Nonconvex Factor Adjustments and the Business Cycle: Do Nonlinearities Matter? Aubhik Khan Julia K. Thomas1 Federal Reserve Bank of Philadelphia Carnegie Mellon University July 2000 1 Please direct correspondence to A. Khan, Research Department, Federal Reserve Bank of Philadelphia, Ten Independence Mall, Philadelphia, PA 19106; tel: 215 574 3905; email: aubhik.khan@phil.frb.org. This paper is the result of a conversation with John Leahy; we thank him for suggesting the topic to us. We would also like to thank Tony Smith for a series of helpful discussions, and Larry Christiano and Jonas Fisher for providing us their data on the relative price of investment goods. All remaining errors are of course our own, and the views expressed in this paper do not necessarily re‡ect those of the Federal Reserve Bank of Philadelphia or of the Federal Reserve System. 1 Introduction We evaluate the aggregate implications of discrete and occasional capital adjustment in an equilibrium business cycle model. In our model economy, nonconvex costs of capital adjustment vary across establishments and lead to periods of investment inactivity. Thus, the model generates a distribution of plants over capital. This distribution evolves over the business cycle in response to changes in productivity which a¤ect not only the levels of investment undertaken by active plants but also the number of plants that are actually engaged in actively adjusting their capital stock. Our objective is to evaluate the contribution of such distributional changes to the aggregate business cycle. Recent studies of establishment level investment provide evidence of lumpy capi- tal adjustment. Examining a 17-year sample of large, continuing U.S. manufacturing plants, Doms and Dunne (1999) …nd that typically more than half of a plant’s cumu- lative investment occurs in a single episode. Long periods of relatively small changes are interrupted by investment spikes. This has been widely interpreted as evidence of (S,s) type investment decisions at the establishment level. Perhaps due to non- convexities in the costs of capital adjustment, plants invest only when their actual capital stock deviates su¢ciently far from a target value. Supporting evidence is provided by Cooper, Haltiwanger and Power (1999) who …nd that the probability of an establishment undergoing a large investment episode is rising in the time since its last such episode. Exploring the aggregate implications of establishment level lumpy investment, Caballero, Engel and Haltiwanger (1995) focus on the e¤ect of interaction between the rising adjustment hazard, the probability of capital adjustment as a function of an establishment’s gap between actual and target capital stocks, and the resultant distri- bution of capital.1 They argue that shifts in the hazard, in response to large shocks to demand or productivity, magnify ‡uctuations in aggregate investment demand and cause a time-varying elasticity of aggregate investment with respect to shocks. 1 See Caballero (1999) for a survey. 1 This emphasis on aggregate nonlinearities arising through micro-level nonconvexities is also found in Cooper, Haltiwanger and Power (1999) who stress that movements in the distribution of capital are important in explaining unusually large deviations in total investment. Moreover, Caballero and Engel (1999) note that “the nonlinear model we estimate has the potential to generate brisker expansions than its linear counterparts. It is also this feature that largely explains its enhanced forecasting properties.” (p. 785, paragraph 1) These and related papers, all of which abstract from the e¤ect of equilibrium price changes, suggest a potentially important role for lumpy investment in propagating the business cycle. However, when Thomas (1999) evaluates the e¤ects of noncon- vex capital adjustment costs in an equilibrium business cycle model, she …nds that standard price movements o¤set the tendency for large changes in the distribution of capital. Solving the model using a system of linear di¤erence equations, she …nds that aggregate quantity responses are virtually una¤ected by the presence of lumpy investment patterns.2 Noting the above emphasis on aggregate nonlinearities, we re-evaluate the equi- librium lumpy investment model of Thomas (1999) using a solution method designed to preserve such phenomenon. Our …rst step is to …x prices and con…rm that the introduction of nonconvex capital adjustment costs does indeed imply aggregate non- linearities in the model. Next, we explore whether these nonlinearities in aggregate investment demand survive equilibrium price determination. Finally, we analyze the aggregate implications of lumpy investment in the context of an equilibrium business cycle model containing an additional source of cyclical ‡uctuations. In addition to the conventional exogenous changes in total factor productivity, we allow for movements in the productivity of investment itself. The recent work of Christiano and Fisher (1998) and Greenwood, Hercowitz and Krusell (2000) suggests that such investment- speci…c productivity shocks are an important source of cyclical ‡uctuations. Since, in the context of a model of lumpy investment, transitory movements in the bene…t from investment expenditures are more likely to shift the adjustment hazard than 2 Veracierto (1998), examining investment irreversibilities, …nds similar results. 2 shocks to total factor productivity, we explore their contribution to the generation of aggregate nonlinearities. The economies we study involve state vectors that are su¢ciently large to make unmodi…ed non-linear solution methods impractical. Therefore, we approximate the aggregate state vector, which involves a distribution of plants across capital, with a smaller object and solve the model using a method that is closely related to the approaches of Den Haan (1996,1997) and Krusell and Smith (1997,1998). In our context, this method itself presents information on the importance of changes in the distribution for the overall business cycle. Despite such e¤orts, our results provide little support for the importance of discrete and occasional investment for the business cycle. This …nding holds for both the original model of Thomas (1999) and the model with separate shocks to output and investment goods production. 2 The Model The model, taken from Thomas (1999), is an extension of the basic equilibrium business cycle model which introduces costs associated with undertaking capital ad- justment. In order to match the observed empirical distribution of investment rates across establishments, we assume a large number of production units, each of which faces time-varying costs of undertaking capital adjustment. Within any period, these costs are …xed at the plant-level, being independent of the level of capital adjustment. Given di¤erences in …xed costs across production units, at any point in time, some plants will adjust their capital while others will not. As a result, there is heterogeneity across productions units, and the model is generally characterized by a distribution of plants over capital.3 At any point in time, a production unit is identi…ed by its capital stock, k, and its current …xed cost of capital adjustment, » 2 [0; B]. This …xed cost is denominated in hours of labor and drawn from a time-invariant distribution G (») common across 3 Given that most available data on establishment level capital adjustment focuses on continuing plants, we abstract from entry and exit by assuming a constant unit meaure of production units. 