Extensive and Intensive Investment over the Business Cycle∗ Boyan Jovanovic and Peter L. Rousseau February 2014
Abstract Investment of U.S. firms responds asymmetrically to Tobin’s Q: investment of established firms — ‘intensive’ investment — reacts negatively to whereas investment of new firms — ‘extensive’ investment — responds positively and elastically to . This asymmetry, we argue, reflects a difference between established and new firms in the cost of adopting new technologies. A fall in the compatibility of new capital with old capital raises measured and reduces the incentive of established firms to invest. New firms do not face such compatibility costs and step up their investment in response to the rise in The model is of an endogenous-growth type and has two aggregate shocks: one to TFP and one to the cost of investment. The model fits the data reasonably well using aggregates since 1900.
1
Introduction
Extensive labor supply — movement in and out of the labor market — is more wage elastic than intensive labor supply — the hours of employed workers. An even starker contrast exists between the response to aggregate Tobin’s of extensive investment — capital formation by entering firms and young firms — and the response of intensive investment — capital formation by older firms and listed firms. Not only is extensive investment more -elastic than intensive investment, but there is also a qualitative difference: When aggregate rises, new firms and young firms raise their investment, whereas older, publicly listed firms reduce theirs. And since large, old firms dominate investment and , the relation between aggregate and investment is negative. ∗ We thank A. Abel and R. Hall for data, G. Alessandria, W. Brainard, J. Lerner, A. Ljungqvist, R. Lucas, V. Midrigan, D. Nandy, G. Violante, L. Wong and D. Xu for comments, S. Bigio, S. Flynn, G. Navarro, H. Tretvoll and V. Tsyrennikov for research assistance, and the National Science Foundation as well as the the Ewing Marion Kauffman Foundation for financial assistance.
1
These facts conflict with the convex adjustment model of investment and its variants. In such models, a positive productivity shock raises both and investment (Prescott 1980, Jovanovic 2009, fig. 4). The same happens in models in which investment is irreversible and in which this constraint sometimes binds (Sargent 1980, fig. 2). In these models the relation between aggregate and investment is positive, and the negative relation between investment and aggregate is a puzzle. To explain the puzzle we use a model in which movements in aggregate are caused by changes in the cost of capital that affect continuing projects but not new ones. Technological change raises the cost of making new capital compatible with capital already in place. Fitting new wiring or new equipment into an old building originally designed for something else is costly. Retraining workers who originally were trained to do something else is also costly. If the arrival of new technology precedes high- periods, then a high is the result of high costs of adjustment that reflect this incompatibility. Our distinction between extensive investment (in new projects) and intensive investment (in continuing projects) is related to one made by Acemoglu, Gancia, and Zilibotti (2012) between innovation and standardization costs. Similarly, Justiniano, Primiceri, and Tambelotti (2011) distinguish shocks to the supply of capital goods from shocks to the (convex) investment adjustment cost for installed capital, and find that the latter play a large role in explaining fluctuations. In a steady state model Chari and Hopenhayn (1991) stress incompatibility between human capital and new technology. Jovanovic (2009) relates the implementation cost to a depletion of ideas that probably occurs as a firm ages. Prusa and Schmitz (1991, 1994) find that firms pursue less radical forms of innovation as they age, presumably because they face such compatibility, adaptability and standardization costs. Relatedly, Asker, Farre-Mensa, and Ljungqvist (2011) argue that as firms age and eventually go public, governance problems impair their investment incentives. Chirinko and Schaller (1995, table 1) find that old firms tend to be in older manufacturing industries which, in the U.S. at least, have been declining, and Carroll and Hannan (2000) find that within industries, old firms make old, standardized products. If aging reduces a firm’s ability to react to new technology, rapid technological change should shift production and value shares toward younger firms. In the U.S. such a displacement is clear during the electrification and diesel epoch, which saw rapid growth of new-technology sectors and a surge of IPOs, particularly in those sectors.1 In our one-sector model, we interpret the investment-specific shock as a cost of making new and old capital compatible, or as a cost of adapting old processes to new technology. Yorukoglu (1998) interprets his investment-cost shock similarly. Such shocks are not likely to be reflected fully in the price of machinery or structures, 1
Figures 14 and 15 in Jovanovic and Rousseau (2005) document the overall rise of IPOs during the electricfiation and IT epochs, and figures 16 and 17 show that IPOs predominated in high-technology sectors at these times.
2
but they will find their way into equilibrium values of firms.2 The model is of an type. The model includes two aggregate shocks: a TFP shock and an investment-cost shock, and in this regard the model has precursors in Greenwood, Hercowicz, and Krusell (1997), Fisher (2006), and Justiniano and Primiceri (2008). Gilchrist and Williams (2000) and Campbell (1998) also estimate vintage-capital models closely related to ours, but in these two models, once implemented, the capital that a project embodies cannot be augmented at a later date. Bilbiie, Ghironi, and Melitz (2012) model extensive investment in new product lines that incumbents as well as entrants can undertake, and intensive investment whereby capital in each project can later be augmented; intensive investment entails diminishing returns and responds less elastically to aggregate shocks than does extensive investment. Papanikolaou (2011) argues that high- firms enjoy high valuations because their focus on producing investment goods makes them better able to implement technologies, which allows them to pay dividends when aggregate consumption is low. We build on these contributions, and our paper adds in three ways. First, it identifies and explains a new puzzle, namely the qualitatively different response — positive for entrants, negative for incumbents — of investment to changes in aggregate . Second, it analyzes investment by venture capitalists and tries to match these and other investment series quantitatively. Third, the model contains two risky assets: public equity and “private equity,” the latter paying dividends based on the IPO performance of entering firms, all of which trade publicly, in a way that roughly corresponds to the institutional characteristics of private and public firms. A privateequity premium or a discount arises depending on whether the investment-cost shock or the TFP shock matters more. Finally, its long-run growth rate is endogenous, and the model is solved analytically. In our model, the value of capital in place is determined solely by the cost of intensive investment and when that cost rises, there is a substitution towards extensive investment (which is not subject to the cost shock) and a rise in the stock market and in . This is the mechanism that explains the asymmetry. Of the two assets, one is a claim to dividends on the portfolio of all continuing projects run by incumbent firms. The other is a private-equity fund that finances all new projects and sells them to incumbent firms or equivalently launches IPOs in the form of new firms. After analyzing the dichotomy between new-firm and incumbent investment, we ask whether it matters for aggregate investment and for policy. While entrants’ investment is small as a fraction of total investment, its standard deviation is nearly 2
As Yorukoglu (1998, fn. 5) puts it, “...when one wants to use advanced software on an old computer, one cannot obtain the full benefit of this advanced software, since typically all of its applications cannot be used with the old computer.” An incompatibility of this kind will not be fully reflected in the price of the new software unless they are produced in perfectly competitive markets and they have the same costs of production. 3
40 percent that of incumbent investment. In the model, investment by new firms is facilitated by a spillover from the capital of incumbents, and this spillover generates underinvestment in equilibrium. The presence of extensive investment signals that the equilibrium rate of growth is lower than a benevolent planner would choose, but we show that the shortfall is small. The next section presents the evidence that motivates the paper, and Section 3 presents the model. Section 4 fits the model to time series of investment — extensive and intensive — and Section 5 contrasts it to the standard model with just one type of investment. Section 6 considers some extensions and discusses the model’s asymmetry assumption, and Section 7 concludes. Three appendixes contain data descriptions, robustness checks and proofs.
