Optimal Monetary Policy with Nominal Rigidities. and Lumpy Investment

Optimal Monetary Policy with Nominal Rigidities and Lumpy Investment Tommy Sveen a, Lutz Weinke b a BI Norwegian Business School b Humboldt-Universität zu Berlin October 31, 2011 Abstract New Keynesian theory generally abstracts from the lumpy nature of plantlevel investment (see, e.g., Woodford 2003, Ch. 5 and 2005). Given the great interest in the role of investment spending for shaping optimal monetary policy, this simplification could be problematic. But is this the case? Our answer is no, which lends additional credibility to a large and growing literature in the field of monetary economics. Keywords: Lumpy Investment, Sticky Prices, Sticky Wages, Welfare. JEL Classification: E22, E31, E32 We thank Jordi Galí and Stephanie Schmitt-Grohé for comments. The usual disclaimer applies. The views expressed in this paper are those of the authors and should not be attributed to Norges Bank. 1

1 Introduction Until very recently, the lumpy nature of plant level investment has not been taken into account in the context of New Keynesian theory. Many times, capital accumulation is abstracted from (see, e.g., Galí 2008, among many others), and if capital accumulation is taken into account then it is common practice to postulate convex adjustment costs of one type or another which makes the model inconsistent with the observed lumpiness in investment (see, e.g., Woodford 2003, Ch. 5 and 2005, or Christiano et al. 2005, among many others). Taken at face value, this might be a surprising aspect of New Keynesian economics: on the one hand, it is clear that firms do not only adjust prices but also make investment decisions. 1 On the other hand, the lumpy nature of investment is an empirical fact which is as uncontroversial as its price-setting counterpart. 2 The latter takes, however, center-stage in New Keynesian theory. Those observations had already motivated our work in Sveen and Weinke (2007). In that model we had assumed that both price-setting and investment decisions are conducted in a time-dependent fashion. Up to a first-order approximation the resulting framework had been shown to be observationally equivalent to the model in Woodford (2003, Ch. 5 and 2005) where he assumes a convex capital adjustment cost. But what are the welfare consequences of lumpy investment? This question is relevant because a meaningful welfare analysis can only be conducted based on a second-order approximation to the welfare criterion. 3 1 And, importantly, it is reasonable to expect that this feature is relevant for the analysis of optimal monetary policy. In the words of Schmitt-Grohé and Uribe (2007): "All the way from the work of Keynes (1936) and Hicks (1939) to that of Kydland and Prescott (1982) macroeconomic theories have emphasized investment dynamics as an important channel for the transmission of aggregate disturbances. It is therefore natural to expect that investment spending should play a role in shaping optimal monetary policy." 2 See, e.g., Caballero and Engel (2007). 3 See, e.g., Schmitt-Grohé and Uribe (2004). 2

Before we turn to our main results in the present paper some more general remarks are in order. In the field of investment theory lumpiness is routinely generated by the presence of stochastic non-convex capital adjustment costs (see, e.g., Thomas 2002, Khan and Thomas 2008 and House 2008). If lumpy investment is, however, introduced in that way into an otherwise conventional New Keynesian model then the latter looses its ability to imply a realistic monetary transmission mechanism. This is the central result in Reiter et al. (2009). 4 Put differently, there is no consensus on the appropriate way of modeling endogenous capital accumulation in the context of monetary economics, and it is unclear whether or not standard investment theory can give us any guidance on how to proceed. It is therefore important to make it clear that the present paper pursues a specific goal. More concretely, we wish to isolate the role of a plausible modeling of lumpy investment in a monetary context for the optimal design of monetary policy. To this end, we start by considering a standard way of modeling endogenous capital accumulation in the context of New Keynesian theory, namely the model of firmspecific capital proposed in Woodford (2003, Ch. 5 and 2005). Otherwise, we follow Erceg et al. (2000). In particular, our model features sticky prices and wages à la Calvo (1983). The welfare criterion is the unconditional expectation of average of household utility, as in Rotemberg and Woodford (1997). We ask to what extent the welfare implications of that model are affected by its abstraction from lumpiness in investment. Given the above mentioned first-order equivalence between the models proposed in Woodford (2003, Ch. 5 and 2005) and Sveen and Weinke (2007), and given the above-mentioned lack of plausible alternative treatments of lumpy investment in the field of monetary economics, it is natural to consider those 4 Johnston (2009) proposes a more optimistic view of state-dependent lumpy investment for monetary economics. However, his result hinges on assuming the quantity theory M t = Y t P t. In a nutshell, his assumption of exogenous money growth combined with enough stickiness in prices must give rise to a relatively smooth equilibrium output process. 3

