Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 812, July 2004Screen Reader Version*
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Abstract:
Changes in monetary policy are typically implemented gradually, an empirical observation known as interestrate smoothing. We propose the explanation that timenonseparable preferences may render interestrate smoothing optimal. We find that when consumers have `` catchingupwiththeJoneses" preferences, optimal monetary policy reacts gradually to shocks to prevent inefficiently fast adjustments in consumption. We also extend our basic model to investigate the effects of capital formation and nominal rigidities on the dynamics of optimal monetary policy. Optimal policy responses continue to be gradual in the presence of capital and sticky prices, with a size and speed that are in line with empirical findings for the U.S. economy. Our results emphasize that gradualism in monetary policy may be needed simply to guide the economy on an optimally smooth path.
Keywords: optimal monetary policy, habit persistence, catching up with the Joneses
JEL Classification: E50, E52
^{*}Email address: [email protected]. This paper is a revised version of Chapter 2 of my doctoral dissertation at the University of Pennsylvania. I thank Bill Dupor, Jesús FernándezVillaverde, and Dirk Krueger for their guidance, Dale Henderson, Rob Martin, John Rogers, Nathan Sheets, and Luis Felipe Zanna for helpful comments, and S. Borağan Aruoba for many helpful discussions. The views expressed here are solely those of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any person associated with the Federal Reserve System. Return to text
Characterizing the nature of optimal monetary policy both in the long run and over the course of business cycles continues to be a fundamental question in macroeconomics. The gradual manner in which monetary policy is usually implemented has received attention in both academic and policy debates of late, including in mid2004 as the Federal Reserve was on the verge of raising its target for the federal funds rate. Rather than a case for gradualism being made solely on the basis of uncertainty on the part of policymakers about the state of the economy, which is an oftencited reason for a cautious approach to monetary policy, much of the case seemed to rest on inducing a smooth transition for both the real economy and financial markets. This paper provides a justification for the latter motivation.
By gradualism in monetary policy, often referred to as interestrate smoothing, we mean that following an exogenous shock that requires policy to respond, the peak response does not occur immediately, but rather with a time lag. Sack (2000) shows that the Greenspanera Fed's peak response does not occur until up to 15 months following some types of shocks, and Sack and Wieland (2000) and Dennis (2003) provide further evidence along the same lines.^{1} The literature has taken the position that such gradualism reflects optimal policysetting, and the challenge is to understand why such persistence is optimal. We take this position here as well.
We offer the explanation that catchingupwiththeJoneses preferences renders interestrate smoothing optimal. Under catchingupwiththeJoneses preferences, each consumer's utility depends not only on the level of his own consumption, but also on the level of lagged aggregate consumption. Thus, preferences display an externality as well as timenonseparability. For the optimal policymaker, who internalizes the consumption externality, this latter feature of preferences acts as internal habit persistence.^{2} Timenonseparability in preferences has been shown in several recent studies to be important in understanding gradual responses of real variables to monetary policy shocks, so our explanation here of the dynamics of optimal policy is closely related.^{3} The intuition for the interestrate smoothing result in our model is the following. With the lagged consumption externality, private agents choose to adjust consumption in response to shocks suboptimally quickly if policy is held fixed. Because the lagged consumption externality acts as internal habit persistence from the perspective of the optimal policymaker, the policymaker chooses a smoother path of consumption than would be chosen by the private sector. In order to support this smoother path of consumption, a smoothly varying nominal interest rate is required.
Consumption externalities and timenonseparability in preferences have also proven useful recently in explaining macroeconomic puzzles in asset pricing (for example, Abel (1990) and Campbell and Cochrane (1999)) and international economics (Uribe (2002)). The implications of consumption externalities for fiscal policy have recently been studied by Ljungqvist and Uhlig (2000) and Abel (forthcoming). We continue building on this recent tradition of investigating the macroeconomic consequences of such preferences.
Of further interest is that the approach we employ is based on the Ramsey formulation of a dynamic stochastic general equilibrium model, with no prior assumptions about the targets of monetary policy. Recent studies that also use this approach include SchmittGrohe and Uribe (2004), Siu (2004), and King and Wolman (1999). To our knowledge, though, we are the first to solve a Ramseytype optimal policy problem with timenonseparable preferences.
The issue of interestrate smoothing has attracted attention in recent years, especially since Clarida, Gali, and Gertler (1999, p. 1702) identified it as an important outstanding issue in monetary policy. Woodford (2003a and 2003b) has recently argued that infusing monetary policy with gradualism is welfareenhancing through its effects on private sector expectations of future policy and hence its effects on longterm interest rates. Our explanation of optimal interestrate smoothing is different from his in two respects: we do not restrict ourselves apriori to a particular class of monetary reaction functions  we instead solve the full Ramsey problem  and, more fundamentally, it is preferences that make such policy optimal in our model. As mentioned above, another strand of the literature shows that gradualism is optimal in the presence of various types of uncertainty facing by policymakers. Our paper demonstrates another possibly important motivation for gradualism.
