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Concentration of capital ownership and investment fluctuations
Review of Economic Dynamics 7 (2004) 668–686 www.elsevier.com/locate/red
Concentration of capital ownership and investment fluctuations Zvi Hercowitz Tel Aviv University, Tel Aviv 61390, Israel Received 22 May 2003; revised 10 November 2003 Available online 20 February 2004
Abstract This paper is motivated by the observation that investment tends to accelerate when output is around trend. The model used to explain this observation is based on the capacity-constrained production setup in Hansen and Prescott [(2001) Manuscript], where capacity is constant over time, and on capital being owned by a fraction of the agents in the economy. When capacity is reached, the capital share increases because its component from capacity ownership becomes positive. The concentration of capital ownership leads then to an acceleration of investment—generated by the desire of capital owners to smooth consumption—as well as to a deceleration of total consumption. The results from the calibrated model contribute, although only partially, to the explanation of the observed behavior. 2004 Elsevier Inc. All rights reserved.
1. Introduction Figure 1 plots investment against GDP—both Hodrick–Prescott filtered—along with a fitted (fifth degree) polynomial. It can be observed that the 0,0 point on each axis is to the right and above the center. This follows from asymmetric cyclical behavior, as there are larger negative deviations than positive ones. In the present context, however, the main feature of the graph is the comovement between the two variables, as illustrated by the fitted line: Around the 0,0 point the slope is higher—suggesting an acceleration of investment that is formally tested later on. This acceleration of investment can not be explained by a standard RBC model, which generates an approximately linear comovement between investment and output. This linear comovement can be rationalized as follows. In that E-mail address: [email protected]. 1094-2025/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.red.2003.12.004
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Fig. 1. Fixed nonresidential investment and GDP (HP filtered) US data 1954:1–2001:2.
model, investment movements follow from consumption smoothing considerations of the representative agent and expected productivity levels. Both considerations depend on the degree of persistence of the productivity shocks, which is constant. The comovement suggested by Fig. 1 is studied here in the context of concentrated ownership of capital and a procyclical capital share. The theoretical framework is an extension of the representative-agent model in Hansen and Prescott (2001)—who study cyclical asymmetries in a framework with constant capacity—to two types of agents. The main characteristics of the present model are that (a) the capital share is more procyclical around the trend than away from it—given that when capacity is reached, the capital share adjust upwards because its component from capacity ownership moves then away from zero, and (b) capital income is concentrated in the hands of agents who have access to the capital market. Given these two characteristics, consumption smoothing of these agents generates an acceleration of investment around trend.1 Section 2 presents the basic, representative agent, model with the production setup from Hansen and Prescott, and Section 3 extends this framework by introducing concentrated ownership of capital. The solution, calibration and simulation of the model are reported in Section 4, and the comparison with actual US data is presented in Section 5. Section 6 1 It is an implication of applying a flexible detrending procedure, as the Hodrick–Prescott filter, which makes
high output levels—those corresponding to capacity—appear as located around trend. This technical aspect is addressed in Section 4.
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concludes, including comments about possible extensions and implications of the present setup.2
2. The representative agent model 2.1. The Hansen–Prescott technology Each firm has M identical locations or plants, with m M of them operated. Locations do not depreciate. In each location, the production function is θ φ y = zk n , φ, θ > 0, φ + θ < 1, if n n, 0, otherwise, where k and n are the movable capital and labor operating in each location and z is a productivity shock following a Markov chain with nz states with bounded values and transition matrix P .3 Given diminishing returns, there is a tendency to operate all locations. However, the restriction n n implies that m = min{M, N/n}, where N is the total input of labor. If N < Mn, some locations are idle. Production at the firm’s level can be seen as using three factors, total movable capital K, total labor input N , and the number of locations M, which is constant. Both K and N can be moved across locations at no cost. The production function of the firm is θ φ K N z m . (2.1) Y = zF (K, N; M) = max m m m=min(M,N/n) In the unconstrained range, where N/n < M, the desired number of plants to operate is smaller than the available number, and thus m = N/n. In this range, the production function takes the form K θ N φ Y =z N/n, N/n N/n or Y = zK θ N 1−θ (n)θ+φ−1 ,
(2.2)
which looks like a standard CRS formulation, with the term in n working as a TFP level. As n decreases, more locations are utilized, and the higher TFP follows from the diminishing 2 An alternative possible explanation of the comovement in Fig. 1 is that investment in capacity may increase as the constraint is approached. If capacity is identified with structures more than with equipment, the nonlinearity in Fig. 1 for structures only should be stronger. However, as reported in Section 5, addressing structures and equipment separately yields for both weaker nonlinear behavior than for total investment. 3 Given that the values of z are bounded, that locations do not depreciate, and a constant cost of installing each new location, Hansen and Prescott derive a constant M endogenously, in the stationary solution. Practically, this setup makes possible to treat M as given, since there is a value of the cost parameter that will make any value of M optimal.
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returns: Spreading the same K and N over a larger number of locations increases total output. In the constrained range, m = M, and the production function has the form θ φ N K M, Y =z M M or Y = zK θ N φ M 1−θ−φ .
(2.3)
Given that φ < 1 − θ , labor’s coefficient in this range is smaller than in the unconstrained range. Diminishing returns come into effect because all M locations are operated.4 A key feature of this technology is labor’s marginal productivity. From (2.2) and (2.3) it follows that θ K (n)θ+φ−1 , if N < Mn, (1 − θ )z N MPN = θ θ+φ−1 K N θ φ−1 1−θ−φ M = φz , if N > Mn. φzK N N M Because 1 − θ > φ, the marginal productivity of labor jumps downwards at N = Mn θ K l (n)θ+φ−1 from MPN = (1 − θ )z N θ K to MPN r = φz (n)θ+φ−1 . N Figure 2 illustrates the MPN schedule as a function of N , with z and K affecting its location, and the upward-sloping line represents a “labor supply schedule.” The figure shows a situation where N = Mn. The key feature of the Hansen–Prescott technology can be shown using this figure: consider, for example, an increase in z—holding K constant for simplicity. Both sections of the MPN schedule shift upwards together, and assume that the supply schedule does not move—as it will be the case with the utility function adopted below. Given the discontinuity in MPN at N = Mn, there is a range of z values for which the increase in z leaves N unchanged at Mn. This is illustrated in Fig. 3, where the flat part of the curve follows from the discontinuity in MPN. However, if z increases beyond a certain value, the supply schedule begins to be crossed by the lower part of the MPN schedule, and correspondingly, N increases with z beyond that value. This is shown in Fig. 3 by the upward sloping part on the right. The upward sloping part on the left is generated in a similar way when z declines below some value. The explanation of the investment behavior shown in the Introduction will be directly related to the nonlinearity in Fig. 3, i.e., to the flat part in the curve whose length depends 4 Note that under this technology, “capacity utilization” is unbounded. More K and N can always be allocated to already utilized locations. One may define N = Mn as “100% utilization,” and compute the average utilization rate in the model. However, the resulting figure would not be comparable with available data on capacity utilization, because the latter is based on 100% as the highest possible value.
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Fig. 2.
Fig. 3.
on 1 − φ − θ . This fraction determines the coefficient of M in output once all location are utilized.5 Finally, the evolution of movable capital has the standard form K = K(1 − δ) + I,
0 < δ < 1,
where I is gross investment. As mentioned above, M does not depreciate. 5 Two main factors determine the degree of cyclical asymmetry in this model: (1) the magnitude of 1 − φ − θ , which affect the length of the flat part in Fig. 3, and (2) the location of the distribution of N relative to Mn. The further to the left this distribution is, the lower the probability that capacity is reached and thus the smaller the degree of asymmetry. In the calibration of the model, the non-stochastic N is set equal to Mn.
