Indeterminacy, Aggregate Demand, and the Real Business Cycle Jess Benhabib Department of Economics New York University jess.benhabib@nyu.edu Yi Wen Department of Economics Cornell University Yw57@cornell.edu Abstract We show that under indeterminacy aggregate demand shocks are able to explain not only aspects of actual uctuations that standard RBC models predict fairly well, but also aspects of actual uctuations that standard RBC models cannot explain, such as the hump-shaped, trend reverting impulse responses to transitory shocks found in US output, investment, and hours. Indeterminacy arises in our model due to capacity utilization and mild increasing returns to scale. We wish to thank Jordi Gali, Stephanie Schmitt-Groh e, Karl Shell, and Martin Uribe for very useful discussions and comments. Technical support from the C.V. Starr Center for Applied Economics at New York University is gratefully acknowledged.
1. Introduction Recently, general equilibrium real business cycle models have been subject to a number of criticisms. A basic criticism is the heavy reliance of such models on technology shocks to explain business cycle facts (for example, Blanchard, 1993; Cochrane, 1994; Evans, 1992; Gordon, 1993; Mankiw, 1989; Summers, 1986). Another is the lack of an endogenous ampli cation and propagation mechanism, which has resulted in the failure of standard RBC models to explain the large hump-shaped, trend-reverting output responses to transitory shocks (Cogley and Nason, 1995, Watson, 1993). Finally, real business cycle models have been criticized for failing to match the forecastable comovements of basic macroeconomic variables observed in the data (see Rotemberg and Woodford, 1996). 1 These problems are not unrelated. Demand shocks are thought to be important for generating the business cycle because components of aggregate demand, such as consumption and investment, may reinforce each other, resulting in a multiplier-accelerator mechanism that can generate highly persistent, trendreverting output dynamics. The mutual reinforcement among components of aggregate demand is possible if there exists unutilized resources that allow for the expansion of output with little or no increase in marginal costs in the short run, meeting or ful lling the rise of total demand. In contrast, all resources in equilibrium business cycle models are fully utilized because prices adjust quickly to clear the markets. Under the assumption of the diminishing marginal product of labor, demand shocks would have a crowdingout e ect, resulting in the negative comovements of the components of aggregate demand and in having minimal impact on aggregate employment and output. Consequently, standard RBC models have relied on supply shocks to explain the business cycle. In this paper we show how to capture the more \Keynesian" features of the demand-shock driven business cycle without abandoning the hypotheses of market clearing and exible prices. There are three essential elements in our model that when acting together, can generate the business cycle facts observed in data, 1 Recently many e orts have been made to nd ways to enrich the internal propagation mechanisms of RBC models driven by technology shocks. Prominent examples include Burnside and Eichenbaum, 1996, Andolfatto, 1996, and Carlstrom and Fuerst, 1997. 2
and meet the criticisms directed at the RBC model. The rst is variable capacity utilization. The second is the presence of demand shocks that exhibit some persistence. The third is the presence of small and empirically reasonable externalities that gives rise to indeterminacy. 2 We show that these three elements are essential and indispensable in order to generate a propagation mechanism that ampli es the impact of temporary demand shocks on output, investment and hours in a hump-shaped manner. 3;4 Variable capacity utilization has the e ect of increasing the marginal product of labor in the short-run by enhancing the output elasticity of labor. Coupled with a mild production externality that is consistent with recent empirical estimates, it makes the model behave as if there were short-run increasing returns to the labor input. This factor is crucial not only because it generates the desired forecastable comovements in consumption, investment, hours and output, but also because it gives rise to the indeterminacy that generates the hump-shaped, trend reverting time series observed in the data by enriching the internal dynamics of the model. We examine three di erent types of aggregate demand shocks: shocks to consumption demand, shocks to government spending, and sunspot shocks to investors' animal spirits. 5 We investigate whether the endogenous propagation mechanism induced by indeterminacy is capable of giving these demand shocks a primary role in explaining the business cycle. Among other things, we nd: a) All three types of demand shocks can each generate positive comovements among output, consumption, investment and hours that are comparable to predictions of standard RBC models under technology shocks. b) Small transitory demand shocks to consumption and government expenditure can generate large, hump- 2 For an excellent and thorough analysis of why models of indeterminacy driven by sunspots alone cannot improve much upon the standard RBC model, see Schmitt-Grohe (2000), and also Schmitt-Grohe, (1997). For a general survey of the literature of indeterminacy and sunspots in macroeconomics, see Benhabib and Farmer (1999). 3 Although small external e ects can give rise to indeterminacy, externalities that do not generate indeterminacy will not give rise to hump-shaped dynamics. In our model, the minimum degree of returns to scale induced by externalities necessary for indeterminacy is in the range of 1:06 1:104; which is plausible even judged by the more recent empirical estimates on returns to scale (e.g., Basu and Fernald, 1997; and Burnside, Eichenbaum, and Rebelo, 1995). 4 The emergence of hump-shaped impulse responses to monetary shocks has been discussed in models with credit constraints when there are investment delays. See Bernanke, Gertler and Gilchrist,1999. 5 Sunspot shocks or extrinsic uncertainty were intoduced into the economics literature by Shell (1977) and by Cass and Shell (1983). 3