The rest of our work is organized as follows. We first present the simplest version of our model, which demonstrates the basic reason why external habit persistence makes interestrate smoothing optimal. In Section 1, we describe this basic model, which features the consumption externality, perfectly competitive product markets, and no physical capital. Here we find two central results  that the Friedman Rule is suboptimal in the presence of catchingupwiththeJoneses and optimal monetary policy responds gradually to technology shocks. In Section 2, we study a more empiricallyrelevant model by adding capital formation and nominal rigidities, the key features of RBC models and New Keynesian models, respectively.^{4} In this full model, the main result of optimal interestrate smoothing remains, and the size and speed of smoothing continues to match fairly well the empirical findings of Sack (2000). Thus, our results are obtained in both models with flexible prices and sticky prices, showing that intertemporal linkages caused by dynamic pricesetting need not be a key factor in explaining monetary policy inertia. In addition, this full model does quite well in matching key business cycle statistics of the U.S. economy. Section 3 concludes.
In this section, we describe the simplest version of our cashinadvance model, an economy that features a consumption externality, no physical capital, and perfectly competitive product markets.
Household preferences are represented by
The household faces a flow budget constraint
(2) 
(3) 
The household maximizes (1) subject to (2), (3), and a sufficiently large negative bound on bond holdings by choosing decision rules for , , , , and , taking as given the prices and as well as lagged aggregate consumption and . The household observes the logtechnology realization , which is the only source of uncertainty in the economy, before making decisions. Associating the Lagrange multiplier with the budget constraint and with the cashinadvance constraint, the firstorder conditions for the household problem are
(4) 
(5) 
(6) 
(7) 
(8) 
Finally, note that combining the firstorder conditions on money holdings and bond holdings and using the optimality conditions for cash consumption and credit consumption gives
(9) 
A representative firm produces the output good by hiring labor in a perfectlycompetitive labor market. The firm's production function is linear in labor,
(10) 
Maximization of profits implies the marginal product of labor is equated to the real wage.
The government has a simple role: it simply puts nominal money into or takes nominal money out of circulation through lumpsum transfers to households. The government budget constraint is
(11) 
A competitive monetary equilibrium for this economy is stochastic processes for , , , , , , and such that the household maximizes utility with the cashinadvance constraint holding with equality, the firm maximizes profit, the labor market clears, the bond market clears every period, so that
(12) 
(13) 
(14) 
To consider optimal policy, we construct a Ramseytype allocation problem. The constraints that competitive monetary equilibrium impose, beyond the resource constraint, on the policymaker are presented in the following Proposition, the proof of which appears in more general form in Appendix A.
, (15)
and the labormarket clearing condition
. (16)
In addition, allocations that satisfy (13), (14), (15), and (16) can be supported as a competitive monetary equilibrium.
Notice that it is the marginal utility of credit good consumption, rather than the marginal utility of cash good consumption as in the implementability constraint of Chari and Kehoe's (1999) monetary model or Siu's (2004) model, that multiplies cash good consumption. This difference arises due to the lumpsum manner, rather than open market operations, through which monetary policy is conducted in our model.
The constraint (16) is needed to ensure labormarket clearing because the government is assumed to not have any instruments that drive a wedge between the wage and the marginal product of labor. In the absence of this constraint, an implicit timevarying labor tax would correct the consumption externality, eliminating the need for monetary policy. To the extent that monetary policy can respond to business cycle events more quickly than labor taxes, we think specifying the policy problem this was is realistic.^{7} The term is the wage expressed in terms of allocations using the household firstorder conditions (5) and (6).
The Ramsey government internalizes the consumption externality and chooses allocations for consumption and labor. That is, in the Ramsey problem, we set and directly in the utility function before optimizing. The Ramsey problem is thus to maximize
(17) 
(18) 
(19) 
Associating the Lagrange multiplier to the resource constraint, the multiplier to the implementability constraint, and the multiplier to the wage constraint, the firstorder conditions for the Ramsey problem with respect to cashgood consumption, creditgood consumption, and labor are
(20) 
(21) 
(22) 
We first describe how we calibrate the model and then present the quantitative results. The period utility function is
(23) 
(24) 
The stochastic process follows the AR(1) process
(25) 
We calibrate the externality parameter so that in the optimal steadystate the nominal interest rate matches the threemonth Treasury bill rate during the period January 1988September 2003, which was 4.6 percent.^{8} Given the rest of our calibration, matching this nominal interest rate requires . This parameter value is a bit lower than but in line with the estimate 0.60 by Ravn, SchmittGrohe, and Uribe (2003).^{9} Table 1 summarizes the calibration of this version of our model.

We now characterize numerically the dynamic properties of the
optimal allocations in this version of our model. Our solution
method is standard loglinear techniques  in particular, we use
Uhlig's (1999) method of undetermined coefficients. We
loglinearize the policymaker's firstorder
conditions (20), (21), and (22), and the resource
constraint (13), around the deterministic steadystate of
these conditions.^{10}
Table 2 presents the optimal steadystate nominal interest
rate for several values of . As soon as the consumption externality rises
above zero, the Friedman Rule of a zero net nominal interest rate
is longer optimal, which is our first central result.