two ways of modeling endogenous capital in order to isolate the welfare relevance of lumpy investment. We start by analyzing the welfare properties of simple interest rate rules. In order to do so we need to compute the relevant second moments of the distribution of firms capital stocks and prices. This is done by using the conceptual innovation introduced in Sveen and Weinke (2009). In that paper we had observed that the rule for price-setting that needs to be computed in Woodford s (2005) method for calculating a first-order approximation to the equilibrium dynamics can be used to obtain the relevant objects. A similar observation can be made in the present context. Interestingly, our first main result shows that the two models under consideration imply optimized interest rate rules which are similar in the following sense: if the optimized rule from one model is used in the other one then the resulting additional welfare loss is negligible. This is an important extension of our earlier work in Sveen and Weinke (2009). In that paper we had shown that optimized interest rate rules, as implied by Woodford s (2003, Ch. 5 and 2005) modelling of firm-specific capital, are very similar (in the same sense as above) to the ones that obtain under the assumption of a rental market for capital (but one needs to adjust the price stickiness in the latter model according to a metric that we had developed in Sveen and Weinke 2005). Much of the related literature on the welfare implications of interest rate rules has adopted the assumption of a rental market for capital. 5 Our first result can therefore also be interpreted as a way of showing that the latter simplification is innocuous for questions related to the optimal design of interest rate rules. Second, we compare the welfare implications of each simple interest rate rule to the corresponding outcome under Ramsey optimal policy. This way we demonstrate that optimal policy is well approximated by optimized simple 5 See, e.g., Schmitt-Grohé and Uribe (2007). 4

interest rate rules. This increases the practical relevance of our first result. 6 The remainder of the paper is organized as follows. Section 2 outlines the model structure. In section 3 the results are presented and section 4 concludes. 2 Model We use a New Keynesian model with complete markets. Shocks to total factor productivity are assumed to be the only source of aggregate uncertainty. There is a continuum of households and a continuum of firms. Each household (firm) is the monopolistically competitive supplier of a differentiated type of labor (type of good) and we assume sticky wages (sticky prices) à la Calvo (1983), i.e., each household (firm) gets to reoptimize its wage (price) with a constant and exogenous probability. Capital accumulation is assumed to take place at the firm level 7 and the additional capital resulting from an investment decision becomes productive with a one period delay. Lumpiness in investment is modeled as in Sveen and Weinke (2007), i.e., there is a Bernoulli draw conditional on which a firm is allowed to invest, or not. 8 Since the details of the model have been discussed elsewhere (see, Erceg et al. 2000 and Sveen and Weinke 2007, 2009) we turn directly to the implied linearized equilibrium conditions. Relying on the insights in Rotemberg and Woodford (1997) we will use this approximation to compute our welfare criterion up to the second order. 6 Special thanks to Stephanie Schmitt-Grohé for making this point. 7 In the context of our model there is no distinction between a firm and a plant. We therefore use those terms interchangeably. 8 This way of modeling lumy investment has originally been proposed in the work by Kiyotaki and Moore (1997). 5

2.1 Some Linearized Equilibrium Conditions We consider a linear approximation around a zero inflation steady state. In what follows variables are expressed in terms of log deviations from their steady state values except for the nominal interest rate, i t, wage inflation, ω t, and price inflation, π t, which denote the level of the respective variable. The consumption Euler equation reads c t = E t c t+1 (i t E t π t+1 ρ), (1) where c t denotes aggregate consumption and E t is the expectation operator conditional on information available trough time t. Moreover, parameter ρ is the time discount rate and the last equation also reflects our assumption of log consumption utility. Up to the first order, aggregate production is pinned down by y t = x t + α k t + (1 α) n t, (2) where y t denotes aggregate output, parameter α indicates the capital share, aggregate capital is k t, aggregate hours are n t and x t represents an exogenous index of technology. The latter is assumed to follow an AR(1) process x t = ρ x x t + ε t, with ρ x (0, 1). The wage inflation equation results from averaging and aggregating optimal wage setting decisions on the part of households, as discussed in Erceg et al. (2000). It takes the following simple form ω t = β E t ω t+1 + λ ω (mrs t rw t ), (3) where rw t denotes the real wage and mrs t c t +ηn t measures the average marginal rate of substitution of consumption for leisure. In the latter definition parameter η indicates the inverse of the (aggregate) Frisch labor supply elasticity. Finally, we 6

have used the definition λ ω (1 βθw)(1 θw) θ w 1 1+ηε N, where parameter θ w denotes the probability that a household is not allowed to reoptimize its nominal wage in any given period, while parameter ε N measures the elasticity of substitution between different types of labor. The price inflation equation takes the form π t = β E t π t+1 + λ mc t, (4) where mc t rw t (y t n t ) denotes the average real marginal cost. We have also used the definition λ (1 βθ)(1 θ) θ 1 α 1 1 α+αε ξ where parameter θ gives the probability that a firm does not get to reoptimize its price in any given period, while parameter ε denotes the elasticity of substitution between the differentiated goods. Finally, parameter ξ is a function of the model s structural parameters, which is computed numerically as in Sveen and Weinke (2007), using the method developed in Woodford (2005). For future reference, let us notice that this method relies on the observation that a firm s price-setting rule can be approximated up to the first order by the following expression p t (i) = ˆp t τ 1 ˆkt (i), with p t (i) p t (i) p t, ˆk t (i) k t (i) k t and p t p t p t, where p t (i) is the price chosen by firm i in period t and p t denotes the aggregate price level associated with the usual Dixit-Stiglitz aggregator. Firm i s relative to average capital stock is given by ˆk t (i). Moreover ˆp t is the average relative price in the group of time t price setters. Finally, parameter τ 1 can be calculated by using the method of undetermined coeffi cients. The law of motion of capital is obtained from averaging and aggregating optimal investment decisions on the part of firms. This implies k t+1 = β E t k t+2 + 1 ɛ ψ [(1 β(1 δ)) E t ms t+1 (i t E t π t+1 ρ)], (5) 7