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2.2. Preferences The preferences adopted have two important advantages in the present context: (a) labor is independent of wealth, and (b) labor varies even when the individual is disconnected from the capital market. Feature (a) facilitates direct aggregation of labor across individuals with different wealth levels. Feature (b) is crucial in the present context, which deals with cyclical fluctuations when a fraction of the agents does not participate in the capital market (Section 3). These preferences are described by 1−σ
∞ η 1+γ t n β ct − (1 − σ ), σ, η, γ > 0, 6 E 1+γ t t =0
where c and n are individual consumption and labor, respectively. With these preferences, the individual’s labor decision in a competitive setup is 1/γ 1 n= , (2.4) w η where w is the real wage. This utility function has the property that labor effort is independent of wealth. In particular, K and z affect labor effort only through the real wage.7,8 2.3. Competitive equilibrium The economy is composed of a large number of identical firms, and a large number of identical households, who are the stock holders. The firms purchase labor and capital services from the households, and pay dividends to them. 6 This utility function is not consistent with stationary labor when z and thus real wages increase over time. However, it can be adapted to allow for stationary n in a growing economy by introducing technological progress also into home activities as follows: (1 − n)1+γ 1−σ
(1 − σ ). u(c, n) = c + z 1+γ
This modification preserves the independence of n from wealth, while the stationarity of n follows from z affecting equally both home and market productivity of labor. 7 Given that z and K do not affect n directly, the labor supply schedule in Fig. 2 under this utility function would not move when z goes up—as it was assumed above in the discussion of Figs. 2 and 3. 8 The standard utility function u(c, 1 − n) = c1−σ v(1 − n)/(1 − σ ), σ > 0, v > 0, v < 0 (King et al., 1988) has undesirable implications in the present context of partial participation in the capital market. For individuals who do not participate, and hence c = w · n, the first-order condition for labor implies that ∂v(1 − n) w 1 = (1 − σ )v(1 − n) = (1 − σ )v(1 − n). ∂(1 − n) c n Therefore, n is constant, and non-participants do not change labor hours over the business cycle. Hansen’s utility function u(c, n) = ln c − γ n, γ > 0, has the same implication. For non-participants, n = 1/γ .
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The household’s problem has the standard form: V (z, k, K; M) = max u r(z, K; M)k + w(z, K; M)n k ,n
+ v(z, K; M)M − k + k(1 − δ), n + βEV (z , k , K ; M) ,
where r(z, K; M)k, w(z, K; M) and v(z, K; M) are the rental price of capital, real wage and dividends, respectively, determined by the aggregate variables. The maximization of profits by the firms implies equalization of w(z, K; M) and r(z, K; M) to the corresponding marginal products, while dividends are 0, if N < Mn, v(z, K; M) = (1 − θ − φ)zK θ N φ M −θ−φ , if N Mn. Equilibrium also requires that
1 w(z, K; M) n= η k = K,
1/γ = N,
K = K(1 − δ) + zF (K, N; M) − C, where C is aggregate consumption.
3. The model with concentration of capital ownership The following distinction between two types of households is now introduced: Type 1 households are similar to those in the basic economy, i.e., work and have access to the capital market (fraction ω of the population). Type 2 households do not own capital and do not have access to the capital market, i.e., their consumption equals their wage income (fraction 1 − ω of the population). This assumption implies that the two types of household cannot interact in the capital market, and that Type 1 agents own both K and M in the economy. Hence, these agents receive all capital income, defined as the sum r(z, K; M)k + v(z, K; M)M.9 The two types have identical utility functions, but the problems they solve are different. Type 1 households face the same problem as those in the basic setup, while Type 2 solve a static problem. The distinction between these two types of households has two components: (1) concentration of capital ownership, and (2) lack of access to the financial market to those who do now own capital. 9 What is really important for the model is that the ownership of M is concentrated on a fraction of the agents,
since dividends are the volatile component of capital income. The assumption that Type 1 agents own also all K is convenient, as then Type 2 agents simply do not participate in the capital market.
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This situation could arise endogenously if the access to the capital market is costly, as in Greenwood and Jovanovic (1990)—where those agents with low income and thus little to gain by smoothing consumption opt for not joining a financial intermediation network. In that model this cost is fixed, and hence with income growth all agents eventually join the financial network. However, if this cost grows over time, say, proportionally to the average real wage, individuals who start off without assets may never find it profitable to join the capital market and own capital. Hours of work of Type 1 are denoted by h, and hours of Type 2 by n. Consumption of the two groups is described by: Type 1: x = r(z, K; M)k + w(z, K; M)h + v(z, K; M)M − k + k(1 − δ). Type 2: c = w(z, K; M)n. The problem of a household of Type 1 is the same as the representative household’s problem above. The problem of a household of Type 2 is
max u w(z, K; M)n, n . n
Similarly as for Type 1, the solution for labor is 1/γ 1 w(z, K; M) . n= η
(3.1)
Substituting into the budget constraint yields 1/γ 1 w(z, K; M)1/γ +1. c= η The remaining equilibrium conditions are the same as in the representative agent setup, except that here 1/γ 1/γ 1 1 + (1 − ω) w(z, K; M) N = ω w(z, K; M) η η 1/γ 1 w(z, K; M) = , (3.2) η K = K(1 − δ) + zF (K, N; M) − ωx − (1 − ω)c. 3.1. Algebraic characterization of output and the capital share The utility function adopted makes possible to compute a closed-form solution for output. The following derivation focuses on the range of output for which labor is constant at N ∗ ≡ Mn (the flat part of the schedule in Fig. 3), which is particularly relevant for confronting the model with the data. Using (3.2), w∗ can be defined as w∗ = η(N ∗ )γ . From the production function in (2.2) and (2.3), when N approaches N ∗ from the left and from the right, the corresponding output levels are, respectively, Y l ≡ zK θ (N ∗ )1−θ (n)θ+φ−1 ,
(3.3)
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and Y r ≡ zK θ (N ∗ )φ M 1−θ−φ .
(3.4)
The K, z plane can be divided into three regions for which (1) N < Mn, (2) N = Mn, and (3) N > Mn. Between regions (1) and (2), the critical condition is MPN l = (1 − θ )zK θ (N ∗ )−θ (n)θ+φ−1 = w∗ . Hence, the combinations of K and z separating regions (1) and (2) satisfy zK θ =
w∗ (N ∗ )θ (n)1−θ−φ . (1 − θ )
Substituting this expression into (3.3) yields Yl =
w∗ N ∗ . (1 − θ )
(3.5)
Between regions (2) and (3), the critical condition is MPN r = φzK θ (N ∗ )φ−1 M 1−θ−φ = w∗ . Therefore, the combinations of K and z separating these two regions are given by zK θ =
w∗ ∗ 1−φ θ+φ−1 (N ) M . φ
Substituting this expression into (3.4) yields Yr =
w∗ N ∗ . φ
(3.6)
From (3.5) and (3.6) it follows that the range of output variation in the flat range of N is given by Yr 1−θ . = l Y φ
(3.7)
The division of output levels into the three ranges Y Y l , Y l < Y < Y r and Y Y r will be useful later on when comparing the model to the data, and it can be used to determine the cyclical behavior of the capital share (income from K and M). The capital share in the lower output region is θ , and in the higher output region is 1 − φ (higher than θ because of decreasing returns). In the intermediate region, the wage bill is constant at w∗ N ∗ = (1 − θ )Y l , but output increases with z and K. Hence, as Y increases between Y l and Y r , the capital share increases from θ to 1 − φ according to (1 − θ )Y l Y − w∗ N ∗ =1− . Y Y
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4. Solution of the model and simulation 4.1. Choice of parameter values 1. Technology. Following Hansen and Prescott (2001): P is computed using Tauchen’s (1986) discrete state-space procedure so as to approximate the process ln zt = ρ ln zt −1 + εt , εt ∼ N(0, σε2 ), with ρ = 0.95, σε = 0.007, nz = 15, where nz is the number of discrete z values, evenly spaced between ±2 standard deviations around zero, corresponding to the non-stochastic steady state. M = 0.315, n = 1, δ = 0.019, θ = 0.38, φ = 0.598. This value is obtained by fitting the simulated asymmetry in output, as measured by the ratio between the average deviations above the HP trend to the average deviation below, to the actual ratio of 0.96 in the 1954:1–2001:2 sample. This, given that the non-stochastic N is set equal to Mn—the point at which all locations are used at the minimum positive level.10 See next for the parameterization of preferences in this respect. 2. Preferences β = 0.99. From Greenwood et al. (1988): σ = 2 (the results are very similar with σ = 1), γ = 0.6, η = 6. This value is chosen so that the non-stochastic N is 0.315—the lowest level of labor for which all locations are utilized. Note that with the current utility function, N is not a fraction of time but represents an index. 3. Fraction of agents who own capital. The parameter ω is calibrated so that the skewness of capital ownership in the model equals actual skewness of stock holdings 10 Correspondingly, the production function used for the non-stochastic solution is that in Eq. (2.2).