With , the policymaker chooses an allocation in which the marginal utilities of the two goods are not equal even though the two goods are technologically identical. The economic reason for the result is straightforward. In the absence of any other tax instruments, the nominal interest rate can be used to tax overall consumption through its effect on cash good consumption. The consumption externality, if uncorrected, causes the consumption index to be too high relative to the Ramsey government's choice, thus taxing consumption via a strictly positive net nominal interest rate brings down the index. Unregulated consumption is not too far above its optimum for small values of because the externality is mild, so the inflation tax rate  controlled via the nominal interest rate  does not need to be very large. As rises, however, the externality problem is more severe and pushes unregulated consumption even further above its optimum, requiring a larger tax. Monetary policy in our model thus operates through the costs of holding money, in contrast to the often ``cashless" environments of ISLMstyle models in which this channel does not exist. This ``fiscal" use of monetary policy is similar in idea to the finding of Ljungqvist and Uhlig (2000) that a labor tax corrects a consumption externality. However, our focus is on the implications for optimal interestrate smoothing, so we now consider the dynamics of the model.
For all dynamic results, here and in Section 2, the economy is assumed to begin in the deterministic steadystate associated with the optimal allocation. Considering an impulse response analysis illustrates the basic idea behind the optimal interestrate smoothing result in the model. Figure 1 presents twentyquarter impulse responses for consumption of each type of good, output, the optimal nominal interest rate, the inflation rate, and the money stock under the optimal policy in response to a one standard deviation positive shock to technology. The peak response of a 39basispoint tightening of the nominal interest rate occurs after four quarters. This magnitude and speed of policy adjustment is in line with that found by Sack (2000) and Dennis (2003) for the Greenspanera Fed. Using a VAR, each finds that the magnitude of the peak response of the nominal interest rate is approximately 20 basis points following a one standard deviation supply shock, which we interpret as the technology shock in our model.^{11}Sack (2000) finds that the peak response occurs after four quarters, while Dennis (2003) finds that the peak response is reached after three quarters. The prediction of our model is thus in line with empirical evidence.
The intuition for why interestrate smoothing is optimal is as described in the introduction, and we now elaborate a bit further. In response to a shock, the policymaker would like consumption to adjust more gradually than the private sector would choose on its own because no individual takes into account the fact that the level of his current consumption, holding all else constant, negatively affects all others (including himself) in the future. A way to think about this is that utility depends not only positively on the level of consumption but also negatively on the growth rate of consumption. This latter aspect, consumers disliking large swings in consumption, gives rise to the terminology `` habit persistence"  consumers are assumed to become accustomed to their level of consumption and dislike changes. However, this concern for consumption growth rates is not taken into account by the individual with catchingupwiththeJoneses preferences, while it is taken into account by the policymaker. This timenonseparability of preferences which is internalized by the policymaker drives the main result of optimal interestrate smoothing both in the basic model and the richer model of Section 2, as well as other versions that we consider in a companion paper, Chugh (2004).
Figure 1 shows that output and consumption of each type of good also reach their peak responses gradually. Because the nominal interest rate is the opportunity cost of holding money, an increase in the nominal interest rate leads to a decrease in consumption of cash goods. Overall, however, consumption, and hence output, rise due to the positive technology shock, in line with standard RBC predictions. In particular, the substitution effect and income effect both work in the same direction so total consumption, the sum of cash consumption and credit consumption, unambiguously rises. Under the optimal policy, inflation tracks the nominal interest perfectly because the real interest rate is constant, thus inflation rises gradually as well.
Figure 1: Twentyquarter impulse responses to a one standard deviation positive technology shock in the economy with no physical capital and perfectly competitive product markets. Variables are measured in percent deviation from their steadystate values. The externality parameter is , and the steadystate policy is .

Next, Table 3 presents key business cycle moments for and . To compute these business cycle statistics, we conduct 1,000 simulations, each of length 100 periods, of the model and report the median first and second moments. As mentioned in Section 1.5, in principle we must impose a constraint on the Ramsey problem that ensures that allocations can be supported as a monetary equilibrium. In solving for the decision rules and conducting these simulations, we drop this noarbitrage constraint and numerically check during the simulations that the noarbitrage constraint never binds. The results we report are for a standard deviation of the productivity shock of , which turns out to be small enough that the noarbitrage constraint is always slack during the simulations. If we increase the volatility to , we still find that the noarbitrage constraint is always slack during the simulations.^{12} Only when we increase the volatility to , ten times our benchmark calibration, we do encounter simulations in which the zero lower bound on the nominal interest rate is reached. Thus, only when the volatility of the technology process is very large does the noarbitrage constraint become a computational issue. With our standard calibration of , this is not an issue, so we seem numerically justified dropping the noarbitrage constraint from the Ramsey problem.
As Table 3 shows, in our model with the consumption externality both the nominal interest rate and the inflation rate are procyclical, consistent with empirical evidence. However, the model's contemporaneous correlations of the nominal interest rate and inflation with output are stronger than those in data  for example, Stock and Watson (1999) report that the contemporaneous correlation of the the threemonth Treasury bill with output is 0.41 and the contemporaneous correlation of the GDP deflatormeasured inflation rate with output is 0.15. The nominal interest rate in our model essentially tracks consumption because its role is to mitigate the consumption externality. Because consumption equals output in a model without investment, the nominal interest rate in the model tracks output, explaining the model's nearperfect correlations between the nominal interestrate and output and between inflation and output.