3 plants. Capital and labor, n, are the sole factors of production, and output at the plant is determined by y = zF (k; n) , where z is stochastic total factor productivity. For convenience, we assume that productivity follows a Markov Chain, z 2 fz1 ; : : : ; zJ g, where ¡ ¢ Pr z 0 = zj j z = zi ´ ¼ij ¸ 0, PJ and j=1 ¼ ij = 1 for each i = 1; : : : ; J. Note that both z and F are common across plants; the only source of heterogeneity in production arises from di¤erences in plant level capital. After production, the plant must decide whether to absorb its current cost in which event it is able to adjust capital. However, it may avoid this cost by setting investment to 0 and allowing capital to passively depreciate. We denote investment by i, and, measuring adjustment costs in units of output using the wage rate, !, summarize the salient features of this choice below.4 ;5 i 6= 0, cost = !», °k0 = (1 ¡ ±) k + i i = 0, cost = 0, °k0 = (1 ¡ ±) k Let capital be de…ned upon K µ R+ and let ¹ : K ! [0; 1] be a Borel measure that represents the distribution of plants over capital in the current period. The aggregate state of the economy is described by (z; ¹), and the distribution of plants evolves over time according to a mapping, ¡, which varies with the aggregate state of the economy, ¹0 = ¡ (z; ¹). We will de…ne this mapping below. In addition to the aggregate state, an establishment is a¤ected by its individual level of capital and adjustment cost. Let v1 (k; »; z; ¹) represent the expected dis- 4 Primes indicate one-period-ahead values. 5 All variables measured in units of output are de‡ated by the trend level of technology, which grows exogenously at the rate ° 1¡µ ¡ 1, where µ is capital’s share of output. For details, see King and Rebelo (1999). 4 counted value of a plant having current capital k and …xed adjustment cost » when the aggregate state of the economy is (z; ¹). We state the dynamic optimization problem for the typical plant using a functional equation, which is de…ned by (1) and (2) below. First we de…ne the beginning of period expected value of a plant, prior to the realization of its …xed cost draw, but after the determination of (k; z; ¹). Z B 0 v (k; z; ¹) ´ v1 (k; »; z; ¹) G (d») (1) 0 Assume that dj (zi ; ¹) is the discount factor applied by plants to their next period expected discounted value if productivity at that time is zj and current productivity is zi . (Except where necessary for clarity, we suppress the index for current productivity below.) Their pro…t maximization problem, which takes as given the evolution of the distribution of plants over capital, ¹0 = ¡ (z; ¹), is then described by the following functional equation. µ 1 v (k; »; z; ¹) = max zF (k; n) ¡ ! (z; ¹) n + (1 ¡ ±) k (2) n 8 0 1 < X J ¡ 0 ¢ + max ¡»! (z; ¹) + max @¡°k0 + ¼ij dj (z; ¹) v0 k ; zj ; ¹0 A ; : k 0 j=1 9 XJ µ ¶=¶ (1 ¡ ±) ¡ (1 ¡ ±) k + ¼ij dj (z; ¹) v 0 k; zj ; ¹0 ° ; j=1 Let nf (k; z; ¹) describe the common choice of employment by all type k plants and kf (k; »; z; ¹) the choice of capital next period by plants of type k with adjustment cost ». The economy is populated by a unit measure of identical households. Households’ wealth is held as one-period shares in plants, which we denote using the measure ¸. They must determine their current consumption, C, hours worked, N, as well as what number of new shares, ¸0 (k) to purchase at price ½ (k; z; ¹). Their lifetime expected utility maximization problem is described below. 5 ³ J X ¡ ¢´ W (¸; z; ¹) = max0 U (C; 1 ¡ N) + ¯ ¼ij W ¸0 ; zj ; ¹0 (3) C;N;¸ j=1 subject to Z Z 0 C+ ½ (k; zi ; ¹) ¸ (dk) · ! (z; ¹) N + v0 (k; z; ¹) ¸ (dk) K K Let c (¸; z; ¹) describe their choice of current consumption, nh (¸; z; ¹) their cur- rent allocation of time to working and ¤ (k; ¸; z; ¹) the quantity of shares they pur- chase in plants that end the current period with capital stock k: A Recursive Competitive Equilibrium is a set of functions ³ ´ !; (dj )J ; ½; v1 ; nf ; kf ; W; c; nh ; ¤ j=1 such that: ¡ ¢ 1. v 1 satis…es 1 - 2 and nf ; kf are the associated policy functions for plants. ¡ ¢ 2. W satis…es 3 and c; nh ; ¤ are the associated policy functions for households. R 3. ¤ (k0 ; ¹; z; ¹) = ¹0 (k0 ) = f(k;») j k0 =kf (k;»;z;¹)g G (d») ¹ (dk). µ ³ ´ ¶ h (¹; z; ¹) = R R f (k; z; ¹) + B »J (1¡±) k ¡ k f (k; »; z; ¹) G (d») ¹(dk), 4. n K n 0 ° where J (x) = 0 if x = 0; J (x) = 1 if x 6= 0. R R Bh ¡ ¢ i 5. c (¹; z; ¹) = K 0 zF k; nf (k; z; ¹) + (1 ¡ ±) k ¡ °kf (k; »; z; ¹) G (d») ¹ (dk) Using C and N, as given by 4 and 5, to now describe the market-clearing values of consumption and hours worked by the household, it is straightforward to show D2 U (C;1¡N) ¯D1 U (C 0 ;1¡N 0 ) that equilibrium requires ! (z; ¹) = D1 U (C;1¡N) and that dj (z; ¹) = D1 U (C;1¡N) . It is then possible to compute equilibrium by solving a single Bellman equation that combines plants’ pro…t maximization problem with the equilibrium implications of household utility maximization. Let p denote the price plants use to value current output, where 6 p (z; ¹) = D1 U (C; 1 ¡ N) , (4) D2 U (C; 1 ¡ N) ! (z; ¹) = . (5) p (z; ¹) A reformulation of (2) yields an equivalent description of a plant’s dynamic problem. Suppressing the arguments of the price functions, µ 1 V (k; »; z; ¹) = max [zF (k; n) ¡ !n + (1 ¡ ±) k] p (6) n 8 0 1 < 0 XJ ³ 0 ´ + max ¡»!p + max @¡°k p + ¯ ¼ij V 0 k ; zj ; ¹0 A ; : k0 j=1 9 XJ µ ¶=¶ (1 ¡ ±) ¡ (1 ¡ ±) kp + ¯ ¼ij V 0 k; zj ; ¹0 ° ; j=1 where Z B 0 V (k; z; ¹) ´ V 1 (k; »; z; ¹) G (d») . (7) 0 Equations 6 and 7 will be the basis of our numerical solution of the economy. This solution exploits several results which we now derive. First, note that plants choose labor n = nf (k; z; ¹) to solve zD2 F (k; n) = ! (¹; z) . Next, we examine the capital choice of establishments undertaking active adjustment decisions. De…ne the value of undertaking such capital adjustment, given the second line of (6), as 0 1 0 J X ³ 0 ´ E (z; ¹) = max @¡°k p + ¯ 0 ¼ij V 0 k ; zj ; ¹0 A , (8) k j=1 and note that the target capital stock which solves the maximization problem will be independent of both k and ». Hence all plants that actively adjust their capital stock choose a common level of capital for the next period, k 0 = k ¤ (z; ¹) given by the right hand side of (8). 7 We now examine the determination of plants’ decision to undertake capital ad- justment. A plant of type k will undertake capital adjustment if its …xed adjustment cost, », falls below some threshold value. Let bk = b (k; z; ¹) describe that level of » » », given current k, which leaves a plant indi¤erent between capital adjustment and allowing its capital stock to passively depreciate. ¡p (z; K) bk ! (z; ¹) + E (z; ¹) » (9) XJ µ ¶ 0 (1 ¡ ±) 0 = ¡p (z; ¹) (1 ¡ ±) k + ¯ ¼ij V k; zj ; ¹ ° j=1 Plants with adjustment costs at or below bk will adjust their capital stock. For- » n n oo mally, de…ne » (k; z; ¹) ´ min B; max 0; bk » so that 0 · » (k; z; ¹) · B. Plants described by the plant-level state vector (k; »; z; ¹) will begin the subsequent period with capital stock given by: 8 < k¤ (z; ¹) if » · » (k; z; ¹), k0 = kf (k; »; z; ¹) = (10) : (1¡±)k if » > » (k; z; ¹). ° Given (10), we are now able to more precisely describe the evolution of the dis- tribution of plants over capital, ¹0 = ¡ (z; ¹). For k 2 K such that k 6= k¤ (z; ¹), · µ µ ¶¶¸ µ ¶ 0 ° ° ¹ (k) = 1 ¡ G » k; z; ¹ ¹ k , (11) 1¡± 1¡± while for k 2 K such that k = k¤ (z; ¹), Z ¡ ¢ ¹0 (k) = G » (k; z; ¹) ¹ (dk) (12) K · µ µ ¶¶¸ µ ¶ ° ° + 1¡G » k; z; ¹ ¹ k . 