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across households in the US. The resulting value is ω = 0.03.11 The basic version with ω = 1 is also computed, as well as an additional version with more moderate ownership concentration, ω = 0.2. 4.2. Solution and simulation procedures The model is solved as follows: For each discrete value of z, a piecewise linear approximation to the dynamic decision rule of Type 1 agents as a function of K is fitted— given that Type 2 agents consume all their wage income, and that the resource constraint of the economy and the equilibrium condition in the labor market are satisfied. The grid for K has 200 points within the range of ±20 percent around the non-stochastic steady state. The criterion of convergence of the solution is that the Euler equation for Type 1 agents holds up to a tolerance error. The model is then used to simulate artificial data. First, 1000 observations are simulated and plotted to illustrate how the model works. Then, 500 simulations of 190 periods— the number of quarters in the 1954:1–2001:2 sample—are computed to obtain average statistics that can be compared with those from the actual data. 4.3. Simulation results Figure 4 shows plots of 1000 simulated data points for the comovement of investment and consumption with output for ω = 1—the basic version—and for the main version with ω = 0.03. Let us start with the basic version on the top two graphs, where capital is equally owned by all agents. One can discern there some steepening of the investment profile and flattening of the consumption profile, approximately between two output levels—which were denoted above as Y l and Y r . This pattern follows from a basic implication of the technology: Between Y l and Y r the share of dividends in output increases from zero to 1 − θ − φ, while the share of wages declines from 1 − θ to φ (for Y > Y r , these shares remain constant at the Y r levels). Correspondingly, between those two levels of output the composition of income changes in favor of a more volatile component. Consumption smoothing leads then to weaker consumption and stronger investment responses. 11 Given population of size 1 and total capital ownership E, the mean equity holdings in the model is ω ·(E/ω) = E, and the variance is (1−ω)(−E)2 +ω(E/ω −E)2 = E 2 ((1 − ω)/ω). Thus, the standard deviation
√ is sdev = E (1 − ω)/ω. Then, skewness = (1 − ω)
3/2 E/ω − E 3 ω 1 − 2ω −E 3 1 − 2ω . +ω = (1 − ω) =√ sdev sdev 1−ω ω2 (1 − ω)ω
The actual skewness is computed using the distribution published in “Equity Ownership in America,” Fig. 7, with 10 groups of households, according to their equity holdings in 1999. Each group is defined as a range of equity holdings: (0), (less than $10,000), ($10,000–25,000), . . . , ($1 million or more). In the present calculation, the middle of the range is taken for each group, except for the top group, whose average holdings is inferred using the overall average holdings of $80,500. The computed skewness is 5.5. Then, ω is obtained by solving the √ equation (1 − 2ω)/ (1 − ω)ω = 5.5. There is only one solution in the positive range, which is ω = 0.03.
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Fig. 4. Basic version: ω = 1. Main version: ω = 0.03.
Concentration of capital ownership, on the bottom two graphs, magnifies the acceleration of investment and deceleration of consumption. The mechanism at work here is that the change in shares between Y l and Y r is accompanied by a concentration of dividends on a small fraction of the agents, exacerbating their consumption smoothing problem and leading to a stronger investment response. At the same time, wage income and thus consumption of most agents remain constant, thereby flattening the profile of aggregate consumption. In the case of no capacity constraint, i.e., when 1 − θ − φ = 0, the comovement of investment and consumption with output (not shown) is practically linear, both for ω = 1 and for ω = 0.03. Hence, concentration of capital ownership generates the nonlinear behavior stressed above only when interacting with procyclical dividend income. 4.4. Detrending cyclically asymmetric data Figure 4 cannot be contrasted directly with actual data because of the nonstationarity of actual variables. The procedure adopted here is common in this context: to HP filter
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Fig. 5. Basic version: ω = 1. Main version: ω = 0.03.
both actual and artificial data and then compare. The filtering of the model’s variables also provides guidance on how to interpret detrended data from an economy subject to a capacity constraint.12 The artificial data shown in Fig. 4 are HP filtered and plotted in Fig. 5. Figure 5 displays the same phenomenon shown in Fig. 4, although it appears here in a different form. Starting with consumption, on the right-side graphs, note the flatter “lines” crossing the steeper “lines” at the 0,0 point. The mechanism generating this pattern is a technical one, but it is important for the comparison of the model’s behavior to the data. In periods when the capacity constraint does not bind—which tend to be prolonged because of the serial correlation of the shock—filtering yields approximately symmetric movements in consumption and output around zero. This is reflected in the longer lines centered on the 0,0 point. In periods when the capacity constraint does bind, output fluctuates most of the time in the range Y l −Y r , in which consumption movements are smaller, as seen in 12 The same would apply to any other flexible detrending procedure.
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Fig. 4. Because these periods also tend to be prolonged, filtering centers those fluctuations around zero as well. Hence, the crossing flatter lines at the 0,0 point represent those periods when the capacity constraint binds and output is in the Y l −Y r range. The cases when output is higher than Y r are likely to be represented by the points away from the crossing lines. Regarding investment, the effect of filtering is similar, but with higher slopes in the constrained periods. The main point from Fig. 5 is, therefore, that in the filtered data, the acceleration of investment and deceleration of consumption appear around the zero points—instead of at high values of output when the data is in levels.
5. Comparison with actual data 5.1. Basic cyclical statistics Table 1 reports basic cyclical statistics of actual and artificial data, both HP filtered, focusing here on the volatility of the main variables—while the nonlinearity is addressed in the next subsection. The statistics from the model are averages over 500 simulations of 190 data points each (the number of quarters in the 1954:1–2001:2 sample). Table 1 shows that the standard deviations in the model are lower than in actual data. The low volatility is due, partly, to the capacity constraint. Eliminating the capacity constraint— i.e., setting φ = 0.62 so that φ + θ = 1—yields the statistics shown in Table 2.13 The standard deviations there are larger than in Table 1, although still lower than in actual data. However, if γ (determining labor elasticity) is considered a free parameter, one can compute the value that equates the volatility of output to that in the data. The result is γ = 0.38—corresponding to an elasticity of labor supply to the real wage of about 2.6. The concentration of capital ownership—moving from ω = 1 to ω = 0.03 in Tables 1 and 2—does not affect much output volatility, but reduces the volatility of investment and increases that of consumption. This follows from the fact that lowering ω increases the fraction of agents who cannot invest. This can be described as the general effect Table 1 Basic statistics Variable
US
Model ω=1
1954:1–2001:2
Output Investment equipment structures Consumption
ω = 0.03
St. dev. %
Corr. w/output
St. dev. %
Corr. w/output
St. dev. %
Corr. w/output
1.63 4.64 5.28 4.76 1.29
1.00 0.79 0.85 0.48 0.88
1.30 3.05 – – 0.76
1.00 0.99 – – 0.98
1.31 2.48 – – 0.96
1.00 0.94 – – 0.96
13 The symbols S and S in Table 2 are defined and used in Section 5.2.1. α
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Table 2 Model without capacity constraint ω=1
Variable
Output Investment Consumption
ω = 0.03
St. dev. %
Corr. w/output
S
Sα /S
St. dev. %
Corr. w/output
S
Sα /S
1.47 3.13 0.92
1.00 1.00 1.00
– 2.13 0.63
– 1.00 1.00
1.47 2.20 1.23
1.00 1.00 1.00
– 1.50 0.83
– 1.00 1.00
of concentrating capital ownership, regardless of whether there is a capacity constraint or not. However, comparing the effects of lowering ω in Table 1 to those in Table 2 shows that when there is a capacity constraint, these effects diminish. In other words, when concentration of capital ownership interacts with procyclical capital income, the acceleration of investment and deceleration of consumption stressed here work in the opposite direction to the general effect of reducing ω. 5.2. Nonlinear cyclical behavior Figure 6 displays the comovement of investment in equipment and structures, separately, with output, along with fitted polynomials. Total investment is displayed in Fig. 1. Figure 7 does the same for consumption. Judging from the fitted lines, it can be observed that: (1) Both types of investment tend to accelerate when output is around the zero level. The behavior of structures is more pronounced in this respect, but the dispersion of the observations around the fitted line is larger, relative to equipment. (2) The slope of the consumption profile declines somewhat when output is around the zero level.