The procyclicality of the nominal interest rate in our model at least stands in contrast to countercyclical nominal interest rates often found in technologyshock driven models, such as Altig et al (2002), but our model must be extended to better quantitatively match the data. With this motivation, as well as in the interest of building a model that can explain key business cycle facts in addition to being suited for an investigation of optimal monetary policy, we turn in Section 2 to the task of extending the model to include both capital formation and nominal rigidities, which are the key features of RBC models and New Keynesian models, respectively. We incorporate these elements because both seem to be important in explaining observed business cycle dynamics and so may reasonably be suspected to have an effect on optimal monetary policy over the business cycle. As we will show, the quantitative predictions regarding interestrate smoothing of this full model with external habit, capital, and sticky prices continue to match U.S. data quite well. Our companion paper, Chugh (2004), studies versions of the model that are intermediate between the basic model of this section and the full model of the next section.
Capital formation and nominal rigidities lie at the heart of RBC models and New Keynesian models, respectively. Most macroeconomic models that include capital formation assume prices are flexible, while most New Keynesian models that feature sluggish price adjustment abstract from capital formation. Both features are likely important in explaining observed business cycle phenomena, however, and bringing these ideas together in macro models is an understudied area of research. The general sense in the literature has been that capital, because it is little correlated with GDP at business cycle frequencies, plays an unimportant role in determining policy over the business cycle and therefore can be safely abstracted from. We know of no work, however, that conclusively shows that capital formation is indeed unimportant for the consideration of optimal monetary policy. Here, we bring together these features along with external habit persistence to study in a richer, and likely more empirically relevant, framework the issue of interestrate smoothing.
We introduce monopolistic competition and nominal rigidities in standard New Keynesian fashion, by segmenting product markets into final goods and differentiated intermediate goods. We sketch how the model is modified to accommodate these elements, show how the optimal policy problem is affected, and then present results.
Cash goods, credit goods, and investment goods are physically indistinguishable  they are all final goods. These final goods are produced in a competitive sector according to a DixitStiglitz CES aggregator
(26) 
(27) 
Each differentiated good is produced by firm using a constantreturnstoscale technology
(28) 
We employ Taylor staggered pricesetting to introduce nominal rigidities. Once set, an intermediate firm's nominal price cannot be changed for periods. The profit maximization problem of each intermediate goods firm is thus a dynamic one,
(29) 
Throughout our analysis, we restrict attention to the case . Imposing and using the demand function (27) to substitute for , the firstorder condition of (29) with respect to is
(30) 
(31) 
We consider only symmetric equilibria in which all intermediate producers in a given pricing cohort make the same decisions, thus subscripts other than time subscripts denote only to which pricing cohort a firm belongs. Given the demand it faces, firm then purchases capital and labor in spot markets to minimize cost. Cost minimization implies that
(32) 
(33) 
With twoperiod Taylor pricesetting, and with symmetry within each pricing cohort, the final goods aggregator can be written as
(34) 
(35) 
(36) 
The household problem is a straightforward extension of that described in Section 1.1. The household chooses aggregate labor and aggregate capital, along with consumption of cash and credit goods as well as asset holdings to maximize (1) subject to the cashinadvance constraint (3), a sufficiently large negative bound on bond holdings, and the flow budget constraint
(37) 
Associating the Lagrange multiplier with the budget constraint and with the cashinadvance constraint, the firstorder conditions for the household problem are the same as expressions (4) (8), with the added optimality condition on capital holdings
(38) 
We now describe the optimal policy problem for this model. The main difference with the basic model of Section 1 is an extra constraint that ensures that the nominal rigidity in product markets is respected. The policymaker maximizes the representative household's lifetime discounted expected utility subject to the resource constraint
(39) 
(40) 
(41) 
(42) 
(43) 
(44) 
(45) 
The household implementability constraint (40) is an extension of that in the basic model, with profit terms arising because the policymaker optimally redistributes them back to households. Because in our dynamic analysis we assume the economy starts in the Ramsey steadystate, the initial capital stock is that implied by the steadystate of the Ramsey optimality conditions for . Thus, we adopt the `` timeless" approach to policy, discussed by Woodford (2003b, p. 538540), in which features of the economy associated with only the startup period are treated as nongeneric events.^{13} With this timeless perspective, all decision rules are timeinvariant. Hence we restrict attention to the firstorder conditions for .
As in the basic model of Section 1, constraints are required to ensure factor markets clear because we assume the government has no tax instruments besides the nominal interest rate. The constraints (42) and (43) ensure labor market clearing and capital market clearing, respectively. Notice that in these constraints we use the aggregates and in the marginal product terms  we can do so because capitallabor ratios are equated across firms, as discussed above.
The stickyprice firm constraint is simply the optimal pricing constraint (31) with prices substituted out using the demand function (27) and gross inflation substituted out using (35). In the first term of the stickyprice constraint, the marginal cost term depends on the type1 firm's capitallabor ratio, and in the second term of the sticky price constraint, the marginal cost term depends on the type2 firm's capitallabor ratio. As noted by (33), though, these capitallabor ratios are common. So we will use marginal cost as a function of the aggregate capitallabor ratio, and doing so ensures that capitallabor ratios are always equated.
Finally, as discussed in Section 1.7, we do not impose the noarbitrage constraint that guarantees the chosen allocations can be supported as a monetary equilibrium because in our simulations the zerolower bound on the nominal interest rate was never reached.
In time period , the policymaker chooses , , , , , , , and to maximize the representative household's lifetime discounted utility. Note that while the policymaker chooses aggregate capital to carry into the next period, the choice about how to divide the current capital stock among the two types of firms is a static one.
