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Durable goods, Investment Shocks and the Comovement Problem Been-Lon Chen† Academia Sinica Shian-Yu Liao‡ National Taiwan University January 2014 Abstract Recent research based on sticky price models suggests that investment shocks are the most important drivers of business cycle fluctuations. Yet, an investment shock generates the comovement problem because there is an intertemporal substitution effect away from consumption and toward investment. This paper resolves the comovement problem by extending the standard neoclassical sticky-price model to a two-sector model with durable consumer services. When investment goods are used as capital investment and durable consumer services, a positive investment shock also generates an intratemporal substitution effect away from durable consumer services toward consumption. As the intratemporal effect dominates the intertemporal effect, consumption increases and thus the comovement problem is resolved. Keywords: Investment shocks; Durables; Sticky prices; Comovement JEL classification: E21; E22; E32 _____________________ † Correspondence author and address: Institute of Economics, Academia Sinica, 128 Academia Road Section 2, Taipei 11529, Taiwan. Phone (886-2)27822791 ext. 309; fax: (886-2)27853946; e-mail: [email protected] ‡ Department of Economics, National Taiwan University, 21 Hsu-Chow Road, Taipei 10055, Taiwan. e-mail: [email protected] 1. Introduction Recent research based on dynamic stochastic general equilibrium models suggests that investment shocks are the most important drivers of business cycle fluctuations in the post-World War II US economy. Justiniano et al. (2010) studied a sticky price-wage, one-sector model with a variety of real and nominal frictions, along the line of Christiano et al. (2005), and several shocks, as in Smets and Wouters (2007), including a shock to total factor productivity (or neutral technology shock), as in the RBC literature; a shock to the marginal productivity of investment (or, for simplicity, investment shock), as in Greenwood et al. (1988); and a shock to desired wage markups (or, equivalently, to labor supply), as in Hall (1997). It found that over 50% of the fluctuations in output and hours, and over 80% of the fluctuations in investment are driven by investment shocks. These shocks may manifest as shocks either to the marginal efficiency of investment, as in Greenwood et al. (1988), or to the investment-specific technology as in Greenwood et al. (1997). Previously, using a structural vector auto regression methodology in a one-sector model, Fisher (2006) also found that investment-specific shocks are the dominant source of business cycles in the US. Despite their quantitative importance in explaining business cycle fluctuations, one difficulty remains in these models: consumption typically falls (or does not rise immediately) after a positive investment shock in the model. Such a comovement problem remains in two-sector models.1 The comovement problem emerges because a positive investment shock generates a substitution effect away from current consumption and toward current investment and thus future consumption which dominates the income effect. As a result, the model does not produce comovement among macroeconomic aggregates in response to an investment shock, unlike observed business cycles in which output, consumption, investment, hours worked, and the real wage all rise and fall together. This lack of comovement is clearly problematic in viewing investment shocks as an important source of business cycles. In this paper, we resolve the comovement problem by extending the standard neoclassical sticky-price model to a model with durable consumer services. In a paper that empirically estimated the link between interest rates and durable consumer services, Mankiw (1985) has stated the importance of knowing fluctuations in consumer durables for understanding economic fluctuations. Baxter (1996) set up a two-sector model with durable consumer services and two types of capital and investigated the effects upon business cycles of a single shock to a Solow residual that is shared by both sectors and two less-than-perfectly-correlated sector shocks to Solow residuals. More recently, Barsky et al. (2003, 2007) studied a two-sector model with durable consumer services and with sticky nondurable prices and envisaged fluctuations in 1 Guerrieri et al. (2010) studied a model with the machinery and non-machinery sectors and found that the multi-factor productivity shock generated a positive comovement between consumption and