1.1 Aggregate Demand/Supply:A Compact Repre- sentation In developing this baseline model,it is useful to keep in mind that what monetary policy can influence is the deviation of economic activity from its natural level.Within our baseline model,the natural level of economic ac- tivity is defined as the equilibrium that would arise if prices were perfectly flexible and all other cyclical distortions were absent.In the limiting case of perfect price flexibility,accordingly,the framework takes on the properties of an real business cycle model.One difference with the latter is that because there is monopolistic competition as opposed to perfect competition,the nat- ural level of economic activity is below the socially efficient level.However, this distinction does not affect the nature of the associated cyclical dynamics of the natural level of economic activity which,within our baseline frame- work,resemble those of an real business cycle model with similar preferences and technology. Aggregate Demand.The aggregate demand relation is built up from the spending decisions of a representative household and a representative firm.In the baseline model,both capital and insurance markets are perfect. Within this frictionless setting,the household satisfies exactly its optimiz- ing condition for consumption/saving decisions.It thus adjusts its expected consumption growth positively to movements in the expected real interest rate.Similarly,with perfect capital markets,the representative firm satisfies exactly its optimizing condition for investment:it varies investment propor- tionately with Tobin's g,the ratio of the shadow value of installed capital to the replacement value. 9
1.1 Aggregate Demand/Supply: A Compact Representation In developing this baseline model, it is useful to keep in mind that what monetary policy can ináuence is the deviation of economic activity from its natural level. Within our baseline model, the natural level of economic activity is deÖned as the equilibrium that would arise if prices were perfectly áexible and all other cyclical distortions were absent. In the limiting case of perfect price áexibility, accordingly, the framework takes on the properties of an real business cycle model. One di§erence with the latter is that because there is monopolistic competition as opposed to perfect competition, the natural level of economic activity is below the socially e¢ cient level. However, this distinction does not a§ect the nature of the associated cyclical dynamics of the natural level of economic activity which, within our baseline framework, resemble those of an real business cycle model with similar preferences and technology. Aggregate Demand. The aggregate demand relation is built up from the spending decisions of a representative household and a representative Örm. In the baseline model, both capital and insurance markets are perfect. Within this frictionless setting, the household satisÖes exactly its optimizing condition for consumption/saving decisions. It thus adjusts its expected consumption growth positively to movements in the expected real interest rate. Similarly, with perfect capital markets, the representative Örm satisÖes exactly its optimizing condition for investment: it varies investment proportionately with Tobinís q, the ratio of the shadow value of installed capital to the replacement value. 9
From the individual spending decisions,it is possible to derive an IS curve-type equation that relates aggregate demand inversely to the short term interest rate,similar in spirit to that arising in a traditional framework. In contrast to the traditional model,however,expectations of the future value of the short term rate matter as well.They do so by influencing long term interest rates and asset prices. In particular,let yt be the percentage gap between real output and its natural level,letrr be the gap between the long term real interest rate and its natural level,and let gt be the corresponding percentage gap in Tobin's g Then by taking log-linear approximations of both the baseline model and the fexible price variant,it is possible to derive an aggregate demand equation that relates the output gap,inversely to the real interest rate gap,r, and positively to the gap in Tobin's g,gt,as follows: 班=-Ye0m+Y初 (1) where Ye and Yi are the shares of consumption and investment,respectively, in steady state output,o is the intertermporal elasticity of substitution,and n is the elasticity of the investment/capital ratio with respect to Tobin's g. In effect,this equation relates the output gap to the sum of two terms.The first corresponds to the consumption gap and the second to the investment gap. In particular,the consumption gap moves inversely with the long term real interest rate gap r.Intuitively,if the long term real rate is above its 5To be clear,we define the natural level of economic activity in any given period period t,conditional on the beginning of period capital stock.Monetary policy has no effect on the natural level of economic activity as we have defined.Monetary policy can affect the path of the capital stock,though these effects are typically small in percentage terms under reasonable parameterizations of the model. 10