shaped, trend-reverting impulse responses for output, investment and hours. c) Combined with shocks to investors' animal spirits, each of these fundamental demand shocks can generate autocorrelation functions for output growth, investment growth, and hours growth that closely match the post-war US economy. 6 The rest of the paper is organized as follows. In section 2, we present further empirical evidence on the trend-reverting dynamics for the US economy. In particular, we show that the responses of consumption, investment, and hours to temporary demand shocks all have hump-shaped features, and they are strongly correlated with output. In section 3, we show that a general equilibrium model featuring indeterminacy is capable of explaining these stylized dynamics with aggregate demand shocks. The model is tested using the method proposed by Cogley and Nason (1995) in section 4. Section 5 concludes the paper. 2. Stylized Responses to Demand Following Blanchard and Quah (1989) and Cogley and Nason (1995), we decompose aggregate output, consumption, investment, and hours into two components, one that pertains to permanent shocks, and the other that pertains to transitory shocks. The transitory components in these variables are interpreted by Blanchard and Quah as uctuations due to aggregate demand shocks. We use the ratio of investment to output as the covariate in a series of bivariate VAR analyses to carry out the Blanchard -Quah decomposition. 7 Balanced growth in RBC models implies that the investment to output ratio is stationary. The demand shocks so identi ed have the natural interpretation of being disturbances that a ect shortrun aggregate savings, such as temporary changes in government spending, in consumers' preferences, and in rms' autonomous investment. These demand shocks are perfectly matched by our general equilibrium model. 8 6 Sunspots cannot by themselves generate the hump-shaped dynamics, as has been demonstrated by Schmitt-Grohe (2000). 7 Namely, the growth rate of output, consumption, investment, and hours are each paired with the investment to output ratio to identify the permanent and transitory components of each series. This is similar to Cochrane (1994) who uses consumption to output ratio as the covariate. 8 Cogley and Nason (1995) use per capita hours as the covariate in carrying out the Blanchard- Quah decomposition. We choose the investment to output ratio also because it seems to be more stationary than the per capita hours series for post-war US economy. The impulse responses identi ed for the two covariates are nevertheless very similar. 4
Figure 1 shows impulse responses to demand from output, consumption, investment, and hours respectively. They all exhibit the familiar hump-shaped, trend-reverting dynamics. These dynamics are very similar to those identi ed by Blanchard and Quah (1989), and the responses of di erent variables move very closely together. To further ensure that the responses so identi ed re ect responses to demand disturbances, we also use the government expenditure to GDP ratio as the covariate in carrying out the Blanchard-Quah decomposition. The identi ed responses to innovations in the government expenditure to GDP ratio are very similar to those identi ed using the investment to output ratio. 9 Figure 2 shows the implied autocorrelation functions (ACF) for growth rates. The transitory components of all four variables exhibit positive serial correlation in growth rates for lags of the rst two quarters, and show negative serial correlation in growth rates for lags beyond three to four quarters. This is consistent with the empirical ndings of Cogley and Nason (1995). As is pointed out by Cogley and Nason, a central weakness of the real-businesscycle paradigm as a convincing explanation of the business cycle is its failure to account for the above stylized business cycle dynamics. Under transitory demand shocks, standard RBC models generate monotonic impulse responses for output, negative serial autocorrelation for output growth, and negative cross correlation of consumption and investment. In other words, contrary to conventional wisdom and postwar US data, demand shocks do not seem to have fully expansionary multiplier e ects in RBC models. This is illustrated in gure 3. 10 9 The data used are log quarterly US real xed investment, consumption expenditure on nondurables and services, and total government expenditure. All data series are from CITIBASE (1960:1-1994:4). Output is de ned as the sum of consumption, investment, and government expenditure. 2 lags are used in the VAR estimation. Both the investment to output ratio and the government spending to GDP ratio have a weak time trend. We used two di erent detrending methods in the VAR estimation to render stationarity for the covariate. One uses a linear time trend and the other uses the Hodrick-Prescott trend (i.e., the HP ltered investment to output ratio was used as the covariate in bivariate VAR analyses). The results are very similar for both typesoftrend.thehp-trendcasesarereportedinfigure1. 10 Figure 3 shows the impulse responses of output and investment in the King-Plosser-Rebelo (1988) model to a consumption demand shock. The responses of the model to a government spending shock are very similar. The consumption demand shock follows an AR(1) process and is modeled as a disturbance to the subsistence level in the utility function, as in Baxter and King (1992). For more detailed description about shocks to consumption demand and government expenditure, see Section 3. 5