1¡± 1¡± It then follows that the market clearing levels of consumption and hours, required to determine p and ! using (4) and (5) are given by 8 Z ³ ³ ´ ¡ ¢h C = zF k; nf (k; z; ¹) ¡ G » (k; z; ¹) °k¤ (z; ¹) K i´ ¡ (1 ¡ ±) k ¹ (dk) (13) Z " Z »(k;z;¹) # f N = n (k; z; ¹) + »G (d») ¹ (dk) . (14) K 0 3 Model Solution Given our focus on nonlinearities arising due to the presence of nonconvex adjustment costs, we adapt existing nonlinear solution methods to solve the model. The solution algorithm involves solving for V 0 by repeated application of the contraction mapping implied by (6) and (7), given the price functions (4) - (5). Numerical approximation of plants’ value functions is accomplished using tensor product splines. Johnson et al (1993) have found multivariate spline approximation to be relatively e¢cient when compared to multilinear grid approximation. These tensor product splines are multivariate functions generated as the product of univariate functions, there is one such univariate function corresponding to each argument of the value function. Each such univariate function is itself a spline constructed piecewise using a grid of values, or knots, on the space of its argument. Each piece of the spline is a polynomial, and adjacent pieces meet at the interior knot points. We use cubic splines constructed using third-order polynomials. Each univariate spline is determined as follows. (i) the spline is required to exactly equal the approximated function at each knot point, and (ii) it must be twice-continuously di¤erentiable at each interior knot point. Two additional conditions, commonly referred to as endpoint conditions, are required to determine all 4 coe¢cients of each polynomial piece. We use the not-a-knot endpoint conditions that require thrice di¤erentiability at the …rst and last interior knot.6 In using these tensor product splines, we increase 6 Additional details on univariate splines are available in De Boor (1978) and Van Loan (2000). De Boor also provides details on implementing the multivariate splines using the B-form; however we implement these using the pp-form by developing the algorithm outlined in Johnson (1989). 9 the number of knots used for each variable until there is no noticeable change in the approximation. A di¢culty with using non-linear methods is that the curse of dimensionality restricts the number of arguments that are feasible. We adopt the method of Krusell and Smith (1997,1998) to approximate the state vector of the economy (z; ¹), which contains a large object, the distribution of production units over capital, with a smaller object (z; m) where m is a vector of elements derived from ¹. For example, Krusell and Smith use statistical moments derived from the original distribution, in particular the mean and standard deviation. For our problem, we have found that it is more e¢cient to a use a set of conditional means. Speci…cally, when m has I elements, they are derived by partitioning the distribution ¹ into I equal-measure parts and then setting m = (m1 ; : : : ; mI ), where mi is the mean of the i¡th partition. Given the discrete nature of our distribution, arising from the uniformity of target capital across adjusting plants, it follows that mI converges monotonically to ¹. Given mI , we assume functional forms that yield current equilibrium prices, p, and next period’s proxy endogenous state, m0 , as functions of the current state, ¡ ¢ b p = p z; m; Âp and for m0 = ¡ (z; m; Âm ) where Âp and Âm are parameters that b l l l l are determined iteratively using a procedure explained below and l indexes these iterations. For the class of utility functions we use, the wage is immediate once p is speci…ed; hence there is no need to assume a wage function. ¡ ¢ b Given p and ¡, the …rst step of the solution method uses Âp ; Âm , having replaced b l l b ¹ with m in (6) - (7) and ¡ with ¡, to solve for V 0 on each point on a grid of values for (k; z; m). In the second step, we simulate the economy for T periods. At each point in time; t = 1; : : : ; T , we record the actual distribution of plants over capital, ¹t , which is a large but …nite-dimensional object in our economy.7 We determine m directly from b b this actual distribution and then use ¡ to specify expectations of m0 = ¡ (z; m; Âm ). l PJ This determines ¯ j=1 ¼ij V 0 (k0 ; zj ; m0 ), and, given any arbitrary current price of output, p, allows us to solve for k¤ (z; ¹) and » (k; z; ¹), as well as nf (k; z; ¹). Fur- e 7 The method is easily extended to cases where ¹ is countable or larger using a polynomial ap- proximation. 10 thermore, this also generates ¹t+1 through (11) - (12). The equilibrium current price of output, p, is determined through (13) as follows. p is that value which leads to plant decision rules, k¤ , nf and » that in turn imply market-clearing levels of con- sumption and hours worked, for the household, such that p = D1 U (C; 1 ¡ N). After the completion of the simulation, the resulting data, (pt ; mt )T is used to re-estimate t=1 ¡ p m ¢ 8 Âl+1 ; Âl+1 using OLS. ¡ ¢ We repeat this two-step process, …rst determining V 0 given Âp ; Âm ; next using l l our solution to plants’ value functions to determine equilibrium decision rules over a simulation, aggregating these rules to obtain (pt ; mt )T , and updating Âp and Âl ; t=1 until these parameters converge. The simulation step may be used to compute errors implied by the use of the set of b b conditional means, m, instead of ¹, and the functional forms p and ¡. In each period, we compare the equilibrium price to the forecasted price and the actual values of the conditional means to their predicted values. Given any functional form, we increase the number of partitions, (the number of conditional means used to approximate the distribution of plants over capital,) until these di¤erences are small. We also experiment with di¤erent functional forms. Below, we report these expectational errors and use them to determine I. 4 Parameter Choices We evaluate the importance of aggregate nonlinearities arising from nonconvex capi- tal adjustments through a series of comparisons. Speci…cally, we contrast the dynamic behavior of the lumpy investment economy with that present in an otherwise iden- tical economy characterized by frictionless investment, using the nonlinear solution approach outlined above. This use of the frictionless neoclassical model as a refer- ence model is appealing both due to its common usage in business cycle studies, and because it provides a benchmark against which to measure nonlinearities, as it has 8 Note that the second step of our solution method, which involves simulation, does not make use b of p. 11 been shown to respond approximately linearly to reasonable-sized shocks.9 Toward our comparison, we specify identical functional forms in utility and production across models. We follow Hansen (1985) and Rogerson (1988) in assuming indivisible la- bor, so that the representative household’s momentary utility function is linear in leisure and additively separable: u(c; L) = log c + sL L. Establishment-level produc- tion functions take a Cobb-Douglas form, zF (k; N) = zkµ N º , as consistent with the observation that capital and labor shares of