G) is continuous, and 28

therefore H : L2 (G) → L2 (G) is also continuous. From Lemma 1 it follows that it is only necessary to show that H is a strongly monotone operator. Since f (z, p) is a nonincreasing function in p for each z ∈ G, it follows that for all p, q ∈ L2 (G) we have hf K 1/2 p − f K 1/2 q, K 1/2 p − K 1/2 qi Z =

[f (z, K 1/2 p(z)) − f (z, K 1/2 q(z))][K 1/2 p(z) − K 1/2 q(z)]dx ≤ 0.

G

It then follows from Lemma 2 that hHp − Hq, p − qi = hp − q − K 1/2 f K 1/2 p + K 1/2 f K 1/2 q, p − qi = ||p − q||2 − hK 1/2 (f K 1/2 p − f K 1/2 q), p − qi = ||p − q||2 − hf K 1/2 p − f K 1/2 q, K 1/2 p − K 1/2 qi ≥ ||p − q||2 , p, q ∈ L2 (G). Hence H is a strongly monotone operator and Hq = 0 has a unique solution in L2 (G), and thus equation A.1.6 has a unique solution. It follows that the solution to the price function p(z−1 ) from (3.14) exists and is unique. It follows that there exist unique household consumption c(i), household labor allocations h(i), nominal wage rate w, and gross return on bonds R as given by equations (3.10), (3.11), (3.6), and (3.7) respectively. Thus the equilibrium from Definition 1 exists and is unique.

Proof of Proposition 2. Define G(Z) as the set of all continuous and bounded functions on Z ⊂ R2 such that g ∈ G and g : R2 → [εg , g¯]. Then by Proposition 1, V h (z−1 , z|g) is unique and is continuous in g. The invariant distribution Γ(z−1 , z) is unique and is continuous independent of household, firm, or monetary authority decisions. By the continuity and uniqueness of both V h and Γ, the ergodic mean of equilibrium household expected utility V (z0 ) defined in (3.17) is continuous in g. Therefore, a maximum of V (z0 ) exists for at least one g ∈ G.

29

A-2

Description of numerical computation

Python and Matlab versions of our code are available upon request. The computation of the stationary optimal monetary equilibrium described in Definition 2 requires the following steps. 1. Calibrate the exogenous parameters of the model β, γ, χ, θ, z, ρ, and σε2 . 2. Discretize the space of possible shock values into n possible values such that z = [z1 , z2 , ...zn ] where zn is the upper bound of z and z1 = −zn is the lower bound of z. Make z into an n-state Markov process that approximates the AR(1) process in (2.19). (a) We use the Tauchen and Hussey (1991) method to generate a Markov transition matrix T that approximates the AR(1) process from (2.19). Each element in the transition matrix Tj,k represents the conditional probability Prob (z 0 = zk |z = zj ). (b) Using this transition matrix T we calculate the stationary (invariant) distribution Γ(z) as any of the rows of limt→∞ T t . The stationary distribution Γ(z) gives the percentage of time that an individual will spend in each state over an infinite life. (c) We calculate a stationary distribution over all two-period histories Γ(z−1 , z) in the following way:   Γ(z1 ) 0 ... 0  0 Γ(z2 ) . . . 0    Γ(z−1 , z) =  .. .. ..  T ⇒ Γ(zj , zk ) = Γ(zj )Tj,k . .  . . . .  0 0 . . . Γ(zn ) 3. Because the optimal money growth rule is continuous and everywhere positive, we can approximate by exponentiating a jth order polynomial of the following form: 2 +...a Pj a1 +a2 z−1 +a3 z+a4 zz−1 +a5 z 2 +a6 z−1 zj j+1+ i i=1 g˜j (z−1 , z) = e Let a = {a1 , a2 , a3 , ...aj+1+Pj i } be the vector of coefficients for the jth order i=1 polynomial in the approximation g˜ of g. 4. Make an initial guess for the coefficients a0 = {a0,1 , a0,2 , a0,3 , ...a0,j+1+Pj i } in i=1 the approximated monetary policy function g˜0 . • A good first guess is a0 = {0, 0, ...0}, which implies g˜ = 1 for all z−1 and z. This is neutral monetary policy. 5. Given the initial guess for the monetary policy function g˜0 , use (3.14) to find the equilibrium price function and use (3.10) and (3.11) to find the equilibrium allocation functions for c and h, respectively. 30