3. Searching for the Propagation Mechanism In this section, we show how to bring the real business cycle paradigm into closer conformity with the data. In particular, we show that indeterminacy arising from variable capacity utilization and mild externalities enables demand shocks to play an essential role in replicating the actual business cycle uctuations. 3.1. The Model This is the one-sector RBC model with variable capacity utilization studied by Wen (1998), based on Greenwood et al. (1989), Burnside and Eichenbaum (1996), Baxter and King (1992), Benhabib and Farmer (1994), and Farmer and Guo (1994). A representative agent in the model chooses sequences of consumption, hours, capacity utilization, and capital accumulation to solve 1! X max E 0 t Ãlog(c t t ) a n1+ t 1+ t=0 subject to c t + i t + g t = t (e t k t ) n (1 ) t ; (3.1) k t+1 = i t +(1 ± t )k t ; where is a random shock to the subsistence level of consumption, which can generate the urge to consume (Baxter and King, 1992); g is government spending and is modeled as an exogenous stochastic process representing a pure resource drain on the economy (Christiano and Eichenbaum, 1992); e 2 [0; 1] denotes capital utilization rate, and is a measure of production externalities and is de ned as a function of average aggregate output which individuals take as parametric: = h (ek) n 1 i ; 0: (3.2) When the externality parameter is zero, the model is reduced to a standard RBC model studied by Greenwood et al (1988). To have an interior solution for e in the steady state, we follow Greenwood et al. by assuming that the capital stock depreciates faster if it is used more intensively: ± t = e µ t ; µ > 1; (3.3) which imposes a convex cost structure on capital utilization. 6
As pointed out by Benhabib and Farmer (1994), the externality model can also be cast in form of a monopolistic competition model with increasing returns to scale at the individual rm level that produces intermediate goods. If the nal output sector's production technology is de ned as Z Y = 1 y(i) di ; and the technology for producing an intermediate commodity y(i) is given by y(i) =[e(i)k(i)] a n(i) b ; then the aggregate production function in the model has the reduced form in a symmetric equilibrium: y =(ek) a n b : If markup is de ned as the logarithm of price-marginal cost ratio, then it can be shown that the markup in the model is given by ¹ =ln 1 1+( 1) ; where ( 1) is the inverse of the price elasticity of demand facing an intermediate goods producing rm. For the monopolistic competition model to be exactly equivalent to the externality model, we simply set a = ; b =1 ; and a + b =1+ : 11 This implies that = 1 = 1 a+b 1+ : The markup is therefore given by ¹ ' : To facilitate comparison with the existing literature, we assume that all shocks to fundamentals follow univariate AR(1) processes in log: " log( t ) log(g t ) # = " 1 ½ 0 0 1 ½ g #" ¹ ¹ g # + " ½ 0 0 ½ g #" log( t 1 ) log(g t 1 ) # + " " t " gt # ; where var à " t " gt! " ¾ 2 = 0 0 ¾g 2 # : 11 See Benhabib and Farmer (1994). 7
Notice that innovations in the fundamental shocks are assumed to be orthogonal. To solve the model, we log-linearize the rst order conditions around the steady state as in King et al. (1988). To allow for the study of investors' animal spirits as a possible source of aggregate demand uncertainty, we arrange the system of linearized equations in a way such that investment rather than the Lagrangian multiplier appears in the state vector. Denoting S as the state vector, and Z as the vector of control variables, the model can be reduced to the following system of linear di erence equations (the hat variables denote percentage deviations from their steady state values): S t+1 = WS t + R t+1 ; Z t = HS t ; where S t = h ^kt ^{ t ^ t ^g t i 0 ; Zt = h ^c t ^n t ^e t i 0 and t+1 is a vector of one-step ahead forecasting errors that satisfy and t+1 = S t+1 E t S t+1 ; E t t+1 =0: Notice that the rst element in t+1 = k t+1 E t k t+1 =0; since k t+1 is known at the beginning of time t; the last two elements in t+1 =[" +1 " gt+1 ] 0 ; and the second element in t+1 = i t+1 E t i t+1. Thus the law of motion for the model's state vector can be written as: 2 3 2 3 2 3 2 3 ^k t+1! 11! 12! 13! 14 ^k ^{ t+1 6 7 4 ^ =! 21! 22! 23! t 0 0 0 2 3 24 6 7 ^{ t t+1 5 4 0 0 ½ 0 5 6 7 4 ^ + r 1 r 2 r " t+1 3 6 7 6 7 4 " t 5 4 1 0 0 5 gt+1 5 (3.4) v ^g t+1 0 0 0 ½ g ^g t 0 1 0 st+1 where v st+1 ^{ t+1 E t^{ t+1 : When the equilibrium is unique, one of the eigenvalues of W should lie outside the unit circle, so that investment i t can be determined uniquely by solving forward the system under the expectation operator E t to obtain a decision rule for i(t) that depends only on current capital stock k t and time t realizations to the fundamentals f t ;g t g : If both eigenvalues of W lie inside the unit circle, however, the steady state is stable. In that case, the model 8
is indeterminate and has multiple equilibria. Hence, the state space of the model includes not only the capital stock and the realizations of the exogenous state variables, f t ;g t g ; but also the time t investment i t. Notice that under indeterminacy the decision rule for the initial investment at time t (starting from the steady state) takes the special form: ^{ t = r 1 " t + r 2 " gt + r 3 v st : Under rational expectations, the forecasting error, v st = i t E t 1 i t ; follows i:i:d processes, implying that rational agents do not make systematic errors in forecasting the future. Since v st can either re ect innovations in fundamentals or be purely extraneous, it can be called sunspots or animal spirits, and can be interpreted as shocks to autonomous investment. We de ne the vector V t =(" t ;" gt ;v st ) 0 ; where v st denotes sunspots. 