investment, but the investment-specific shock did not. 1 nondurables and durable consumer services in response to monetary policy shocks. This brings about a surge of research interests in studying the comovement between nondurables and durable consumer services in response to monetary policy shocks; see Erceg and Levin (2006), Monacelli (2009), Carlstrom and Fuerst (2010), Huang et al. (2013) and Chen and Liao (2014). Except for Mankiw (1985) and Baxter (1996), all these papers of consumer durables envisage sticky-price models with more sticky nondurable prices.2 Our model may be thought of as an extension of the line of research led by Barsky et al. (2003, 2007) and studies the effects of a positive investment shock on the comovement between consumption and investment. Specifically, our model includes one sector that produces consumption goods and the other sector that produces investment goods. Consumption goods are nondurable and are used only for consumption. Investment goods are durable and used as investment that accumulates the stock of capital and investment that accumulates the stock of durable consumer services. Durable prices are more flexible than nondurable prices. Capital investment is subject to a positive shock. The household obtains utilities from an index of consumption which is a composite of nondurable consumption and durable consumer services. Without durable consumer services, our model is a standard two-sector neoclassical sticky-price model. Like existing papers, in such a model, a positive investment shock results in the comovement problem as it gives a higher relative price of investment goods and generates an intertemporal substitution effect away from consumption and toward investment and thus future consumption. In contrast, with durable consumer services, a higher relative price of investment goods also generates an intratemporal substitution effect away from durable consumer services and toward nondurable consumption. Since the intratemporal substitution effect dominates the intertemporal substitution effect, nondurable consumption rises. As a result, the comovement problem is resolved. Our study complements three recent studies which resolved the comovement problem using alternative mechanisms. First, Kahn and Tsoukalas (2011) solved the comovement problem by studying a one-sector sticky price-wage model of variable capacity utilization with the cost of higher capital utilizations in terms of a higher depreciation rate of capital, as in Greenwood et al. (1988). Since a higher utilization rate increases the marginal productivity of labor, this generates a substitution effect, away from leisure and toward consumption. Next, Eusepi and Preston (2009) also studied a model of variable capacity utilizations in terms of a higher depreciation rate of capital. They found that the heterogeneity in labor supply and consumption of employed and unemployed workers can generate comovement in response to investment shocks since individual consumption is affected by the number of hours worked with employed consuming more on average than unemployed and 2 Bils and Klenow (2004) have documented that the price of durable goods changes more frequently than that of the overall consumption goods. 2 changes in the employment rate then affects aggregate consumption. Finally, Furlanetto and Seneca (2010) envisaged a model of variable capacity utilizations in terms of the maintenance cost of capital, as in Christiano et al. (2005), and they reconciled the comovement problem when variable capacity utilizations are combined with nominal rigidities (in prices and wages) and nonseparable preferences with a zero wealth effect in labor supply, as in Greenwood et al. (1988). Unlike these papers, the mechanism to resolve the comovement problem in our paper does not rely on variable capacity utilizations. In particular, the innovation in our paper is to consider durable consumer services so that a positive investment shock generates an intratemporal substitution effect away from durable consumer services toward consumption. This paper is organized thus. In Section 2, we set up basic two-sector sticky-price models. In Section 3, we calibrate the model and envisage the impulse responses of a positive investment shock. Finally, concluding remarks are offered in Section 4. 2. Two-sector sticky-price models with and without consumer durable goods Two two-sector neoclassical sticky-price models with and without consumer durable goods are analyzed. We start with the benchmark model with durable consumer goods. The economy includes two sectors, the consumption and investment sectors. The consumption sector produces nondurable goods only for consumption, while the investment sector produces durable goods for capital investment which accumulates the stock of capital and consumer investment which accumulates the stock of durable consumer services. 