2
Evidence on Investment and Q
The patterns that we outline above can be seen in both aggregate and firm-level data. Here we summarize that evidence. Investment of Compustat firms is negatively related to aggregate .–Standard and Poor’s Compustat database consists of public firms that we generally think of as incumbents. The investment rates of these exchange-listed firms depend positively on their own ’s but negatively on aggregate . Figure 1 illustrates this, where the left panel shows a mildly positive relation between the log of firm-level investment rates and the log of firm-specific ’s in a pooled ordinary least squares regression using data from 1962 through 2006.3 This is consistent with the standard -theory of investment, which suggests that high- firms will have higher desired investment rates than low- firms. Interestingly, the right panel shows our new fact that investment rates of the same Compustat firms respond negatively to aggregate , with the observations on the horizontal axis ordered by the value-weighted annual averages of the firm-specific ’s.4 Table B1 in Appendix B shows that the relationships in figure 1 are robust to the inclusion of fixed effects for firms, years, and four-digit standard industry classifications (SICs), and to estimation by GMM to control for potential measurement error in Finally, figures 10 and 11 in Appendix B schematically depict the two panels of figure 1. 3
T-statistics based on robust standard errors are in parentheses. See Appendix A for the sources and methods used to construct all data items used in our empirics. 4
For the two slopes to have opposite signs, cross-section variation in must be sufficiently larger than the time-series variation in aggregate . The extension of our model in Section 6.1 illustrates this point, and we show how it can arise statistically in Appendix B, part 2 (see especially figures 10 and 11). The negative relation between a firm’s investment and aggregate Q remains strong when we regress the log of investment on the level of Q, and retains its negative sign but is not always statistically significant when, as in Hayashi (1982), all variables enter the regression in levels. 4
Figure 1: Firm-level investment rates and Q, Compustat sample, 1962-2006. The same contrast emerges in another set of regressions that use the same Compustat data displayed in figure 1. Firm ’s investment in year as a percentage of its capital stock at the end of the previous year is denoted by −1 , aggregate at the end of the previous year is denoted by −1 , and firm ’s log deviation from it is ln(−1 )−ln(−1 ). Estimation is again by OLS. The estimates and t-statistics are ln( −1 ) = 2519 − 0167 ln(−1 ) + 0573[ln (−1 ) − ln(−1 )] (2426)
(−111)
(913)
(1)
with 2 =0.10 and 141,956 firm-level observations.5 The effects of fluctuations in aggregate on firm investment are large. For example, an increase in aggregate −1 from a standard deviation below its mean to a standard deviation above it implies a more than 50 percent reduction in the firm’s investment ratio .6 In other words, such a change in aggregate cuts the investment rate of the average firm approximately in half. 5
When we regress the investment rate on ln(−1 ) and ln(−1 ) separately to relax the subtractive linear constraint, the coefficient on ln(−1 ) becomes even more strongly negative at -0.740 with a t-statistic of -46.2. 6
We compute this as + = −
¶−0167−0573 µ µ¯ ¶−074 + 139 + 063 = = 0483 ¯− 139 − 063
5
Table 1 Investment Regressions, 1962-2006 and 1978-2006 Dependent variable: log of firm investment as percent of capital (I /K−1 ) 1962-2006 1978-2006 log(Q −1 )
0.599 (83.6)
log(Q −1 )
0.582 (82.4) -0.421 (26.7)
-0.785 (49.7)
log(Q −1 )-log(Q −1 )
0.591 (76.5) -0.203 (13.3)
0.578 (75.3) -0.422 (26.7)
-0.767 (44.1)
0.582 (82.4)
-0.189 (10.9) 0.578 (75.3)
4-digit SIC effects
yes
yes
yes
yes
yes
yes
yes
yes
Year effects
yes
n.a.
n.a.
n.a.
yes
n.a.
n.a.
n.a.
0.180
0.062
0.142
0.142
0.183
0.062
0.143
0.143
2 No. of observations
141,956
117,317
Note. — T-statistics based on robust standard errors are in parentheses. The independent variables Q −1 are firm-specific Tobin’s Qs measured at the end of the previous year, and the Q −1 are annual market value-weighted means of Tobin’s Qs for all Compustat firms. Estimation is by OLS.
With fixed effects for two-digit SIC industries, the coefficient for −1 falls to -0.203, suggesting an even stronger negative elasticity of investment to aggregate , and the coefficient for ln(−1 )−ln(−1 ) rises slightly to 0.582.7 Both coefficients remain highly significant statistically and the 2 rises to 0.14. Table 1 shows that the positive own -elasticity and negative aggregate -elasticity of investment found for the 1962-2006 Compustat sample remains when we begin the sample in 1978, which corresponds to when the coverage of firms in Compustat widens substantially.8 7
Tables 1, 2, and B1 include specifications with fixed effects for four-digit SIC industries because the difference between firm ages is clearly correlated with industry (i.e., young firms tend to be in high-technology industries), and so the fixed effects aim to capture differences in responses or measurement errors by industry. 8
All coefficients in Tables 1 and 2 remain statistically significant at the one percent level when we cluster standard errors by firm. For example, the T-statistics in the second row of Table 1 become 17.1, 32.1, 8.5, 18.3, 29.3, and 7.2, and those in the 6
Table 2 Investment Regressions Based on Years Since Compustat Listing, 1962-2006 Dependent variable: log of firm investment as percent of capital (I /K−1 )
≤ 2 years
2 years
≤ 3 years
3 years
log(Q −1 )
0.152 (4.0)
0.096 (2.4)
-0.271 (16.6)
-0.276 (16.8)
0.143 (4.5)
0.112 (3.4)
-0.327 (19.4)
-0.325 (19.2)
log(Q −1 )-log(Q −1 )
0.545 (40.9)
0.532 (36.2)
0.553 (79.7)
0.573 (73.1)
0.546 (48.9)
0.542 (43.7)
0.552 (75.4)
0.577 (69.8)
no
yes
no
yes
no
yes
no
yes
0.095
0.150
0.092
0.868
0.094
0.144
0.092
0.142
4-digit SIC effects
2 No. of observations
27,548
114,408
39,153
102,803
Note. — T-statistics based on robust standard errors are in parentheses. The independent variable Q −1 is the value-weighted mean of Tobin’s Qs for all Compustat firms measured at the end of the previous year, while the log(Q −1 )-log(Q −1 ) are differences of individual firms’ Qs and their market value-weighted mean. Estimation is by OLS.
Investment by young firms is positively related to aggregate .–Although all Compustat firms are what we would call incumbents, firms that listed recently are most likely to exhibit investment behavior similar to new firms. To us, this means that the investment rates of recent listings might well be positively related to −1 even though investment rates of the entire pool of incumbents are negatively related to it. Table 2 shows that this is indeed the case, with columns 1-2 and 5-6 indicating a strong and positive elasticity of investment with respect to aggregate −1 for firms listed for two years or less and three years or less, respectively. And as we would expect, the complement of firms listed less recently in columns 3-4 and 7-8 have an even more deeply negative aggregate -elasticity than the entire set of Compustat firms. This elasticity is also most negative for the narrowest definition of “old” firms (i.e., more than three years since listing). Because firm-specific TFP terms are not included, the coefficients of may all be biased upwards but the asymmetry of responses is still highly significant. Indeed, not only does the asymmetry remain, but in spite of a possible upward bias in the response to of old firms’ investment, its response to aggregate remains negative, and this magnifies the puzzle further. Nevertheless, first row of Table 2 become 3.4, 2.1, 10.4, 10.8, 3.5, 2.7, 12.1, and 12.3.
7
we shall use data on venture-capital investments to confirm the conclusion that young firms respond positively to aggregate . Venture capital flows into young firms.–A systematic source of micro evidence on entering firms is Thomson’s VentureXpert sample of venture-backed firms. Venture capitalists (VCs) invest almost exclusively in start-ups. One cannot compute a firmspecific for these firms, but one can use the that prevails for listed firms in particular sectors. Gompers et al. (2008, tables 4 and 5, pp. 11-12) show that VC investments respond elastically to for their sector using annual data from 1975 to 1998. In particular, OLS regressions of the annual number of investments in each of 69 VentureXpert sectors on one-year lags of their average sectoral ’s (computed as market-to-book ratios from Compustat) deliver coefficients on of 0.330 in a specification without year or industry fixed effects and 0.172 with year and industry effects included. They also obtain a coefficient on of 0.043 when regressing firmlevel investment on sectoral ’s with fixed effects for industries and years. In sum, aggregate data as well as micro evidence from Compustat and VentureXpert show a clear asymmetry in the response of investment to changes in aggregate : Entering and young firms respond positively, while older firms respond negatively.