12 There are potentially three types of aggregate demand disturbances in the model: innovations to government spending " g, innovations to consumption demand " ; and innovations to autonomous investment v s. Hence, sunspots can act as an equilibrium selection device. The covariance matrix for V is given by: = 2 6 4 ¾ 2 0 ¾ s 0 ¾ 2 g ¾ gs ¾ s ¾ gs ¾ 2 s where the correlations between sunspots and innovations in the fundamentals can take values in the interval [ 1; 1] : Since the linear version of the model imposes no restrictions on the variance and covariance of sunspots, they need to be speci ed exogenously. Notice that in the decision rule for ^{ t ; the rst two terms including r 1 and r 2 are completely determined by preferences and technologies of the model, but the third term is arbitrary, depending on the variance of sunspots. We can therefore take the normalization r 3 =1: To see how the model responds to shocks under indeterminacy, take the simpler case where the shocks are all i:i:d (½ = ½ g =0): Starting from the steady state at time t, wehave: ³^kt 1 ;^{ t 1 ; ^ t 1 ; ^g t 1 =0: In addition, we have ^kt =0: Time t innovations (" t ;" gt ;v st ) thus a ect the control vector Z t through their impact on (i t ; t ;g t ) : Therefore, autonomous investment is clearly important in determining 12 When v s contains innovations to the fundamental shocks, it implies that the sunspots are correlated with fundamental shocks. 3 7 5 ; 9
the initial responses of the model. However, how the initial impact of shocks gets propagated through time does not depend so much on autonomous investment. Rather, it depends crucially on the eigenvalues of the matrix, " #!11! = 12 :! 21! 22 This is so because impulse responses j periods ahead are determined by the autoregressive equation: " ^kt+j ^{ t+j # = j " ^kt ^{ t # ; j =1; 2; 3; :::; 1: The eigenvalues of therefore govern the patterns of propagation. If the eigenvalues are complex, for example, the model economy will oscillate around the steady state after an initial disturbance, showing profoundly di erent propagation mechanism from the cases where the eigenvalues are real valued. If the model is determinate, however, it has only monotonic transitional dynamics (in the standard RBC case where!>0) because the law of motion for the state vector reduces to ^k t+1 =!^k t : This illustrates why indeterminacy can profoundly change the propagation mechanism of a RBC model. To quantify the model's eigenvalues, we calibrate the structural parameters of the model as follows. We set the capital share to 0:3; the time discount factor to 0:99; the inverse elasticity of labor supply to 0; and choose µ such that the rate of capital depreciation in the steady state is 10 percent a year. The externality parameter is most crucial in the model and is left free for experiments. Table 1 reports the model's eigenvalues when the externality takes on different values. The rst column lists the values of ; the second column lists the corresponding eigenvalues, and the third column lists the modulus of the complex eigenvalues, which indicate indeterminacy if the modulus is less than one. 10
Table 1. Ext. ( ) Eigenvalues Modulus 0.000 0.925, 1.092 0.100 0.731,1.901 0.102 0.671,3.311 0.103 0.616, 28.74 0.104 0.434, 0.187 0.105 0.548 0:377i 0.665 0.106 0.696 0:375i 0.790 0.107 0.771 0:351i 0.847 0.108 0.816 0:327i 0.879 0.11 0.868 0:289i 0.915 0.20 0.990 0:085i 0.994 0.30 0.995 0:060i 0.997 0.40 0.996 0:049i 0.998 It is seen from Table 1 that indeterminacy and complex roots emerge under mild externalities. The minimum degree of the externality required for indeterminacy is 0:104, 13 and that for complex roots is 0:105, implying a markup around 0:1 or a degree of aggregate returns to scale around 1:10 1:11, which is very plausible even when judged by more recent empirical studies (e.g., Basu and Fernald, 1997; and Burnside et al., 1995). 14 Notice that the aggregate labor demand curve is downward sloping as in Benhabib and Farmer (1996) when indeterminacy arises in the model, which is in sharp contrast to the earlier model of Benhabib and Farmer (1994). 15 3.2. Impulse responses to demand shocks As a benchmark, we assume that is diagonal. The parameters associated with exogenous variables are chosen as follows. We set the persistence parameter for g 13 One of the eigenvalues changes sign from +1 to 1 between =0:1029 and =0:1030: 14 The degree of markup or externality required for indeterminacy can be reduced even further if the time discount factor is larger or if the steady-state depreciation rate is higher. For example, when = 0:995; the minimum value of for indeterminacy reduces to 0:057. Complex eigenvalues arise for = 0:058: 15 The slope of the aggregate labor demand curve (in logs) is given by (1 )(1 + ) 1: For example, when =0:108; the slope is 0:22: 11
as ½ g =0:9 (a magnitude similar to that estimated by Christiano and Eichenbaum, 1992), and the persistence parameter for as ½ =0:9 (a magnitude similar to that estimated by Baxter and King, 1992). In order to highlight the dramatic e ect of indeterminacy, two di erent values of are used in generating impulse responses of the model. Figure 4 corresponds to the case of =0:1 where the steady state is a saddle and the equilibrium is unique. It shows the impulse responses of output, consumption, investment and hours to one standard deviation increases in " (solid lines) and " g (dashed lines) respectively. Figure 5 corresponds to the case of = 0:108 where the steady state is a sink and the equilibrium is indeterminate. It shows impulse responses of income, consumption, investment and hours to a one standard deviation increase in " (solid lines), in " g (long dashed lines), and in " s (short dashed lines) respectively. Several features of the model that generate Figure 4 deserve mention. First, capacity utilization and a mild externality can generate positive comovements in the model under shocks to demand. This is in stark contrast with the standard RBC model examined by Baxter and King (1992), where an extremely high externality is necessary to generate positive comovements. Second, the impulse responses of all variables to demand shocks are still monotonic, as opposed to hump-shaped, when the steady state is a saddle, despite the presence of capacity utilization and increasing returns. Figure 5 shows a completely di erent picture when the externality is slightly increased to a level such that the \capacity-utilization augmented" short-run returns to labor are greater than one and indeterminacy is present. 