Fig. 6.
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Fig. 7. Consumption and output (HP filtered). US data 1954:1–2001:2.
These features can be summarized quantitatively by estimating two slopes for each variable: One for the entire range of output variation and the other for some range around zero to be determined below, and testing whether the two are different. Then, these results can be compared with those from the model. 5.2.1. Statistics of nonlinear behavior The comovement of x—HP filtered log(X)—with output in the present context can be characterized by the regression coefficients of x on y over the entire sample and in a reduced sample in which |y| < α. These slope statistics are denoted as S(x) and Sα (x), respectively. An indication about appropriate values for α can be obtained from Eq. (3.7), which says that Y l and Y r are separated by the factor (1 − θ )/φ = 0.62/0.598 ≈ 1.037. Hence, the corresponding value for α is about 0.0184, half of the percentage difference. The slopes coefficients can be estimated by running regressions of the type: x on a constant, y, dα and dα ∗ y, where dα is a dummy variable for |y| < α. The coefficient of y corresponds to S(x) and the coefficient of dα ∗ y to Sα (x) − S(x). The results below correspond to α = 0.0165, which within the range 0.015−0.02 yields the highest significance levels for the coefficients of dα ∗ y in the actual data. This value of α is then applied also to the model. The OLS estimates of the slopes with US quarterly data, and the corresponding coefficients from the model, are reported in Table 3. The actual data estimates are consistent with the hypothesis of accelerating investment, but their statistical significance is likely to be overstated given serial correlation in the regression residuals. To overcome this problem, the regression is reestimated using annual averages of the quarterly filtered data. Specifically, the quarterly variables x, y and dα are converted to annual frequency, implying that dα has now five possible values instead of two (1: if |y| < α in the four quarters of the year; 0.75: if |y| < α in three quarters of the year, and so on). The number
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Table 3 Slope statistics—quarterly frequency Variable
US
Model ω=1
1954:1–2001:2 Investment equipment structures Consumption
ω = 0.20
ω = 0.03
S
Sα /S
S
Sα /S
S
Sα /S
Sα
Sα /S
2.11 2.65 1.20 0.73
1.38 (0.010) 1.22 (0.054) 1.84 (0.027) 0.86 (0.125)
2.38 – – 0.57
1.03 – – 0.95
2.01 – – 0.71
1.06 – – 0.94
1.92 – – 0.74
1.09 – – 0.93
Note. In parentheses: significance level of Sα − S. Table 4 Slope statistics—annual averages Variable
US
Model ω=1
1954–2000 Investment equipment structures Consumption
ω = 0.20
ω = 0.03
S
Sα /S
S
Sα /S
S
Sα /S
Sα
Sα /S
2.02 2.61 1.01 0.76
1.74 (0.041) 1.45 (0.082) 2.98 (0.079) 0.90 (0.644)
2.24 – – 0.59
1.08 – – 0.89
1.76 – – 0.75
1.18 – – 0.85
1.66 – – 0.78
1.21 – – 0.85
Note. In parentheses: significance level of Sα − S.
of observations is obviously reduced, but time aggregation internalizes residual serial correlation within the year, while part of the quarterly information is preserved by the annual average of dα .14 Table 4 reports the corresponding slope statistics from the data and from the model. The acceleration of investment in actual data—represented by the ratios Sα (x)/S(x)— is quantitatively large. This result is statistically significant at the 5% level for total investment, although not for equipment and structures separately. This seems consistent with the present mechanism, which focuses on investment in general as vehicle for consumption smoothing. Regarding consumption, the deceleration is far from statistical significance. The investment-output plot with annual averages—counterpart of Fig. 1— is shown in Fig. 8, where the triangles represent the fitted polynomial. The investmentoutput comovement in this figure is similar to that with quarterly data. For the model, Table 4 shows the Sα (x)/S(x) statistics for ω = 1, ω = 0.2 and ω = 0.03.15 The model does contribute to the explanation of the observed behavior, but the magnitudes are smaller than in actual data. The results for the model in the table quantify the characteristics shown graphically in Figs. 4 and 5. Part of the acceleration 14 Two alternative procedures were attempted to deal with the serial correlation problem. One was to model residual serial correlation directly, as in a standard structural regression. The estimates of Sα (x) − S(x), using AR(2) or ARMA(1,1) processes to achieve white noise residuals, had the expected signs, but were statistically insignificant. The other was to conduct the tests on filtered annual data—rather than using the annual averages of quarterly filtered data. This procedure did not eliminate residual serial correlation in the regressions. 15 The statistics from the model are averages of 500 simulations of 190 periods.
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Fig. 8. Fixed nonresidential investment and output (HP filtered—annual averages). US data 1954–2000.
of investment is due to procyclical capital income itself—column ω = 1—which yields Sα (x)/S(x) = 1.08. Concentration of capital ownership, in columns ω = 0.2 and ω = 0.03, magnifies this effect to Sα (x)/S(x) ratios of 1.18 and 1.21, respectively, compared to 1.74 in actual data. Concentration of capital ownership—i.e., reducing ω—by itself does not generate nonlinear investment/output comovement. Table 2 shows the Sα (x)/S(x) ratios for the constant-returns economy (θ + φ = 1), which equal 1 for both ω = 1 and ω = 0.03.
6. Concluding comments This paper is motivated by the observation, shown in Fig. 1, that investment tends to accelerate when output is around the HP smoothed level. This acceleration is statistically significant at the 5% level for total investment, while separately for equipment and structures it is significant only at the 10% level. The deceleration of consumption is not statistically significant. The model used to explain this observation is based on the capacity-constrained production setup in Hansen and Prescott (2001), and on capital being owned by a fraction of the agents in the economy. In the model, when output increases beyond the critical point where all available capacity is utilized at the minimal level, dividends become positive and the real wage begins to lag behind labor’s marginal productivity. Beyond that point, concentration of capital ownership leads to an acceleration of investment—generated by smoothing consumption considerations of capital owners—and to a deceleration of total consumption—given that for a large fraction of the agents, consumption equals wage income, which slows down. When the model’s data is HP filtered, this behavior appears around the zero level, similarly as in the filtered actual data.
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Quantitatively, the model explains only a fraction of the observed behavior. However, the data simulated from the model mimics qualitatively quite well the acceleration of investment in actual data stressed in Figs. 1 and 6. The mechanism studied here may have interesting implications for policies aimed at providing alternative ways, maybe Pareto improving, of smoothing consumption of capital income earners. If the production function includes public infrastructure, which complements private capital, it may be interesting to consider the welfare implications of taxing capital income in booms to finance infrastructure investment. Compared to accelerated private investment in booms, which reduces the return on private investment later on, public investment would do the opposite. This consideration is of particular interest given the observed procyclicality of public investment, which seems to contradict the Keynesian-type wisdom that public investment should be used as a countercyclical tool.
Acknowledgments I thank Fatih Guvenen, Jeremy Greenwood, Yona Rubinstein, Matthew Shapiro, two referees and the editor of this journal for helpful discussions and comments, Pavel Livertovsky for research assistance, and specially Dmitri Byzalov for programming. Financial support from the Armand Hammer Fund at Tel Aviv University is also gratefully acknowledged.