3
Model
The output of the final good depends on capital, , and a technology shock : output = The law of motion for capital is 0 = (1 − ) + +
(2)
where represents capital produced in continuing projects, and is capital produced in new projects. () Continuing projects.–Continuing projects can be enlarged at the gross rate of return of 1, which is the same over all continuing projects. That is, an investment today of units of the consumption good in an existing project yields one unit of new capital. Then if the capital created via existing projects is , the total cost is . We assume a representative incumbent firm with capital stock . () New projects.–A project uses as inputs a unit of the consumption good and an idea. As output it delivers units of capital ready for use in the next period. The quality of the project, , is known at the start. New projects are born each period, and their quality is distributed with cumulative distribution () and density (). Ideas arrive in proportion to the size of the capital stock. Thus the unnormalized distribution of new ideas is (), where is their arrival rate. The presence of reflects a technological spillover from incumbents’ capital creation to the generation 8
of new ideas.9 Ideas cannot be stored. Idea quality is evaluated privately by venture funds — an agent does not know the of his own idea. A loose interpretation of () is that of a TFP distribution among start-up firms in the “extensive investment goods” sector. Let us note two differences in how investment in new and old projects is treated. Contracting between agents and venture funds.–Capital created though new projects is sold to incumbent firms or floated through IPO at a price equal to the cost, , of making capital via the incumbents’ technology. Each new-idea owner submits his idea to a venture fund for evaluation. A fund pays the idea’s owner only if it uses the idea to float an IPO or sells the capital privately to an incumbent firm, either way receiving the price of per unit. There are no long-term contracts, and contracts cannot condition on . Advance payments by the venture fund cannot be made because anyone could pretend to have an idea and collect the payment. Ex post, payment cannot be made conditional on because the fund would always claim that the idea was of the lowest acceptable quality. Therefore the fund pays the minimum that it takes to get the idea owner to develop the project, and that payment is unity. The owner of the idea ends up with zero rents. Since the fund knows it can get for the idea, its profit from the project is − 1 and so it will implement all projects with 1 ≥ m =
(3)
so that m is the quality of the marginal project. The fund collects revenue , where = is given by Z ∞ () (4) () = 1
Private-equity dividends.–The profits of the fund are () , where Z ∞ ( − 1) () () =
(5)
1
and profits are paid out as dividends to the households that own the funds. Income identity.–Let be consumption, = , and = Output is divided between consumption and the two forms of investment so that the income identity reads = − − () (6)
where, if (3) holds,
∙ µ ¶¸ 1 () ≡ 1 − (7) is the cost of the new projects and is the cost of the continuing projects, with each 9
We may interpret along the lines of Arrow (1962) which posits a positive external effect of investment, or Luttmer (2007) where an entering firm tries to imitate a randomly-sampled incumbent. 9
supply of investment in continuing projects
y(q)
q
i(z, q) q2 x1
q1 0
x2 y1 y2
i2
i1
i
Figure 2: The determination of investment. term in (6) measured per unit of . Moreover, (3) and (7) imply that the marginal costs of capital entailed in the two investment margins are equal:10 0 () =
(8)
The determination of investment.–Figure 2 illustrates the effect of a rise in when is held constant. Investment, ≡ + , is determined by households’ savings decisions (to be described presently).11 Households’ demand for investment is negatively sloped because a rise in raises the cost of future consumption and reduces the demand for it (see eq. (13) below). The investment rate (i.e., supply) of entering capital is determined by (8) and increasing in ; this is the positively-sloped function in Figure (2). Incumbent investment takes up the slack between the households’ demand for and the supply of . As rises, two things happen: first, total investment declines so that falls from 1 to 2 ; second, the supply of entrants rises from 1 to 2 . Therefore falls by more than does.12 As rises, on the other hand, the h
³
´i
1 Formally, () ≡ 1 − () where () is the inverse of the function defined on 0 the LHS of (4). Then () = − (m ) m and m = 1m (m ) 10
11
Since = + , the income identity (6) can be writen as + = + ()
12
These implications remain valid under a more general specification: suppose that the resource constraint in (6) was = − − () with 6= 0. Then instead of (8) the FOC would be 0 () = 1− and the conclusions illustrated in figure 2 would 10
downward-sloping curve in Figure 2 shifts to the right (again, see eq. (13) below). This causes both and to rise. ª ©P∞ Preferences.–Preferences are 0 ( ) , with =0 () =
1− 1−
Let ≡ ( ) be i.i.d. with distribution (). Contemporaneously, and may be correlated. The state of the economy is ( ), but since returns are constant and preferences homothetic, affects neither interest rates nor investment rates. Assets.–The household may own two assets: () Private equity.–We assume that the limited partners’ stakes in the privateequity (PE) fund are tradable.13 The projects mature in a single period and PE pays dividends every period. The price is () per share in the PE fund, and a share pays the dividend () . The fund manages only new projects and sells them right away, and benefits through an external effect from the growth of . () Public equity.–The number of firms is fixed and the size of the representative firm is proportional to .14 The firm finances its purchases of new capital (consisting of units created by the consumption-good-conversion technology at the cost of per unit, and of obtained by a purchase from the private-equity funds at the price of per unit) by withdrawing ( + ) from its earnings. This reduces the firm’s dividends, but it leads to an exactly offsetting capital gain because the new capital is valued at per unit. Thus the total return of the public-equity fund is regardless of its investment activity. We assume that production occurs before depreciation, but that trade of equities occurs at the end of the period so that since is the firm’s beginning-of-period capital stock, the firm’s end-of period price per share is (1 − ) . The household’s budget constraint.–After dividing both sides by , the constraint is 0 + 0 + = ( + ) + ( + (1 − )) remain qualitatively intact so long as 1. That is, would be increasing in and would be decreasing in . If, on the other hand, = 1, would be a constant depending on neither nor and would solve the equation 0 () = 1. But would still be decreasing in . 13
By “private equity” we mean limited partners’ stakes in PE funds, and not shares in privately held companies — all capital is publicly owned in our model. The finance literature treats PE as being illiquid — e.g., Sorensen, Wang and Yang (2012), but since the limited partners are typically large entities that also hold liquid securities, they may require only a small liquidity premium, in which our “as if” approach should yield reasonable results. 14
Returns to scale are constant and firm size is indeterminate.
11
where and are the shares in the private and public equity funds that the household carried over from the previous period. In equilibrium = = 1. As in Lucas (1978), we substitute these conditions into the budget constraint. The equilibrium price of capital.–We shall assume that, as in Figure 2, the size of entering ideas is too small to affect today’s market price of the marginal unit of capital, i.e., that is always below the point where () intersects ( ) In that case, the price of capital must equal . Any price above would draw forth an infinite supply of or, at any rate, an investment level that is larger than the supply of savings on the part of households. At the price , all the s above 1 are implemented, yielding entering capital () and yielding ( ) ≡ ( ) − () as the residual. The resulting public-equity price per unit of is Z µ ¶ = ( 0 + (1 − ) 0 ) (0 ) (9) 0 If ever strayed above the point where () intersects ( ), then would be zero and the equilibrium price of capital would be below . We stress that (9) is valid only if the range of is sufficiently narrow such that this never happens. The price of private equity.–The PE price is Z µ ¶ () = (1 − + ()) ( (0 ) + (0 )) (0 ) 0
(10)
The multiplicative factor (1 − + ) = 0 in (10) corrects for the growth of the capital stock, to which prices and dividends are proportional. It is absent from (9) because the capital gains it implies are offset exactly by the cost of investment. The two assets have very different income streams. Public equity pays dividends that equal earnings, , less investment; these are rents stemming from the quasi-fixity of . Private equity pays dividends () that depend on alone; these are rents drawn on the ’s
3.1
Equilibrium: Definition and properties
The state of the economy is ( ), but constant returns and homothetic preferences suggest that consumption, investment, and share prices of incumbents and privateequity will all be proportional to , thereby reducing the state to Equilibrium.–The functions () () and () are defined in terms of primitives, and () = () − () is defined in terms of (). Therefore we define equilibrium as consisting of three functions: () () and () that solve (6), (9) and (10) for all .