16 The impulse response of output to fundamental demand shocks becomes hump-shaped, revealing a signi cant change in the model's dynamic structure. The output response to sunspots (long dashed lines), although continuing to show positive comovements with other variables, nevertheless fails to exhibit the initial hump. This observa- 16 The social returns to labor is only 1:108(1 ) =0:7756: Variable capacity utilization makes the model behave as if the short-run output elasticity of labor is greater than one. According to Wen (1998), the "capacity utilization augmented" short-run elasticity of labor is µ (1 )(1 + ) µ (1 + ) =1:016: See Wen (1998c) for detailed analyses on the dynamic e ect of capacity utilization on returns to labor. 12
tion, that sunspots cannot generate hump-shaped impulse responses for output, has been discussed by Schmitt-Grohe (2000). 17 So far we have not allowed the model to respond simultaneously to both fundamental shocks and sunspots. Figure 5 suggests that mixing fundamental shocks and sunspots shocks should not destroy the hump-shaped impulse response patterns of the model as long as the variance of sunspots is not extremely large relative to that of fundamental shocks. Figure 6 compares several possible response functions of output to a fundamental shock (" ) and sunspots (" s ) when the correlation between " and " s takes ve possible values: ¾ s = f1; 0:5; 0; 0:5; 1g : We choose the standard deviation of " to be one, and that of the sunspots to be 0:5. 18 It is seen in gure 6 that regardless the correlations between sunspots and fundamentals, the impulse response functions of output are always hump-shaped. As a matter of fact, if the relative standard deviation of sunspots is allowed to vary instead of being xed at 0:5, the model can generate any one of those impulse response functions shown in gure 6 for any given ¾ s : This means that adding sunspots will certainly improves the model's empirical t since the model tends to generate too much hump under fundamental shocks alone (Figure 5). 19 The hump-shaped propagation mechanism that arises under indeterminacy can be understood as follows. In a standard RBC model without indeterminacy, the only channel of propagation is capital accumulation. This, however, is an extremely weak channel for transmitting the impact of shocks, because capital is a stock while investment is a ow. Given that the steady-state investment to capital ratio is small, even a large change in the current investment ow induced by a technology shock would have little impact on the next period capital stock. In addition, since higher consumption in the next period is accompanied by higher leisure, labor supply would immediately fall after the impact period, which would 17 It is worth emphasizing that the hump-shaped impulse responses of output to the fundamental demand shocks do not depend on the persistence of these shocks. Even when they are i:i:d processes, the impulse responses are still hump-shaped, except that the humps are too sharp and are less smooth than the data. 18 To compute the impulse responses of output to multiple shocks, we draw two i:i:d processes from standard normal distributions (taking the rst process as " t ), and mix them in various degrees to form the sunspots variable " st so that the correlation between sunspots and the rst series equal exactly to the pre-speci ed values. We then generate arti cial time series of output by simulating the model. The impulse response functions of output for each possible case in gure 6 are estimated using the corresponding arti cial data sample. 19 Sunspots help here because they change the magnitude of initial responses of the model, so as to make them less hump-shaped. 13
e ectively terminate the possibility of continuous increases in output. For the same reason, a demand shock can only generate a negative comovement between and consumption and investment. Therefore, in the standard RBC model, output dynamics mirrors the dynamics of technology shocks (for more detailed analyses of the propagation mechanism of standard RBC models, see Wen, 1995). By contrast, in our model, variable capacity utilization o ers an additional channel of propagation that can propagate not only supply shocks but also demand shocks in a manner that better conforms to the data. Variable capacity utilization coupled with a mild externality on labor mimics increasing returns to labor, inducing consumption and labor to comove in response to a demand shock. Therefore, even in response to a demand shock, output can rise su±ciently to permit positive investment and a higher capital stock in the next period. Since consumption follows permanent income, a higher future capital stock, coupled with the persistence of demand shocks, leads to higher future consumption, and therefore to higher future labor supply and higher future output. Continuous output expansion is thus sustainable by aggregate demand. In other words, the propagation mechanism in our model is powerful not because more capital stock leads to more output, but because more capital stock leads to more consumption demand (the permanent-income or wealth e ect), which in turn leads to higher capacity utilization, more employment and more output (the returns-to-scale effect) in subsequent periods. This is a cumulative process of expansion, similar to a classic multiplier-accelerator mechanism. As the expansion continues, the rising marginal cost of capacity utilization, and the diminishing marginal product of capital dictate a tapering o in the increases of output and a decline in the growth rate of the capital stock. The result is that sooner or later, the capital stock and output must stop growing, and hence consumption must begin to fall. A reduction in consumption demand then triggers a contraction in labor supply and in output. As the contraction begins, the propagation mechanism reverses direction, leading to a cumulative process of contraction. When the capital stock falls, however, its marginal product starts to rise, and a new round of expansion eventually follows. The non-monotonic and cumulative dynamics of output growth therefore depends crucially on the richer two-dimensional dynamics of indeterminacy that allows these oscillatory hump-shaped impulse responses. 14