3 Each sector has a continuum of firms which produce and sell final goods at competitive prices and has a continuum of producers which produce and sell intermediates at monopolistic prices. The economy also consists of a continuum of households with a unit mass. Households supply labor elastically, consume nondurable goods and durable consumer services and offer capital. 2.1 Final Goods Producers In each sector, there is a perfectively competitive final goods producer which produces Yj, j=C, I. The final goods producer in sector j produces Yj by combining a continuum of intermediate goods Yjt(z), z[0, 1], according to the following technology. 1 Y jt [ (Y jt ( z )) 0 ( j 1)/ j j /( j 1) dz ] , j=C, I, (1a) where εj>1 is the elasticity of substitution between intermediates. Consumption goods YC are only used for nondurable consumption C. Investment goods YI are used either for capital investment IK 3 As in Baxter (1996), durable consumer goods are regarded as a type of investment goods in the business cycle literature. See also Cooley and Prescott (1995), Christiano et al. (2005) and Del Negro et al. (2007). 3 which forms capital K or for consumer investment ID which forms durable consumption services D. The laws of motions for capital and durable consumption services will be specified in the household’s problem below. Maximization of profits in sector j gives the demand for the intermediate z in sector j. Y jt ( z ) ( Pjt ( z ) j Pjt ) Y jt , z[0, 1], j=C, I, (1b) where Pjt(z) is the price of the intermediate z in sector j. Denote PCt as the price of consumption goods and PIt the price of consumption goods. A zero profit of final goods in sector j gives the following price of final goods j 1 1 j Pjt [ ( Pjt ( z )) 0 2.2 1/( 1 j ) dz ] . Intermediate Goods Producers In each sector, there is a continuum of intermediate producers of a unit mass indexed by z[0, 1]. A producer rents capital and hires labor to produce intermediate goods according to the following technology. 1 j Y jt ( z ) A j K jt ( z ) j L jt ( z ) , j=C, I, (2a) where Kjt(z) is capital and Ljt(z) is labor used by an intermediate producer z in sector j. Parameter αj(0, 1) is the capital share and Aj>0 is a productivity coefficient in sector j. An intermediate firm z in sector j sells intermediates to final goods producers in sector j at a monopolistic price. Following Rotemberg (1982), in setting its price Pjt(z), an intermediate producer z faces the following adjustment cost. ( Pjt ( z )) j 2 P (z) ( Pjtjt1 ( z ) 1) 2 PjtY jt , (2b) where ϑj measures the degree of nominal rigidities in sector j.4 In each period, the representative intermediate firm z has to decide how much labor to hire, how much capital to rent and what prices to set. Thus, managers of the firm maximize the value to its owners which is the present discounted value of all current and future expected cash flows. E0 0 t [ Pjt ( z )Y jt ( z ) Wt L jt ( z ) Qt K jt ( z ) ( Pjt ( z ))] , j=C, I, t (3) t 0 where Wt is the nominal wage and Qt is the nominal capital rentals. In the objective, Et is conditional expectations at any given period t, and β(0, 1) and Λt are, respectively, the discount factor and the period-t marginal utility of nominal income of the representative household who owns the firm. 4 An alternative method of price adjustments is random price durations based on Calvo (1983). According to Galí (2008, p.41), these two methods generate the same inflation dynamics. 4 Given the technology (2a), managers choose {Ljt(z), Kjt(z), Pjt(z)} t0 to maximize the cash flows in (3) subject to the demand for intermediates in (1b). Let λjt(z) denote the current-valued Lagrange multiplier of the demand for intermediate z in (1b), j=C, I. Moreover, denote MPjtL(z) and MPjtK(z) as the period-t marginal product of labor and capital for z in sector j, respectively. The first-order conditions for Ljt(z), Kjt(z) and Pjt(z) are wt jt ( z ) qt jt ( z ) [1(1 (z)) ]( jt j Pjt (z) j Pjt Pjt ( z ) PC t Pjt ( z ) PCt MPjtL ( z ), (4a) MPjtK ( z ), (4b) P (z) ) j ( jt (z) 1) Pjt1jt(z) jt (z)[ jtYjt ( jtPjt ) j ] Yjt Et [tt1 j ( jt1(z) 1) P Y (z) jt1(z) Pj,t1 Ct1 Pjt (z) j,t1 Y ] 0, (4c) where πjt(z)≡Pjt(z)/Pjt-1(z) is the gross inflation of intermediate z in sector j. In (4a) and (4b), wt≡Wt/PCt and qt≡Qt/PCt are, respectively, the real wage and the real capital