12
Existence and characterization.–If15 ¶1− Z µ 1 () ≤ 1 1−+
(11)
then Appendix C proves the following result: Proposition 1 (homogeneous x). There exist positive constants 0 and 0 defined in (40) and (41), such that in state ( ) along with (4) and (5), we have + () + (1 − ) 1 + 1−1
(12)
( + ()) −1 − (1 − ) 1 + 1−1
(13)
=
= and
= (1 − + )
(14)
Corollaries of Proposition 1: ¡ ¢−1 1. The marginal propensity to consume out of is 1 + 1−1 and is decreasing in if 1 2. Stock returns and investment are positively correlated: Earnings are and 0 price appreciation is ( 0 − ) , so the measured return is = + − 1. And ¤ £ 1 1 + . Both (13) can be re-written as = ( + ) − (1 − ) and are increasing in and decreasing in , and the latter is i.i.d. Thus the correlation is positive. 3. Using (13) and (4), = − is increasing in and decreasing in . Special cases a) = 1, and = 0–With log utility and ≡ 0, (40) yields = (1 − ) so that (12) and (13) become = (1 − ) ( + (1 − ) ) = − (1 − ) (1 − )
and (15)
b) = 0.–The solutions (12) and (13) are not valid for = 0, and not likely to be valid for close to zero because the interiority assumption on which (9) depends will fail if there is significant variation in or . See footnote 15. 15
Condition (11) holds at = 1 and it holds for all 1 if the support of under the measure is entirely above The latter is true for our sample in which the lowest value of the ratio is 0.12. Condition (11) fails for close to zero, however, because has a mean of 0.41 and in this model, for some feasible investment policies, expected utility is infinite. 13
3.2
Other implications
The model has implications for five open questions in macroeconomics. First, are stock prices a proper measure of wealth and welfare; second, is Tobin’s a sufficient statistic for investment; third, should consumption follow a random walk; fourth what determines the private-equity premium and fifth, on what factors does the long-run growth rate depend. Wealth rises with , but welfare declines.–In current consumption units, wealth = + = [1 + (1 − + [ ]) ] ≈
(16)
The approximate equality in (16) holds if, as is true in our baseline calibration with = 00085, private-equity dividends () are small compared to so that is a small number. Although is an adverse shock that reduces lifetime utility, it raises wealth. This seeming paradox arises because real rates fall more than the reduction in future dividends. Real rates fall because a rise in raises , and the term ( 0 ) rises in the pricing kernel of the RHS of (9). Future dividends decline because the rise in causes to grow more slowly, as we saw from (13). Thus lifetime utility falls but wealth rises. In other words the conclusion that welfare and stock prices go hand in hand (e.g., Barro 2006) does not hold in our model. is sufficient for but not for or .–That Tobin’s is, in general, not a sufficient statistic investment generally was noted early on by Sargent (1980). In the model where investment is subject to convex costs of adjustment, Tobin’s is an endogenous variable that responds to underlying shocks. Since aggregate consumption is a function of such shocks, so is aggregate investment and generally is not a sufficient statistic for investment. This holds true for and here as well, but is driven only by as (8) and figure 2 show. Table 4 shows no consistent evidence that affects . is not a random walk.–We normalize to unity so that = ( ) and 0 = ( 0 0 ) (1 − + ). Then ∆ ln ≈ − + where if ( ) is i.i.d., = ln 0 −ln is MA(1). On the other hand, since is monotone in , the pair ( ) are sufficient statistics, consistent with eq. (8) of Hall (1978) which projects +1 on and the S&P 500 index. The private-equity premium.–Private equity dividends, (), are high when is high and, according to (12), this also occurs when is high. On these grounds private equity must offer a premium over public equity. On the other hand, () does not depend on and, since it is a small part of the representative agent’s portfolio, is essentially uncorrelated with aggregate consumption, whereas dividends of public equity firms are just . On these grounds PE should offer a smaller return than public equity. The two forces appear to offset each other: Kaplan and Schoar (2005) report 14
that for the period 1980-2001, PE returns net of fees were roughly equal to the return on the S&P 500. Long-run growth.–The long run growth rate of the economy’s and, hence, of its output is roughly Z LR growth = ∆ ln = − + () () (17) with () given in (13). Equilibrium growth is below the efficient rate because of the positive external effect of in the function (). In Section 5 we discuss this issue in detail.
4
Fit to aggregate data
We first describe the data and how we use them, and then discuss the parametrization.
4.1
Data
We measure six variables: Two exogenous shocks ( ) and four endogenous variables ( ). This subsection describes the measures in outline only; Appendix A contains detailed descriptions of the sources. 4.1.1
Relationship between the shock and measured
We shall use lower case and its variants for theoretical concepts, and upper case and its variants for empirical measures of Tobin’s . The relationship between the two is independent of the presence of , so we temporarily set = 0 to make = 0, and the model reads = +
0 = (1 − ) +
and
Let be today’s price of a unit of capital in place tomorrow. The FOC for investment — a re-statement of (9) — then states that the cost of a unit of capital must equal its value = (18) The market value of the economy’s capital stock is and its replacement cost is . Therefore the model counterpart of average Tobin’s is TRUE =
= 1
With replacement cost not directly observed, aggregate Tobin’s as we measure it (see Appendix A) is ˆ = market value of equity and bonds total private fixed assets 15
(19)
ˆ are aggregates for the non-financial corporate sector from Both components of the Federal Reserve Board’s Flow of Funds Accounts. The numerator is the market value of equity and bonds and is the equivalent of . The denominator in (19) is a measure of physical capital; our three sources (Hall 2001, Wright 2004, Abel and Eberly 2011) have different degrees of inclusiveness thereby causing us to splice the series, but all are highly correlated with .16 Therefore the denominator in (19) is really proportional to , and not to . In that case, ˆ = =
(20)
ˆ as the measure of the shock . using (18), and we can use 4.1.2
The series for and
The series and .–Data for (the stock of private fixed assets at the end of the previous year) are from the Bureau of Economic Analysis (BEA, 2006) for 1925-2005 and Goldsmith (1955) for 1901-1924. Data for (private domestic investment over the year divided by ) are from the BEA for 1929-2005 and Kendrick (1961) for 19011928. We adjust the series for inflation and population growth over each year; see the discussion preceding (37) in Appendix A. The series .–Since output is , we measure as the ratio of private output over the course of a given year to private fixed assets at the end of the previous year. We adjust the series for inflation and population growth over each year as well. Figure 4 shows the time series for ( ). 4.1.3
Two alternative series for and
¡ L L¢ We construct¡ two pairs of series, the long series are based on IPOs and the ¢ short series S S are based on venture investment. The details are in Appendix A; we summarize them briefly here, first for and then for . The series L .–IPO values relative to the capital stock are interpreted as the market value of the new capital brought in by entrants. Division by yields We then correct for the upward trend in stock market capitalization, as described in (35) and (36), and adjust for population growth over the course of a year to get L . The series S .–This series measures investment in new companies by venture capitalists for an unbalanced panel of about 20,000 startup firms. Investment is normalized by commitments (all VC funds’ credit lines from their limited partners), which are highly trended, rising by a factor of twenty from 0.006 percent of in 1969 to 0.11 percent in 2005. We correct for this trend too, as described in (38) and (39), and then for population growth over each year to get S . 16
For instance, for the component used for the period 1950-1999, the correlation between the denominator of ˆ and our measure of is 0.993. 16
Each series has advantages. If extensive investment is undertaken only by new firms, then the short series, S , is a better counterpart to (4) in that it contains only investments in start-ups. Some of these startups are destined to become part of incumbent firms; they do not experience IPOs but are bought directly by incumbents from independent VCs and their capital is converted to incumbents’ internal use. If their price is per unit of capital sold, then the model interprets these sales in the same way as it interprets IPOs.17 It is undeniable, however, that incumbents do build new plants and that they do create new product lines. For instance, Bernard, Redding, and Schott (2010) find that product entry and exit in surviving firms exhibits many of the same features as the entry and exit of firms themselves. If some incumbent investment is extensive, and if a corporation pays profits from all sources as dividends, its dividends will include a term such as (), and (9) will no longer characterize the equation for the price of the firm’s capital. The model can handle a broader definition of “extensive” investment if corporateventure capital (CVC) divisions are arms-length entities that trade separately and pay dividends (), and if the transfer of from the CVC entity to the parent firm entails a payment equaling the (correct) transfer price . Such securities will be indistinguishable from private equity and will trade at the price given in (10). The series S includes CVC investments, and this is appropriate for two reasons: First, since 1995, CVC investment averages 8.9 percent of total VC investment, but responds highly elastically to , reaching 15 percent of the total in 1999 and 2000 before falling back to its 15-year average. Second, corporations tend to separate their CVC division to keep it at arms length, sometimes even setting up a dedicated external fund outside the corporation, or as a limited partner in another VC firm (Gompers 2002, p. 2).18 Some investment in new companies is not VC-backed, and neither is every IPO. Thus S is a narrow measure of new investment. The long series L provides a broader measure that covers all IPOs on the NYSE, Amex and Nasdaq since 1901. The average age of IPO-ing firms has varied over the past 110 years, as figure 1 of Jovanovic and Rousseau (2001a) shows, but these firms are considerably younger than the median publicly-listed incumbent.19 And when a young firm invests, it does not 17
One interesting feature is that IPOs, and hence L , peak in 1999, whereas venture investments and, hence S , peak in 2000. There is some inertia in the venture investment series probably because instead of just one period as the model assumes, projects take several periods to mature and investments made in the late 1990s were sunk into projects that needed additional funding in 2000 to reach maturity. 18
Table 6 of Gompers (2002) shows that the performance of CVC-backed investments is in most respects similar to that of independent VC-backed enterprises. 19
The median firm age since IPO on Compustat from 2000 to 2006 is 7 years. The annual median starts at five years in 2000 and because there were many IPOs in the late 1990’s rises to nine years in 2005 and 2006. 17
face the compatibility costs that an incumbent must confront, and this is why in the model the young firms’ investment rises with . The series L and S .–A decomposition of into and depends on how is defined. These series, respectively, are L ≡ − L and S ≡ − S , with the superscripts denoting “long” (1901-2005) and “short” (1969-2005), and are described in Appendix A.