6. Use equilibrium c and h given monetary policy function guess g˜0 to calculate the equilibrium indirect utility function u(c, h). Then use value function iteration
h g0 . on the equilibrium Bellman equation (3.15) to find V z−1 , z|˜ 7. Use the ergodic distribution Γ(z−1
, z) over all two-period histories and the household value function V h z−1 , z|˜ g0 given monetary policy guess g˜0 to calculate the ergodic mean of lifetime utility V (z0 |˜ g0 ) given g˜0 . Z Z
V (z0 |˜ g0 ) = Γ(z−1 , z)V h z−1 , z|˜ g0 dz−1 dz z

z−1

8. Use a Newton method to update the guess of polynomial coefficients a1 , thereby updating the guess of policy function g˜1 to maximize V (z0 |˜ g ). 9. Repeat steps 5 through 8 until convergence. From Proposition 2, we only know that an optimal policy rule exists. We do not know if it is unique. There is no guarantee that the monetary policy that we find using the algorithm above is the global optimal policy, or even one of the global optimal policies. For this reason, we run the algorithm above with multiple different starting guesses and confirm that the solutions are the same. If they are not, we choose the solution with the largest criterion value.

31

References Abreu, Dilip, “On the Theory of Infinitely Repeated Games with Discounting,” Econometrica, March 1988, 56 (2), 383–396. , David Pearce, and Ennio Stacchetti, “Toward a Theory of Discounted Repeated Games with Imperfect Monitoring,” Econometrica, September 1990, 58 (5), 1041–1063. Ad˜ ao, Bernardino, Isabel Correia, and Pedro Teles, “Gaps and Triangles,” Review of Economic Studies, October 2003, 70 (4), 699–713. Azariadis, Costas, Intertemporal Macroeconomics, Cambridge, Massachusetts: Blackwell Publishers, Inc., 1993. Barro, Robert J. and David B. Gordon, “Rules, Discretion, and Reputation in a Model of Monetary Policy,” Journal of Monetary Economics, January 1983, 12 (1), 101–121. Calvo, Guillermo A., “Staggered Prices in a Utility-maximizing Framework,” Journal of Monetary Economics, September 1983, 12 (3), 383–398. Chari, V. V. and Patrick J. Kehoe, “Modern Macroeconomics in Practice: How Theory is Shaping Policy,” Journal of Economic Perspectives, Fall 2006, 20 (4), 3–28. , , and Ellen R. McGrattan, “New Keynesian Models: Not Yet Useful for Policy Analysis,” American Economic Journal: Macroeconomics, January 2009, 1 (1), 242–266. Christiano, Lawrence J., Martin Eichenbaum, and Sergio Rebelo, “When is the Government Spending Multiplier Large?,” Journal of Political Economy, February 2011, 119 (1), 78–121. Clower, Robert W., “A Reconsideration of the Microfoundations of Monetary Theory,” Western Economic Journal, December 1967, 6 (1), 1–8. Deimling, Klaus, Nonlinear Functional Analysis, Springer-Verlag, 1985. Dixit, Avinash K. and Joseph E. Stiglitz, “Monopolistic Competition and Optimum Product Diversity,” American Economic Review, June 1977, 67 (3), 297–308. Fischer, Stanley, “Rules versus Discretion in Monetary Policy,” in Benjamin M. Friedman and Frank H. Hahn, eds., Handbook of Monetary Economics, Vol. 2 of Handbooks in Economics, Elsevier, 1990, chapter 21. Ireland, Peter N., “Sustainable Monetary Policies,” Journal of Economic Dynamics and Control, November 1997, 2 (1), 87–108.