rental. By imposing the symmetry conditions of Kjt(z)=Kjt, Ljt(z)=Ljt, Pjt(z)=Pjt, Yjt(z)=Yjt and λjt(z)=λjt for all z, (4a)-(4c) give, respectively, wt jt Pjt PCt MPjtL , j=C, I, (5a) qt jt Pjt PCt MPjtK , j=C, I, (5b) 1 jt j j j , t , j=C, I, where j ,t 1 j ( j ,t 1) j ,t Et [ tt1 j ( j ,t 1 1) ( j ,t 1 ) 2 Y j ,t 1 C ,t 1 Y j ,t (5c) ], j=C, I. The multiplier λjt stands for the real marginal cost of intermediates in sector j in period t, and its inverse is the markup over the marginal cost. In determining the demand for labor and capital, the real marginal cost of intermediates is needed in order to fill the wedge between the real wage and the real marginal product of labor in (5a) and the wedge between the real rental and the real marginal product of capital in (5b). Data indicates that durable prices are more flexible than nondurable prices (cf. Bils and Klenow, 2004). To simplify the analysis, we focus on the case wherein durable prices are flexible and nondurable prices are sticky. A flexible durable price gives ϑI=0, and thus ΩIt=1 for all t. In this case, the markup 1/λIt=εI/(εI-1) is constant for all t. Moreover, in a steady state, πjt=πjt+1=1 for both j=C, I, so Ωjt=Ωjt+1=1. Then, the markup 1/λj=εj/(εj-1) is constant in a steady state for both j=C, I. 2.3 Households In each period t, the representative household consumes an index of consumption Xt. Following Baxter (1996), Barsky et al. (2007) and Monacelli (2009), the index of consumption is defined as 5 1 1 1 1 X t [(1 ) (Ct ) ( Dt ) ] 1 , (6a) where Ct is nondurable consumption and Dt is the stock of consumer durables which offers services to the household. The parameter μ>0 is the share of durable consumer services, and η≥0 is the elasticity of substitution between nondurable consumption and durable consumer services. The household maximizes the following expected lifetime utility E0 ( )t log X t t 0 ( Lt )1 1 , (6b) where Lt is the hours of work. Parameter ν>0 is the coefficient associated with the disutility of work, and ϕ>0 is the inverse of the Frisch labor supply elasticity. The instantaneous utility in (6b) is separable in consumption and leisure with the logarithmic form of consumption. The utility function is consistent with the balanced growth path (e.g., King and Rebelo, 1999). In each period t, the representative household faces the following budget constraint. Ct pt I Dt pt I Kt bt Rt 1 btCt1 wt Lt qt K t Tt PC t Ft PC t , (7) where Kt is the household’s capital holding in the beginning of period t and pt≡PIt/PCt is the relative price of investment goods. πCt≡PCt/PCt-1 and πIt≡PIt/PIt-1 are the gross inflation of nondurable goods and the gross inflation of investment goods, respectively. Bt-1 is the household’s nominal bond holdings in the end of period t-1, with bt-1≡Bt-1/PCt-1 being its real value. Rt-1 is the gross nominal interest rate on a bond holding between t-1 and t. Tt is nominal lump-sum transfers in t, and Ft is nominal profits/dividends remitted from firms. Thus, in each period t, with real income flows from returns to bond, wage, returns to capital, lump-sum transfers and dividends, the household chooses consumption, consumer investment, capital investment and bond holdings. While capital investment accumulates the stock of capital, consumer investment forms the stock of durable consumer services. The stock of capital and the stock of durable consumer services evolve according to K t 1 (1 ) K t t I Kt , (8a) Dt (1 ) Dt 1 I Dt , (8b) where δ is the depreciation rate. In (8a), ξt is a source of exogenous variations in the efficiency with which capital investment in period t can be transformed into the stock of capital in period t+1, and thus into next period’s capital input.5 Following Greenwood et al. (1988), we assume that the investment-specific technology 5 According to Justiniano et al. (2011), this variation might stem from technological factors specific to the production of investment goods, as in Greenwood et al. (1997), and from disturbances to the process by which these investment goods are turned into productive capital. Like these authors, we ignore that distinction and 6 change ξt follows the first-order stochastic process as follows. log t log t 1 et , (9) where the innovation et is assumed to be independent and identically distributed normally with mean 0 and variance σ2. With the laws of motion in (8a) and (8b), the representative household’s problem is to maximize the lifetime utility (6b) subject to the budget constraint (7). We denote UCt, UDt and ULt, respectively, as the marginal utility of nondurable consumption, durable consumer services and labor