From the individual spending decisions, it is possible to derive an IS curve-type equation that relates aggregate demand inversely to the short term interest rate, similar in spirit to that arising in a traditional framework. In contrast to the traditional model, however, expectations of the future value of the short term rate matter as well. They do so by ináuencing long term interest rates and asset prices. In particular, let yet be the percentage gap between real output and its natural level, let rre l t be the gap between the long term real interest rate and its natural level, and let qet be the corresponding percentage gap in Tobinís q 5 : Then by taking log-linear approximations of both the baseline model and the áexible price variant, it is possible to derive an aggregate demand equation that relates the output gap, yet ; inversely to the real interest rate gap , rre l t , and positively to the gap in Tobinís q, qet , as follows: yet = c rre l t + i qet (1) where c and i are the shares of consumption and investment, respectively, in steady state output, is the intertermporal elasticity of substitution, and is the elasticity of the investment/capital ratio with respect to Tobinís q. In e§ect, this equation relates the output gap to the sum of two terms. The Örst corresponds to the consumption gap and the second to the investment gap. In particular, the consumption gap moves inversely with the long term real interest rate gap rre l t . Intuitively, if the long term real rate is above its 5To be clear, we deÖne the natural level of economic activity in any given period period t, conditional on the beginning of period capital stock. Monetary policy has no e§ect on the natural level of economic activity as we have deÖned. Monetary policy can a§ect the path of the capital stock, though these e§ects are typically small in percentage terms under reasonable parameterizations of the model. 10
natural value,households will be induced to save more than in the natural equilibrium and,hence,consumption will be lower.Similarly,if g is above its natural value,firms will be induced to invest more than they would under flexible prices. In order to link aggregate demand to monetary policy,it is useful to define the short term real interest rate gap,rrt,as the difference between the short term real rate and its natural equilibrium value,rr,that is rTt三(rt-EtTt+1)-Tr where rt is the short term nominal interest rate and t+1 is the rate of inflation from t to t+1.We note next: (i)rdepends positively on current and expected future values of rrt. (ii)gt depends inversely on current and expected future values of rrt. Proposition(i)emerges from the link between long term interest rates and current and expected short term interest rates implied by the expectation hypothesis of the term structure Proposition (ii)arises Tobin's g depends on the discounted returns to capital investment,where the discount rates depend on the expected path of short term real interest rates. Thus the mechanism through which monetary policy influences aggregate demand works can be thought of as working as follows.Given the sluggish adjustment of prices,by varying the short term nominal rate the central bank is able to influence the short-term real interest rate and,hence,the corresponding real interest rate gap.Thus,through its current and expected future policy settings it is able to affect the corresponding path of rr and, in turn,influence the long term real rate gap,rr,and the gap in Tobin's 9,9t- 11
natural value, households will be induced to save more than in the natural equilibrium and, hence, consumption will be lower. Similarly, if q is above its natural value, Örms will be induced to invest more than they would under áexible prices. In order to link aggregate demand to monetary policy, it is useful to deÖne the short term real interest rate gap, rre t , as the di§erence between the short term real rate and its natural equilibrium value, rrn t , that is rre t (rt Ett+1) rrn t where rt is the short term nominal interest rate and t+1 is the rate of ináation from t to t + 1. We note next: (i) rre l t depends positively on current and expected future values of rre t : (ii) qet depends inversely on current and expected future values of rre t : Proposition (i) emerges from the link between long term interest rates and current and expected short term interest rates implied by the expectation hypothesis of the term structure Proposition (ii) arises Tobinís q depends on the discounted returns to capital investment, where the discount rates depend on the expected path of short term real interest rates. Thus the mechanism through which monetary policy ináuences aggregate demand works can be thought of as working as follows. Given the sluggish adjustment of prices, by varying the short term nominal rate the central bank is able to ináuence the short-term real interest rate and, hence, the corresponding real interest rate gap. Thus, through its current and expected future policy settings it is able to a§ect the corresponding path of rre t and, in turn, ináuence the long term real rate gap, rre l t , and the gap in Tobinís q, qet . 11
As in the traditional models,the framework can incorporate exogenous fluctuations in government purchases or other aggregate demand compo- nents.These fluctuations influence both the natural level of output and the natural real interest rate.However,the form of the aggregate demand equation is not affected,since this relation is expressed in terms of gap vari- ables. Finally,we note that the compact form of the aggregate demand curve depends on the assumption of perfect capital markets,so that both the per- manent income hypothesis for consumption and the g theory for investment are valid.As we discuss later,recent work relaxes the assumption of perfect capital markets. Aggregate Supply:The aggregate supply relation evolves from the price setting decisions of individual firms.To capture nominal price inertia, it is assumed that firms set prices on a staggered basis:each period a subset of firms set their respective prices for multiple periods.Under the most common formulation,due to Calvo (1983),each period a firm adjust its price with a fixed probability that is independent of history.This assumption is not an unreasonable approximation of the evidence(Nakamura and Steinsson (2007)and Alvarez (2007)). Under flexible prices,during each period firms set price equal to a constant markup over nominal marginal cost.With staggered price setting,firms that are able to adjust in a given period set price equal to a weighted average 6The idea of using staggering to introduce nominal inertia is due to Fischer(1997) and Taylor(1980),who used it to describe nominal wage setting.A virtue of the Calvo formulation is that is facilitates aggregation.Because the adjustment probability is inde- pendent of how long a firm has kept its price fixed,it is not necessary to keep track of when different cohorts of firms adjusted their prices. 12