References Equity ownership in America, 1999. Publication of the Investment Company Institute and the Securities Industry Association. Greenwood, J., Jovanovic, B., 1990. Financial development, growth, and the distribution of income. Journal of Political Economy 98, 1076–1107. Greenwood, J., Huffman, G.W., Hercowitz, Z., 1988. Investment, capacity utilization and the real business cycle. American Economic Review 78, 402–417. Hansen, G.D., Prescott, E.D., 2001. Capacity constraints, asymmetries, and the business cycle. Manuscript. King, R.G., Plosser, C.I., Rebelo, S.T., 1988. Production, growth and business cycles I. Journal of Monetary Economics 21, 195–232. Tauchen, G., 1986. Finite state Markov-chain approximations to univariate and vector autoregressions. Economic Letters 20, 177–181.
Concentration of capital ownership and investment fluctuations Zvi Hercowitz Tel Aviv University, Tel Aviv 61390, Israel Received 22 May 2003; revised 10 November 2003 Available online 20 February 2004
Abstract This paper is motivated by the observation that investment tends to accelerate when output is around trend. The model used to explain this observation is based on the capacity-constrained production setup in Hansen and Prescott [(2001) Manuscript], where capacity is constant over time, and on capital being owned by a fraction of the agents in the economy. When capacity is reached, the capital share increases because its component from capacity ownership becomes positive. The concentration of capital ownership leads then to an acceleration of investment—generated by the desire of capital owners to smooth consumption—as well as to a deceleration of total consumption. The results from the calibrated model contribute, although only partially, to the explanation of the observed behavior. 2004 Elsevier Inc. All rights reserved.
1. Introduction Figure 1 plots investment against GDP—both Hodrick–Prescott filtered—along with a fitted (fifth degree) polynomial. It can be observed that the 0,0 point on each axis is to the right and above the center. This follows from asymmetric cyclical behavior, as there are larger negative deviations than positive ones. In the present context, however, the main feature of the graph is the comovement between the two variables, as illustrated by the fitted line: Around the 0,0 point the slope is higher—suggesting an acceleration of investment that is formally tested later on. This acceleration of investment can not be explained by a standard RBC model, which generates an approximately linear comovement between investment and output. This linear comovement can be rationalized as follows. In that E-mail address: [email protected]. 1094-2025/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.red.2003.12.004
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Fig. 1. Fixed nonresidential investment and GDP (HP filtered) US data 1954:1–2001:2.
model, investment movements follow from consumption smoothing considerations of the representative agent and expected productivity levels. Both considerations depend on the degree of persistence of the productivity shocks, which is constant. The comovement suggested by Fig. 1 is studied here in the context of concentrated ownership of capital and a procyclical capital share. The theoretical framework is an extension of the representative-agent model in Hansen and Prescott (2001)—who study cyclical asymmetries in a framework with constant capacity—to two types of agents. The main characteristics of the present model are that (a) the capital share is more procyclical around the trend than away from it—given that when capacity is reached, the capital share adjust upwards because its component from capacity ownership moves then away from zero, and (b) capital income is concentrated in the hands of agents who have access to the capital market. Given these two characteristics, consumption smoothing of these agents generates an acceleration of investment around trend.1 Section 2 presents the basic, representative agent, model with the production setup from Hansen and Prescott, and Section 3 extends this framework by introducing concentrated ownership of capital. The solution, calibration and simulation of the model are reported in Section 4, and the comparison with actual US data is presented in Section 5. Section 6 1 It is an implication of applying a flexible detrending procedure, as the Hodrick–Prescott filter, which makes
high output levels—those corresponding to capacity—appear as located around trend. This technical aspect is addressed in Section 4.
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concludes, including comments about possible extensions and implications of the present setup.2
2. The representative agent model 2.1. The Hansen–Prescott technology Each firm has M identical locations or plants, with m M of them operated. Locations do not depreciate. In each location, the production function is θ φ y = zk n , φ, θ > 0, φ + θ < 1, if n n, 0, otherwise, where k and n are the movable capital and labor operating in each location and z is a productivity shock following a Markov chain with nz states with bounded values and transition matrix P .3 Given diminishing returns, there is a tendency to operate all locations. However, the restriction n n implies that m = min{M, N/n}, where N is the total input of labor. If N < Mn, some locations are idle. Production at the firm’s level can be seen as using three factors, total movable capital K, total labor input N , and the number of locations M, which is constant. Both K and N can be moved across locations at no cost. The production function of the firm is θ φ K N z m . (2.1) Y = zF (K, N; M) = max m m m=min(M,N/n) In the unconstrained range, where N/n < M, the desired number of plants to operate is smaller than the available number, and thus m = N/n. In this range, the production function takes the form K θ N φ Y =z N/n, N/n N/n or Y = zK θ N 1−θ (n)θ+φ−1 ,
(2.2)
which looks like a standard CRS formulation, with the term in n working as a TFP level. As n decreases, more locations are utilized, and the higher TFP follows from the diminishing 2 An alternative possible explanation of the comovement in Fig. 1 is that investment in capacity may increase as the constraint is approached. If capacity is identified with structures more than with equipment, the nonlinearity in Fig. 1 for structures only should be stronger. However, as reported in Section 5, addressing structures and equipment separately yields for both weaker nonlinear behavior than for total investment. 3 Given that the values of z are bounded, that locations do not depreciate, and a constant cost of installing each new location, Hansen and Prescott derive a constant M endogenously, in the stationary solution. Practically, this setup makes possible to treat M as given, since there is a value of the cost parameter that will make any value of M optimal.
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returns: Spreading the same K and N over a larger number of locations increases total output. In the constrained range, m = M, and the production function has the form θ φ N K M, Y =z M M or Y = zK θ N φ M 1−θ−φ .
(2.3)
Given that φ < 1 − θ , labor’s coefficient in this range is smaller than in the unconstrained range. Diminishing returns come into effect because all M locations are operated.4 A key feature of this technology is labor’s marginal productivity. From (2.2) and (2.3) it follows that θ K (n)θ+φ−1 , if N < Mn, (1 − θ )z N MPN = θ θ+φ−1 K N θ φ−1 1−θ−φ M = φz , if N > Mn. φzK N N M Because 1 − θ > φ, the marginal productivity of labor jumps downwards at N = Mn θ K l (n)θ+φ−1 from MPN = (1 − θ )z N θ K to MPN r = φz (n)θ+φ−1 . N Figure 2 illustrates the MPN schedule as a function of N , with z and K affecting its location, and the upward-sloping line represents a “labor supply schedule.” The figure shows a situation where N = Mn. The key feature of the Hansen–Prescott technology can be shown using this figure: consider, for example, an increase in z—holding K constant for simplicity. Both sections of the MPN schedule shift upwards together, and assume that the supply schedule does not move—as it will be the case with the utility function adopted below. Given the discontinuity in MPN at N = Mn, there is a range of z values for which the increase in z leaves N unchanged at Mn. This is illustrated in Fig. 3, where the flat part of the curve follows from the discontinuity in MPN. However, if z increases beyond a certain value, the supply schedule begins to be crossed by the lower part of the MPN schedule, and correspondingly, N increases with z beyond that value. This is shown in Fig. 3 by the upward sloping part on the right. The upward sloping part on the left is generated in a similar way when z declines below some value. The explanation of the investment behavior shown in the Introduction will be directly related to the nonlinearity in Fig. 3, i.e., to the flat part in the curve whose length depends 4 Note that under this technology, “capacity utilization” is unbounded. More K and N can always be allocated to already utilized locations. One may define N = Mn as “100% utilization,” and compute the average utilization rate in the model. However, the resulting figure would not be comparable with available data on capacity utilization, because the latter is based on 100% as the highest possible value.
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Fig. 2.
Fig. 3.
on 1 − φ − θ . This fraction determines the coefficient of M in output once all location are utilized.5 Finally, the evolution of movable capital has the standard form K = K(1 − δ) + I,
0 < δ < 1,
where I is gross investment. As mentioned above, M does not depreciate. 5 Two main factors determine the degree of cyclical asymmetry in this model: (1) the magnitude of 1 − φ − θ , which affect the length of the flat part in Fig. 3, and (2) the location of the distribution of N relative to Mn. The further to the left this distribution is, the lower the probability that capacity is reached and thus the smaller the degree of asymmetry. In the calibration of the model, the non-stochastic N is set equal to Mn.