in t. The first-order conditions for Ct, Lt, bt, Kt+1 and Dt are U Ct t , (10a) U Lt U Ct (10b) wt , U Ct Et U Ct 1 CtRt1 , (10d) U Ct pt U Dt Et [U Ct 1 pt 1 (1 )], (10e) U Ct pt t (10c) Et U Ct 1 qt 1 ptt11 (1 ) , along with limt→∞(β)tΛtbt=0, limt→∞(β)tΛtKt+1/ξt=0 and limt→∞(β)tΛtDt=0 which ensure that there is no “Ponzi scheme.” Note that Λt is the current-valued Lagrange multipliers of the constraint (7) and is thus the marginal utility of nominal income of the representative household. In these conditions, (10a) and (10b) are standard, and (10c) is the consumption-Euler equation which equates the marginal utility of nondurable consumption in this period to the discounted expected marginal utility of shifting one unit of nondurable consumption to the next period. Condition (10d) determines the demand for capital in the next period. The marginal cost of increasing one unit of capital investment to the next period is the investment-shock-adjusted relative price of investment goods in terms of the marginal utility of consumption in this period. The marginal benefit is the discounted expected sum of the capital rental and the investment-shock-adjusted relative price of investment goods of the undepreciated part of capital in terms of the marginal utility of consumption in the next period. Finally, (10e) determines the demand for durable consumer services in this period. The condition equalizes the real marginal cost of investment in durable consumer services in this period and the real marginal benefit from a marginal unit of durable consumer investment. The real marginal cost in this period is the relative price of investment goods in terms of the marginal utility of nondurable consumption in this period. The real marginal benefit is the sum of the immediate marginal utility of durable consumer services in this period and the discounted expected relative maintain an agnostic stance on the ultimate source of these disturbances. 7 price of investment goods of the undepreciated part of durable consumer services in terms of the marginal utility of nondurable consumption in the next period. Through (10e), variations in the relative price of investment goods are expected to affect the demand for nondurable consumption as analyzed in the next section. 2.4 Equilibrium In equilibrium, the consumption goods and the investment goods markets are clear. YCt 2C ( Ct 1) 2 YCt Ct , (11a) YIt 2I ( It 1) 2 YIt I Kt I Dt . (11b) We abstract from redistribution via the fiscal policy, and hence Tt=0. Moreover, the capital, the labor and the bond markets are all clear. K t KCt K It , (12a) LCt LIt Lt , (12b) bt 0. (12c) The model is closed by the following monetary policy rule. Rt R ( t ) , χ>1, (13) where πt≡(πCt)1-μ(πIt)μ is a composite inflation index with the weight for investment goods inflation being the share of durable consumer services in consumption μ, and both R and π are steady-state values. We assume that χ is sufficiently large in order to ensure equilibrium determinacy. 2.4 The model without durable consumer goods A special case is a two-sector model without durable consumer services. Even with two sectors, because of no durable consumer services, as will be seen below, a positive investment-specific technology shock leads to an increase in investment goods and a decrease in consumption goods, thus the comovement problem, as in Fisher (2006) and Justiniano et al. (2010). In the model without durable consumer goods, the utility function in (6b) is reduced to logCt-ν(Lt)1+ϕ/(1+ϕ). Moreover, (10e) is not an equilibrium condition. As equilibrium conditions (7) and (11a) involve durable consumer goods, they are also modified. 3. Comparisons of the both models In this section, we study the effects of investment-specific technology shocks on the impulse responses of aggregate macro and other relevant variables. 8 3.1 Calibration The time frequency is quarters. The steady-state real rate of return R is pinned down by households’ discount factor β. We choose the real rate of return per annum of 4%. This implies a quarterly discount factor of β=0.99. For production, the total factor productivity in the consumption- and investment-goods sectors are normalized to unity, so AC=AI=1. We follow Acemoglu and Guerrieri (2008) and set the capital shares of the investment-goods and consumption-goods sectors equal to αI=0.27 and αC=0.47, respectively, to match with the average capital share in the respective sector in 1987-2005.6 Following Monacelli (2009), the elasticity of substitution between intermediate varieties in the final goods production is εC=εI=6, which implies a steady-state