32

Jr., Robert E. Lucas, “Equilibrium in a Pure Currency Economy,” Economic Inquiry, April 1980, 18 (2), 203–280. Khan, Aubhik, Robert G. King, and Alexander L. Wolman, “Optimal Monetary Policy,” Review of Economic Studies, October 2003, 70 (4), 825–60. King, Robert G. and Alexander L. Wolman, “Monetary Dicretion, Pricing Complementarity, and Dynamic Multiple Equilibria,” Quarterly Journal of Economics, November 2004, 119 (4), 1513–1553. Klein, Paul, Per Krusell, and Jos´ e-V´ıctor R´ıos-Rull, “Time-consistent Public Policy,” Review of Economic Studies, July 2008, 75 (3), 789–808. Kydland, Finn E. and Edward C. Prescott, “Rules Rather than Discretion: The Inconsistency of Optimal Plans,” Journal of Political Economy, June 1977, 85 (3), 473–92. Lahart, Justin, “Steel Yourself for Inflation?,” CNN/Money, Dec. 4, 2003. http://money.cnn.com/2003/12/04/markets/scrapsteel/. Li, Fuyi, Yuhua Li, and Zhanping Liang, “Existence of solutions to nonlinear Hammerstein integral equations and applications,” Journal of Mathematical Analysis and Applications, November 2006, 323 (1), 209–227. Ljungqvist, Lars and Thomas J. Sargent, Recursive Macroeconomic Theory, 2nd ed., Cambridge, Massachusetts: The MIT Press, 2004. Marcet, Albert and Ramon Marimon, “Recursive Contracts,” CEP Discussion Paper 1055, Centre for Economic Performance June 2011. McCallum, Bennett T., “Robustness Properties of a Rule for Monetary Policy,” Carnegie-Rochester Conference Series on Public Policy, 1988, 29, 173–203. , “Targets, Indicators, and Instruments of Monetary Policy,” in William S. Haraf and Philip Cagan, eds., Monetary Policy for a Changing Financial Environment, AEI Press, 1990, pp. 44–70. and Edward Nelson, “Performance of Operational Policy Rules in an Estimated Semiclassical Structural Model,” in John B. Taylor, ed., Monetary Policy Rules, Vol. 31 of Business Cycles Series, University of Chicago Press, 1999, pp. 15–45. Phelan, Christopher, “Opportunity and Social Mobility,” Review of Economic Studies, April 2006, 73 (2), 487–504. Smets, Frank and Rafael Wouters, “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach,” American Economic Review, June 2007, 97 (3), 586–606. Stokey, Nancy L. and Robert E. Lucas Jr., Recursive Methods in Economic Dynamics, Harvard University Press, 1989. 33

Tauchen, George and Robert Hussey, “Quadrature-based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models,” Econometrica, March 1991, 59 (2), pp. 371–396. Taylor, John B., “Aggregate Dynamics and Staggered Contracts,” Journal of Political Economy, February 1980, 88 (1), 1–24. , “Discretion versus Policy Rules in Practice,” Carnegie-Rochester Conference Series on Public Policy, December 1993, 39, 195–214.

34

TECHNICAL APPENDIX T-1

Derivation of Monopolistic Competition Individual Demand and Aggregate Price

In this section, we derive the expressions for individual differentiated-good demand and aggregate price shown in (2.5) and (2.6) in Section 2.2. This approach was first proposed by Dixit and Stiglitz (1977). We assume that consumers only care about aggregate consumption ct in each period and that the elasticity of substitution among differentiated goods is a constant θ. Let i be the index for the unit measure of differentiated firms. As in (2.3), the Dixit-Stiglitz CES consumption aggregator is defined as the following, Z ct ≡

1

ct (i)

θ−1 θ

θ  θ−1

∀t and θ > 1

di

(T.1.1)

0

where θ > 1 represents the elasticity of substitution among all differentiated goods. The elasticity of substitution has a value of θ > 1 because firm profits must be greater than or equal to zero and less than infinity. Under this specification, price is a markup over expected marginal cost. And the markup is θ/(θ − 1). The individual demand equations for each differentiated good ct (i) for all i result from minimizing the cost of consuming some aggregate level of consumption Ct which is less than or equal to aggregate consumption ct by choosing the optimal consumption bundle ct (i) given individual prices pt (i).19 1

Z

pt (i)ct (i) di subject to Ct ≤

min ct (i)

1

Z

ct (i)

θ−1 θ

θ  θ−1

di

(T.1.2)