4.2
Parametrization
The parameters are , and . We now describe how their values are chosen. Choice of .–Since ( ) are i.i.d. and stationary, we choose to fit their long-run frequency distribution. Over the century, ( ) have a cross correlation of 0.256. Choice of .–We assume a Pareto form for : µ ¶− () = 1 − (21) 0 for ≥ 0 0 and 1. Then for −1 0 , (4) and (5) become () =
0 −1 −1
and () =
0 −1
(22)
Parameter choices for were as follows. () The quality of the worst idea 0 : When is given in (21), and as shown in (22) identify only the product 0 , and not and 0 separately; we set 0 = 0296 so that the economy always has a marginal , even at its highest value of , which is 3376 in the year 1999. This ensures that the functional forms in (22) are valid for all . () The Pareto tail parameter : The lifetime value of an idea is , the lifetime value of the discounted income that additional units of capital will provide. Thus () is the CDF of IPO values and values of other equivalent idea sales mediated outside the stock market. If the new projects were all VC-backed ventures, we could use table 1 of Jovanovic and Szentes (2013) who fit a Pareto distribution to the distribution of VCs’ “exit values” (i.e., values obtained via IPO or private sale) and obtain = 155.20 But that would be an underestimate of because VC is a relatively small part of the economy and one would expect ideas not mediated by VCs that are in the lower-tech sectors to have a much thinner tail. Indeed, figure 5 20
The projects matured in the late 1990s, a time when was changing and the estimate is for a mixture, leading to the conclusion that the true was lower than 1.55. See Feuerverger and Hall (1999) on the estimation of mixtures of Pareto distributions.
18
of Eeckhout and Jovanovic (2002) shows that a log-normal distribution (which has a thin tail compared to Pareto distributions) fits very well the book values of firms on the Compustat, and figure 9 of Jovanovic and Rousseau (2001b) shows that a log-normal right tail is not a bad approximation of most decadal IPO market-value distributions. The set ∈ [155 ∞) is large, and so we appeal to firm span-of-control estimates to set = 3.21 Choice of .–We choose so that average ( ) fits the sample averages of L and S respectively. This yields L = 0096 and S = 00042. That L is an order of magnitude larger than S is largely because VC-backed investment is a small fraction of all start-up investment. Choice of and .–We set = 095, and = 4. The value = 0037 enables the model to fit the sample average of L in the case when ( ) are autocorrelated. Thus the parameters are Table 3 0.95 4
4.3
0 L S 0.037 3.0 0.296 0.096 0.0042
Fit
Model fit to the long series.–Figure 3 shows the resulting fit for the long series. The post-1947 correlations between the model and data are 0.818 for L and 0.470 for L , which we regard as a success considering that the model is simple and that the data are unfiltered. The correlations fall to 0.586 and 0.195 respectively for the full 1901-2005 period. The model predicts a negative only in 1999, and then just barely (1999 = −0014). It turns out that the model fits both the long and short data series for better than the series for (see also Figure 5), the simulated series for being both too high and too volatile. Though not a target in the estimation, the ratio ¯L ¯ L is 00620623 = 00996, and the model predicts it to be 00602. This fit will be a bit closer when we allow for autocorrelation in and in Section 6.2. Finally, allowing ( ) to be correlated also stabilizes ; under the i.i.d. assumption on , if is high this year but is expected to revert to its average value next year, then it is more tempting to postpone investment for a year. We shall look at these departures in Section 6.2. Finally, a higher value of would stabilize and lower ¯ and the implied rate of growth as well. But in order to fit ¯ and the growth rate exactly, even in the 21
When the firm is a price taker and has a span-of-control parameter , the tail of the employment distribution is 1 − times the tail of the TFP distribution. With a value of = 064 we would obtain an employment tail of 106 that Axtell (2001) and others have reported. Atkenson, Khan and Ohanian (1996, Sections 5 and 6) explain why this value of is reasonable in light of various types of evidence. 19
yL (right scale) 0.04
data
0.4
model
0.02 0
xL (left scale) model
0.2
data
0 1900 '10
'20
'30
'40
'50
'60
'70
'80
'90 2000
Figure 3: The L and L series and their simulated values, 1901-2005.
q (right scale)
3 2 1
0.6
0
z (left scale)
0.5 0.4 0.3 1900 '10
'20
'30
'40
'50
'60
'70
'80
'90 2000
Figure 4: The two shocks, 1901-2005
20
yS (right scale)
0.002
data
model 0.001
0.4
0
xS (left scale) model data
0.2
0 1969
1974
1979
1984
1989
1994
1999
2004
Figure 5: The S and S series and their simulated values, 1969-2005. autocorrelated case we shall need = 14 . We give details in Section 6.2, as well as discussion of the curvature issue. A discrepancy for between model and data that no reasonable version of the model can seem to fix arises in the 1990s and beyond. As figure 4 shows, was well above average in the 1990s and remains above its historical average today. The model interprets this as a high cost of capital and, as a result, is below average. Model fit to the short series.–Figure 5 shows the fit to the short series. The top panel of figure 5 displays S , which represents investment in roughly 20,000 VCbacked young companies. The bottom panel depicts incumbent investment computed as S = − S , and the simulated series remains positive throughout. We mentioned earlier that the “gestation” lag in VC investment is longer than a year, and the model likely misses by a year the year-2000 peak in L for that reason. The model also misses the high investment of the early 1980s, for reasons not clear to us. The correlations between the model and data are 0.539 for S and 0.207 for S . For the short series too, the model generates an S that is too high. But in order to fit ¯ and the growth rate exactly, we shall again need a higher but this time it needs to be raised only to = 65. See Section 6.2 for more details. Corollaries 2 and 3 state that and should both decrease with and increase in Since ( ) are exogenous, we can test these predictions by regression. We report the results in table 4. The regressions indicate that both and are increasing in 21
Table 4 Investment Regressions Using Annual Time Series Data
ln L 19012005
ˆ ln
ln L
ln
19472005
19622005
19012005
19472005
19622005
19012005
19472005
19622005
-0.111 -0.194
0.935 (4.5)
1.301 (4.9)
1.736 (4.1)
(0.3)
-0.083 (2.3)
-0.111
(4.1)
-0.271 (4.1)
-0.018
(1.7)
ln
3.407 (11.9)
1.143 (4.1)
1.561 (3.8)
2.954 (3.3)
1.758 (1.1)
-1.007 (0.4)
3.371 (13.9)
1.248 (5.9)
1.387 (4.5)
Trend
0.005 (5.1)
0.002 (1.4)
0.002 (0.8)
-0.004 (1.5)
0.001 (0.1)
-0.005 (0.4)
0.004 (4.8)
0.002 (2.3)
0.002 (1.0)
Const.
0.204 (0.7)
-1.657 -1.209
-2.462 -4.127 -6.393
-1.526 -1.323
(7.6)
(4.0)
(2.8)
(3.7)
(3.4)
0.324 (1.4)
(9.4)
(5.9)
2
0.631
0.362
0.461
0.292
0.561
0.577
0.698
0.421
0.467
105
59
44
105
59
44
105
59
44
(2.3)
Note. — T-statistics based on robust standard errors are in parentheses. Data are from figure 3 and figure 4, where is the sum of L and L . Estimation is by OLS.
for the 1901-2005, 1947-2005, and 1962-2005 periods, while and are decreasing in as the model predicts. Finally, raises in all three samples and the prediction of Proposition 1 that is sufficient for holds up in the post war samples, but not for the century as a whole.22 Even here, though, effect of on is smaller than its effect on and .