output have remained roughly constant in US time series. Our solution of each model economy also requires the speci…cation of several pa- rameters governing preferences and technology. We …x the length of a period to correspond to one year; this allows us to use evidence on establishment-level invest- ment in the parameterization of the adjustment cost function below. The model’s parameters are selected to ensure agreement between the reference model and ob- served long-run values for key postwar US aggregates. In particular, we choose the mean growth rate of technological progress, °, to imply a 1.6 percent average annual growth rate of real per-capita output, the discount factor, ¯, to yield an average annual interest rate of 6.5 percent, (King and Rebelo 1999,) and the rate of capital depreciation to match an average investment-to-capital ratio of 7.6 percent (Coo- ley and Prescott 1995). Given these values, capital’s share of output is determined such that the average capital-to-output ratio is 2.6 (Prescott 1986). Labor’s share of output is consistent with direct estimates from postwar data, while the parameter governing the preference for leisure, sL , is taken to imply an average of 20 percent of available time spent in market work (King, Plosser and Rebelo 1988). To complete our calibration of the reference model, we …rst estimate parameters for a continuous shock and then assume an equivalent discretized shock process. Speci…cally, we assume an exogenous productivity process of the form, 0 z 0 = z ½ e" ; " » n(0; ¾2 ), " 9 This follows from Christiano (1990), who shows that the LQ approximation of Kydland and Prescott (1982) is highly accurate for this class of models. 12 selecting the persistence term ½ and the variability of the log normal innovations, ¾" , to be consistent with measured Solow residuals from the US economy 1953-1997, using the Stock and Watson (1999) data set. Next, we discretize this productivity process, using a grid of 5 possible shock realizations. We select this grid of values, along with the transition matrix ¦ (with typical element ¼ij ´ pr(z 0 = zj j z = zi ),) to match the required shock persistence and variability, following a method developed by Rouwenhorst (1995). Table 1 and equation (15) together summarize the parameter set for the reference model. Table 1 ° ¯ ± µ º sL ½ ¾" 1:016 :954 :06 :325 :58 3:614 :9225 :0134 Z = [:9328 :9658 1:0000 1:0354 1:0720] (15) 2 3 0:8537 0:1377 0:0083 0:0002 0:0000 6 7 6 7 6 0:0344 0:8579 0:1035 0:0042 0:0001 7 6 7 6 7 ¦ = 6 0:0014 0:0690 0:8593 0:0690 0:0014 7 6 7 6 7 6 0:0001 0:0042 0:1035 0:8579 0:0344 7 4 5 0:0000 0:0002 0:0083 0:1377 0:8537 As this set of parameters is also used for the lumpy investment model, only the properties of adjustment costs remain to be determined. We assume that adjust- » ment costs are uniformly distributed, with cumulative distribution G(») = B. The distribution’s upper support, B, is selected to maximize the model’s agreement with three stylized facts noted in Doms and Dunne’s (1998) study of establishment-level investment: (i) In the average year, plants raising their real capital stocks by more than 30 percent, (lumpy investors,) are responsible for 25 percent of aggregate invest- ment, (ii) these lumpy investors constitute 8 percent of plants, while (iii) 80 percent of plants are low-level investors exhibiting annual capital growth below 10 percent. Setting B = :002 roughly matches these observations, with lumpy investments com- 13 prising 27 percent of aggregate investment, and lumpy investors (low-level investors) representing 6 percent (78 percent) of plants. 5 Results In this section, we examine the dynamic implications of establishment-level lumpy investment, with particular emphasis on aggregate nonlinearities. As indicated above, we present companion results for the frictionless investment counterpart throughout as a reference against which to isolate these e¤ects. In some cases, results for an identically parameterized traditional partial adjustment model are also included to aid in our comparisons. This partial adjustment model is distinguished by a convex ¡i ¢2 adjustment cost function, Á k ¡ ¸ k. Here, ¸ represents the economy’s steady state 2 investment-to-capital ratio, and deviations from this average investment rate entail the payment of a quadratic cost of capital adjustment. Following Kiyotaki and West (1996), we set the parameter Á governing the magnitude of this quadratic cost at Á = 2:2, which implies a steady-state elasticity of the investment-to-capital ratio to Tobin’s marginal q of 5.98. In all other respects, this alternative model is identical to our reference model. Before proceeding further, we stress one feature of steady state that will be helpful in understanding the behavior of aggregate investment demand below. An immediate and important implication of the model is the rising adjustment hazard described by Caballero, Engel and Haltiwanger (1995). It is simple to show that V 0 is increasing in k, plant level capital. It follows that J X µ ¶ (1 ¡ ±) E (z; ¹) ¡ (1 ¡ ±) kp + ¯ ¼ij V 0 k; zj ; ¹0 ° j=1 ¯ ¯ ¯ ¯ is increasing in ¯ (1¡±) k ¡ k¤ (z; ¹)¯. In other words, the larger is the di¤erence between ° non-adjusted capital and target capital, the greater is the net value of adjustment. It then follows from (6) and (8) that » (k; z; ¹) is also increasing in the gap between nonadjusted and target capital. Hence the probability that a production unit of type ¡ ¢ k undertakes capital adjustment, G » (k; z; ¹) , is increasing in its capital deviation, 14 as seen in the upper panel of …gure 1. Notice that the hazard is centered at the °k¤ (z;¹) capital level associated with target capital, 1¡± , and as capital deviates to the left or right of this value, probabilities of adjustment monotonically rise. Note further that, in steady state, all plants are positioned along the left ramp of the hazard, given depreciation and trend technological progress, having capital levels at or below that associated with the target. The implication of this is a monotonically rising steady state distribution of plants that peaks at the target, as shown by the solid curve of the …gure’s lower panel. The lower, dashed, curve depicts the measure of plants at each capital level that do not adjust their capital stocks. Thus, the area between represents the steady state measure of adjusting plants, here roughly 30 percent. 