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2.2. Preferences The preferences adopted have two important advantages in the present context: (a) labor is independent of wealth, and (b) labor varies even when the individual is disconnected from the capital market. Feature (a) facilitates direct aggregation of labor across individuals with different wealth levels. Feature (b) is crucial in the present context, which deals with cyclical fluctuations when a fraction of the agents does not participate in the capital market (Section 3). These preferences are described by 1−σ
∞ η 1+γ t n β ct − (1 − σ ), σ, η, γ > 0, 6 E 1+γ t t =0
where c and n are individual consumption and labor, respectively. With these preferences, the individual’s labor decision in a competitive setup is 1/γ 1 n= , (2.4) w η where w is the real wage. This utility function has the property that labor effort is independent of wealth. In particular, K and z affect labor effort only through the real wage.7,8 2.3. Competitive equilibrium The economy is composed of a large number of identical firms, and a large number of identical households, who are the stock holders. The firms purchase labor and capital services from the households, and pay dividends to them. 6 This utility function is not consistent with stationary labor when z and thus real wages increase over time. However, it can be adapted to allow for stationary n in a growing economy by introducing technological progress also into home activities as follows: (1 − n)1+γ 1−σ
(1 − σ ). u(c, n) = c + z 1+γ
This modification preserves the independence of n from wealth, while the stationarity of n follows from z affecting equally both home and market productivity of labor. 7 Given that z and K do not affect n directly, the labor supply schedule in Fig. 2 under this utility function would not move when z goes up—as it was assumed above in the discussion of Figs. 2 and 3. 8 The standard utility function u(c, 1 − n) = c1−σ v(1 − n)/(1 − σ ), σ > 0, v > 0, v < 0 (King et al., 1988) has undesirable implications in the present context of partial participation in the capital market. For individuals who do not participate, and hence c = w · n, the first-order condition for labor implies that ∂v(1 − n) w 1 = (1 − σ )v(1 − n) = (1 − σ )v(1 − n). ∂(1 − n) c n Therefore, n is constant, and non-participants do not change labor hours over the business cycle. Hansen’s utility function u(c, n) = ln c − γ n, γ > 0, has the same implication. For non-participants, n = 1/γ .
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The household’s problem has the standard form: V (z, k, K; M) = max u r(z, K; M)k + w(z, K; M)n k ,n
+ v(z, K; M)M − k + k(1 − δ), n + βEV (z , k , K ; M) ,
where r(z, K; M)k, w(z, K; M) and v(z, K; M) are the rental price of capital, real wage and dividends, respectively, determined by the aggregate variables. The maximization of profits by the firms implies equalization of w(z, K; M) and r(z, K; M) to the corresponding marginal products, while dividends are 0, if N < Mn, v(z, K; M) = (1 − θ − φ)zK θ N φ M −θ−φ , if N Mn. Equilibrium also requires that
1 w(z, K; M) n= η k = K,
1/γ = N,
K = K(1 − δ) + zF (K, N; M) − C, where C is aggregate consumption.
3. The model with concentration of capital ownership The following distinction between two types of households is now introduced: Type 1 households are similar to those in the basic economy, i.e., work and have access to the capital market (fraction ω of the population). Type 2 households do not own capital and do not have access to the capital market, i.e., their consumption equals their wage income (fraction 1 − ω of the population). This assumption implies that the two types of household cannot interact in the capital market, and that Type 1 agents own both K and M in the economy. Hence, these agents receive all capital income, defined as the sum r(z, K; M)k + v(z, K; M)M.9 The two types have identical utility functions, but the problems they solve are different. Type 1 households face the same problem as those in the basic setup, while Type 2 solve a static problem. The distinction between these two types of households has two components: (1) concentration of capital ownership, and (2) lack of access to the financial market to those who do now own capital. 9 What is really important for the model is that the ownership of M is concentrated on a fraction of the agents,
since dividends are the volatile component of capital income. The assumption that Type 1 agents own also all K is convenient, as then Type 2 agents simply do not participate in the capital market.
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This situation could arise endogenously if the access to the capital market is costly, as in Greenwood and Jovanovic (1990)—where those agents with low income and thus little to gain by smoothing consumption opt for not joining a financial intermediation network. In that model this cost is fixed, and hence with income growth all agents eventually join the financial network. However, if this cost grows over time, say, proportionally to the average real wage, individuals who start off without assets may never find it profitable to join the capital market and own capital. Hours of work of Type 1 are denoted by h, and hours of Type 2 by n. Consumption of the two groups is described by: Type 1: x = r(z, K; M)k + w(z, K; M)h + v(z, K; M)M − k + k(1 − δ). Type 2: c = w(z, K; M)n. The problem of a household of Type 1 is the same as the representative household’s problem above. The problem of a household of Type 2 is
max u w(z, K; M)n, n . n
Similarly as for Type 1, the solution for labor is 1/γ 1 w(z, K; M) . n= η
(3.1)
Substituting into the budget constraint yields 1/γ 1 w(z, K; M)1/γ +1. c= η The remaining equilibrium conditions are the same as in the representative agent setup, except that here 1/γ 1/γ 1 1 + (1 − ω) w(z, K; M) N = ω w(z, K; M) η η 1/γ 1 w(z, K; M) = , (3.2) η K = K(1 − δ) + zF (K, N; M) − ωx − (1 − ω)c. 3.1. Algebraic characterization of output and the capital share The utility function adopted makes possible to compute a closed-form solution for output. The following derivation focuses on the range of output for which labor is constant at N ∗ ≡ Mn (the flat part of the schedule in Fig. 3), which is particularly relevant for confronting the model with the data. Using (3.2), w∗ can be defined as w∗ = η(N ∗ )γ . From the production function in (2.2) and (2.3), when N approaches N ∗ from the left and from the right, the corresponding output levels are, respectively, Y l ≡ zK θ (N ∗ )1−θ (n)θ+φ−1 ,
(3.3)
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and Y r ≡ zK θ (N ∗ )φ M 1−θ−φ .
(3.4)
The K, z plane can be divided into three regions for which (1) N < Mn, (2) N = Mn, and (3) N > Mn. Between regions (1) and (2), the critical condition is MPN l = (1 − θ )zK θ (N ∗ )−θ (n)θ+φ−1 = w∗ . Hence, the combinations of K and z separating regions (1) and (2) satisfy zK θ =
w∗ (N ∗ )θ (n)1−θ−φ . (1 − θ )
Substituting this expression into (3.3) yields Yl =
w∗ N ∗ . (1 − θ )
(3.5)
Between regions (2) and (3), the critical condition is MPN r = φzK θ (N ∗ )φ−1 M 1−θ−φ = w∗ . Therefore, the combinations of K and z separating these two regions are given by zK θ =
w∗ ∗ 1−φ θ+φ−1 (N ) M . φ
Substituting this expression into (3.4) yields Yr =
w∗ N ∗ . φ
(3.6)
From (3.5) and (3.6) it follows that the range of output variation in the flat range of N is given by Yr 1−θ . = l Y φ
(3.7)
The division of output levels into the three ranges Y Y l , Y l < Y < Y r and Y Y r will be useful later on when comparing the model to the data, and it can be used to determine the cyclical behavior of the capital share (income from K and M). The capital share in the lower output region is θ , and in the higher output region is 1 − φ (higher than θ because of decreasing returns). In the intermediate region, the wage bill is constant at w∗ N ∗ = (1 − θ )Y l , but output increases with z and K. Hence, as Y increases between Y l and Y r , the capital share increases from θ to 1 − φ according to (1 − θ )Y l Y − w∗ N ∗ =1− . Y Y
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4. Solution of the model and simulation 4.1. Choice of parameter values 1. Technology. Following Hansen and Prescott (2001): P is computed using Tauchen’s (1986) discrete state-space procedure so as to approximate the process ln zt = ρ ln zt −1 + εt , εt ∼ N(0, σε2 ), with ρ = 0.95, σε = 0.007, nz = 15, where nz is the number of discrete z values, evenly spaced between ±2 standard deviations around zero, corresponding to the non-stochastic steady state. M = 0.315, n = 1, δ = 0.019, θ = 0.38, φ = 0.598. This value is obtained by fitting the simulated asymmetry in output, as measured by the ratio between the average deviations above the HP trend to the average deviation below, to the actual ratio of 0.96 in the 1954:1–2001:2 sample. This, given that the non-stochastic N is set equal to Mn—the point at which all locations are used at the minimum positive level.10 See next for the parameterization of preferences in this respect. 2. Preferences β = 0.99. From Greenwood et al. (1988): σ = 2 (the results are very similar with σ = 1), γ = 0.6, η = 6. This value is chosen so that the non-stochastic N is 0.315—the lowest level of labor for which all locations are utilized. Note that with the current utility function, N is not a fraction of time but represents an index. 3. Fraction of agents who own capital. The parameter ω is calibrated so that the skewness of capital ownership in the model equals actual skewness of stock holdings 10 Correspondingly, the production function used for the non-stochastic solution is that in Eq. (2.2).