4. An Econometric Test Cogley and Nason demonstrate that standard RBC models cannot explain two stylized facts about US GDP: positive serial correlation of GDP growth and a hump-shaped, trend reverting impulse response function to innovations in the temporary component. Accordingly, this section employs the same econometric procedure proposed by Cogley and Nason to test whether indeterminacy has the potential to account for the aspects of actual uctuations which the standard realbusiness-cycle paradigm cannot explain. Speci cally, we calibrate and estimate the model's parameters and ask how likely it is that the actual US sample time series are generated from a world that is equivalent to our model. We calibrate most of the parameters following the existing literature. Specifically, we set the capital share to 0:3; thetimediscountfactor to 0:99; the steady-state rate of capital depreciation ± to 0:025, the inverse labor supply elasticity to 0 (Hansen, 1985), the persistence parameter for government spending ½ g to 0:96 (estimated by Christiano and Eichenbaum, 1992), the persistence parameter for consumption shock ½ to 0:97 (estimated by Baxter and King, 1992), and the externality parameter to 0:108 (consistent with indeterminacy). We arbitrarily set the standard deviation of sunspots, ¾ s =0:01; and choose parameters, f¾ ;¾ g g ; so that each of them acting alone with the sunspots can generate autocorrelation functions of output growth that matches the data as closely as possible. 20 To show the robustness of our empirical results, we let the correlation between sunspots and fundamental shocks take di erent values between 1 and 1; so as to show that the model's successes do not depend on whether and how sunspots are correlated with fundamentals. These parameterizations are reported in Table 2 (the pre-speci ed values for cross correlation between sunspots and fundamentals are listed on the last two columns). 20 This is equivalent to an informal estimation procedure called methods of moments widely adopted in the RBC literature. Notice in table 2 that as the correlation parameter changes, ¾ changes accordingly but ¾ g barely changes at all. 15
Table 2. Calibration ¾ ¾ g ¾ s ¾ s ¾ gs Model 0:01 Model A 1 0:009 0:01 1 Model A 2 0:012 0:01 0:5 Model A 3 0:015 0:01 0 Model A 4 0:022 0:01 0:5 Model A 5 0:022 0:01 1 Model B 1 0:0042 0:01 1 Model B 2 0:0042 0:01 0:5 Model B 3 0:0042 0:01 0 Model B 4 0:0042 0:01 0:5 Model B 5 0:0042 0:01 1 Model is our benchmark that has sunspots shocks only. Then we add consumption demand shocks (models A 1 A 5 ) and government shocks (models B 1 B 5 ) respectively into the environment. We test each of these versions of the model with respect to impulse response functions and autocorrelation functions by computing the generalized Q test statistics, which are de ned as follows: Q =(^c c) 0 ^V 1 c (^c c) : The vector ^c is the sample impulse response function or autocorrelation function, and c is the mean of N (= 500) simulated impulse response functions or autocorrelation functions computed from simulated time series of the same length as the actual data, namely, 140 quarters. 21 That is, c = 1 N P Ni=1 c i. The covariance matrix, ^V c ; is estimated by ^V c = 1 N NX (c i c)(c i c) 0 : i=1 The test statistic Q has approximate  2 distribution with degrees of freedom equal to the number of elements in c. Following Cogley and Nason, the number of lags 21 Since all shocks in the model are transitory demand shocks (there are no permanent demand shocks), we do not use the Blanchard-Quah decomposition to estimate conditional impulse response functions. Instead, a univariate VAR is su±cient to obtain the conditional impulse response functions for each variable. We use a lag length of four in the univariate VARs. The autocorrelatiion functions are simply computed by their formula. 16
chosen is 8, which gives a critical value of 20:1 at the 1% signi cance level and a critical value of 15:5 at the 5% signi cance level. The test results for autocorrelation function of growth are reported in table 3. The benchmark model driven by sunspots only perform poorly in matching the data, consistent with the ndings of Schmitt-Grohe (2000). The various versions of the model (A 1 B 5 ) driven by a combination of one of the fundamental demand shocks and sunspots, however, perform extremely well with respect to all variables except consumption growth. The results indicate that the covariance between sunspots and fundamental shocks do not matter, as long as both types of shocks arepresent. Wehavealsoaddedintable3(lastrow)theversion(ModelC)in which all three types of shocks are present, so as to see whether it can improve the match on consumption growth. Figure 5 suggests that the 3 types of shocks have very di erent e ect on consumption, suggesting that a mixture of them can generate a hump-shaped consumption response function similar to the data. The last row in table 3 con rms this conjecture. Table 3. Q Statistics Q acf y i n c Model 120:9 113:3 156:2 1:9 A 1 2.1 7.2 12.6 55.7 A 2 1.7 2.6 10.6 58.3 A 3 1.8 4.9 11.9 73.9 A 4 1.8 5.3 9.7 66.4 A 5 2.1 9.2 13.0 78.3 B 1 2.1 6.1 12.6 160.1 B 2 2.1 5.6 12.7 145.0 B 3 2.3 5.7 13.0 136.7 B 4 2.2 5.9 13.0 155.6 B 5 2.2 5.9 12.8 140.8 Model C 2:1 5:6 12:2 11:9 The test statistics for the output impulse response function show similar results. Namely, the models driven by fundamental shocks (either " or " g )along with sunspots perform much better than the models driven by sunspot shocks alone. To see this graphically, gures 7, 8, and 9 show the output impulse response functions of three di erent versions of the model respectively. Figure 7 corresponds to model ; gure 8 corresponds to model A (version 3); and gure 8 17
