Nội dung

Chapter 7 DYNAMIC STOCHASTIC GENERAL-EQUILIBRIUM MODELS OF FLUCTUATIONS Our analysis of macroeconomic fluctuations in the previous two chapters has developed two very incomplete pieces. In Chapter 5, we considered a full intertemporal macroeconomic model built from microeconomic foundations with explicit assumptions about the behavior of the underlying shocks. The model generated quantitative predictions about fluctuations, and is therefore an example of a quantitative dynamic stochastic general-equilibrium, or DSGE, model. The problem is that, as we saw in Section 5.10, the model appears to be an empirical failure. For example, it rests on large aggregate technology shocks for which there is little evidence; its predictions about the effects of technology shocks and about business-cycle dynamics appear to be far from what we observe; and it implies that monetary disturbances do not have real effects. To address the real effects of monetary shocks, Chapter 6 introduced nominal rigidity. It established that barriers to price adjustment and other nominal frictions can cause monetary changes to have real effects, analyzed some of the determinants of the magnitude of those effects, and showed how nominal rigidity has important implications for the impacts of other disturbances. But it did so at the cost of abandoning most of the richness of the model of Chapter 5. Its models are largely static models with one-time shocks; and to the extent their focus is on quantitative predictions at all, it is only on addressing broad questions, notably whether plausibly small barriers to price adjustment can lead to plausibly large effects of monetary disturbances. Researchers’ ultimate goal is to build a model of fluctuations that combines the strengths of the models of the previous two chapters. This chapter starts down that path. But we will not reach that goal. The fundamental problem is that there is no agreement about what such a model should look like. As we will see near the end of the chapter, the closest thing we have to a consensus starting point for a micro founded DSGE model with nominal rigidity has core implications that appear to be grossly counterfactual. There are two possible ways to address this problem. One is to modify the 309 310 Chapter 7 DSGE MODELS OF FLUCTUATIONS baseline model. But a vast array of modifications and extensions have been proposed, the extended models are often quite complicated, and there is a wide range of views about which modifications are most useful for understanding macroeconomic fluctuations. The other possibility is to find a different baseline. But that is just a research idea, not a concrete proposal for a model. Because of these challenges, this chapter moves us only partway toward constructing a realistic DSGE model of fluctuations. The bulk of the chapter extends the analysis of the microeconomic foundations of incomplete nominal flexibility to dynamic settings. This material vividly illustrates the lack of consensus about how best to build a realistic dynamic model of fluctuations: counting generously, we will consider seven distinct models of dynamic price adjustment. As we will see, the models often have sharply different implications for the macroeconomic consequences of microeconomic frictions in price adjustment. This analysis shows the main issues in moving to dynamic models of price-setting and illustrates the list of ingredients to choose from, but it does not identify a specific ‘‘best practice’’ model. The main nominal friction we considered in Chapter 6 was a fixed cost of changing prices, or menu cost. In considering dynamic models of price adjustment, it is therefore tempting to assume that the only nominal imperfection is that firms must pay a fixed cost each time they change their price. There are two reasons not to make this the only case we consider, however. First, it is complicated: analyzing models of dynamic optimization with fixed adjustment costs is technically challenging and only rarely leads to closed-form solutions. Second, the vision of price-setters constantly monitoring their prices and standing ready to change them at any moment subject only to an unchanging fixed cost may be missing something important. Many prices are reviewed on a predetermined schedule and are only rarely changed at other times. For example, many wages are reviewed annually; some union contracts specify wages over a three-year period; and many companies issue catalogues with prices that are in effect for six months or a year. Thus price changes are not purely state dependent (that is, triggered by developments within the economy, regardless of the time over which