where ms t rw t (k t n t ) represents the average real marginal return to capital. The average real marginal return to capital is measured in terms of marginal savings in labor costs since firms are demand-constrained in our model. Moreover, parameter β denotes the subjective discount factor, while parameter δ is the rate of depreciation. Finally, parameter ɛ ψ measures the smoothness in aggregate capital accumulation. It is a function of the model s structural parameters which is evaluated as in Sveen and Weinke (2007), where we observe that a firm s investment rule can be approximated up to the first order by the following expression k t+1 (i) = ˆk t+1 τ 2 ˆp t (i), with k t+1 (i) k t+1 (i) k t+1, k t+1 k t+1 k t+1 and ˆp t (i) p t (i) p t where k t+1 (i) is the capital stock chosen by firm i in period t. Moreover k t+1 is the average capital stock in the group of time t investors and ˆp t (i) is firm i s relative price in period t. Parameter τ 2 can be calculated by using the method of undetermined coeffi cients. The value of parameter ɛ ψ in (5) depends crucially on the probability θ k with which a firm is not allowed to invest in a given period. Let us also note that ɛ ψ would be the parameter measuring the convex capital adjustment cost if capital accumulation was modeled in the way proposed by Woodford (2003, Ch. 5 and 2005). Finally, let us observe that the value of λ in (4) coincides with its counterpart in Woodford (2005) if parameter θ k is chosen in such a way that the laws of motion of capital coincide in the two models. This is the reason why the two models are observationally equivalent, up to the first-order. 9 The goods market clearing condition reads y t = ζ c t + 1 ζ δ 9 For details, see Sveen and Weinke (2007). [k t+1 (1 δ) k t ], (6) 8

where ζ 1 δα ρ+δ denotes the steady state consumption to output ratio. Notice that the frictionless desired markup, µ ε, does not enter the latter definition. ε 1 The reason is our assumption of a wage subsidy which makes the steady state of our model effi cient. This allows us to approximate our welfare criterion up to the second order relying on the first-order approximation that we have just considered. Next we discuss the welfare criterion that will be used to assess the desirability of alternative arrangements for the conduct of monetary policy. 2.2 Welfare As in Erceg et al. (2000), the policymaker s period welfare function is assumed to be the unweighted average of households period utility W t U (C t ) + 1 0 V (N t (h)) dh = U (C t ) + E h {V (N t (h))}. (7) The assumptions of complete asset markets combined with a utility function which is separable in its two arguments, consumption and hours worked, imply that heterogeneity in hours worked (a consequence of wage dispersion) does not translate into consumption heterogeneity. This is reflected in (7), where U (C t ) indicates consumption utility and V (N t (h)) denotes the disutility associated with household h decision to supply N t (h) hours. The notation E h is meant to indicate a crosssectional expectation integrating over households. We also follow Erceg et al. (2000) in expressing period welfare as a fraction of steady state Pareto optimal consumption, i.e., we consider Wt W, where a bar indicates the steady state value of the U C C original variable and U C is the marginal utility of consumption. The second-order approximation to period welfare is calculated based on the method by Rotemberg 9

and Woodford (1997). 10 Our welfare criterion is the unconditional expectation of period welfare which we write in the way proposed by Svensson (2000) W lim β 1 E t { (1 β) k=0 β k ( Wt+k W U C C ) }. We derive the following expression for that welfare criterion in the context of the lumpy investment model outlined above (LI for short) W LI Ω 1 E { } yt 2 + Ω2 E { } c 2 t + Ω3 E { } kt 2 + Ω4 E { } n 2 t +Ω 5 E { } ω 2 t + Ω6 E { } π 2 t + Ω7 E { ( k t ) 2} + Ω 8 E { } zt 2, (8) where the symbol is meant to indicate that an approximation is accurate up to the second order. We have also used the definition z t k t + π t and parameters Ω 1 to Ω 8 are functions of the structural parameters, as defined in the appendix. For the model where capital is firm-specific as in Woodford (2003, Ch. 5 and 2005) (F S for short) the welfare criterion is calculated along the lines of Sveen and Weinke (2009). In order to derive (8) one needs to calculate the cross-sectional variances of prices and capital holdings at each point in time as well as their covariance. This is shown in the appendix. In a nutshell, we note that the above mentioned linearized rules for price-setting and for investment can be used to compute the relevant second moments with the accuracy that is needed for our second-order approximation to the welfare criterion. 10 The proof that the method of Rotemberg and Woodford (1997) can be applied to the problem at hand carries over from Edge s (2003) to our work because the relevant steady states properties of the two models are identical. 10