5.1 Dynamics under …xed prices We begin with a series of …xed price experiments designed to gauge lumpy invest- ment’s potential for generating aggregate nonlinearities. In these examples, we study aggregate factor demand responses to aggregate productivity ‡uctuations under the assumption that wages and interest rates faced by the economy’s establishments re- main …xed at their steady state values. We view this as a useful way of exploring the ability that our model has for producing the sorts of features uncovered by pre- vious partial equilibrium studies, as discussed in section 1 above. Perhaps more importantly, this series of examples helps us to clarify the mechanism through which heterogenous capital adjustments may act to produce such features. In the following stylized example, we consider the e¤ects of temporary shocks to productivity. The …rst panel of …gure 2 displays an initial adjustment hazard for the lumpy investment model, centered around the capital value associated with steady state target capital. In the face of a one standard deviation rise in productivity that is expected to persist, establishments’ desired capital holdings increase sharply; this re-centers the adjustment hazard, shifting it rightward. Recall that, on average, most plants are positioned along the left ramp of the hazard, due to capital depreciation and trend productivity growth. When those plants associated with initial capital holdings below 1.18 experience a rise in desired capital, they …nd that their current 15 capital lies su¢ciently far below their (raised) target that they are willing to su¤er large adjustment costs to correct this shortfall. In this particular example, as the economy begins at its deterministic steady state, all plants lie along the left ramp on the initial hazard. Here, the rise in productivity generates such a large rise in desired capital that even the highest adjustment cost draw does not dissuade such plants from investing, and adjustment probabilities rise to 1. Thus, in …gure 2’s lower panel, we see that all establishments adjust away from their initial capital holdings, with the total measure of adjustors rising dramatically from .295 to 1. Next, consider the converse, the e¤ects of a one standard deviation drop in pro- ductivity, as depicted in …gure 3. In this case, the fall in target capital implies a substantial leftward shift in the adjustment hazard. Those plants with very low cap- ital holdings, initially associated with high adjustment probabilities, now …nd their current capital much closer to the desired value, and hence are less likely to undertake costly adjustment. At the same time, plants with current capital roughly between 1 and 1.2 …nd that, rather than having a minor capital shortfall, they now have substantial excess. For these plants, adjustment probabilities rise. On balance, this left-shifting adjustment hazard implies only a minor rise in the number of active cap- ital adjustors, from .295 to .308, as depicted in the lower panel of the …gure. Thus, while positive productivity shocks have the potential to generate substantial external- margin e¤ects on aggregate investment demand, negative productivity shocks do not. The signi…cance of this distributional asymmetry becomes apparent in …gure 4. Here, we consider deviations from trend growth rates in response to the positive shock of …gure 2, occurring in period 6, followed by 14 periods of average productivity dur- ing which the economy resettles, and then the negative shock of …gure 3. From panel A, we see that target capital’s deviation from steady state behaves roughly symmet- rically, and matches the approximately linear reference model closely. However, in response to the positive shock, the rise in target capital is substantially ampli…ed by a large rise in the measure of investors, as is apparent from panel B, where the growth rate of aggregate capital demand under lumpy investment rises roughly 18 percent more than in the reference model. By contrast, when the negative shock occurs, 16 changes along the external margin play only a minor role for the lumpy investment economy. There, the fall in target capital is mitigated by the fact that only about 30% of establishments actually disinvest to the new target; consequently, the growth rate of aggregate capital demand exhibits less than half the decline seen in the refer- ence model (where all plants disinvest). We conclude from this example that: (i) our model of lumpy investment does have the potential to generate aggregate nonlinear- ities; (ii) these nonlinearities may take the form of asymmetric responses to shocks - sharper expansions and dampened contractions - as suggested by the …ndings of previous authors; (iii) these features result entirely from the asymmetric e¤ects of rightward and leftward shifts in the adjustment hazard upon the total number of ad- justors, and hence subsequent distributions of plants; (iv) the dynamics of adjustment along the intensive margin are roughly una¤ected by the presence of nonconvexities in plant-level adjustment technologies. The discussion above illustrates the powerful distributional e¤ects possible in the lumpy investment model. We next assess these e¤ects over a 2500 period simulation of the economy, again holding prices …xed. Figures 5 and 6, along with table 2, summarize the results. First, in the upper panels of …gure 5, we rank the deviations in aggregate investment relative to trend for the lumpy investment and reference models. Here, we use the horizontal axis to represent 5 broad categories of invest- ment episodes, ranging from extremely low to extremely high, with the vertical axis measuring the fraction of dates spent in each of these ranges. From the upper left panel, note that, within the reference economy, the fraction of investment periods away from near-trend is distributed perfectly evenly. By contrast, the lumpy in- vestment economy displays a disproportionate fraction of extremely high, relative to high, investment episodes, and has fewer extremely low, relative to low, observa- tions. Speci…cally, while times of near-average investment occur with roughly equal frequency, the inclusion of nonconvex capital adjustments shifts 2.5 percent of very low investment realizations upward into the low range, while nearly 2 percent (50 periods) of high investment episodes are pushed into the extremely high range. This evidence of lumpy investment’s dampened contractions and heightened expansions 17 over the simulation is summarized in the …gure’s lower right-hand panel, which iso- lates the di¤erences between the two upper panel histograms. Finally, at lower left, we align these histograms alongside results for the partial adjustment model, which by comparison displays far less dispersion in investment, given the convexifying force of quadratic adjustment costs. For a more detailed examination of the …xed price simulation results, we next construct series containing di¤erences in the relative deviations in investment and capital from trend, between the lumpy investment versus reference economies, for each date in the simulation. Table 2 provides several measures of the absolute values of the gaps in investment, along with the results for the Partial Adjustment versus Reference. Table 2: Reference deviations in Investment Demand minimum mean median maximum Lumpy Inv. .00010 .463 .1590 2.053 Partial Adj. .00008 .690 .0907 11.630 Note that average di¤erences from the reference economy are substantial, 46 percent for lumpy investment and 69 percent for partial adjustment. We present a cumulative ranking of the proportion of all observations represented by each di¤erence, (Lumpy Inv. minus Reference,) for capital in the upper panel of …gure 6. Note that this ranking is highly asymmetric around zero. To the left, we see substantial mass for dates where lumpy investment’s percentage deviations from trend lie between zero and 25 percent below the reference model. By contrast, the right tail, re‡ecting higher capital growth in the lumpy investment economy, is both long and thin, with fewer observations distributed over a much wider range. These features are particularly apparent when contrasted with the near-perfect symmetry of the partial adjustment versus reference model di¤erences in the …gure’s lower panel. From these closer inspections of the …xed price simulation results, we conclude that lumpy establishment-level investments can substantially reshape the distribution of investment and capital growth rates, relative to economies with smooth underlying 18 investment patterns yielding approximately linear aggregate dynamics, when move- ments in factor demands are unconstrained by changes in prices. 