markup rate of 20%. We follow Hansen (1985) and set the depreciation rate of capital equal to δ=0.025 which implies a 10% annual depreciation rate.7 For the preference, with hours of work L=1/3, we use the consumption-leisure tradeoff condition in (10b) to calibrate the parameter of leisure in preference and obtain ν=5.573. The elasticity of substitution between nondurable consumption and durable consumer services is set to be η=1, implying a Cobb-Douglas form for the consumption index. We choose the share of durable consumer services in the consumption index at μ=0.2 in order to match with 20% share of the spending of consumer durables in total private spending in the US. We set the inverse of the Frisch labor supply elasticity at ϕ=1, which is within the range of values used in existing literature.8 For the degree of price-stickiness, we set ϑI=0 so durable prices are flexible, as shown in Bils and Klenow (2004), among others. We target the stickiness of nondurable prices at four quarters and pin down ϑC=58.25.9 As for the monetary policy rule, we set the coefficient of the inflation in the policy rule at χ=1.5, which is a standard value in the literature regarding Taylor rules. Finally, the autocorrelation of the investment shock and the standard deviation of error to the investment shock is set to be ρ=0.72 and σ=0.0603, respectively, which are within the range of the 6 Baxter (1996) estimated αI=0.32 and αC=0.52 over the period 1948-1985. Our results are the same if we set αI=0.32 and αC=0.52. 7 This value of the depreciation rate gives a quarterly capital-to-output ratio of 9.5 which is in the range of the data. Alternatively, if we use a lower value of δ=0.02 or a higher value of δ=0.03, the quarterly capital-to-output ratio would be 11.2 and 8.2, respectively. Our results are robust under these different values. 8 The results are robust under ϕ[0.01, 10]. 9 To obtain the value of ϑ , we use the log-linearization of the optimal pricing condition in (5c) to yield the C slope of the Phillips curve equal to (εC -1)/ϑC. Then, we equate the slope to the slope of the New Keynesian Phillips curve in the standard Calvo-Yun model, which is (1-θC)(1-βθC)/θC, where θC is the probability of not resetting prices for nondurables. Thus, we obtain ϑC=(εC -1)θC/[(1-θC)(1-βθC)]. See Appendix A. By setting the stickiness of nondurable prices at four quarters so 1-θC=0.25, with εC=6 and β=0.99, we obtain ϑC=58.25. 9 estimated values in literature, such as Smets and Wouters (2007), Justiniano et al. (2010) and Khan and Tsoukalas (2011). Parameter values in the benchmark model are summarized in Table 1. In the Table, we also list parameter values used in the model without consumer durable services wherein μ=0. [Insert Table 1 here] 3.2 Effects of a positive investment shock We carry out a positive investment shock in the same way as Justiniano et al. (2010). Specifically, the innovation of the investment shock et is increased by one standard deviation in the stochastic process with the initial value of the investment shock ξt normalized at unity. The results of impulse responses are illustrated in Figure 1. An increase in et raises the marginal efficiency of investment. As a result of a positive shock to the marginal efficiency of investment, there is an increase in the demand for investment goods on impact (cf. Panel C). Since durable prices are more flexible than nondurable prices, the relative price of investment goods increases (cf. in Panel G). The nominal interest rate and inflation also rise (cf. Panels H and I). [Insert Figure 1 here] 3.2.1 The Model without Consumer Durables To see the impulse responses of other variable, we begin by envisaging the results in a neoclassical sticky-price model without durable consumer services. The results are delineated by dashed red lines in Figure 1. A higher relative price of investment goods generates an intertemporal substitution effect away from consumption and toward investment and thus future consumption. The higher return on currently available resources would, at the same time, operate to persuade individuals to postpone leisure. Consequently, there is an expansion in hours worked and output. Moreover, given the fixed stock of capital, an increase in working hours lowers the marginal product of labor on impact. Therefore, all real variables increase except for consumption and the labor productivity, measured as average output per hour worked (cf. dashed red lines in Panels B and F). As a result, the model fails to generate the comovement between investment and consumption in response to an investment shock. 