3 Results As already mentioned, the notation LI is used for the lumpy investment model considered above and the notation F S for the model in which capital is firm-specific as in Woodford (2003, Ch. 5 and 2005). We first consider a conventional family of monetary policy rules and analyze constrained optimal rules, i.e., we restrict attention to a particular subset of possible parameter values that parametrize the rule. It is useful to note that rational expectations equilibrium is locally unique (i.e., determinate) under the constrained optimal polices. The simple interest rate rules under consideration below are therefore also implementable. In a second step we ask how well those simple optimal rules approximate the Ramsey optimal policy. 3.1 Baseline Parameter Values Let us start by mentioning the values which are assigned to the model parameters in our quantitative analysis. 11 We consider a quarterly model. The capital share, α, is set to 0.36. The elasticity of substitution between goods, ε, takes the value 11. The rate of capital depreciation, δ, is assumed to be equal to 0.025. We set the elasticity of substitution between different types of labor, ε N, equal to 6. Our choice for the Calvo price stickiness parameter, θ, is 0.75 and the value for the wage stickiness parameter, θ w, is 0.75. This implies an average expected duration of price- and wage contracts of one year. Finally, the coeffi cient of autocorrelation in the process of technology, ρ x, is assumed to take the value 0.95. Those parameter values are justified in Sveen and Weinke (2007, 2009), Erceg et al. (2000) and the references therein. Finally, we set the lumpiness parameter θ k = 0.955. The latter 11 To solve the dynamic stochastic system of equations we use Dynare (www.dynare.org). Matlab code for our implementation of Woodford s (2005) method is available upon request. We also thank Junior Maih for code that has been used in the computation of the optimized coeffi cients of the interest rate rules under consideration. 11

choice makes our model consistent with the evidence in Khan and Thomas (2008) according to which 18% of plans undertake lumpy investments in any given year. Conditional on our baseline choices for the remaining parameters, this results in ɛ ψ = 6.359, a value implying a plausible degree of smoothness in aggregate capital accumulation. 12 3.2 The Welfare Consequences of Simple Implementable Rules We consider a family of simple interest rate rules of the form r t = ρ + τ r (r t 1 ρ) + τ s [τ ω ω t + (1 τ ω ) π t ] + τ y ỹ t, (9) where parameter τ s measures the overall responsiveness of the nominal interest rate to changes in inflation, whereas τ ω is the relative weight put on wage inflation. The weight on price inflation is therefore given by (1 τ ω ). Parameter τ r denotes the interest rate smoothing coeffi cient and parameter τ y indicates the quantitative importance attached to a measure of real economic activity which is taken to be the output gap, ỹ t. To be precise, ỹ t y t yt nat, where yt nat is natural output which is computed à la Neiss and Nelson (2003), i.e., the equilibrium output that would obtain in the absence of nominal frictions. Only positive parameter values are considered and parameter τ ω is required to be less or equal to one. 3.2.1 Wage-Price Rule We compare optimized interest rate rules under LI and F S. Our first set of results regards interest rate rules according to which the nominal interest rate is only adjusted in response to changes in price- and wage inflation, i.e., parameter τ y is set 12 As we have mentioned already, ɛ ψ is simply the parameter measuring the convex capital adjustment cost in Woodford (2003, Ch. 5 and 2005). 12

to zero. For each model under consideration we report the optimized coeffi cients entering the interest rate rule as well as the the corresponding welfare measure. We follow Erceg et al. (2000) and consider the business cycle cost of nominal rigidities. Specifically, we compute our welfare measure in each model under consideration under the assumption of flexible prices and wages and subtract the resulting expression from the value in the corresponding model with the nominal rigidities being present. As they do, we divide this difference by the productivity innovation variance and refer to the resulting number as a being a welfare loss. 13 We also state the welfare loss obtaining in LI if the optimized rule from F S is used in that model. The results are shown in Table 1. [Table 1 about here] The welfare losses under the respective optimized rules are very similar in LI and F S and the differences between the corresponding optimized parameter values, as implied by the two models, are also small. Moreover, the additional welfare loss is tiny if the optimized rule from F S is used in LI (compared with the outcome that obtains under the optimized rule in LI). In other words, a central banker would only make a small mistake by abstracting from lumpiness in investment when optimizing a wage-price interest rate rule. Those are interesting and surprising results, given the differences between LI and F S, which are relevant for the computation of the respective welfare losses. First, and most obviously, heterogeneity in F S originates only in the price-setting decisions, whereas in LI investment and price-setting decisions are two independent sources of heterogeneity. But there is also a second, more subtle, difference between the two models. This difference regards the respective 13 Let us give a concrete example for the interpretation of the welfare loss numbers in our tables. Suppose the productivity innovation variance is 0.01 2. Then, a welfare loss of 10 means that the representative household would be willing to give up 10 0.01 2 100 = 0.1 percentage points of steady state (Pareto optimal) consumption in order to avoid the business cycle cost associated with the presence of the nominal rigidities. 13

versions of LI and F S under the assumption of flexible prices and wages. In this case, there is a first-order difference between the two models for the following reason. The law of motion of capital, as implied by LI, is different from its counterpart in F S because price stickiness affects the smoothness of aggregate capital accumulation in the former model but not in the latter, as analyzed in Sveen and Weinke (2007). 14 It is therefore interesting and surprising that the two models have very similar implications for the optimal design of wage-price rules. But is this striking similarity between the welfare results in LI and F S specific to this type of interest rate rule, or does this statement hold more generally? We turn to this next. 3.3 Taylor-Type Rule Once again, we compare welfare implications in LI and F S. The structure of the comparison is the same as in the preceding section, but we now consider interest rate rules according to which the nominal interest rate is only adjusted in response to changes in price inflation and the output gap, i.e., parameter τ ω is set to zero. Once again, our analysis brings out a clear message. This is shown in Table 2. [Table 2 about here] There are only very small differences between the welfare losses under the respective optimized rules in LI and F S and the same is true for the respective coeffi cients parametrizing those rules. Moreover, the additional welfare loss is once again small if the optimized rule from F S is used in LI (compared with the outcome that obtains under the optimized rule in LI). We have already mentioned that the law of motion of natural capital is different in the two models because price stickiness affects the smoothness of aggregate capital accumulation in LI, but not in F S. This 14 In a nutshell the reason is that lumpy investors take into account to what extent they are able to create demand over the expected lifetime of the capital stock and the latter is affected by the degree of price stickiness. For details, see Sveen and Weinke (2007). 14