5.2 Equilibrium dynamics b We begin this section with a discussion of the accuracy of the forecasting rules, p and b ¡, used by agents. Table 3 displays the equilibrium forecasting functions, conditional on current productivity, when the distribution is approximated by only a single par- tition.10 ,11 The standard errors and R2 s associated with each regression indicate that the statistical mean alone is an e¢cient proxy for the distribution. This is con…rmed in table 4, where we re-solve the economy using two partitions to approximate the distribution. Note that there are only marginal reductions in the standard errors on equilibrium price regressions, indicating little additional relevant information. Since it is di¢cult to draw inferences from the relative magnitudes of the errors in fore- casting future conditional means, as neither m0 nor m0 in table 4 corresponds to 1 2 the mean in table 3, we use …gure 7 to present the aggregate capital series from each lumpy investment economy over the same 2500 period history. We …nd no discernible di¤erences, and take this as strong evidence that we need not partition the distribu- tion further.12 A comparison of table 3 with corresponding results from an economy whose distribution is exactly its mean is still more compelling. Speci…cally, when we solve for equilibrium forecasting functions in the reference economy, we …nd minimal changes in the regression coe¢cients and standard errors. As an illustration of this, the reference economy’s standard errors for m0 and p are 1:22e ¡ 4 and 2:63e ¡ 5 re- 1 spectively when productivity is at its highest value z5 , and 2:49e ¡ 4 and 5:36e ¡ 5 for z = z3 . Comparing these values with the corresponding errors of table 3 foreshadows the remaining results of this section. 10 Partitions here refer to I, the number of elements in m. 11 We have experimented with a variety of functional forms, including, for example, higher order terms. These produce similar results to the log linear form reported here. In the extension of the model, in section 6 below, we use quadratic forms. 12 The maximum di¤erence in these series is 2:1e ¡ 4. However, except where explicitly noted otherwise, the lumpy investment results below correspond to the the 2-partition economy. 19 We now re-examine the productivity simulation of our …xed price experiments in general equilibrium. We begin with an overview of second moments in table 5. Panel A displays percentage standard deviations in the growth rates of output, invest- ment, consumption, employment, wages and interest rates across model economies. From these results, it is clear that the variability under lumpy investment is virtually identical to the reference economy, regardless of whether we use one partition of the distribution (row 3), or two (row 2). This similarity is further emphasized by compar- ison with the partial adjustment model, where the cycle is dampened by the sluggish responsiveness of investment demand. The similarities between Lumpy Investment and Reference economies are also evident in the comovements with output reported in panel B. By contrast, aggregate quantities move more closely with the cycle in the partial adjustment results. From table 5, it is evident that lumpy investment fails to reshape the aggregate cycle in equilibrium. In what follows, we explore this further. In …gure 8, we present histograms of the relative deviations in investment from trend over the simulation, the equilibrium counterpart to …gure 5. Two features of this …gure are noteworthy. First, investment in both the lumpy investment and reference economies exhibits far less dispersion than was evident in …gure 5, as changes in factor prices largely o¤set the swings in investment demand seen under …xed prices. Second, while the reference economy’s investment series continues to be approximately symmetric around zero, the distribution is now closer to the Normal. Here again, the e¤ects of equilibrium price movements o¤set plants’ desires for large capital adjustments, shifting substan- tial mass away from extreme investment episodes inward toward levels corresponding to more moderate changes. This same force removes the lumpy investment economy’s tendency for sharp expansions, shifting mass from the highest investment deviations downward. As a result, the di¤erences in these two histograms essentially disappear in equilibrium; the largest di¤erence is in the zero band, where the lumpy investment economy displays about 0.5 percent fewer realizations than the reference economy. From the results presented thusfar, it is apparent that lumpy investment does not produce the stronger expansions and dampened recessions suggested by the …xed 20 price results of …gure 4, at least on average. Table 6 indicates that di¤erences in the Lumpy Investment versus Reference investment series are never of quantitative signi…cance in equilibrium, reaching only 0.3 percent at their maximum. We also see that the gaps present in the second row are reduced when price changes are present to dampen ‡uctuations in the Reference investment series. Table 6: Reference deviations in Equilibrium Investment minimum mean median maximum Lumpy Inv. 1.6 e-7 5.9 e-4 4.8 e-4 .0030 Partial Adj. 3.6 e-6 3.3 e-2 2.5 e-2 .1672 In …gure 9, we display cumulative rankings of the di¤erences between capital deviations relative to trend for both Lumpy Investment and Partial Adjustment with Reference economies, the equilibrium analogue of …gure 6. The distributions here exhibit greater symmetry around zero than was the case with prices held …xed. Much more importantly, though, note the scale of the horizontal axis in the …gure’s upper panel. Capital’s percentage deviations from trend in the lumpy investment economy are never so much as 0.1 percent away from those of the reference economy. We take this as further evidence that the implications of nonconvex establishment-level capital adjustment for aggregate dynamics are unimportant. Based on the discussion above, it would appear that changes in extensive-margin capital adjustment within the lumpy investment economy must be quite minor in equilibrium. We close this section by verifying this lack of distributional shifts for our original example. In …gure 10, we reconsider the asymmetric shock history which illustrated lumpy investment’s potential for nonlinearities when real wages and inter- est rates were constant. We …nd that, even in this example, the equilibrium lumpy investment economy exhibits no greater evidence of an asymmetric response than does the approximately linear reference economy. 