the developments have occurred); they are partly time dependent (that is, triggered by the passage of time). Because time-dependent models are easier, we will start with them. Section 7.1 presents a common framework for all the models of this part of the chapter. Sections 7.2 through 7.4 then consider three baseline models of time-dependent price adjustment: the Fischer, or Fischer-Phelps-Taylor, model (Fischer, 1977; Phelps and Taylor, 1977); the Taylor model (Taylor, 1979); and the Calvo model (Calvo, 1983). All three models posit that prices (or wages) are set by multiperiod contracts or commitments. In each period, the contracts governing some fraction of prices expire and must be renewed; expiration is determined by the passage of time, not economic developments. The central result of the models is that multiperiod contracts Chapter 7 DSGE MODELS OF FLUCTUATIONS 311 lead to gradual adjustment of the price level to nominal disturbances. As a result, aggregate demand disturbances have persistent real effects. The Taylor and Calvo models differ from the Fischer model in one important respect. The Fischer model assumes that prices are predetermined but not fixed. That is, when a multiperiod contract sets prices for several periods, it can specify a different price for each period. In the Taylor and Calvo models, in contrast, prices are fixed: a contract must specify the same price each period it is in effect. The difference between the Taylor and Calvo models is smaller. In the Taylor model, opportunities to change prices arrive deterministically, and each price is in effect for the same number of periods. In the Calvo model, opportunities to change prices arrive randomly, and so the number of periods a price is in effect is stochastic. In keeping with the assumption of time-dependence rather than state-dependence, the stochastic process governing price changes operates independently of other factors affecting the economy. The qualitative implications of the Calvo model are the same as those of the Taylor model. Its appeal is that it yields simpler inflation dynamics than the Taylor model, and so is easier to embed in larger models. Section 7.5 then turns to two baseline models of state-dependent price adjustment, the Caplin-Spulber and Danziger-Golosov-Lucas models (Caplin and Spulber, 1987; Danziger, 1999; Golosov and Lucas, 2007). In both, the only barrier to price adjustment is a constant fixed cost. There are two differences between the models. First, money growth is always positive in the Caplin-Spulber model, while the version of the Danziger-GolosovLucas model we will consider assumes no trend money growth. Second, the Caplin-Spulber model assumes no firm-specific shocks, while the DanzigerGolosov-Lucas model includes them. Both models deliver strong results about the effects of monetary disturbances, but for very different reasons. After Section 7.6 examines some empirical evidence, Section 7.7 considers two more models of dynamic price adjustment: the Calvo-withindexation model and the Mankiw Reis model (Mankiw and Reis, 2002; Christiano, Eichenbaum, and Evans, 2005). These models are more complicated than the models of the earlier sections, but appear to have more hope of fitting key facts about inflation dynamics. The final sections begin to consider how dynamic models of price adjustment can be embedded in models of the business cycle. Section 7.8 presents a complete DSGE model with nominal rigidity the canonical three-equation new Keynesian model of Clarida, Galí, and Gertler (2000). Unfortunately, as we will see, the model is much closer to the baseline real-business-cycle model than to our ultimate objective. Like the baseline RBC model, it is elegant and tractable. But also like the baseline RBC model, the evidence for its key ingredients is weak, and we will see in Section 7.9 that together the ingredients make predictions about the macroeconomy that appear to be almost embarrassingly incorrect. Section 7.10 therefore discusses elements of other DSGE models with monetary non-neutrality. Because of the models’ 312 Chapter 7 DSGE MODELS OF FLUCTUATIONS complexity and the lack of agreement about their key ingredients, however, it stops short of analyzing other fully specified models. 