observation is particularly relevant for the analysis of Taylor-type interest rate rules because natural output is used in the construction of the output gap. However, as our results clearly show, the quantitative importance of this effect is also very limited in this context. Next we analyze how well the simple and implementable interest rate rules considered so far approximate Ramsey optimal policy. 3.4 Ramsey Optimal Policy We now compute Ramsey optimal welfare losses in LI and F S. 15 In each case the result is compared with the corresponding outcome under the optimized Taylor-type rule that has been considered before. This is shown in Table 3. [Table 3 about here] In each model the welfare outcome under Ramsey optimal policy is well approximated by the corresponding optimized interest rate rule. Taken together the present paper therefore conveys a positive message. The fact that monetary economists have focused almost exclusively on the role of lumpy price adjustment on the part of firms while abstracting from the lumpiness in investment appears to be inconsequential for the normative results obtained. In fact, the conclusions for the desirability of alternative arrangements for the conduct of monetary policy are strikingly similar whether or not lumpiness in investment is taken into account. 3.5 Sensitivity Analysis Let us challenge the results obtained so far. Our first sensitivity analysis regards the notion of the output gap. In the context of a model featuring endogenous capital accumulation, Woodford (2003, Ch. 5) proposes to refine that notion in the following way. He uses the equilibrium output that would obtain if the nominal rigidities were 15 To compute Ramsey optimal policy we use Dynare (www.dynare.org). 15

absent and expected to be absent in the future but taking as given the capital stock resulting from optimizing investment behavior in the past in an environment with the nominal rigidities present. Woodford argues that this measure of natural output is more closely related to equilibrium determination than the alternative measure which has been proposed by Neiss and Nelson (2003) and which we have used so far in our analysis. We have stated already that under their definition natural output is the equilibrium output that would obtain if nominal rigidities were absent. But this means that they are not only currently absent and expected to be absent in the future but that they had also been absent in the past. This distinction turns out, however, to be of negligible importance for the results in the present paper. 16 This is shown in the next table. 17 [Table 4 about here] Our second sensitivity analysis is concerned with the following question. What are the additional welfare losses implied by deviations from the optimized interest rate rules that have been analyzed so far in the context of LI? Let us therefore reconsider the general interest rate rule stated in equation (9). The left-hand side of figure 1 shows by how much the welfare loss increases if the policy parameters deviate from their optimal values in the case of the wage-price rule considered above (i.e., the specification where parameter τ y is set to zero). The right-hand side of figure 1 shows the corresponding analysis for the case of the Taylor-type rule (i.e., the specification where parameter τ ω is set to zero). Each panel of the figure illustrates 16 Let us also mention here that our results would not change in any quantitatively important way if we used output growth as our measure of real economic activity. Also in this case, the presence of lumpiness in investment does not alter in any important way the conclusions regarding the optimal design of monetary policy that one obtains. The same is true under more general specifications of the interest rate rule. Those results are available upon request. 17 A related question is whether or not it would matter for the results obatined if natural output entering the definition of the output gap in one model was computed using the natural law of motion of capital from the other model. The answer is that this would also have negligible consequences from a quantitative point of view. Those results are available upon request. 16

the welfare loss associated with a change in one policy parameter at a time while holding the remaining parameters constant at their optimal values. In each case, the figure also shows the corresponding indeterminacy region in the parameter space. [Figure 1 about here] From an economic point of view our sensitivity analysis with respect to the output gap response in the Taylor-type rule deserves special attention. It is shown that small deviations from the optimal rule result in negligible welfare losses. Only to the extent that the central bank attaches very little importance to the output gap in setting the nominal interest rate does the resulting rule imply a large welfare loss with respect to the optimal policy. 4 Conclusion In a seminal paper Erceg et al. (2000) had asked how monetary policy should be conducted in an economic environment with sticky prices and wages. Their analysis has been extended in Sveen and Weinke (2009) by allowing for endogenous capital accumulation. In that paper, we had, however, abstracted from lumpiness in investment. This motivates the present paper. As in our 2009 paper, we augment the model in Erceg et al. (2000) by a standard textbook treatment of endogenous capital accumulation in the context of New Keynesian theory, namely the firm-specific investment decision considered in Woodford (2003, Ch. 5 and 2005). The latter model abstracts, however, from lumpiness in investment, which is an uncontroversial empirical regularity. The present paper shows that this abstraction appears to be innocuous from a policy design point of view. In fact, the implications of Woodford s (2003, Ch. 5 and 2005) modelling of capital accumulation are shown to be surprisingly similar to the ones that emerge in the presence of a lumpy investment 17

decision of the form considered in Sveen and Weinke (2007). Under both alternative assumptions on capital accumulation, we find that simple implementable interest rate rules generate welfare results that are surprisingly similar. Moreover the additional welfare loss associated with using the optimized rule from one model in the other one is very small. Finally, optimized simple interest rate rules are shown to approximate very well the corresponding outcomes under Ramsey optimal policy. We had already mentioned that there is no consensus on the way in which endogenous capital accumulation should be modeled in the context of monetary models. Clearly, this issue is high on our research agenda, and, needless to say, alternative ways of modeling lumpiness in investment would not only change the implied monetary transmission mechanism but also the model s welfare implications. This caveat notwithstanding, it is important to realize, we believe, that a modeling of lumpy investment which is plausible in a monetary context does not lead to changes in the welfare relevant results with respect to standard textbook treatments of endogenous capital in the New Keynesian theory. 18