5
Comparison to the one-investment model
If = 0, our model is a standard one-sector model and a special case of Greenwood, Hercowitz, and Krusell (2000) which stresses the negative correlation of investment with the price of equipment. What difference does = L = 0096 make to such a world? 22
We choose the period from 1901 to 2005 to correspond with our long-series simulation. The 1947-2005 period covers the post World War II part of our sample, while the 1962-2005 period corresponds with the firm-level regressions in Section 2. 22
0.012
i λ - i0
0.008
cλ - c0
0.004
0
-0.004 1900 '10
'20
'30
'40
'50
'60
'70
'80
'90 2000
Figure 6: The model economy with =0.096 and with =0, 1901-2005: differences relative to Holding the other parameters the same as in table 3, figure 6 reports the changes in and that the increase in from zero to 0096 brings about. When = 0, the income identity is + = whereas when 0 it reads + = +
(23)
where is given in (5). The identity (23) implies that − 0 + ( − 0 ) = () which is closely related to the vertical summation of the two series plotted in figure (6). As ranges between 0.402 in 1920 and 3.376 in 1999, () ranges between 0.0001 and 0.0479, with a sample mean of 2005 1 X ( ) = 0034 105 =1901
(24)
All these numbers are relative to , and to convert them into fractions of output one multiplies by 1/¯z = 2.627. This is an income effect or wealth effect, but it implies no substitution effect towards investment because the marginal project still costs . The rents come from the inframarginal projects. In other words, if investment were 23
unchanged, consumption would rise in each subsequent period by the amount () but the desire to smooth consumption means that − 0 is negative during the low periods, and positive during the high- periods. The positive effect on can be shown analytically: From (40) it is clear that is decreasing in which means in (12) that ( ) 0 for all ( ) (25) which explains why − 0 is positive at all dates in Figure 6. The effect of on cannot be signed because in low- periods () is low and the added consumption occurs at the expense of lower investment.
5.1
and long-run growth
The focus of this paper is on high-frequency activity, but for completeness let us compare the long-run growth implications of our model to that of the = 0 economy. Equilibrium growth when ( ) are constant.–To save space we shall discuss only the case = 0. With ( ) constant, the long-run growth rate solves the investment optimality condition that equates the cost of the investment, , to its discounted return, ∞ X = (1 + )− (26) =1 −
because 0 ( ) 0 (0 ) = (1 + ) dix C)
. Solving the geometric sum yields (see Appen-
µ ∙ ¸¶1 1+ = 1+ (27) Note that when = 1, = as given in (15) when the latter is evaluated at = 0. Equilibrium vs. optimal growth.–A positive raises optimal growth above its equilibrium level: is the number of projects that entrants get to choose from. Thus creates a spillover from incumbents to entrants. Since is raised by , and since incumbents are not compensated for the ideas that they create, is below its optimal level. Moreover, too is below its optimal level: Entrants would raise if they implemented more s, and thus the socially optimal marginal project has a value below 1. If lump-sum taxation were possible, it would be optimal to subsidize capital formation by entrants and incumbents alike. The difference between optimum and equilibrium, however, turns out to be very small, and therefore not a policy concern. A rough idea emerges when we solve (still for the case with = 0) the optimal growth problem with output + = + 24
and capital evolution 0 = + This ignores the distinction between and , but the technology delivers rents yet does not affect the quality of the marginal investment. The optimal growth rate ∗ that maximizes the representative agent’s discounted utility is then easily calculated as a no-externality competitive equilibrium but with replaced by + : Proposition 2 Optimal growth ∗ satisfies µ ∙ ¸¶1 + ∗ 1+ = 1+
(28)
and to a first approximation around = 0, ∗ − ≈ (1 + )1−
(29)
The deviation from optimality turns out to be small. From (24) the average value of is .0034, and that of is 1.147. If = 015 and = 4 (1 + )1− = 0956. With = 095 ∗ − ≈ (956) (95) (0034) 4−1 (1147)−1 = 673 × 10−4 At these values, the right-hand side is less than one tenth of a percent. If an economy were to grow at the rate ∗ , after a century, say, its income would exceed the economy that grew at the rate by only about seven percent.
6
Robustness
This section briefly discusses two modifications of the model. First, the relaxation of homogeneity among incumbents’ investment projects and, second, allowing for autocorrelation in ( )
6.1
ˆ Incumbent heterogeneity and the determination of
Recall that lower case and its variants refers to theoretical concepts, and that upper case and its variants refer to for empirical measures of Tobin’s . We have so far assumed that all continuing projects can augment their capital at a unit cost ˆ as defined in (19) depends only on — indeed, (20) states of . Then the measure ˆ = , and has no role in explaining . ˆ Now we show that by introducing that heterogeneity among incumbents’ projects, the model easily accommodates a role for without overturning its asymmetry implications for investment. This generalization also leads to investment equations of the type shown in (1) and table 1 in which incumbent firms respond positively to own and negatively to aggregate . Suppose, then, that incumbents and their continuing projects are also heterogeneous. This implies that the equilibrium price of capital in place will now depend on both and as in the standard model with convex adjustment costs. Suppose 25
also that each project delivers a unit of capital tomorrow, but that the cost, , is project-specific, and that the distribution of is Pareto: à µ ¶−1 ! ; ≥ # of available continuing projects less costly than = 1 − (30) The most efficient incumbent project costs units of consumption, and is an index of the heterogeneity of continuing projects. As → 0 continuing-projects become homogeneous with each unit of capital costing . If all incumbents’ projects were implemented the capital stock would double. One could add a normalizing constant to (30) analogously to in (4). The equilibrium price of shares, call it ˆ, now depends on both and . Instead of (20) we now have ˆ = ˆ. (31) We do not have an analytic solution for it, but ˆ and Appendix C proves the following result: Proposition 3 (heterogeneous x). When (30) holds, the income identity reads à ! 1 − (1 − )1− = + () + (32) 1− where the function (·) is defined in (4), and the investment function is µ ¶−1 ˆ ( ˆ) = 1 − (33) ³ ´ 1− As → 0, 1−(1−) → so that (32) becomes the same as (6). And the 1− price of shares, ˆ, converges to for all ( ). Figure 7 shows a simulated version of figure 3, and then adds two supply curves for covering the case 0. Each curve pertains to a hypothetical realization of the shock . When 0 one must distinguish ˆ from , and so the vertical axis now measures ˆ. The supply functions ( ˆ) and (ˆ ) are taken from (33) and (22) respectively. We draw for ∈ {1 2} and ∈ {0 2}, and we draw (ˆ ) from (22) ) plots (13) evaluated at the values of 0 , and given in Table 1. The function (ˆ at the sample mean value of = 0381, but we stress that this schedule becomes correctly places only as → 0; the important point is that for sufficiently small, (ˆ ) is downward sloping in ˆ.23 23
Referring to Figure 7, eq. (13) shows the investment function for the case when incumbent projects are all the same. This is what is plotted. We expect the ( ) to be a continuous function of , in which case ( ) → () pointwise in as → 0. We need ( ) to intersect ( ˆ) at a level of investment higher than the level at which it intersects ˆ (). At the chosen parameter values we know that this will be the case if is small enough. 26
Ê
q
Ê
y(q) from eq.(23)
3
Ê
x(q,q) – from eq.(34) q=σ=2 q = 2, σ = 0
2
Ê
i(q) – from eq.(13) q = 1, σ = 2 q = 1, σ = 0
1
0 0
0.025
0.05
0.075
0.1
0.125
0.15
investment Figure 7: The effect of heterogeneous projects. The asymmetric response of investment to ˆ remains.–Figure 7 shows that regardless of the slope of the the incumbent supply curve, a rise in shifts the supply curve to the left because the entire distribution of ’s shifts to the right, and total investment must fall and that of entrants must rise. The equilibrium ˆ = ˆ ( ) solves + = and the model’s predictions go through if the supply of incumbent investment is flatter and below the supply of entering investment. From (4) and (22) we see that () must be very steep because, relative to , the measure of new projects, = 0096 is small compared to the measure of incumbents’ projects which is normalized to one. Thus if the variances of the potential projects in the two populations are roughly equal, the variance of the implemented projects will be far larger among the new projects than that among the continuing ones.24 ˆ But now also affects .–Figure 7 also show that the role of in determining ˆ depends inversely on the investment-supply elasticity. The latter depends on heterogeneity, . The larger is , the larger the effect of on ˆ. Conversely, the smaller is , the flatter is the incumbents’ supply curve, the closer ˆ is to and the closer (18) is to the truth. This shows that making incumbents’ projects heterogeneous brings 24
A referee suggested that the asymmetry could be softened by letting the distribution of ’s also depend on . The dependence of the two project distributions on could then be parametrized and estimated.