As in the traditional models, the framework can incorporate exogenous áuctuations in government purchases or other aggregate demand components. These áuctuations ináuence both the natural level of output and the natural real interest rate. However, the form of the aggregate demand equation is not a§ected, since this relation is expressed in terms of gap variables. Finally, we note that the compact form of the aggregate demand curve depends on the assumption of perfect capital markets, so that both the permanent income hypothesis for consumption and the q theory for investment are valid. As we discuss later, recent work relaxes the assumption of perfect capital markets. Aggregate Supply: The aggregate supply relation evolves from the price setting decisions of individual Örms. To capture nominal price inertia, it is assumed that Örms set prices on a staggered basis: each period a subset of Örms set their respective prices for multiple periods. Under the most common formulation, due to Calvo (1983), each period a Örm adjust its price with a Öxed probability that is independent of history.6 This assumption is not an unreasonable approximation of the evidence (Nakamura and Steinsson (2007) and Alvarez (2007)). Under áexible prices, during each period Örms set price equal to a constant markup over nominal marginal cost. With staggered price setting, Örms that are able to adjust in a given period set price equal to a weighted average 6The idea of using staggering to introduce nominal inertia is due to Fischer (1997) and Taylor (1980), who used it to describe nominal wage setting. A virtue of the Calvo formulation is that is facilitates aggregation. Because the adjustment probability is independent of how long a Örm has kept its price Öxed, it is not necessary to keep track of when di§erent cohorts of Örms adjusted their prices. 12
of the current and expected future nominal marginal costs.The weight on a given future nominal marginal cost depends on the likelihood that the firm's price will have remained fixed until that particular period,as well as on the firm's discount factor.The firms that do not adjust prices in the current period simply adjust output to meet demand,given that the price is not below marginal cost.Thus,the nominal price rigidities permit output to fluctuate about its natural level.Given that firms'supply curves slope upwards,further,these demand induced fluctuations lead to countercyclical markup behavior. By combining the log-linear versions of the optimal price setting decision, the price index and the labor market equilibrium,one can obtain the following structural aggregate supply relation: Tt=B ETt+1+K班+t (2) where,following Clarida,Gali,and Gertler (1999),uz is interpretable as a "cost push shock."The equation has the flavor of a traditional Phillips curve in the sense that it relates inflation mt to excess demand as measured by y and also a term that reflects inflation expectations,in this case B E. In sharp contrast to the traditional Phillips curve,however,the opti- mization based-approach here places tight structure on the relation.The coefficient on expected inflation,B,is the household's subjective discount factor.The slope coefficient on excess demand k,in turn,is a function of two sets of model primitives.The first set reflects the elasticity of marginal cost with respect to output.The less sensitive is marginal cost to output (i.e.,the flatter are supply curves),the less sensitive will price adjustment be to movements in output (i.e.,the smaller will be K).The second set 13
of the current and expected future nominal marginal costs. The weight on a given future nominal marginal cost depends on the likelihood that the Örmís price will have remained Öxed until that particular period, as well as on the Örmís discount factor. The Örms that do not adjust prices in the current period simply adjust output to meet demand, given that the price is not below marginal cost. Thus, the nominal price rigidities permit output to áuctuate about its natural level. Given that Örmsí supply curves slope upwards, further, these demand induced áuctuations lead to countercyclical markup behavior. By combining the log-linear versions of the optimal price setting decision, the price index and the labor market equilibrium, one can obtain the following structural aggregate supply relation: t = Ett+1 + yet + ut (2) where, following Clarida, Gali, and Gertler (1999), ut is interpretable as a "cost push shock." The equation has the áavor of a traditional Phillips curve in the sense that it relates ináation t to excess demand as measured by yet and also a term that reáects ináation expectations, in this case Ett+1: In sharp contrast to the traditional Phillips curve, however, the optimization based-approach here places tight structure on the relation. The coe¢ cient on expected ináation, ; is the householdís subjective discount factor. The slope coe¢ cient on excess demand , in turn, is a function of two sets of model primitives. The Örst set reáects the elasticity of marginal cost with respect to output. The less sensitive is marginal cost to output (i.e., the áatter are supply curves), the less sensitive will price adjustment be to movements in output (i.e., the smaller will be ). The second set 13