Appendix: Welfare with Lumpy Investment Throughout the appendix we use the notation and the definitions that are introduced in the text. We approximate our utility based welfare criterion up to the second order. In what follows, we make frequent use of two rules A t A A a t + 1 2 a2 t, (A1) where is meant to indicate that an approximation is accurate up to the second order. We have also used the definition a t ln ( ( A t ) 1 ) 1 A. Moreover, if At = A t (i) γ γ di then 0 a t E i a t (i) + 1 2 γv ar ia t (i), (A2) with V ar i indicating a cross-sectional variance. As we have already mentioned in the text, the policymaker s period welfare function reads W t U (C t ) 1 0 V (N t (h)) dh = U (C t ) E h {V (N t (h))}. (A3) It follows that W t W + U C C ( c t + 1 ) 2 c2 t V N NE h {n t (h) + 12 } n t (h) 2 + 1 2 U CCC 2 c 2 t 1 2 V NNN 2 E h { nt (h) 2}. The Pareto optimality of the steady state implies: V N N = U C C 1 α. We therefore ζ have W t W U C C ( c t + 1 ) 2 c2 t 1 2 c2 t 1 α E h {n t (h)} ζ 1 1 α { (1 + φ) E h nt (h) 2}. (A4) 2 ζ 19

Next we show how the linear terms in consumption and employment in the last expression can be approximated up to the second order. We start by analyzing the consumption portion of welfare. To this end we invoke the resource constraint. The Consumption Portion of Welfare The resource constraint reads Y t = C t + K t+1 (1 δ) K t, where K t 1 0 K t (i) di and K t (i) is the amount of composite goods used in firm i s production. It follows that c t + 1 2 c2 t 1 ζ ( y t + 1 ) 2 y2 t + 1 ζ ζ 1 δ δ The Labor Portion of Welfare 1 ζ ( 1 k t+1 + 1 ) ζ δ 2 k2 t+1 ( k t + 1 2 k2 t ). (A5) Aggregate labor supply is given by N t rule we can write Aggregate labor demand reads L t 1 0 n t E h n t (h) + 1 ɛ N 1 V ar h n t (h). 2 ɛ N L t (i) di = 1 0 ( 1 N 0 t (i) ɛn 1 ) ɛ N ɛ N 1 ɛ N di. Using the second ( ) 1 Yt (i) 1 α 1 di = B X t K t (i) α 1 α t, where B t ( 1 0 B t (i) 1 1 α di ) 1 α and Bt (i) Yt(i) α X tk t(i). Clearing of the labor market 20

implies that N t = L t. We therefore have n t = 1 1 α b t. Invoking the second rule we obtain b t E i b t (i) + 1 1 2 1 α V ar ib t (i), and we also note that b t (i) = y t (i) x t α k t (i). The last result implies E i b t (i) = E i y t (i) x t αe i k t (i), V ar i b t (i) = V ar i y t (i) + α 2 κ t 2αCov i (y t (i), k t (i)), with V ar i denoting the cross sectional variance and Cov i the covariance operator. We can therefore write b t E i y t (i) x t αe i k t (i) + 1 1 [ V ari y t (i) + α 2 κ t 2αCov i (y t (i), k t (i)) ]. 2 1 α Let us also observe that E i k t (i) k t 1 2 V ar ik t (i), E i y t (i) y t 1 2 ε 1 V ar i y t (i). ε 21

Combining the last three results we arrive at b t y t x t α k t + 1 α 2 1 α V ar ik t (i) + 1 1 2 ε α 1 α Cov i (y t (i), k t (i)). ( 1 α + αε 1 α ) V ar i y t (i) It follows that E h n t (h) 1 1 α (y t x t αk t ) + 1 α 2 (1 α) 2 V ar ik t (i) + 1 1 1 α + αε 2 ε (1 α) 2 V ar i y t (i) α (1 α) 2 Cov i (y t (i), k t (i)) 1 ε N 1 V ar h n t (h). 2 ε N Using the demand functions for goods and labor services we obtain V ar i y t (i) ε 2 t, Cov i (y t (i), k t (i)) εψ t, V ar h n t (h) ε 2 Nλ t, where t V ar i p t (i), ψ t Cov i ( p t (i), k t (i)) and λ t V ar h ŵ t (h), with ŵ t (h) w t (h) w t. In the latter definition w t (h) is household h s nominal wage and w t denotes the aggregate wage level associated with the corresponding Dixit-Stiglitz aggregator. We therefore have E h n t (h) 1 1 α (y t x t αk t ) + 1 2 + 1 α 2 (1 α) 2 κ t + 1 α + αε (1 α) 2 ε t αε (1 α) 2 ψ t 1 2 (ε N 1) ε N λ t, (A6) 22