For brevity, we do not present the firstorder conditions of the optimal policy problem, except to note that they now include both forwardlooking terms and backwardlooking terms. The former arise because of the timenonseparability of preferences, capital formation, and the nominal rigidity, while the latter arise because of the nominal rigidity. We again proceed by loglinearizing these firstorder conditions around their deterministic steadystate and present the results in the next section.
With the exception of the externality parameter , we use the same parameter values presented in Table 1. We also now have a few new parameter values to choose. The production function is assumed to be CobbDouglas, , with capital's share of output to match the annual capital to output ratio of 2.8 observed in the U.S. The quarterly depreciation rate of capital is set to . We set the static gross markup of intermediate goods producers at , which is consistent with the empirical evidence of Basu and Fernald (1997) and is a commonlyused value in the New Keynesian literature. Finally, we use the same strategy discussed in Section 1.6 to calibrate a value for . Matching an optimal steadystate annual nominal interest rate of 4.6 percent here requires .
First, we discuss the steadystate inflation rate in the absence of habit (i.e., with ). A result in many stickyprice models is that zero inflation is optimal because of the resource misallocation cost that would result from any nonzero level of inflation. In our model, zero inflation is not optimal even in the absence of habit persistence. The reason is that there is an explicit cost of holding money in our model, which we model through the cashinadvance constraint, in contrast to the often `` cashless" environments studied in many monetary policy models. In the presence of an explicit cost of holding money but absent nominal rigidities or any other market imperfections, a wellknown result is that the optimal monetary policy is the Friedman Rule  deflate at the rate of time preference. With both nominal rigidities and costs of holding money, a natural conjecture is that, absent habit persistence, the optimal inflation rate would be between the Friedman deflation and price stability. This is in fact what we find, as Table 4 shows. Table 4 shows that with , the optimal steadystate gross annual inflation rate is with an associated optimal gross annual nominal interest rate of . Optimal monetary policy balances the opportunity cost of holding money against the resource misallocation cost arising from nonzero inflation, which is consistent with those studies, such as Khan, King, and Wolman (2003), that do employ both nominal rigidities and costs of holding money.

Next, we characterize the dynamic optimal monetary policy response. As before, we conduct an impulse response analysis to a onetime, onestandarddeviation positive shock to . Figure 2 shows that the peak response of the nominal interest rate is a tightening of 96 basis points, which occurs seven quarters after the shock. The time lag with which the peak response occurs is longer than in the basic model, a result driven by the presence of capital formation. Jermann (1998) and Lettau and Uhlig (2000) show that in the presence of both capital and habits, consumption is excessively smooth compared to data and compared to RBC models without habit, and our result seems closely related. As we discussed in Section 1.7, the nominal interest rate essentially tracks consumption in our model because it supports the optimal consumption path. The slow optimal response of consumption in the presence of capital is being supported by a slow optimal response of the nominal interest rate. Nevertheless, the sevenquarter lag of the peak response is not grossly out of line with the three or fourquarter lag Sack (2000) and Dennis (2003) find. The peak response of optimal policy in this model is higher than in the basic model due to the stronger degree of habit persistence required to calibrate the steadystate. Comparison of Figure 2 with Figure 3, which uses  our calibrated externality parameter in the basic model of Section 1.6  illustrates the role of the larger in driving the peak response of policy higher. With , the steadystate optimal policy is and the peak response is a tightening of 46 basis points, which occurs six quarters after the shock.
Figure 2: Fortyquarter impulse responses to a one standard deviation positive technology shock in the economy with physical capital and sticky prices. Variables are measured in percent deviation from their steadystate values. The externality parameter is , the steadystate gross markup is , and the steadystate policy is .
Figure 3: Fortyquarter impulse responses to a one standard deviation positive technology shock in the economy with physical capital and sticky prices. Variables are measured in percent deviation from their steadystate values. The externality parameter is , the steadystate gross markup is , and the steadystate policy is .
In both Figure 2 and Figure 3, we see that factor use and hence production by the firm currently setting price, the type1 firm, falls while factor use and hence production by the firm not currently setting price, the type2 firm, rises. This result is similar to that described by Woodford (2003b, Chapter 3)  when there is an asymmetry across sectors, substitutability allows the possibility that one sector's output rises and another sector's falls in response to a common shock. Here, the asymmetry across the two pricing cohorts is that in steadystate they produce different quantities  if inflation is strictly positive, the price set by priceadjusting firms is higher than the price of nonadjusters and so they produce less. Under both the parameter values and , this is exactly the case, so the optimal policymaker takes advantage of this and assigns nonadjusting firms larger output following the shock. Also interesting to note is that in both Figure 2 and Figure 3 we see that investment has a humpshaped profile, a feature of business cycles that Christiano, Eichenbaum, and Evans (2001) call an important one for business cycle models to replicate. They replicate this feature by introducing adjustment costs of capital, while we obtain this result without such costs.