corresponds to model B (version 3): 22 It is seen that allowing for demand shocks to preferences or government expenditure in the model produces very good matches to the data with regard to output dynamics (the Q test statistics are 4:1 and 2:5 respectively), while the benchmark model (driven by sunspots only) fails to generate the initial hump in the response function of output, resulting in an autocorrelation function of output growth that falls short in matching the data. Hence, persistent AR(1) demand shocks to fundamentals are essential for explaining the US business cycles. 23 The impulse response functions for investment and hours, even though they have very similar shapes to those of the data (see Figure 10), fail the Q test. This is so because our model does not generate the exact relative volatilities for investment and hours. 24 The Q test statistics are very sensitive to even such small level di erentials and therefore tend to magnify the di erence between model and data. Hence, despite the very close matches for the impulse response functions between our model and the data, our model fails the Q test with respect to investment and hours. Finally, we check whether the model is able to explain aspects of actual uctuations that standard RBC models predict fairly well, with respect to the small number of second moments stressed by Kydland and Prescott (1982). Table 4 shows that our model is comparable to the standard model of King, Plosser and Rebelo (1988). 25 22 It makes little di erence which version of type A or type B to use. 23 Notice that persistent AR(1) shocks do not generate hump-shaped impulse responses in standard RBC models (see gure 3 and Cogley and Nason, 1995). 24 The variance of the shocks in our model are adjusted to match the response of output to theimpulseattheperiodofimpact. Analternativestrategymaybetochoosethevarianceof the shocks to match a weighted sum of the initial responses of these key variables to the impulse so as to further improve their Q statistics. 25 The common parameters of the models are the same. Notice that, as expected, consumption is too smooth under the government and sunspot demand shocks, but not under the preference shock. A combination of the three demand shocks and the technology shock, properly weighted, can be used to further improve the match between the data and the model. 18
Table 4. Selective Moments ¾ c =¾ y ¾ i =¾ y ¾ n =¾ y corr(c; y) corr(i; y) corr(n; y) US Economy (1960:4-1994:4) 26 0:65 4:16 1:07 0:91 0:89 0:85 RBC Model: Technology Shock 27 0:61 3:60 0:67 0:75 0:95 0:74 EBC Model: Consumption Shock 28 0:65 3:79 0:96 0:64 0:93 0:998 EBC Model: Government Shock 0:06 3:90 0:97 0:54 0:95 0:998 EBC Model: Sunspot Shock 0:03 4:65 0:99 0:36 0:999 0:999 EBC Model: Consumption + Sunspot 0:57 4:02 0:97 0:56 0:95 0:998 EBC Model: Government + Sunspot 0:05 4:09 0:97 0:49 0:96 0:999 4.1. Comovements of Output, Hours, and Consumption Rotemberg and Woodford (1996) highlight another characteristic of economic uctuations that is di±cult for the real-business-cycle paradigm to explain. They show that, in US data, forecastable changes in output, hours, investment and consumption are positively correlated, whereas standard RBC models imply that hours and consumption move in opposite directions in approaching the steady state after the impact period of a permanent technology shock. They also show that the problem can be solved within the standard RBC framework if the preferences are such that the degree of intertemporal substitution and the elasticity of labor supply are both substantially reduced. These modi cations however signi cantly worsen the model's ability to explain the moments of the data along other dimensions. Speci cally, the volatility of hours becomes too low and that of consumption becomes too high in comparison with the data. Our model addresses the co-movement problem by mimicking short-run increasing returns to labor, which induce hours and consumption to move together 26 The data are detrended by the HP lter. 27 The AR(1) autocoe±cient is 0:9. 28 EBC stands for our endogenous business cycle model. 19
along the model's transition path towards the steady state. The whole issue can be illustrated using the labor market equilibrium condition given by u n (1 n) =u c w where the left-hand side is the marginal cost of labor supply measured in utility terms and the right-hand side is the marginal bene t of labor supply measured in utility terms. In a standard RBC model capital moves little in the short run. As hours go up, the real wage, w, goes down, so the right-hand side of the equation decreases. On the other hand, since leisure goes down, the marginal utility of leisure goes up. So the left-hand side increases. To balance the equation, consumption has to decrease in order to increase the marginal utility u c : RBC models solve the co-movement problem by introducing technology shocks that increase the real wage. However, as is noted by Rotemberg and Woodford, this solves the problem at the impact period, but at least with permanent shocks, negative comovements are likely during the periods of transition towards the steady state. In the reduced form of our model the real wage is an increasing function of hours due to the combination of variable capacity utilization and external e ects, and therefore the labor market equation above implies a positive co-movement between consumption and hours. 5. Conclusion In this paper we have demonstrated that an equilibrium real business cycle model incorporating variable capacity utilization, persistent demand shocks and indeterminacy is capable of explaining aggregate uctuations. In such a model demand shocks can generate not only positive comovements among consumption, investment, hours and output, but also the trend reverting hump-shaped time series exhibited by these macroeconomic variables. It should be possible to further enrich the model by combining it with equilibrium models that introduce a role for money. 20