with κ t V ar iˆkt (i). Finally, we note that E h { nt (h) 2} can be written as E h { nt (h) 2} ε 2 N λ t + n 2 t. (A7) The Welfare Function Combining (A4), (A5), (A6) and (A7) we arrive at the following approximation to period welfare W t W U C C 1 ζ x t 1 α ζ ρ + δ [k t+1 + (1 + ρ) k t ] + 1 1 2 ζ y2 t 1 2 c2 t 1 1 ζ 1 2 ζ δ k2 t+1 + 1 1 ζ 1 δ kt 2 2 ζ δ 1 1 α (1 + φ) n 2 t 1 ε 1 α + αε t 2 ζ 2 ζ 1 α 1 1 α 2 ζ 1 α κ t 1 (1 α) ε N (1 + φε N ) λ t 1 αε 2 ζ ζ 1 α ψ t. (A8) The first linear term in the last expression is proportional to the level of aggregate (log) technology, x t, which is exogenous. The remaining linear terms are proportional to current and next period s aggregate capital, k t and k t+1. Next we compute { ( k=0 βk W t+k W )}, which allows us to invoke a result by Edge (2003). 1 β U C C E t As in her model the terms in aggregate capital cancel except for the initial one. Following the lead of Svensson (2000) we consider the limit for β 1. This allows us to abstract from initial conditions. 18 W LI Ω 1 E { } yt 2 + Ω2 E { } c 2 t + Ω3 E { } kt 2 + Ω4 E { } n 2 t +Ω λ E {λ t } + Ω E { t } + Ω κ E {κ t } + Ω ψ E {ψ t }, (A9) 18 For a formal proof, see Dennis (2007). 23

with Ω 1 1 1 2 ζ, Ω 2 1 2, Ω 3 1 1 ζ, 2 ζ Ω 4 1 1 α (1 + φ), Ω λ 1 (1 α) ε N 2 ζ 2 ζ Ω 1 ε 1 α + αε, Ω κ 1 1 α 2 ζ 1 α 2 ζ (1 + φε N ), 1 α, Ω ψ 1 ζ αε 1 α. Next we derive recursive formulations for the cross-sectional variances of wages, prices and capital holdings as well as the welfare relevant covariance of prices and capital holdings. As in Erceg et al. (2000), the cross-sectional variance of wages, λ t, can be written in the following way λ t θ w λ t 1 + θ w 1 θ w (ω t ) 2, and the unconditional expected value is E {λ t } θ w (1 θ w ) 2 E { } ω 2 t. Next we derive recursive formulations for t, κ t and ψ t. To this end we start by invoking the pricing rule and the capital accumulation rule p t (i) F O p t τ 1 kt (i), ln P t (i) F O ln P t τ 1 (k t (i) k t ), k t+1 (i) F O k t+1 τ 2 (ln P t (i) ln P t ), where F O is used to indicate that the approximation holds up to the first order. Moreover, we define p t E i ln P t (i), 24

and in an analogous manner p t. We should note that p t reference, we first derive the following two expressions F O ln P t. For future p t p t 1 = E i [ln P t (i) p t 1 ] = θ p E i [ln P t 1 (i) p t 1 ] + (1 θ p ) E i [ln P t (i) p t 1 ], F O (1 θp ) ( p t p t 1 ), k t+1 k t = E i [k t+1 (i) k t ] = θ k E i [k t (i) k t ] + (1 θ k ) E i [ k t+1 (i) k t ], F O (1 θk ) ( k t+1 k t ). We rewrite the price dispersion term, t, in the way proposed by Woodford (2003, Ch. 6) t = E i [ (ln Pt (i) p t 1 ) 2] ( p t p t 1 ) 2, = θ p E i [ (ln Pt 1 (i) p t 1 ) 2] + (1 θ p ) E i [ (ln P t (i) p t 1 ) 2] ( p t p t 1 ) 2. After invoking the price-setting rule we obtain the following recursive formulation for the measure of price dispersion t [ t θ p t 1 + (1 θ p ) E i ( p t τ 1 (k t (i) k t ) p t 1 ) 2] ( p t p t 1 ) 2 [ ( ) ] 2 1 θ p t 1 + (1 θ p ) E i ( p t p t 1 ) τ 1 (k t (i) k t ) ( p t p t 1 ) 2 1 θ p ( ) 1 θ p t 1 + (1 θ p ) τ 2 1V ar i k t (i) + 1 ( p t p t 1 ) 2 1 θ p θ p t 1 + (1 θ p ) τ 2 1κ t + θ p 1 θ p π 2 t. 25