27
the model closer to the convex-adjustment-cost model in which also plays a role in determining the equilibrium price of installed capital. Defining and .–Interpreting the results in (1) and table 2 requires that we define firm-level for entrants and incumbents. We shall do so briefly, in a stylized way. An entrant invests one unit and gets ˆ for it, and so taking the ratio of the two yields = ˆ An incumbent invests units and gets a unit of investment goods valued at ˆ. If he had no other capital his Q would be = ˆ. Interpreting Eq. (1) and Table 1.–Entrant invests if ≥ 1 so that both ˆ and the deviation of from ˆ raise entrant ’s investment. Let us now show that ˆ and positively to − . ˆ Substituting firm ’s investment responds negatively to the equilibrium ˆ = ˆ ( ˆ ( )) into (33), we get the fraction of firms that invest, ³ ´−1 be firm ’s i.e., the percentile of the marginal incumbent. Now let = 1 − percentile. Then ½ 1 if ˆ (34) = 0 if ≥ ˆ When rises, so does ˆ, whereas ˆ falls, as figure 7 shows. This means that is stochastically decreasing in ˆ the latter being a proxy for −1 in (1). And is also decreasing in . But since a rise in lowers the firm’s own dividend, its value and, hence, its , conditional on ˆ ( ) are positively correlated. Qualitatively, then, we have the result in (1).25
6.2
Autocorrelation in ( )
Agents in the i.i.d. version of model assume that ( ) is i.i.d., with a contemporaneous cross correlation of 0.256. This allows us to solve explicitly in eq. (12), (13) and (14), which greatly adds to intuition. In fact, however, the autocorrelations of the shock series are ( −1 ) = 077 and ( −1 ) = 087. When agents assume that ( ) follows a first-order Markov process with the autocorrelation properties listed above, we can discretize the ( ) process and simulate the model. This version of the model implies the same ( ) series as in (22); solves (8), which is a static decision involving only. It also does not depend on which we shall vary here. Therefore there is no change in the top panel of figure 3, at least conditional on the parameters remaining the same. 25
A rigorous study of the panel data would need to adjust for size in both groups. For instance, as things stand, all entrants invest exactly one unit regardless of and all incumbents create exactly one unit of capital regardless of . This can be fixed by assuming that new projects and incumbent projects are each characterized by tuples (1 2 ) and ( 1 2 ) denoting the cost investment, 1 or 1 , and the output (in units of ) of the project, 2 or 2 . Then the quality of the entrants’ projects would be ≡ 2 1 , and the cost the incumbents’ projects would be ≡ 1 2 , with projects being implemented iff 1 ≤ 2 i.e., iff ≥ , and iff 1 ≤ 2 i.e., iff ≤ and we could proceed as stated in the text. 28
xL, γ=4 model
0.2 0.1
data 0
xL, γ=14 0.2
model
data
0.1 0 1900 '10
'20
'30
'40
'50
'60
'70
'80
'90 2000
Figure 8: The L series and its simulated values with autocorrelation in and , 1901-2005.
xS, γ=4 0.2
model 0.1
data 0.2
0
xS, γ=6.5 model data
0.1 0 1969
1974
1979
1984
1989
1994
1999
2004
Figure 9: The S series and its simulated values with autocorrelation in and , 1969-2005. 29
On the other hand, comparing the upper panel of figure 8 and the lower panel of figure 3, we see that the volatility of falls substantially when shocks are autocorrelated. This is because the agent does not see himself as needing to smooth consumption as much. With every shock interpreted as transitory, the agent wants to save and invest more when is high and invest less when is low. When this perceived need to smooth is diminished, investment becomes less volatile with the standard deviation of dropping from 0.069 in the i.i.d. case to 0.057 in the autocorrelated case. A similar reduction in volatility of S also occurs — compare the bottom panel of figure 5 to the top panel of figure 9. Unless we raise above 4, the model predicts too high a value for ¯ and, therefore, it also overpredicts the long-run growth rate by the same amount. So, keeping all other parameter values as listed in table 3, we seek the value of that allows us to fit ¯ That value is L = 14 for the long series and S = 65 for the short series. The resulting prediction for S is shown in the bottom panel of figure 8. Changing does not affect the series at all — see (22). Finally, since the long-run growth rate in the model is () − and the sample average of investment over the century is ¯ = 069, when the model fits ¯, with = 0037 the long-run growth of per capita output is 3.2 percent. The cross-correlation between and +1 is 0.202, a statistic that one would expect to find in a full vintage capital model in which, when a new technology appears, firms have the option of investing in capital that embodies it. TFP then rises only when such an investment has taken place. In Solow (1960) technology is embodied in capital and all vintages are equally costly to install (same ) but some are more productive than others (vintage-specific s). If one were to add to Solow’s model the assumption that a productive type of new capital was also more costly, then the new capital (built at the higher cost ) would be more productive (i.e., have a higher ) than the old capital. In terms of the identity 0 = (1 − ) + + , the higher would not multiply all of 0 , but only + . This is why, in a general vintage-capital model, the arrival of a new technology would precede high- periods and high- periods.
6.3
On the asymmetric effect of
The immunity of entrants to the cost shocks and lack thereof for incumbents is at the heart of our explanation for the asymmetric response of the two groups’ investments as summarized in figures 2 and 7. An entrant’s innovative ability is whereas an incumbent’s innovative ability is 1 The entrant loses his innovative advantage after one period. In reality this transition is probably gradual. Along these lines, Acemoglu, Akcigit, Bloom and Kerr (2013) assume that a firm loses its innovative ability after entry, but unlike us, they assume that this event occurs at a random age determined by a Poisson process. They provide their own evidence for this assumption in their figure 1, and cite other similar evidence in their footnote 3. In the introduction we cited other evidence about the asymmetry in innovativeness 30
or investment of young and older firms. Figures 14 and 15 of Jovanovic and Rousseau (2005) highlight two epochs of major technological change; the first was the electrification and combustion engine epoch (1892—1930) and the second is the IT epoch that began in the early 1970s. Compared to other times during the 1870—2003 period, IPOs as a percentage of stock market value and as a percentage of private investment were considerably higher. This directly ties IPOs to the advent of technology and venture-capital financing (our S series is itself a product of rapid technological change). Some incumbents are able to adapt, especially through venture activities, and in the business literature, Chesbrough (2002) shows that investment by new firms flourished in times of aggregate investment slowdowns, at least for some of the big companies, which may strengthen our main point. Asymmetries due to finance.–Another asymmetry, highlighted by Fazzari, Hubbard, and Petersen (1988) is that incumbents’ investment is more elastic with respect to changes in firm-specific Q, and that this arises because large firms have better access to financing. This difference does show up in table 2, with differences in the response elasticity ranging between 0.006 and 0.017, although they are not statistically significant. Now, if is correlated with collateral values, banks will lend more when is high. Araujo and Minetti (2012) model this effect, and Becker and Ivashina (2011) find that small firms benefit more than established firms when bank lending rises. A related argument is in Baker, Stein, and Wurgler (2003), who show that the sensitivity of investment to for equity-dependent firms is more than triple that of other firms. Their theory rests on the idea that firms will over-invest when is high for nonfundamental reasons, and they provide additional tests that are consistent with this view. These arguments may explain a positive response of to . But they do not explain the negative response of to . The following argument does contain a possible demand-based and finance-related explanation for the asymmetric response of investment to technology shocks under convex adjustment costs: When a new technology arrives it raises the ’s of young firms more than the ’s of old firms. The resulting positive response of investment by the young firms would raise the equilibrium interest rate, resulting in lower investment among existing firms. Note that this argument also invokes an asymmetry, namely on how the ’s of the new firms change versus those of old firms. It also requires that the cost of capital faced by older firms rises significantly as a result of the increased investment of young firms, and this effect is probably small. Moreover, correlation (1901-2005) between and the commercial paper rate is -0.0014, which is small and not significant. An alternative, implied by results of Justiniano, Primiceri, and Tambalotti (JPT, 2011), is that itself reflects financial shocks since an adverse investment shock is accompanied by a rise in credit spreads. This may shed light on what affects , but again does not explain why only old firms would reduce their investment in response to a rise in . 31
Asymmetries due to a two-staged investment.–Acemoglu Gancia, and Zilibotti (2012) decompose investment into a creative followed by an implementative component in a model with no aggregate risk. JPT decompose it into an intermediate-goodcapital investment and a final-good-capital investment: there are no rents in the first stage, which is true of incumbent investment in our model, and rents to investment appear at the second stage, which is true for our entrants. So there is some similarity to these models except that our two investment types are simultaneous whereas JPT’s two investments are sequential in that one builds on the other. 6.3.1
Microfoundations for incumbents’ technological inertia
There are several models in which an incumbent is less likely to adopt new technology. First, Reinganum (1983) shows that an incumbent firm will not look for new products as hard as a challenger will, because, if it succeeds, it would merely augment an existing market position, whereas a successful challenger could gain the entire market. Second, there is the sunk investment argument or simply vintage human capital: Chari and Hopenhayn (1991), Jovanovic and Nyarko (1996, Proposition 3.1), and Grossman and Shapiro (1985) model specialization that can lead a firm into a competency trap. A new technology will then draw in entrants who are not committed to the old methods. Prusa and Schmitz (1994) make a related “sampling bias” argument: Entrants achieve their status by having a good idea through a lucky draw, after which regression to the mean takes over. In our terminology, entrants by definition are firms with ideas that have sufficiently high ’s. Once they become incumbents, they draw low ’s or simply no ’s at all, and are better off implementing the ideas of others, something they will find harder to do or of which they remain unaware until it is too late. Third, organization capital depreciates with age and vested interests grow. Martimort (1999) shows that as a firm ages, its insiders can more easily collude against the owners. The insiders then exert less effort, including effort in making investments that would adapt to a technology shock of the type that embodies.