Table 5 presents business cycle moments. Comparing these with the moments reported by King and Rebelo (1999, p. 938) for the U.S. economy, we see that our model performs reasonably well. Regarding real variables, it performs best on firstorder autocorrelations and less so on contemporaneous correlations with output. The main reason for this is that with strong habit persistence, consumption optimally moves very slowly, so its correlation with output is reduced, leaving investment to drive more of the fluctuations in output. In our models, the correlation of investment with output is higher than found in data and the correlation of consumption with output is much lower than found in data. Regarding nominal variables, the contemporaneous correlation of inflation with output is in line with that found by Stock and Watson (1999, p. 32) for the U.S. economy, while that of the nominal interest rate is a bit higher than they find.

Thus, optimal interestrate smoothing of empirically plausible size and speed emerges in our model with habits, capital accumulation, and sticky prices. The time lag of the peak response is a bit longer than empirically observed but not grossly so. We conjecture that the policy response can be sped up by the introduction of adjustment costs of capital, just as Chugh (2004) does in a model with capital accumulation but flexible prices. However, this does not seem to be a firstorder concern in this case. Furthermore, the business cycle properties are in line with empirical facts, providing further encouragement to researchers to study models with both capital and sticky prices.
The result that gradualism in monetary policy may optimally arise due to concern by policymakers to move the economy along a smooth path is robust to many features of the economic environment, as the results here and in Chugh (2004) show. We believe this motivation for interestrate smoothing is one that the literature on monetary policy has largely ignored. The existing literature has focused on policymakers' uncertainty about data and/or parameters of the economy as the main driving force behind gradualism. While uncertainty may be one reason behind this approach to policy, we have shown that concern for a smooth path for the economy may be another important reason. In fact, our models match fairly well empiricallyobserved sizes and speeds of policy responses despite full certainty on the part of policymakers about parameters and shocks.
In addition to addressing the issue of optimal interestrate smoothing, we aimed to build a model which may be useful in explaining both optimal monetary policy and key business cycle facts. Capital formation and nominal rigidities are widely viewed as two important sources of persistence in the economy. Understanding the properties of optimal monetary policy in a model that features both thus seems crucial to better guide policy, but relatively little work has been done in this area to date.
On a broader note, the study of the nature of optimal monetary policy in the presence of habit persistence, both internal and external, is receiving growing attention. For example, Amato and Laubach (2004) have begun exploring this issue as well, albeit in a somewhat different class of models.^{14} SchmittGrohe and Uribe (2004) note that they intend to characterize optimal monetary and fiscal policy in a business cycle model along the lines of Christiano, Eichenbaum, and Evans (2001), which includes, among other features, habit persistence. Our work here makes a contribution to the beginning of this field.
Consumption externalities and the often closelyrelated idea of habit persistence are not yet widelyaccepted features of the representative agent's preferences in macroeconomic models, but they have proven useful recently in explaining macroeconomic puzzles in the monetary transmission mechanism, asset pricing, and international economics. We have found here that such preferences also explain gradualism in monetary policy. The study of optimal monetary policy can thus be conducted in macroeconomic models that are able to address these other issues as well. Even more generally, we believe it worthwhile to study the consequences for optimal macroeconomic policy of other richer forms of preferences as well, as Backus, Routledge, and Zin (2004) suggest.
Here we present the Ramsey problem for the most general flexibleprice version of our model, one that features consumption externalities, physical capital, and monopolistic competition. We specialize to the specific models considered here and in Chugh (2004) as we proceed.
We wish to prove that any competitive monetary equilibrium (CME) satisfies the resource constraint
(46) 
(47) 
(48) 
(49) 
(50) 
First we prove that a CME satisfies (46), (47), (48), (49), and (50). Any equilibrium must satisfy the resource constraint (46). Because we restrict attention to monetary equilibria in which the net nominal interest rate must be nonnegative, (47) must also hold. The conditions (48) and (49) must hold because the government does not have access to instruments that tax labor or capital, thus these conditions ensure factormarket clearing.
To show that a CME must also satisfy (50), we first present again the household problem. Begin with the household budget constraint
(51) 
(52) 
(53) 
(54) 
(55) 
(56) 
(57) 
(58) 
(59) 
Next, multiply by and sum over to get
(60) 
(61) 
(62) 
We now show that any allocation of consumption, capital, and labor that satisfies (46), (47), (48), (49), and (50) can be supported as a CME. To establish this, we need to construct prices, policies, money holdings, and bond holdings.
First multiply the household budget constraint (51) by , and sum over time following period to get
(63) 
(64) 
(65) 
(66) 
The last remaining item to consider is bond holdings. The previous summation can be used to obtain an expression for bond holdings at time , which depends, as in Chari and Kehoe (1999), on future allocations. It is difficult to see from the resulting expression, however, that bond holdings are zero, which they must be because they are privatelyissued bonds. An easier way to understand why the bond market clears is to construct the household budget constraint from the resource constraint. Because factor markets clear and the technology is constantreturnstoscale, we have
(67) 
(68) 
The Ramsey problem for the model with staggered pricesetting is a straightforward extension of the Ramsey problem for the model with flexible prices described in Appendix A. The only difference is in the equilibrium expression for aggregate profits. With symmetry in each pricing cohort,
(69) 
(70) 
Abel, Andrew B. 1990. Asset Prices under Habit Formation and Catching up with the Joneses. American Economic Review. 80. 3842.
Abel, Andrew B. Forthcoming. Optimal Taxation When Consumers Have Endogenous Benchmark Levels of Consumption. Review of Economic Studies.