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across households in the US. The resulting value is ω = 0.03.11 The basic version with ω = 1 is also computed, as well as an additional version with more moderate ownership concentration, ω = 0.2. 4.2. Solution and simulation procedures The model is solved as follows: For each discrete value of z, a piecewise linear approximation to the dynamic decision rule of Type 1 agents as a function of K is fitted— given that Type 2 agents consume all their wage income, and that the resource constraint of the economy and the equilibrium condition in the labor market are satisfied. The grid for K has 200 points within the range of ±20 percent around the non-stochastic steady state. The criterion of convergence of the solution is that the Euler equation for Type 1 agents holds up to a tolerance error. The model is then used to simulate artificial data. First, 1000 observations are simulated and plotted to illustrate how the model works. Then, 500 simulations of 190 periods— the number of quarters in the 1954:1–2001:2 sample—are computed to obtain average statistics that can be compared with those from the actual data. 4.3. Simulation results Figure 4 shows plots of 1000 simulated data points for the comovement of investment and consumption with output for ω = 1—the basic version—and for the main version with ω = 0.03. Let us start with the basic version on the top two graphs, where capital is equally owned by all agents. One can discern there some steepening of the investment profile and flattening of the consumption profile, approximately between two output levels—which were denoted above as Y l and Y r . This pattern follows from a basic implication of the technology: Between Y l and Y r the share of dividends in output increases from zero to 1 − θ − φ, while the share of wages declines from 1 − θ to φ (for Y > Y r , these shares remain constant at the Y r levels). Correspondingly, between those two levels of output the composition of income changes in favor of a more volatile component. Consumption smoothing leads then to weaker consumption and stronger investment responses. 11 Given population of size 1 and total capital ownership E, the mean equity holdings in the model is ω ·(E/ω) = E, and the variance is (1−ω)(−E)2 +ω(E/ω −E)2 = E 2 ((1 − ω)/ω). Thus, the standard deviation
√ is sdev = E (1 − ω)/ω. Then, skewness = (1 − ω)
3/2 E/ω − E 3 ω 1 − 2ω −E 3 1 − 2ω . +ω = (1 − ω) =√ sdev sdev 1−ω ω2 (1 − ω)ω
The actual skewness is computed using the distribution published in “Equity Ownership in America,” Fig. 7, with 10 groups of households, according to their equity holdings in 1999. Each group is defined as a range of equity holdings: (0), (less than $10,000), ($10,000–25,000), . . . , ($1 million or more). In the present calculation, the middle of the range is taken for each group, except for the top group, whose average holdings is inferred using the overall average holdings of $80,500. The computed skewness is 5.5. Then, ω is obtained by solving the √ equation (1 − 2ω)/ (1 − ω)ω = 5.5. There is only one solution in the positive range, which is ω = 0.03.
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Fig. 4. Basic version: ω = 1. Main version: ω = 0.03.
Concentration of capital ownership, on the bottom two graphs, magnifies the acceleration of investment and deceleration of consumption. The mechanism at work here is that the change in shares between Y l and Y r is accompanied by a concentration of dividends on a small fraction of the agents, exacerbating their consumption smoothing problem and leading to a stronger investment response. At the same time, wage income and thus consumption of most agents remain constant, thereby flattening the profile of aggregate consumption. In the case of no capacity constraint, i.e., when 1 − θ − φ = 0, the comovement of investment and consumption with output (not shown) is practically linear, both for ω = 1 and for ω = 0.03. Hence, concentration of capital ownership generates the nonlinear behavior stressed above only when interacting with procyclical dividend income. 4.4. Detrending cyclically asymmetric data Figure 4 cannot be contrasted directly with actual data because of the nonstationarity of actual variables. The procedure adopted here is common in this context: to HP filter
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Fig. 5. Basic version: ω = 1. Main version: ω = 0.03.
both actual and artificial data and then compare. The filtering of the model’s variables also provides guidance on how to interpret detrended data from an economy subject to a capacity constraint.12 The artificial data shown in Fig. 4 are HP filtered and plotted in Fig. 5. Figure 5 displays the same phenomenon shown in Fig. 4, although it appears here in a different form. Starting with consumption, on the right-side graphs, note the flatter “lines” crossing the steeper “lines” at the 0,0 point. The mechanism generating this pattern is a technical one, but it is important for the comparison of the model’s behavior to the data. In periods when the capacity constraint does not bind—which tend to be prolonged because of the serial correlation of the shock—filtering yields approximately symmetric movements in consumption and output around zero. This is reflected in the longer lines centered on the 0,0 point. In periods when the capacity constraint does bind, output fluctuates most of the time in the range Y l −Y r , in which consumption movements are smaller, as seen in 12 The same would apply to any other flexible detrending procedure.
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Fig. 4. Because these periods also tend to be prolonged, filtering centers those fluctuations around zero as well. Hence, the crossing flatter lines at the 0,0 point represent those periods when the capacity constraint binds and output is in the Y l −Y r range. The cases when output is higher than Y r are likely to be represented by the points away from the crossing lines. Regarding investment, the effect of filtering is similar, but with higher slopes in the constrained periods. The main point from Fig. 5 is, therefore, that in the filtered data, the acceleration of investment and deceleration of consumption appear around the zero points—instead of at high values of output when the data is in levels.
5. Comparison with actual data 5.1. Basic cyclical statistics Table 1 reports basic cyclical statistics of actual and artificial data, both HP filtered, focusing here on the volatility of the main variables—while the nonlinearity is addressed in the next subsection. The statistics from the model are averages over 500 simulations of 190 data points each (the number of quarters in the 1954:1–2001:2 sample). Table 1 shows that the standard deviations in the model are lower than in actual data. The low volatility is due, partly, to the capacity constraint. Eliminating the capacity constraint— i.e., setting φ = 0.62 so that φ + θ = 1—yields the statistics shown in Table 2.13 The standard deviations there are larger than in Table 1, although still lower than in actual data. However, if γ (determining labor elasticity) is considered a free parameter, one can compute the value that equates the volatility of output to that in the data. The result is γ = 0.38—corresponding to an elasticity of labor supply to the real wage of about 2.6. The concentration of capital ownership—moving from ω = 1 to ω = 0.03 in Tables 1 and 2—does not affect much output volatility, but reduces the volatility of investment and increases that of consumption. This follows from the fact that lowering ω increases the fraction of agents who cannot invest. This can be described as the general effect Table 1 Basic statistics Variable
US
Model ω=1
1954:1–2001:2
Output Investment equipment structures Consumption
ω = 0.03
St. dev. %
Corr. w/output
St. dev. %
Corr. w/output
St. dev. %
Corr. w/output
1.63 4.64 5.28 4.76 1.29
1.00 0.79 0.85 0.48 0.88
1.30 3.05 – – 0.76
1.00 0.99 – – 0.98
1.31 2.48 – – 0.96
1.00 0.94 – – 0.96
13 The symbols S and S in Table 2 are defined and used in Section 5.2.1. α
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Table 2 Model without capacity constraint ω=1
Variable
Output Investment Consumption
ω = 0.03
St. dev. %
Corr. w/output
S
Sα /S
St. dev. %
Corr. w/output
S
Sα /S
1.47 3.13 0.92
1.00 1.00 1.00
– 2.13 0.63
– 1.00 1.00
1.47 2.20 1.23
1.00 1.00 1.00
– 1.50 0.83
– 1.00 1.00
of concentrating capital ownership, regardless of whether there is a capacity constraint or not. However, comparing the effects of lowering ω in Table 1 to those in Table 2 shows that when there is a capacity constraint, these effects diminish. In other words, when concentration of capital ownership interacts with procyclical capital income, the acceleration of investment and deceleration of consumption stressed here work in the opposite direction to the general effect of reducing ω. 5.2. Nonlinear cyclical behavior Figure 6 displays the comovement of investment in equipment and structures, separately, with output, along with fitted polynomials. Total investment is displayed in Fig. 1. Figure 7 does the same for consumption. Judging from the fitted lines, it can be observed that: (1) Both types of investment tend to accelerate when output is around the zero level. The behavior of structures is more pronounced in this respect, but the dispersion of the observations around the fitted line is larger, relative to equipment. (2) The slope of the consumption profile declines somewhat when output is around the zero level.