21 6 Extension: Investment Shocks The results of the preceding section indicate that nonconvex capital adjustments can generate important nonlinearities in an environment with unchanging prices, but fail to do so when markets clear. It is tempting, then, to conclude that lumpy investment is not particularly important to the business cycle. However, this conclusion may rely on the assumption that business cycles are generated by a single driving force - an aggregate productivity shock that a¤ects all production units in the economy. Recent work by Christiano and Fisher (1998) and Greenwood, Hercowitz and Krusell (2000) suggests that, in fact, ‡uctuations in the price of investment goods may explain a substantial portion of the business cycle. Identifying the relative price of new equipment as a measure of the price of investment goods, Greenwood et al present evidence that shocks shifting the price of investment above and below its long-run downward trend can account for 30 percent of the cyclical variation in output. Measuring investment good prices more broadly, Christiano and Fisher …nd that investment-speci…c shocks explain 75 percent of output ‡uctuations at business cycle frequencies.13 The importance of investment-speci…c shocks in the economy raises questions about the generality of the results of the previous section. We reason as follows. A shock that is speci…c to the productivity of investment necessarily a¤ects only those establishments currently adjusting capital stocks, unlike the shock to overall produc- tion examined above. All plants bene…t from the e¤ects of a positive total factor productivity shock, regardless of whether they expand their factors of production; to better exploit these bene…ts, some plants increase capital. By contrast, a positive investment-speci…c shock provides a more direct incentive for capital adjustment, since it bene…ts only those establishments that invest. Thus such shocks have the potential to yield much larger shifts in the economy’s adjustment hazard, which may be su¢cient to overcome the convexifying forces of equilibrium. To explore this pos- 13 Fluctuations in the price of investment goods may be interpreted as the result of shocks to the productivity of investment, or investment-speci…c technology shocks. Throughout this section, we follow this interpretation. 22 sibility, we now extend our previous description of the lumpy investment model, (as well as the reference model,) to allow for exogenous ‡uctuations in the productivity of investment. Our extension of the model incorporates a one-sector hybrid of the approaches taken by Christiano and Fisher (1998) and Greenwood et al (2000), which involves the following modi…cations to our previous speci…cation.14 We assume that investment- speci…c productivity follows a …rst order Markov process with average growth rate G ¡ 1. Plant-level capital accumulation is now governed by Âk 0 = (1 ¡ ±)k + ³i, where ³ denotes the current level of detrended investment-speci…c productivity, and 1  ¡ 1 denotes the long-run growth rate of aggregate capital, which is (°G) 1¡µ ¡ 1.15 The exogenous aggregate state is given by z; ³, and we follow the previous authors in assuming that shocks to total factor productivity and investment-speci…c productivity are independently distributed. Transition probabilities are ¡ ¢ ¼ij ´ Pr z 0 = zj j z = zi ¡ ¢ ¿ ls = Pr ³ 0 = ³ s j ³ = ³ l : With these alterations, equations (6) and (7) describing the plant’s dynamic problem in section 2 change as shown below. µ· ¸ 1 (1 ¡ ±) k V (k; »; z; ³; ¹) = max zF (k; n) ¡ !n + p (16) n ³ 8 0 1 < Âk 0 X ³ 0 ´ + max ¡»!p + max @¡ p+¯ ¼ij ¿ ls V 0 k ; zj ; ³ s ; ¹0 A ; : k0 ³ j;s 9 X µ ¶=¶ (1 ¡ ±) k (1 ¡ ±) ¡ p+¯ ¼ij ¿ ls V 0 k; zj ; ³ s ; ¹0 ³  ; j;s 14 We analogously adapt the reference model. 15 As before, all variables denominated in units of output are de‡ated by trend output. With the ¡ ¢ 1 inclusion of investment-speci…c productivity growth, and trend output now grows at rate °G µ 1¡µ ¡ 1 1, rather than ° 1¡µ ¡ 1. We recalibrate the household’s discount factor ¯ to maintain the steady state interest rate at 6.5 percent. 23 Z B 0 V (k; z; ³; ¹) ´ V 1 (k; »; z; ³; ¹) G (d») . (17) 0 Equations (8) and (9) determining the target capital and threshold adjustment costs change accordingly, z being replaced with z; ³, and  replacing °, and the evolution of the distribution of plants over capital ¹0 = ¡(z; ³; ¹) is given by the following. · µ µ ¶¶¸ µ ¶ 0   ¹ (k) = 1 ¡ G » k; z; ³; ¹ ¹ k ; 8k 2 K s.t. k 6= k ¤ (z; ³; ¹) 1¡± 1¡± (18) Z 0 ¡ ¢ ¹ (k) = G » (k; z; ³; ¹) ¹ (dk) (19) K · µ µ ¶¶¸ µ ¶   + 1¡G » k; z; ³; ¹ ¹ k ; k = k¤ (z; ³; ¹) 1¡± 1¡± The equations describing equilibrium consumption and hours, (13) - (14), are simi- larly modi…ed. Finally, we assume that shocks to the investment productivity process take the form, 0 ³ 0 = ³ ½³ e"³ ; "³ » n(0; ¾2³ ), " (20) then discretize the state space using the procedure outlined above in section 4.16 Note that the output-growth-de‡ated relative price of a unit of investment is simply 1 . As such, we are able to use a relative price series for aggregate investment, ³ from 1982 through 1998, to directly estimate the parameters of (20), ½³ and ¾"³ , as well as the trend growth parameter G.17 This yields G = 1:022 , b³ = :706, and ½ b ¾"³ = :017. We now examine the e¤ect of lumpy investment in an economy subject to both ‡uctuations in aggregate productivity and investment-speci…c shocks.18 In table 6, we see that adding the investment shock raises overall volatility in both the lumpy 16 For this exercise, we discretize the [z; ³] state space on a 3 £ 3 grid of values. 17 We follow Christiano and Fisher (1998), section 2.2, closely in constructing this price series, adapting their method only as required to translate the quarterly series to an annual frequency. 18 As standard errors in forecasting regressions continue to be small, averaging roughly 6e ¡ 4 for m1 and 8e ¡ 4 for p, we present results only for the I = 1 partition economy. 24 investment and reference economies. This is particularly true for investment and employment, and consequently interest rates, whose relative standard deviations rise substantially, as is consistent with our reasoning above. Note that there is a more pronounced di¤erence between the two economies, though the results remain very close; this is also true for the output correlations shown in the table’s lower panel. Turning to …gure 11, we examine the histograms of investment deviations relative to trend. In comparison to …gure 8, where only the TFP shock was present, there are no more pronounced di¤erences with the inclusion of the investment shock; this is evident from the scale of the lower right panel. We have hypothesized that the investment shock is more likely to a¤ect the dis- tribution of plants than does the total productivity shock. Thus, it is possible that the latter is acting to dampen nonlinearities. In table 7 and …gure 12, we explore this possibility by eliminating variation in total factor productivity. Beginning with the table, it is apparent that the investment shock alone is insu¢cient to drive the cycle; output variability is reduced nearly half relative to the results of table 5. However, now relative volatilities in investment and employment are at their highest; in the case of employment, the rise is dramatic. In both panels, we begin to see slightly larger di¤erences between R and L rows, particularly for investment, as expected. Nonetheless, these di¤erences remain negligible. It is worthy of note that, beyond the di¢culty of reduced output volatility in both models, the consumption, wage and interest rate series have become countercyclical in absence of the TFP shock. Hence, while this example may be useful in studying the aggregate e¤ects of lumpy investment, it is not a plausible model for business cycle analysis.19 Finally, examining …gure 12, we …nd that the histograms for investment deviations do exhibit greater di¤erences when the productivity shock is removed. This is clearest when they are viewed together in the lower left panel, and con…rmed by the di¤erences plotted at lower right. Nonetheless, the variations across the lumpy investment versus 19 Christiano and Fisher (1998) avoid these problems within our reference model by allowing for two sectors in the economy and assuming that labor input must be determined prior to the investment shock’s realization. 