Altig, David, Lawrence J. Christiano, Martin S. Eichenbaum, and Jesper Linde. 2002. Technology Shocks and Aggregate Fluctuations. Federal Reserve Bank of Cleveland.
Backus, David, Bryan Routledge, and Stanley Zin. 2004. Exotic Preferences for Macroeconomists. NBER Working Paper.
Basu, Susanto and J. G. Fernald. 1997. Returns to Scale in U.S. Production: Estimates and Implications. Journal of Political Economy. 105. 249283.
Boivin, Jean and Marc Giannoni. 2003. Has Monetary Policy Become More Effective? NBER Working Paper 9459.
Caballero, Ricardo. 1999. Investment. Handbook of Macroeconomics. John B. Taylor and Michael Woodford, Vol. 1B. Elsevier.
Campbell, John Y. and John H. Cochrane. 1999. By Force of Habit: A ConsumptionBased Explanation of Aggregate Stock Market Behavior. Journal of Political Economy. 107. 205251.
Chari, V.V., Lawrence Christiano, and Patrick Kehoe. 1991. Optimal Monetary and Fiscal Policy: Some Recent Results. Journal of Money, Credit, and Banking. 23. 519539.
Chari V. V., and Patrick J. Kehoe. 1999. Optimal Fiscal and Monetary Policy. Handbook of Macroeconomics. John B. Taylor and Michael Woodford, Vol. 1C. Elsevier.
Chari, V.V., Patrick J. Kehoe, and Ellen R. McGrattan. 2000. Sticky Price Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem? Econometrica. 68. 11511179.
Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans. 2001. Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy. Northwestern University.
Chugh, Sanjay K. 2004. Gradualism in Monetary Policy: The Roles of Habits, Capital, and Sticky Prices. University of Pennsylvania.
Clarida, Richard, Jordi Gali, and Mark Gertler. 1999. The Science of Monetary Policy: A New Keynesian Perspective. Journal of Economic Literature. 37. 16611707.
Clarida, Richard, Jordi Gali, and Mark Gertler. 2000. Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory. Quarterly Journal of Economics. 115. 147180.
Cooley, Thomas F. and Edward C. Prescott. 1995. Economic Growth and Business Cycles. Frontiers of Business Cycle Research. Thomas F. Cooley. Princeton.
Dennis, Richard. 2003. Inferring Policy Objectives from Economic Outcomes. Federal Reserve Bank of San Francisco Working Paper.
Dynan, Karen E. 2000. Habit Formation in Consumer Preferences: Evidence from Panel Data. American Economic Review. 90. 391406.
English, William B., William R. Nelson, and Brian P. Sack. 2003. Interpreting the Significance of the Lagged Interest Rate in Estimated Monetary Policy Rules. Contributions to Macroeconomics. 3. 116.
Ferson, Wayne E. and George M. Constantinides. 1991. Habit Persistence and Durability in Aggregate Consumption: Empirical Tests. Journal of Financial Economics. 29. 199240.
Friedman, Milton. 1969. The Optimum Quantity of Money. The Optimum Quantity of Money and Other Essays. Chicago: Aldine. 150.
Fuhrer, Jeffrey C. 2000. Habit Formation in Consumption and Its Implications for MonetaryPolicy Models. American Economic Review. 90. 367390.
Fuhrer, Jeffrey C. 1995a. Inflation Persistence. Quarterly Journal of Economics. 110. 127160.
Fuhrer, Jeffrey C. and George Moore. 1995b. Monetary Policy TradeOffs and the Correlation between Nominal Interest Rates an Real Output. American Economic Review. 85. 219239.
Hall, Robert E. 1997. Macroeconomic Fluctuations and the Allocation of Time. Journal of Labor Economics. 15. S223S250.
Holden, Steinar and John C. Driscoll. 2003. Inflation Persistence and Relative Contracting. American Economic Review. 93. 13691372.
Khan, Aubhik, Robert G. King and Alexander L. Wolman. 2003. Optimal Monetary Policy. Review of Economic Studies. 70. 825860.
King, Robert G. and Sergio T. Rebelo. 1999. Resuscitating Real Business Cycles. Handbook of Macroeconomics. John B. Taylor and Michael Woodford, Vol. 1B. Elsevier.
King, Robert G. and Alexander L. Wolman. 1999. What Should the Monetary Authority Do When Prices Are Sticky? Monetary Policy Rules. John B. Taylor. NBER Conference on Research in Business.
Lettau, Martin and Harald Uhlig. 2000. Can Habit Formation Be Reconciled with Business Cycle Facts? Review of Economic Dynamics. 3. 7999.
Ljungqvist, Lars and Harald Uhlig. 2000. Tax Policy and Aggregate Demand Management Under Catching Up with the Joneses. American Economic Review. 90. 356366.
Lucas, Robert E. Jr. and Nancy L. Stokey. 1987. Money and Interest in a CashinAdvance Economy. Econometrica. 55. 491513.
Mankiw, N. Gregory and Ricardo Reis. 2002. Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve. Quarterly Journal of Economics. 117. 12951328.
Ravn, Morten, Stephanie SchmittGrohe, and Martin Uribe. 2003. Deep Habits. Duke University.
Rudebusch, Glenn D. 2002. Term Structure Evidence on Interest Rate Smoothing and Monetary Policy Inertia. Journal of Monetary Economics. 49. 11611187.