References [1] D. Andolfatto, 1996, Business cycles and labor-market search, American Economic Review 86 (March), 112-132. [2] S. Basu and J. Fernald, 1997, Returns to scale in US production: estimates and implications, Journal of Political Economy 105 (April), 249-283. [3] M. Baxter and R. King, 1992, Productive externality and cyclical volatility. Working paper 245, University of Rochester. [4] J. Benhabib and R. Farmer, Indeterminacy and increasing returns, Journal of Economic Theory 63 (1994), 19-41. [5] J. Benhabib and R. Farmer, 1996, Indeterminacy and sector-speci c externalities, Journal of Monetary Economics 37, 421-443. [6] J. Benhabib and R. Farmer, 1999, Indeterminacy and sunspots in macroeconomics, Handbook for Macroeconomics, eds. John Taylor and Michael Woodford, North-Holland, New York, volume 1A, 387-448. [7] B. S. Bernanke, M. Gertler and S. Gilchrist, 1999, The nancial accelerator in a quantitative business cycle framework, in Handbook of Macroeconomics, eds. J. Taylor and M. Woodford, Elsevier, volume 1C, 1341-1393. [8] O. Blanchard, 1993, Consumption and the recession of 1990-1991, American Economic Review 93 (May), 270-273. [9] O. Blanchard and D. Quah, 1989, The dynamic e ects of aggregate demand and supply disturbances, American Economic Review 79, 655-673. [10] C. Burnside and M. Eichenbaum, 1996, Factor hoarding and the propagation of business cycle shocks, American Economic Review 86 (December), 1154-1174. [11] C. Burnside, M. Eichenbaum and S. Rebelo, 1993, Labor hoarding and the business cycle, Journal of Political Economy 101, 245-273. [12] C. Carlstrom and T. Fuerst, 1997, Agency costs, net worth, and business uctuations: A computable general equilibrium analysis, American Economic Review 87 (December), 893-910. 21
[13] D. Cass and K. Shell, 1983, Do sunspots matter? Journal of Political Economy 91, 193-227. [14] L. Christiano and M. Eichenbaum, 1992, Current real-business-cycle theories and aggregate labor-market uctuations, American Economic Review 82, 430-50. [15] J. Cochrane, 1994, Shocks, Carnegie-Rochester Conference Series on Public Policy 41, 295-364. [16] T. Cogley and J. Nason, 1995, Output dynamics in real-business-cycle models, American Economic Review 85, 492-511. [17] C. Evans, 1992, Productivity shocks and real business cycles, Journal of Monetary Economics 29, 191-208. [18] R. Farmer and J. T. Guo, 1994, Real business Cycles and the animal spirits hypothesis, Journal of Economic Theory 63, 42-72. [19] R. Gordon, 1993, Are procyclical productivity uctuations a gment of measurement error? Manuscript, Northwestern University. [20] J. Greenwood, Z. Hercowitz and G. Hu man, 1988, Investment, capacity utilization, and the real business cycle, American Economic Review 78, 402-417. [21] G. Hansen, 1985, Indivisible labor and the business cycle, Journal of Monetary Economics 16, 309-325. [22] King, R., C. Plosser and S. Rebelo, 1988, Production, growth and business cycles: I. The basic neoclassical model, Journal of Monetary Economics 21, 195-232. [23] G. Mankiw, 1989, Real business cycles: A new Keynesian perspective, Journal of Economic Perspectives 3, 79-90. [24] R. Perli, 1998, Indeterminacy, home production, and the business cycle: A calibrated analysis, Journal of Monetary Economics 41. 22
[25] J. Rotemberg and M. Woodford (1996), Real-business-cycle models and the forecastable movements in output, hours, and consumption, American Economic Review 86 (March), 71-89. [26] S. Schmitt-Grohe, 1997, Comparing four models of aggregate uctuations due to self-ful lling expectations, Journal of Economic Theory 72, 96-147. [27] S. Schmitt-Grohe, 2000, Endogenous business cycles and the dynamics of output, hours, and consumption, American Economic Review, forthcoming. [28] K. Shell, 1977, Monnaie et allocation intertemporelle, Mimeo, Seminaire d'econometrie Roy-Malinvaud, Centre National de la Recherche Scienti que, Paris. [29] L. Summers, 1986, Some skeptical observations on real business cycle theory, Federal Reserve Bank of Minneapolis Quarterly Review, Fall, 23-27. [30] M. Watson, 1993, Measures of t for calibrated models, Journal of Political Economy 101, no.6, 1011-1041. [31] Y. Wen, 1995, Where's the beef? The trivial dynamics of real business cycle models, Manuscript, The University of Iowa. [32] Y. Wen, 1998, Capacity utilization under increasing returns to scale, Journal of Economic Theory 81, 7-36. 23
Fig. 1. Responses to Demand Shocks (US 1960:1-1994:4) 24
Fig. 2. Autocorrelation Function of Growth Rates (US 1960:1-1994:4) 25
Fig. 3. Predictions of a Standard RBC model (Solic Symbols). First row: Impulse responses to demand shocks. Second row: ACF of growth rates. 26
Fig. 4. Impulse Responses ( = 0:1). Solid lines: responses to a consumption shock. Dashed lines: responses to a government shock. 27
Fig. 5. Impulse Responses ( = 0:108). Solid lines: responses to consumption shock. Long dashed lines: responses to government shock. Short dashed lines: responses to sunspots. 28
Fig. 6. Impulse Responses of Output when the covariances between sunspots and fundamental shocks take di erent values. 29
Fig. 7. Predictions of Model (solid symbols). Upper window: Impulse Response Function. Lower window: Autocorrelation Function. 30
Fig. 8. Predictions of Model A (solid symbols). Upper window: Impulse Response Function. Lower window: Autocorrelation Function. 31
Fig. 9. Predictions of Model B (solid symbols). Upper window: Impulse Response Function. Lower window: Autocorrelation Function. 32
Fig. 10. Impulse responses of model (solid lines) and data (dashed lines). 33