For the cross-sectional variance of capital holdings, κ t+1, we obtain κ t+1 = E i [ (kt+1 (i) k t ) 2] (E i k t+1 (i) k t ) 2 [ = θ k E i (kt (i) k t ) 2] [ (k ) ] 2 + (1 θ k ) E i t+1 (i) k t (k t+1 k t ) 2 [ (k θ k κ t + (1 θ k ) E i t+1 k t τ 2 (ln P t (i) ln P t ) ) ] 2 (k t+1 k t ) 2 [ (k (1 θ k ) t+1 k t τ 2 (E i ln P t (i) ln P t ) ) ] 2 + 1 (k t+1 k t ) 2 1 θ k θ k κ t + (1 θ k ) τ 2 2 t + θ k 1 θ k (k t+1 k t ) 2. Finally, the covariance between prices and capital holdings, ψ t, takes the following form ψ t = E i [(k t (i) k t 1 ) (ln P t (i) ln P t 1 )] [(E i k t (i) k t 1 ) (E i ln P t (i) ln P t 1 )]. = θ p θ k E i [(k t 1 (i) k t 1 ) (ln P t 1 (i) ln P t 1 )] +θ p (1 θ k ) E i [(k t (i) k t 1 ) (ln P t 1 (i) ln P t 1 )] + (1 θ p ) θ k E i [(k t 1 (i) k t 1 ) (ln P t (i) ln P t 1 )] + (1 θ p ) (1 θ k ) E i [(k t (i) k t 1 ) (ln P t (i) ln P t 1 )] (k t k t 1 ) π t θ p θ k ψ t 1 + θ p (1 θ k ) E i [(k t k t 1 τ 2 (ln P t 1 (i) ln P t 1 )) (ln P t 1 (i) ln P t 1 )] + (1 θ p ) θ k E i [(k t 1 (i) k t 1 ) (ln Pt τ 1 (k t 1 (i) k t 1 + k t 1 k t ) ln P t 1 )] + (1 θ p ) (1 θ k ) E i (k t k t 1 ) π t (k t k t 1 τ 2 (ln P t 1 (i) ln P t 1 )) (ln P t ln P t 1 τ 1 (k t k t ) + τ 1 τ 2 (ln P t 1 (i) ln P t 1 )) θ p θ k ψ t 1 τ 2 (1 θ k ) [θ p + τ 1 τ 2 (1 θ p )] t 1 τ 1 (1 θ p ) θ k κ t 1 (k t k t 1 ) π t. 26

We therefore have E {λ t } θ w (1 θ w ) 2 E { } ω 2 t, E { t } θ p E { t } + (1 θ p ) τ 2 1E {κ t } + θ p E { } π 2 t, 1 θ p E {κ t } θ k E {κ t } + (1 θ k ) τ 2 2E { t } + θ k 1 θ k E { ( k t ) 2}, E {ψ t } θ p θ k E {ψ t } τ 2 (1 θ k ) [θ p + τ 1 τ 2 (1 θ p )] E { t } τ 1 (1 θ p ) θ k E {κ t } E {( k t ) π t }, or, equivalently B E { t } E {κ t } E {ψ t } C E {π 2 t } E { ( k t ) 2} E {z 2 t }, with B and C (1 θ p ) (1 θ p ) τ 2 1 0 (1 θ k ) τ 2 2 (1 θ k ) 0 τ 2 (1 θ k ) [θ p + τ 1 τ 2 (1 θ p )] τ 1 (1 θ p ) θ k (1 θ p θ k ) θ p 1 θ p 0 0 0 1 2 θ k 1 θ k 0 1 2 1 2, or, equivalently E { t } E {κ t } E {ψ t } A E {π 2 t } E { ( k t ) 2} E {z 2 t }, with A B 1 C and z t = k t + π t. We can therefore rewrite our welfare criterion 27

as W LI Ω 1 E { } yt 2 + Ω2 E { } c 2 t + Ω3 E { } kt 2 + Ω4 E { } n 2 t +Ω 5 E { ω 2 t } + Ω6 E { π 2 t } + Ω7 E { ( k t ) 2} + Ω 8 E { z 2 t }, (A10) with Ω 5 Ω λ θ w (1 θ w ) 2, Ω 6 [Ω A 11 + Ω κ A 21 + Ω ψ A 31 ] Ω 7 [Ω A 12 + Ω κ A 22 + Ω ψ A 32 ], Ω 8 Ω ψ A 33. The last equation is (8) in the text. 28

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Table 1: Wage-Price Rule Parameter FS LI τ r 1.0352 1.0353 τ s 0.8399 0.5837 τ ω 0.3416 0.3295 Welfare 9.1104 ( 9.1342 1 ) 9.1264 Table 2: Taylor-Type Rule with Neiss and Nelson Output Gap Parameter FS LI τ r 1.0423 0.9911 τ π 0.0365 0.0093 τ y 0.0283 0.0074 Welfare 9.1914 ( 9.4744 1 ) 9.1639 1 Welfare in the lumpy model with F S rule. 32

Table 3: Ramsey Optimal Policy FS LI Welfare 8.9311 8.9157 Table 4: Taylor-Type Rule with Woodford Output Gap Parameter FS LI τ r 1.0305 1.0080 τ π 0.0545 0 τ y 0.0227 0.0343 Welfare -9.1935 ( 9.3054 1 ) 9.0641 1 Welfare in the lumpy model with F S rule. 33

Welfare Welfare Welfare Welfare Welfare Welfare 0 5 10 15 Wage rule 0.4 0.6 0.8 1 1.2 τ r 5 10 0 Taylor rule 15 0.8 1 1.2 τ r 0 5 10 15 0 0.5 1 0 5 10 τ s 15 0 0.5 1 τ w 0 5 10 15 0 0.1 0.2 0 5 10 τ s 15 0 0.1 0.2 τ y Figure 1: Robustness Analysis of Simple Implementable Interest Rate Rules. 34