Fig. 6.
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Fig. 7. Consumption and output (HP filtered). US data 1954:1–2001:2.
These features can be summarized quantitatively by estimating two slopes for each variable: One for the entire range of output variation and the other for some range around zero to be determined below, and testing whether the two are different. Then, these results can be compared with those from the model. 5.2.1. Statistics of nonlinear behavior The comovement of x—HP filtered log(X)—with output in the present context can be characterized by the regression coefficients of x on y over the entire sample and in a reduced sample in which |y| < α. These slope statistics are denoted as S(x) and Sα (x), respectively. An indication about appropriate values for α can be obtained from Eq. (3.7), which says that Y l and Y r are separated by the factor (1 − θ )/φ = 0.62/0.598 ≈ 1.037. Hence, the corresponding value for α is about 0.0184, half of the percentage difference. The slopes coefficients can be estimated by running regressions of the type: x on a constant, y, dα and dα ∗ y, where dα is a dummy variable for |y| < α. The coefficient of y corresponds to S(x) and the coefficient of dα ∗ y to Sα (x) − S(x). The results below correspond to α = 0.0165, which within the range 0.015−0.02 yields the highest significance levels for the coefficients of dα ∗ y in the actual data. This value of α is then applied also to the model. The OLS estimates of the slopes with US quarterly data, and the corresponding coefficients from the model, are reported in Table 3. The actual data estimates are consistent with the hypothesis of accelerating investment, but their statistical significance is likely to be overstated given serial correlation in the regression residuals. To overcome this problem, the regression is reestimated using annual averages of the quarterly filtered data. Specifically, the quarterly variables x, y and dα are converted to annual frequency, implying that dα has now five possible values instead of two (1: if |y| < α in the four quarters of the year; 0.75: if |y| < α in three quarters of the year, and so on). The number
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Table 3 Slope statistics—quarterly frequency Variable
US
Model ω=1
1954:1–2001:2 Investment equipment structures Consumption
ω = 0.20
ω = 0.03
S
Sα /S
S
Sα /S
S
Sα /S
Sα
Sα /S
2.11 2.65 1.20 0.73
1.38 (0.010) 1.22 (0.054) 1.84 (0.027) 0.86 (0.125)
2.38 – – 0.57
1.03 – – 0.95
2.01 – – 0.71
1.06 – – 0.94
1.92 – – 0.74
1.09 – – 0.93
Note. In parentheses: significance level of Sα − S. Table 4 Slope statistics—annual averages Variable
US
Model ω=1
1954–2000 Investment equipment structures Consumption
ω = 0.20
ω = 0.03
S
Sα /S
S
Sα /S
S
Sα /S
Sα
Sα /S
2.02 2.61 1.01 0.76
1.74 (0.041) 1.45 (0.082) 2.98 (0.079) 0.90 (0.644)
2.24 – – 0.59
1.08 – – 0.89
1.76 – – 0.75
1.18 – – 0.85
1.66 – – 0.78
1.21 – – 0.85
Note. In parentheses: significance level of Sα − S.
of observations is obviously reduced, but time aggregation internalizes residual serial correlation within the year, while part of the quarterly information is preserved by the annual average of dα .14 Table 4 reports the corresponding slope statistics from the data and from the model. The acceleration of investment in actual data—represented by the ratios Sα (x)/S(x)— is quantitatively large. This result is statistically significant at the 5% level for total investment, although not for equipment and structures separately. This seems consistent with the present mechanism, which focuses on investment in general as vehicle for consumption smoothing. Regarding consumption, the deceleration is far from statistical significance. The investment-output plot with annual averages—counterpart of Fig. 1— is shown in Fig. 8, where the triangles represent the fitted polynomial. The investmentoutput comovement in this figure is similar to that with quarterly data. For the model, Table 4 shows the Sα (x)/S(x) statistics for ω = 1, ω = 0.2 and ω = 0.03.15 The model does contribute to the explanation of the observed behavior, but the magnitudes are smaller than in actual data. The results for the model in the table quantify the characteristics shown graphically in Figs. 4 and 5. Part of the acceleration 14 Two alternative procedures were attempted to deal with the serial correlation problem. One was to model residual serial correlation directly, as in a standard structural regression. The estimates of Sα (x) − S(x), using AR(2) or ARMA(1,1) processes to achieve white noise residuals, had the expected signs, but were statistically insignificant. The other was to conduct the tests on filtered annual data—rather than using the annual averages of quarterly filtered data. This procedure did not eliminate residual serial correlation in the regressions. 15 The statistics from the model are averages of 500 simulations of 190 periods.
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Fig. 8. Fixed nonresidential investment and output (HP filtered—annual averages). US data 1954–2000.
of investment is due to procyclical capital income itself—column ω = 1—which yields Sα (x)/S(x) = 1.08. Concentration of capital ownership, in columns ω = 0.2 and ω = 0.03, magnifies this effect to Sα (x)/S(x) ratios of 1.18 and 1.21, respectively, compared to 1.74 in actual data. Concentration of capital ownership—i.e., reducing ω—by itself does not generate nonlinear investment/output comovement. Table 2 shows the Sα (x)/S(x) ratios for the constant-returns economy (θ + φ = 1), which equal 1 for both ω = 1 and ω = 0.03.
6. Concluding comments This paper is motivated by the observation, shown in Fig. 1, that investment tends to accelerate when output is around the HP smoothed level. This acceleration is statistically significant at the 5% level for total investment, while separately for equipment and structures it is significant only at the 10% level. The deceleration of consumption is not statistically significant. The model used to explain this observation is based on the capacity-constrained production setup in Hansen and Prescott (2001), and on capital being owned by a fraction of the agents in the economy. In the model, when output increases beyond the critical point where all available capacity is utilized at the minimal level, dividends become positive and the real wage begins to lag behind labor’s marginal productivity. Beyond that point, concentration of capital ownership leads to an acceleration of investment—generated by smoothing consumption considerations of capital owners—and to a deceleration of total consumption—given that for a large fraction of the agents, consumption equals wage income, which slows down. When the model’s data is HP filtered, this behavior appears around the zero level, similarly as in the filtered actual data.
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Quantitatively, the model explains only a fraction of the observed behavior. However, the data simulated from the model mimics qualitatively quite well the acceleration of investment in actual data stressed in Figs. 1 and 6. The mechanism studied here may have interesting implications for policies aimed at providing alternative ways, maybe Pareto improving, of smoothing consumption of capital income earners. If the production function includes public infrastructure, which complements private capital, it may be interesting to consider the welfare implications of taxing capital income in booms to finance infrastructure investment. Compared to accelerated private investment in booms, which reduces the return on private investment later on, public investment would do the opposite. This consideration is of particular interest given the observed procyclicality of public investment, which seems to contradict the Keynesian-type wisdom that public investment should be used as a countercyclical tool.
Acknowledgments I thank Fatih Guvenen, Jeremy Greenwood, Yona Rubinstein, Matthew Shapiro, two referees and the editor of this journal for helpful discussions and comments, Pavel Livertovsky for research assistance, and specially Dmitri Byzalov for programming. Financial support from the Armand Hammer Fund at Tel Aviv University is also gratefully acknowledged.
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