7.1 Building Blocks of Dynamic New Keynesian Models Overview We will analyze the various models of dynamic price adjustment in a common framework. The framework draws heavily on the model of exogenous nominal rigidity in Section 6.1 and the model of imperfect competition in Section 6.5. Time is discrete. Each period, imperfectly competitive firms produce output using labor as their only input. As in Section 6.5, the production function is one-for-one; thus aggregate output and aggregate labor input are equal. The model omits government purchases and international trade; thus, as in the models of Chapter 6, aggregate consumption and aggregate output are equal. Households maximize utility, taking the paths of the real wage and the real interest rate as given. Firms, which are owned by the households, maximize the present discounted value of their profits, subject to constraints on their price-setting (which vary across the models we will consider). Finally, a central bank determines the path of the real interest rate through its conduct of monetary policy. Households There is a fixed number of infinitely lived households that obtain utility from consumption and disutility from working. The representative household’s objective function is ∞  β t [U (C t ) − V (L t )], 0 < β < 1. (7.1) t =0 As in Section 6.5, C is a consumption index that is a constant-elasticity-ofsubstitution combination of the household’s consumption of the individual goods, with elasticity of substitution η > 1. We make our usual assumptions about the functional forms of U (•) and V(•):1 1−θ U (C t ) = V(L t ) = 1 B γ Ct 1−θ γ Lt , , θ > 0, B > 0, γ > 1. The reason for introducing B in (7.3) will be apparent below. (7.2) (7.3) 7.1 Building Blocks of Dynamic New Keynesian Models 313 Let W denote the nominal wage and P denote the price level. Formally, P is the price index corresponding to the consumption index, as in Section 6.5. Throughout this chapter, however, we use the approximation we used in the Lucas model in Section 6.9 that the log of the price index, which we will denote p, is simply the average of firms’ log prices. An increase in labor supply in period t of amount dL increases the household’s real income by (Wt /Pt ) dL. The first-order condition for labor supply in period t is therefore V  (L t ) = U (C t ) Wt Pt . (7.4) Because the production function is one-for-one and the only possible use of output is for consumption, in equilibrium C t and L t must both equal Yt . Combining this fact with (7.4) tells us what the real wage must be given the level of output: Wt Pt = V (Yt ) U (Yt ) . (7.5) Substituting the functional forms in (7.2) (7.3) into (7.5) and solving for the real wage yields Wt Pt θ + γ −1 = BYt . (7.6) Equation (7.6) is similar to equation (6.58) in the model of Section 6.5. Since we are making the same assumptions about consumption as before, the new Keynesian IS curve holds in this model (see equation [6.8]): 1 ln Yt = a + ln Yt +1 − rt . θ (7.7) Firms Firm i produces output in period t according to the production function Yi t = L i t , and, as in Section 6.5, faces demand function Yi t = Yt (Pi t /Pt )−η . The firm’s real profits in period t, R t , are revenues minus costs:     Pi t Wt Rt = Yi t − Yi t Pt Pt (7.8)     −η  Pi t 1−η Wt Pi t − . = Yt Pt Pt Pt Consider the problem of the firm setting its price in some period, which we normalize to period 0. As emphasized above, we will consider various assumptions about price-setting, including ones that imply that the length 314 Chapter 7 DSGE MODELS OF FLUCTUATIONS of time a given price is in effect is random. Thus, let q t denote the probability that the price the firm sets in period zero is in effect in period t. Since the firm’s profits accrue to the households, it values the profits according to the utility they provide to households. The marginal utility of the representative household’s consumption in period t relative to period 0 is β t U  (C t )/U  (C 0 ); denote this quantity λ t . The ∞ firm therefore chooses its price in period 0, P i , to maximize E t = 0 q t λt R t ≡ A, where R t is the firm’s profits in period t if P i is still in effect. Using equation (7.8) for R t , we can write A as ⎡ ⎤  1−η   −η ∞  Pi Wt Pi ⎦ q t λ t Yt − A= E⎣ Pt Pt Pt t =0 (7.9) ⎡ ⎤ ∞  η −1 1−η −η  q t λ t Yt Pt Pi − Wt P i ⎦ . =E⎣ t =0 The production function implies that marginal cost is constant and equal to Wt , and the elasticity of demand for the firm’s good is constant. Thus the price that maximizes profits in period t, which we denote P t∗ , is a constant times Wt (see equation [6.57]). Equivalently, Wt is a constant times P t∗ . Thus we can write the expression in parentheses in (7.9) as a function of just P i and P t∗ . As before, we will end up working with variables expressed in logs rather than levels. Thus, rewrite (7.9) as ⎡ ⎤ ∞  η −1 q t λ t Yt Pt F ( p i , pt∗ )⎦ , (7.10) A= E⎣ t =0 where p i and p ∗t denote the logs of P i and P t∗ . One can say relatively little about the P i that maximizes A in the general case. Two assumptions allow us to make progress, however. The