3.2.2 The Model with Consumer Durables Next, we come to an otherwise the same model except for durable consumer services. Durable consumer goods are a substitute for nondurable consumption. Now, there is another substitution effect. A higher relative price of investment goods also generates an intratemporal substitution effect away from durable consumer services and toward nondurable consumption. As the intratemporal 10 substitution effect dominates the intertemporal substitution effect, nondurable consumption goes up (cf. solid blue lines in Panel B in Figure 1). As shown in Panels A–E, nondurable consumption co-moves with output, investment, hours and real wage. Moreover, the labor productivity also increases because output rises more than hours (cf. Panel F). In particular, it should be noted that with durable consumer services, our model amplifies the impulse responses of real variables to a positive investment shock. To evaluate the extent to which our model reproduces the key quantitative features of business cycles, Table 2 provides statistics summarizing contemporaneous correlations of key macro variables with output. Column 1 in the table reports the US quarterly data over the period 1947:Q1–2013:Q3.10 It is clear to see that all key real variables in the US data are procyclical in the postwar era. Columns 2 and 3 of this table compare simulated correlations of key macro variables with output in the models with and without durable consumer services to those in the data. Column 3 is the simulated correlations for the model without durable consumer services. The results in column 3 indicate a weak correlation of the labor productivity with output and, in particular, the correlation between consumption and output is negative rather than positive. In contrast, as seen in column 2, by adding durable consumer services into an otherwise standard neoclassical sticky-price model, the simulated correlation between consumption and output can match well with the data and all other variables move with output procyclically. [Insert Table 2 here] 3.4 Why does the model with durable goods resolve the comovement problem? This subsection explains the underlying reasons why, in an otherwise standard two-sector neoclassical sticky-price model with more flexible durable prices, adding durable consumer services can resolve the comovement problem. At the center of the analysis is the fact that in a two-sector model with durable consumer services, the demand for durable consumer services changes the household’s expenditure behavior. First, we analyze the model without durable consumer services. By (10b) and (5a), we can obtain U Lt U Ct Lt Ct jt Pjt PCt MPjtL , j=C, I. (14) With standard preferences and technology, the marginal rate of substitution (-ULt/UCt) depends positively on consumption and working hours, while the marginal product of labor MPjtL is 10 The source of data is the Federal Reserve Economic Data, Federal Reserve Bank of St. Louis. We use the seasonally adjusted data, deflated by the GDP deflator, taken logarithms and de-trended with the Hodrick-Prescott filter with a smoothing parameter of 1,600. Following Del Negro et al. (2007), consumption corresponds to personal consumption expenditures on nondurables and services, while investment is the sum of gross private domestic investment and personal consumption expenditures on durables. Following Hansen (1985), the labor productivity is measured by real output per hour. 11 decreasing in working hours. For ease of exposition, we focus on a one-sector model with perfectly competitive markets, so λjt=1 and Pjt/PCt=1 for j=C, I. As a result, any shock that increases working hours and thus, decreases the marginal product of labor, on impact must also generate a fall in consumption in order for (14) to hold at the new equilibrium. This is exactly what happens in response to an investment shock in the model without durable consumer services, as was described above. Even though we have extended a one-sector model with nondurable consumption and durable investment to a two-sector model with nondurable consumption goods and durable investment goods, the mechanisms are still the same and thus consumption falls in response to a positive investment-specific technology shock. By contrast, in our model with consumer durables, there is an additional optimization condition which is a household’s shadow value of durable consumer services in period t given by (10e). In optimum, this shadow value is equal to the real marginal cost of investment in durable consumer services. If we denote Vt≡UCtpt, the household’s shadow value of durable