Sack, Brian P. 2000. Does the Fed Act Gradually? A VAR Analysis. Journal of Monetary Economics. 46. 229256.
Sack, Brian P. and Volker Wieland. 2000. InterestRate Smoothing and Optimal Monetary Policy: A Review of Recent Empirical Evidence. Journal of Economics and Business. 52. 205228.
SchmittGrohe, Stephanie and Martin Uribe. 2004. Optimal Fiscal and Monetary Policy Under Sticky Prices. Journal of Economic Theory. 114. 198230.
Siu, Henry E. 2004. Optimal Fiscal and Monetary Policy with Sticky Prices. Journal of Monetary Economics. 51. 576607.
Stock, James H. and Mark W. Watson. 1999. Business Cycle Fluctuations in U.S. Macroeconomic Time Series. Handbook of Macroeconomics. John B. Taylor and Michael Woodford, Vol. 1A. Elsevier.
Uhlig, Harald. 1999. A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily. Computational Methods for the Study of Dynamic Economies. Ramon Marimon and Andrew Scott. Oxford University Press.
Uribe, Martin. 2002. The PriceConsumption Puzzle of Currency Pegs. Journal of Monetary Economics. 49. 533569.
Woodford, Michael. 2003a. Optimal Interest Rate Smoothing. Review of Economic Studies. 70. 861886.
Woodford, Michael. 2003b. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton: Princeton University Press.
1. Empirical Taylorstyle monetary policy rules also suggest, through ubiquitously large positive and significant estimated coefficients on lagged interest rates, that there is a great deal of persistence in monetary policy during the Greenspan era. See, for example, Clarida, Gali, and Gertler (2000), Rudebusch (2002), and English, Nelson, and Sack (2003). Estimated coefficients on the lagged interest rate are typically in the range 0.40 to 0.80. However, Taylor rules, typically specified with inflation as an explanatory variable, will not be a good way to consider the conduct of optimal policy in our model because inflation will track the nominal interest rate perfectly. Return to text
2. As such, we will sometimes refer to our model as displaying external habit persistence, which is used synonymously in the literature with the term catchingupwiththeJoneses. A further point about terminology: ``catching up with the Joneses" has come to mean something different from ``keeping up with the Joneses:" the latter refers to a contemporaneous externality, while the former refers to a lagged externality. We use ``keeping up" in our title because it is the more common idiom, but it is the catchingup specification that we use throughout. Return to text
3. Recent studies that argue that timenonseparability in preferences is important to understanding dynamics in monetary economies are Fuhrer (2000), Christiano et al (2001), Altig et al (2002), and Boivin and Giannoni (2003). Return to text
4. A companion paper, Chugh (2004), studies the separate effects of capital formation, monopolistic competition, and sticky prices on optimal interestrate smoothing in the presence of catchingupwiththeJoneses preferences. Return to text
5. An assumption made particularly awkward by the usual interpretation, which we follow, that cash goods and credit goods are physically indistinguishable. Return to text
6. Technically, they have a common price both because of the unit marginal rate of transformation and because both credit sales and cash sales in period result in cash available for spending at the beginning of period . See Lucas and Stokey (1987, p. 493494) for more discussion. Return to text
7. See Chari and Kehoe (1999, p. 1680) for more discussion on this. Return to text
8. We choose this sample period because the empirical Taylor Rule literature has noted that estimated monetary policy reaction functions during this period differ statistically from those estimated using preGreenspan periods or longer periods that include the Greenspan era. See, for example, Taylor (1999, p. 330). This finding suggests that monetary policy has been conducted fundamentally differently in the past 17 years than in the preGreenspan era. Return to text
9. Ravn, SchmittGrohe, and Uribe (2003) model consumption externalities in a way fairly closer to our way and estimate a parameter which is comparable to our parameter of 0.60. They also allow the subutility function over the quasidifference in consumption to have coefficient of relative risk aversion , whereas we have a coefficent of relative risk aversion of 1 due to our log specification. However, their estimate is , so their results seem to be fairly comparable to our specification. Also note that the empirical works of Ferson and Constantinides (1991) and Fuhrer (2000), using macroeconomic data, and Dynan (2000), using microeconomic data, both assume internal habit persistence and so are not directly relevant here. Return to text
10. We do not prove that a Ramsey steadystate exists, or even that a solution to the Ramsey problem exists. Furthermore, as is also a common problem in the Ramsey literature, we cannot guarantee that even if a Ramsey steadystate exists that the decision rules imply convergence to it. We do find numerically a steadystate of the Ramsey firstorder conditions. Return to text
11. Sack (2000) uses the Federal Funds rate and Dennis (2003) uses the threemonth Treasury bill rate. Return to text
12. With and , the steadystate nominal interest rate is 4.6 percent, and the lowest nominal interest rate ever realized during our simulations was 1.2 percent. Return to text
13. This `` timeless" issue does not arise in the model without capital. In the model without capital, there is no asymmetry between the decision rules for and the decision rules for because there are no initial assets. Return to text
14. They consider internal habit, whereas we study external habit. Furthermore, they focus attention on the use of monetary feedback rules, whereas we solve a Ramseytype optimal taxation problem. Nonetheless, we view our work as complementary to theirs. Return to text
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