first, and most important, is that inflation is low and that the economy is always close to its flexible-price equilibrium. The other is that households’ discount factor, β, is close to 1. These assumptions have two important implications η −1 is negligible relabout (7.10). The first is that the variation in λ t Yt Pt ative to the variation in q t and p ∗t . The second is that F (•) can be well approximated by a second-order approximation around p i = p ∗t .2 Period-t profits are maximized at p i = p ∗t ; thus at p i = p ∗t , ∂F ( p i , p ∗t )/∂p i is zero and 2 2 ∂ F ( p i , p ∗t )/∂ p i is negative. It follows that F ( p i , p ∗t )  F ( p ∗t , p ∗t ) − K ( p i − p ∗t )2 , K > 0. (7.11) 2 These claims can be made precise with appropriate formalizations of the statements that inflation is small, the economy is near its flexible-price equilibrium, and β is close to 1. 7.1 Building Blocks of Dynamic New Keynesian Models 315 This analysis implies that the problem of choosing Pi to maximize A can be simplified to the problem, min pi ∞  q t E [( p i − p ∗t )2 ] t =0 = ∞  (7.12) ∗ 2 ∗ q t {( p i − E [ p t ]) + Var ( p t )}, t= 0 where we have used the facts that q t and pi are not uncertain and that the expectation of the square of a variable equals the square of its expectation plus its variance. Finding the first-order condition for pi and rearranging gives us pi = ∞  ωt E [ p ∗t ], (7.13) t =0  where ωt ≡ q t / ∞ τ =0 q τ . ωt is the probability that the price the firm sets in period 0 will be in effect in period t divided by the expected number of periods the price will be in effect. Thus it measures the importance of period t to the choice of pi . Equation (7.13) states that the price firm i sets is a weighted average of the profit-maximizing prices during the time the price will be in effect. In two of the models we will consider in this chapter (the Calvo model of Section 7.4 and the Christiano Eichenbaum Evans model of Section 7.7), prices are potentially in effect for many periods. In these cases, the assumption that the firm values profits in all periods equally is problematic, and so it is natural to relax the assumption that the discount factor is close to 1. The extension of (7.12) to a general discount factor is min pi ∞  q t β t {( pi − E [ p ∗t ])2 + Var ( p ∗t )}. (7.14) t=0 The resulting expression for the optimal pi analogous to (7.13) is pi = ∞  t=0 ~t E [ p ∗ ], ω t t ~t ≡  β qt ω . ∞ τ τ =0 β q τ (7.15) Finally, it will often be useful to substitute for p ∗t in equation (7.13) (or [7.15]). A firm’s profit-maximizing real price, P ∗/P , is η/(η − 1) times the real wage, W/P . And we know from equation (7.6) that wt equals pt + ln B + (θ + γ − 1)yt (where wt ≡ ln Wt and yt ≡ ln Yt ). Thus, the profitmaximizing price is p ∗ = p + ln[η/(η − 1)] + ln B + (θ + γ − 1)y. (7.16) 316 Chapter 7 DSGE MODELS OF FLUCTUATIONS Note that (7.16) is of the form p ∗ = p + c + φy, φ > 0, of the static model of Section 6.5 (see [6.60]). To simplify this, let m denote log nominal GDP, p + y, define φ ≡ θ +γ −1, and assume ln[η/(η−1)]+ln B = 0 for simplicity.3 This yields p ∗t = φm t + (1 − φ)pt . (7.17) Substituting this expression into (7.13) gives us pi = ∞  ωt E 0 [φm t + (1 − φ)pt ]. (7.18) t =0 ~ (see In the case of a general discount factor, the ω’s are replaced by the ω’s [7.15]). The Central Bank Equation (7.18) is the key equation of the aggregate supply side of the model, and equation (7.7) describes aggregate demand for a given real interest rate. It remains to describe the determination of the real interest rate. To do this, we need to bring monetary policy into the model. One approach, along the lines of Section 6.4, is to assume that the central bank follows some rule for how it sets the real interest rate as a function of macroeconomic conditions. This is the approach we will use in Section 7.8 and in much of Chapter 12. Our interest here, however, is in the aggregate supply side of the economy. Thus, along the lines of what we did in Part B of Chapter 6, we will follow the simpler approach of taking the path of nominal GDP (that is, the path of m t ) as given. We will then examine the behavior of the economy in response to various paths of nominal GDP, such as a one-time, permanent increase in its level or a permanent increase in its growth rate. As described in Section 6.5, a simple interpretation of the assumption that the path of nominal GDP is given is that the central bank has a target path of nominal GDP and conducts monetary policy to achieve it. This approach allows us to suppress not only the money market, but also the new Keynesian IS equation, (7.7). 