consumer services in (10e) can be rewritten as Vt U Dt (1 ) Et (Vt 1 ) Et (1 ) U Dt , (15) 0 in which the second equality follows from the law of iterated expectations. The condition above indicates that a household’s shadow value of durable consumer services in this period is equal to the expected discounted sum of the marginal utility of undepreciated durable consumer services from period t onward. The household’s shadow value of durable consumer services is quasi-constant, since variations in consumer durable flows in this period have little effects on the stock of durable consumer services upon which the marginal utility of durable consumer services depends. Further, the shadow value of durable consumer services depends for an important part on the marginal utility of durable consumer services in the distant future, which is even less sensitive to temporary shocks. With a quasi-constant shadow value of durable consumer services for households on the right-hand side of (15), the relative price of investment goods pt and the marginal utility of nondurable consumption UCt in this period on the left-hand side of (15) move in opposite directions. As a positive investment shock increases the relative price of investment goods, a rise in the relative price of investment goods is accompanied by a decrease in the marginal utility of nondurable consumption which is associated with an increase in expenditure on nondurable consumption. As a result, nondurable consumption rises. 4. Concluding remarks Recent research based on sticky price models suggests that investment shocks are the most 12 important drivers of business cycle fluctuations in the post-World War II US economy. Despite their quantitative importance in explaining business cycle fluctuations, the comovement problem emerges because a positive investment shock generates an intertemporal substitution effect away from consumption and toward investment. Thus, investment increases but consumption decreases. This lack of comovement is problematic in viewing investment shocks as an important source of business cycles. In this paper, we resolve the comovement problem by extending the standard neoclassical sticky-price model to a two-sector model with durable consumer services. With durable consumer services, a positive investment shock also generates an intratemporal substitution effect away from durable consumer services toward consumption. 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By equating these two slopes, we can obtain the coefficient of the cost of price adjustment ϑj. 15 Table 1: Parameter Setting Description Parameter TFP in consumption and investment sector Elasticity of sub. between nondurables and durables Elasticity of sub. between intermediates Share of durable services Inverse elasticity of labor supply Discount factor Hours of work Autocorrelation of the investment shock Standard deviation of error to the investment shock Coefficient of inflation rates in Taylor rule Capital share in investment sector Capital share in consumption sector Depreciation rate Parameter of labor in utility Coefficient of price adjustment in investment sector Coefficient of price adjustment in consumption sector A C, A I η εC, εI μ ϕ β L ρ σ χ αI αC δ ν ϑI ϑC (frequency: quarterly) Model with Model without consumer consumer durables durables 1 1 1 1 6 6 0.2 0 1 1 0.99 0.99 1/3 1/3 0.72 0.72 0.0603 0.0603 1.5 1.5 0.27 0.27 0.47 0.47 0.025 0.025 5.573 5.794 0 0 58.25 58.25 Table 2: Contemporaneous Correlations with Output Model Data with consumer without consumer 1947-2013 durables durables Output 1.00 1.00 1.00 Consumption 0.76 0.75 -0.01 Investment 0.84 0.95 0.85 Hours 0.85 0.92 0.86 Wages 0.89 0.98 0.98 Labor productivity 0.41 0.95 0.12 16 A. Output B. Consumption .14 C. Investment .03 .35 .30 .12 .02 .25 .10 .20 .01 .08 .15 .06 .00 .10 .04 .05 -.01 .02 .00 -.02 .00 5 10 15 -.05 5 D. Hours 10 15 5 E. Real wage .08 10 15 F. Labor productivity .10 .07 .07 .06 .08 .06 .05 .05 .04 .06 .04 .03 .03 .02 .04 .02 .01 .01 .00 .02 .00 -.01 .00 -.01 5 10 G. Relative price of investment goods .05 -.02 5 15 10 15 H. Nominal interest rate 10 15 I. Inflation .020 .014 .012 .016 .04 5 .010 .012 .008 .03 .008 .006 .02 .004 .004 .002 .01 .000 .00 .000 -.004 5 10 15 -.002 5 10 15 5 10 15 The model with consumer durables (our model) The model without consumer durables Figure 1. Impulse responses to a positive investment shock Note: The horizontal axis is divided in quarters; the vertical axis represents percentage deviations from a steady state. Investment is the sum of durable consumer investment and capital investment. 17