0

0

The Lagrangian is the following: 1

Z L=

"

Z

1

pt (i)ct (i) di + λt Ct − 0

ct (i)

θ−1 θ

θ #  θ−1

di

(T.1.3)

0

Because the Lagrange multiplier λt has the interpretation of being the marginal cost of an extra unit of aggregate consumption, λt is the price of aggregate consumption pt . That is, pt is the price index of all differentiated goods consumed. The Lagrangian can then be rewritten with the aggregate price pt instead of the multiplier λt . " Z  θ # Z 1

L=

1

pt (i)ct (i) di + pt Ct − 0

ct (i)

θ−1 θ

θ−1

di

(T.1.4)

0

19

The dual problem of maximizing the level of aggregate consumption subject to a budget constraint of expenditures being less than or equal to the currency held at the time of exchange does not yield the same result because the multiplier on the budget constraint does not have the interpretation as the price of an extra unit of aggregate consumption.

35

Because the constraints always bind, the first order conditions are: Z

1

ct (i)

pt (i) = pt

θ−1 θ

1  θ−1

di

1

ct (i)− θ

∀ t, i

(T.1.5)

0 1

Z

ct (i)

ct =

θ−1 θ

θ  θ−1

∀t

di

(T.1.6)

0

Solving for ct (i) in (T.1.5) substituting in the constraint (T.1.6) which is the definition of the CES consumption aggregator gives the following expression for the individual demand for differentiated good i.  ct (i) =

pt (i) pt

−θ ct

∀t, i

(T.1.7)

This is the individual demand function given in equation (2.5). As the price of the individual differentiated good pt (i) increases relative to the aggregate price index pt , demand for the individual differentiated good decreases relative to total consumption. Plugging (T.1.7) back into (T.1.6) and solving for pt gives the expression for the price of aggregate consumption given in (2.6). Z pt =

1

pt (i)1−θ di

1  1−θ

∀t

(T.1.8)

0

The aggregate price index pt is an increasing function of the individual prices pt (i).

36

T-2

Problem with Taylor rule being function of inflation π in this model

In Section 4, we presented a Taylor rule of the following form, where g¯, Y ∗ , and p∗ are the long-run averages of the money growth rate, output level, and price level, respectively, from the optimal monetary equilibrium model. 

Y g(p, Y ) = g¯ Y∗

 φ1 

p p∗

 φ2 (4.1)

However, most Taylor rules, starting with Taylor (1993), are formulated with the interest rate as a function of deviations of output from long-run output and inflation (not price level) from some inflation target. Because the monetary instrument in our model is the money growth rate gt , and the interest rate Rt is an equilibrium function of the underlying shocks and the money
growth rate rule R z−1 , z|g(·) in (3.7), it is fine to write the left-hand-side of the Taylor rule as in (4.1). But an approach more in line with the existing Taylor rule literature would be to specify our Taylor rule in the following way, 

Y g(π, Y ) = g¯ Y∗

φ1 

π  φ2 π∗

(T.2.1)

with current period inflation π and target inflation π ∗ replacing current period price level p and the target price level p∗ , respectively. If the policy rule were g(π, Y ) as in (T.2.1), the equilibrium pricing rule would have to be a function not only of the previous period productivity level z−1 but also of the two-periods previous productivity level z−2 . The reason comes from the equilibrium difference equation in the price function.  " #!−1    −γ gt+1 θ χ  ∀t Et−1 gt e−zt Et 1−γ (3.13) pt = β θ−1 pt+1 A policy rule g(π, Y ) must be mapped into a function of the underlying productivity levels in order to solve for the p function in (3.13). Because inflation equals (p − p−1 /p−1 ), the money growth rate rule can only be determined as a function of at least z−1 and z−2 . If we must know the two period history (z−2 , z−1 ) to determine g(π, Y ), the prices must at least be a function of that two period history (z−2 , z−1 ). But prices being a function of two period histories means that the policy rule can only be determined with a three-period history (z−3 , z−2 , z−1 ) because of the inflation term π. The argument is circular and implies that the state space using inflation π in the Taylor-type money growth rate rule makes the equilibrium state the entire history of productivity levels. This is neither computationally tractable nor theoretically appealing. 37