7.2 Predetermined Prices: The Fischer Model We can now turn to specific models of dynamic price adjustment. Before proceeding, however, it is important to emphasize that the issue we are interested in is incomplete adjustment of nominal prices and wages. There are many reasons involving uncertainty, information and renegotiation costs, 3 It was for this reason that we introduced B in (7.3). 7.2 Predetermined Prices: The Fischer Model 317 incentives, and so on that prices and wages may not adjust freely to equate supply and demand, or that firms may not change their prices and wages completely and immediately in response to shocks. But simply introducing some departure from perfect markets is not enough to imply that nominal disturbances matter. All the models of unemployment in Chapter 11, for example, are real models. If one appends a monetary sector to those models without any further complications, the classical dichotomy continues to hold: monetary disturbances cause all nominal prices and wages to change, leaving the real equilibrium (with whatever non-Walrasian features it involves) unchanged. Any microeconomic basis for failure of the classical dichotomy requires some kind of nominal imperfection. Framework and Assumptions We begin with the Fischer model of staggered price adjustment.4 The model follows the framework of the previous section. Price-setting is assumed to take a particular form, however: each price-setter sets prices every other period for the next two periods. And as emphasized above, the model assumes that the price-setter can set different prices for the two periods. That is, a firm setting its price in period 0 sets one price for period 1 and one price for period 2. Since each price will be in effect for only one period, equation (7.13) implies that each price (in logs) equals the expectation as of period 0 of the profit-maximizing price for that period. In any given period, half of price-setters are setting their prices for the next two periods. Thus at any point, half of the prices in effect are those set the previous period, and half are those set two periods ago. No specific assumptions are made about the process followed by aggregate demand. For example, information about m t may be revealed gradually in the periods leading up to t ; the expectation of m t as of period t − 1, E t −1 m t , may therefore differ from the expectation of m t the period before, E t −2 m t . Solving the Model In any period, half of prices are ones set in the previous period, and half are ones set two periods ago. Thus the average price is pt = 12 ( p1t + p 2t ), 4 (7.19) The original versions of the Fischer and Taylor models focused on staggered adjustment of wages; prices were in principle flexible but were determined as markups over wages. For simplicity, we assume instead that staggered adjustment applies directly to prices. Staggered wage adjustment has qualitatively similar implications. The key difference is that the microeconomic determinants of the parameter φ in the equation for desired prices, (7.17), are different under staggered wage adjustment (Huang and Liu, 2002). 318 Chapter 7 DSGE MODELS OF FLUCTUATIONS where p1t denotes the price set for t by firms that set their prices in t − 1, and p 2t the price set for t by firms that set their prices in t − 2. Our assumptions about pricing from the previous section imply that p1t equals the expectation as of period t − 1 of p ∗t , and p2t equals the expectation as of t − 2 of p ∗t . Equation (7.17) therefore implies p1t = Et−1 [φm t + (1 − φ) pt ] = φ Et−1 m t + (1 − φ) 12 ( p1t + p2t ), p2t = Et−2 [φm t + (1 − φ) pt ] = φ Et−2 m t + (1 − φ) 12 (E t−2 p1t + p2t ), (7.20) (7.21) where E t−τ denotes expectations conditional on information available through period t − τ . Equation (7.20) uses the fact that p2t is already determined when p1t is set, and thus is not uncertain. Our goal is to find how the price level and output evolve over time, given the behavior of m. To do this, we begin by solving (7.20) for p1t ; this yields p1t = 2φ 1+φ Et−1 m t + 1−φ 1+φ p2t . (7.22) Since the left- and right-hand sides of (7.22) are equal, the expectation as of t − 2 of the two sides must be equal. Thus, Et−2 p1t = 2φ 1+φ Et−2 m t + 1−φ 1+φ p2t , (7.23) where we have used the law of iterated projections to substitute Et−2 m t for Et−2 Et−1 m t . We can substitute (7.23) into (7.21) to obtain   2φ 1 1−φ 2 2 2 (7.24) Et−2 m t + pt + pt . p t = φ Et−2 m t + (1 − φ) 2 1+φ 1+φ Solving this expression for p2t yields simply p2t = Et−2 m t . (7.25) We can now combine the results and describe the equilibrium. Substituting (7.25) into (7.22) and simplifying gives p1t = Et−2 m t + 2φ 1+φ (Et−1 m t − Et−2 m t ). (7.26)