25 reference economies continue to be small, with only about 2.8 percent more dispersion away from near-average investment episodes in the former than the latter. From this and the previous set of results, we conclude that the conjecture which prompted our inclusion of an investment-speci…c driving process was correct, but quantitatively irrelevant. 7 Concluding Remarks To be added. 26 References [1] De Boor, C. (1978) A practical guide to splines. Springer-Verlag [2] Caballero, R. J. (1999) “Aggregate Investment,” chapter 12 in M. Woodford and J. Taylor (eds.) Handbook of Macroeconomics IB Elsevier Science. [3] Caballero, R. J and E. M. R. A. 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Prentice Hall. 29 Figure 1: THE STATIONARY ADJUSTMENT HAZARD 1 0.8 0.6 0.4 0.2 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 ADJUSTMENT ACROSS THE DISTRIBUTION (TOTAL = 0.295) 0.25 0.2 0.15 µ 0.1 0.05 0 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 k Figure 2: RISE IN PRODUCTIVITY: HAZARD SHIFTS RIGHT 1 0.8 0.6 0.4 0.2 0.8 1 1.2 1.4 1.6 1.8 2 2.2 ADJUSTMENT ACROSS THE DISTRIBUTION (TOTAL = 1) 0.25 0.2 0.15 µ 0.1 0.05 0 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 k Figure 3: FALL IN PRODUCTIVITY: HAZARD SHIFTS LEFT 1 0.8 0.6 0.4 0.2 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 ADJUSTMENT ACROSS THE DISTRIBUTION (TOTAL = 0.308) 0.25 0.2 µ 0.15 0.1 0.05 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 k Figure 4A: Target Capital in Reference and Lumpy Inv. models 0.6 0.4 0.2 target 0 -0.2 -0.4 0 5 10 15 20 25 30 Figure 4B: Aggregate Capital in Reference and Lumpy Inv. models 0.6 0.4 Lumpy aggregate capital 0.2 0 -0.2 -0.4 0 5 10 15 20 25 30 date Figure 5: Distribution of Investment Deviations 1 1 0.8 0.8 Lumpy Inv. Model Reference Model 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -10 -5 0 5 10 -10 -5 0 5 10 Reference, Lumpy Inv. and Partial Adj. Differences 1 0.03 0.02 Lumpy Inv. - Reference 0.8 0.01 0.6 0 0.4 -0.01 0.2 -0.02 0 -0.03 -10 -5 0 5 10 -10 -5 0 5 10 Figure 6: Ranked Fixed Price Capital Differences from Reference 1 0.8 Lumpy Inv. 0.6 0.4 0.2 0 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 1 0.8 Partial Adj. 0.6 0.4 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Figure 7: Aggregate Capital in 1 Partition v. 2 Partition Lumpy Inv. Models 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0 500 1000 1500 2000 2500 date Figure 8: Distribution of Equilibrium Investment Deviations 0.5 0.5 The Lumpy Investment Model 0.4 0.4 Reference Model 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 Reference, Lumpy Inv. and Partial Adj. -3 Differences x 10 0.7 4 0.6 Lumpy Inv. - Reference 2 0.5 0.4 0 0.3 -2 0.2 -4 0.1 0 -6 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 Figure 9: Ranked Equilibrium Capital Differences from Reference 1 0.8 Lumpy Inv. 0.6 0.4 0.2 0 -10 -8 -6 -4 -2 0 2 4 6 8 -4 x 10 1 0.8 Partial Adj. 0.6 0.4 0.2 0 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Figure 10: The Absence of Nonlinearities in Equilibrium 0.015 0.01 0.005 Aggregate Capital 0 -0.005 -0.01 -0.015 0 5 10 15 20 25 30 date Figure 11: Distribution of Equilibrium Investment Deviations (w/ z and ζ shocks) 0.5 0.5 Lumpy Investment Model 0.4 0.4 Reference Model 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 Reference and Lumpy Inv. -3 Differences x 10 0.5 3 2 0.4 1 0.3 0 0.2 -1 -2 0.1 -3 0 -4 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 Figure 12: Distribution of Equilibrium Investment Deviations (w/ only ζ shock) 0.7 0.7 0.6 0.6 Lumpy Investment Model Reference Model 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 Reference and Lumpy Inv. Differences 0.7 0.01 0.6 0 0.5 0.4 -0.01 0.3 0.2 -0.02 0.1 0 -0.03 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 Table 3: Forecasting Rules with One Partition log(y) = [log(m1)] β1 + β2 [ 2 z1 (161 obs) β1 β2 SE R m1’ -0.0195 +0.8255 1.8739e-004 0.999950 p +1.2034 -0.4855 1.0004e-004 0.999958 2 z2 (647 obs) β1 β2 SE R m1’ -0.0071 +0.8251 2.3579e-004 0.999936 p +1.1810 -0.4807 8.3050e-005 0.999977 2 z3 (903 obs) β1 β2 SE R m1’ +0.0054 +0.8224 2.5248e-004 0.999933 p +1.1584 -0.4787 7.0519e-005 0.999985 2 z4 (626 obs) β1 β2 SE R m1’ +0.0181 +0.8206 2.1888e-004 0.999942 p +1.1355 -0.4768 6.8577e-005 0.999983 2 z5 (163 obs) β1 β2 SE R m1’ +0.0308 +0.8201 1.2563e-004 0.999977 p +1.1123 -0.4736 4.2475e-005 0.999992 Table 4: Forecasting Rules with Two Partitions log(y) = [log(m1)] + β3 [ β1 + β2 [ [log(m2)] 2 z1 (161 obs) β1 β2 β3 SE R m1’ -0.1041 +0.4827 +0.3115 1.0291e-003 0.998355 m2’ -0.1864 +0.4305 +0.4314 1.0662e-003 0.998509 p +0.8664 -0.2488 -0.2368 7.2610e-005 0.999978 2 z2 (647 obs) β1 β2 β3 SE R m1’ -0.0750 +0.4252 +0.3749 8.8943e-004 0.999033 m2’ -0.1904 +0.4670 +0.3864 1.0688e-003 0.998772 p +0.8474 -0.2479 -0.2330 7.4008e-005 0.999982 2 z3 (903 obs) β1 β2 β3 SE R m1’ -0.0632 +0.3389 +0.4391 7.2543e-004 0.999392 m2’ -0.1804 +0.5580 +0.3145 8.8844e-004 0.999257 p +0.8263 -0.2469 -0.2318 6.3122e-005 0.999988 2 z4 (626 obs) β1 β2 β3 SE R m1’ -0.0528 +0.3625 +0.4141 9.1506e-004 0.998874 m2’ -0.1715 +0.5659 +0.3072 7.9795e-004 0.999306 p +0.8053 -0.2495 -0.2272 5.4232e-005 0.999989 2 z5 (163 obs) β1 β2 β3 SE R m1’ -0.0241 +0.3216 +0.4653 9.0684e-004 0.998743 m2’ -0.1763 +0.5945 +0.2640 8.8826e-004 0.998964 p +0.7844 -0.2484 -0.2252 3.2380e-005 0.999996 Table 5: Business Cycle Moments Standard Deviations Output Investment Consumption Employment Wage Interest Rate R 1.906 6.373 0.935 1.101 0.935 0.793 L2 1.906 6.386 0.933 1.102 0.933 0.793 L1 1.905 6.373 0.933 1.100 0.933 0.795 PA 1.547 3.458 1.094 0.473 1.094 1.068 Contemporaneous Correlations with Output Output Investment Consumption Employment Wage Interest Rate R 1.000 0.971 0.924 0.946 0.924 0.685 L2 1.000 0.972 0.925 0.947 0.925 0.683 L1 1.000 0.972 0.926 0.947 0.926 0.681 PA 1.000 0.990 0.995 0.972 0.995 0.545 R = Reference; L2 =Lumpy Investment w/ I=2; L1 = Lumpy Investment w/ I=1; PA = Partial Adjustment. All series log HP-filtered. Table 6: Business Cycle Moments: Shocks to Both Production and Investment Technology Active Standard Deviations Output Investment Consumption Employment Wage Interest Rate R 2.172 8.575 1.374 2.251 1.374 1.410 L1 2.182 8.670 1.373 2.265 1.373 1.407 Contemporaneous Correlations with Output Output Investment Consumption Employment Wage Interest Rate R 1.000 0.884 0.258 0.808 0.258 0.243 L1 1.000 0.885 0.253 0.810 0.253 0.242 R = Reference; L1=Lumpy Investment (1 partition) All series log HP-filtered. Table 7: Business Cycle Moments: Only Investment-Specific Shock Active Standard Deviations Output Investment Consumption Employment Wage Interest Rate R 1.128 7.037 1.042 2.021 1.042 1.140 L1 1.152 7.222 1.062 2.060 1.062 1.156 Contemporaneous Correlations with Output Output Investment Consumption Employment Wage Interest Rate R 1.000 0.953 -0.733 0.936 -0.733 -0.203 L1 1.000 0.952 -0.732 0.936 -0.732 -0.200 R = Reference; L1=Lumpy Investment (1 partition) All series log HP-filtered.