7
Conclusion
We document an asymmetry in the response of investment to aggregate : Small and young firms respond positively to it, while large and old firms respond negatively. We argue that this occurs because a high is a signal of low compatibility of old capital with the new and, hence, of high implementation costs specific to incumbents. Entrants do not face compatibility problems because they start de novo and raise their investment when the cost is high. Variations in implementation costs are driven by technological change. On this view, aggregate stock-price volatility is caused by technological change, a property 32
that our model shares with others that feature shocks to the cost of investment. Rising stock prices reflect unmeasured implementation shocks and, other things constant, high stock prices are bad news for welfare. This feature may have gone unnoticed because stock prices have tended to be high when TFP too has been high, as was true in the 1920s and the 1990s, and these two outcomes have opposing effects on welfare. The dichotomy changes the standard one-aggregate investment model both in terms of behavior and in terms of optimal policy. Entrants’ investment is small but highly volatile. Moreover, the model recognizes a spillover from incumbents to entrants and indicates that the equilibrium growth is below its socially optimal rate, but the implied growth difference is small. Finally, we showed that private-equity should pay a premium or discount depending on whether the investment-cost shock or the TFP shock dominates.
Appendix A. Data and Methods In this appendix we document the data sources and methods used to construct the series depicted in our figures and included in the empirical analysis. Figure 1 .–The investment rate of an individual firm (i.e., −1 ) is its annual expenditures on property, plant, and equipment (PPE, Compustat item 30) as a percentage of its PPE stock at the end of the previous year (item 7). We remove observations with zero investment before taking logs. We compute the firm-specific using year-end data from Compustat. The numerator is the value of a firm’s common equity at current share prices (the product of Compustat items 24 and 25), to which we add the book values of preferred stock (item 130) and of long and short-term debt (items 9 and 34). We use book values of preferred stock and debt because prices of preferred stocks are not available from Compustat and we do not have information on issue dates for debt with which to estimate its market value more accurately. Book values of these components are reasonable approximations of market values so long as interest rates do not vary excessively. The denominator of is computed in the same way except that the book value of common equity (Compustat item 60) is used in place of market value. Our micro-based measures of thus reflect the value of a firm’s outstanding securities and implicitly assume that the proceeds of their issue are fully applied to the formation of capital, both physical and intangible. The aggregate used in the regression shown in the right panel are market value-weighted annual averages of the micro-based measures. Thus the aggregate are calculated from the same set of Compustat firms used in the left panel of figure 1. Figure 3 .–To construct the L series shown in the upper panel of figure 3, we begin with the real value of IPOs. This is measured as the aggregate year-end market value of the common stock of all firms that enter the University of Chicago’s Center 33
for Research in Securities Prices (CRSP) files in each year from 1925 through 2005, excluding American Depository Receipts. The CRSP files include listings only for the New York Stock Exchange (NYSE) from 1925 until 1961, with American Stock Exchange and NASDAQ firms joining in 1962 and 1972 respectively. This generates large entry rates in 1962 and 1972 that for the most part do not reflect initial public offerings. Because of this, we linearly interpolate between entry rates in 1961 and 1963 and between 1971 and 1973, and assign these values to the years 1962 and 1972 respectively. For 1901-1924 we obtain market values of firms that list for the first time on the NYSE using our pre-CRSP database of stock prices, par values, and book capitalizations that we collected for all common stocks traded on the NYSE using the The Commercial and Financial Chronicle, Bradstreet’s, The New York Times, and The Annalist (see Jovanovic and Rousseau, 2001a, b). We then divide by the implicit price deflator for GDP and aggregate (described below) to obtain L in constant 1996 dollars. Because stock-market capitalization has grown rapidly over the century (from 8.8 percent of in 1901 to 62.4 percent in 2005) and IPOs along with it, we correct L for the trend by constructing L
L = ¯ predicted MKTcap−1
where 1 ¯ =
2005 X
=2005−
MKTcap
(35)
(36)
The MKTcap are from CRSP and our backward extension of it measured at the end of each year. The are end-year stocks of private fixed assets from the BEA (2006, table 6.1, line 1) for 1925 through 2005. For 1900-1924, we use annual estimates from Goldsmith (1955, vol. 3, table W-1, col. 2, pp. 14-15) that include reproducible, tangible assets (i.e., structures, equipment, and inventories), and then subtract government structures (col. 3), public inventories (col. 17) and monetary gold and silver (col. 18), and join the result with the BEA series. The resulting time series average of their ratio, , ¯ turns out to be 0.228 for 1901-2005. We then obtain predictions of real market capitalization as the fitted values from an OLS regression of its log on the log of real , with both variables deflated using the implicit price deflator. Since IPO investments are made throughout the year and we measure the denominator of L at the start of each year, we adjust L to reflect population growth. To do this, we average growth factors in mid-year total populations from the Census Bureau over the two years that overlap , and use the square root as a deflator to yield the final L series. To build the L shown in the lower panel of figure 3, we start with the aggregate investment rate, , constructed as annual gross private domestic investment from the BEA (2006,table 5.2.5, line 4) for 1929-2005, to which we join estimates from 34
Kendrick (1961, table A-IIb, col. 5, pp. 296-7) for 1901-1928 and then divide by −1 . We then adjust for inflation during year , which was high in the later 1970s for example, by averaging the annual inflation factors across the two years that overlap , and then using its square root as a deflator. Finally, after adjusting for population growth as described above for the L series, we form L = − L
(37)
as a residual. Figure 4 .–For , we use private output, defined as GDP less government expenditures on consumption and investment from the BEA (2006) for 1929-2005, to which we join Kendrick’s (1961, table A-IIb, pp. 296-7, col. 11) estimates of gross national product less government for 1901-1928. We then divide the result by −1 and correct for inflation and population growth in year as described for the series above. For aggregate , we use fourth quarter observations underlying Hall (2001) for 1950-1999, and then join them with estimates underlying Abel and Eberly (2011) for 1999 to 2005. These authors derive aggregate Tobin’s from the Federal Reserve Board’s Flow of Funds Accounts as the ratio of total market value of equity and bonds to private fixed assets in the non-financial corporate sector, which includes unlisted corporations that are not in Compustat. We then bring the aggregate series back to 1901 by ratio splicing the “equity ” measure underlying Wright (2004). Hall’s measure of exceeds Wright’s by factor of more than 1.5 in 1950, when the splice occurs, producing ’s before 1950 that are considerably higher than Wright’s published estimates. Figure 5 .–To construct the S series shown in the upper panel of figure 5, we begin with the value of venture capital investment for the numerator, S . This is from the National Venture Capital Association (2011, fig. 3.08, p. 27) for 1985-2005, to which we join annual venture capital disbursements from Kortum and Lerner (2000, table 1, p. 679) after converting all quantities into constant 1996 dollars using the implicit price deflator for GDP. Because the venture capital investment and commitments have grown so rapidly over the 1969-2005 period, we correct S for the trend by constructing S =
S predicted VCC−1
where 1 =
2005 X
=2005−
VCC
(38)
(39)
The VCC are annual commitments to U.S. venture funds from the National Venture Capital Association (2011, fig. 2.01, p. 20) for 1985-2005, to which we join commitments from Gompers (1994, Figure 2A, p. 11) for 1969-1984, again converting all 35
values into constant 1996 dollars. The are end-year stocks of private fixed assets from the BEA (2006, table 6.1, line 1). The resulting time series average of their ratio, , turns out to be 0.00053 for 1969-2005. We then obtain predictions of real venture capital commitments as the fitted values from an OLS regression of its log on the log of real . To build the S shown in the lower panel of figure 5, we start with the aggregate investment rate constructed as annual gross private domestic investment from the Bureau of Economic Analysis (BEA 2006, table 5.2.5, line 4) for 1969-2005 divided by −1 . We then form by subtracting from .
Appendix B. Robustness of Results in Figure 1 7.1
Alternative Specifications and Estimations
Table B1 shows the robustness of the regressions in figure 1 to the inclusion of fixed effects for firms, years, and four-digit SIC industries, as well as to estimation with GMM using higher-order moments. The GMM estimations use the method developed for investment regressions by Erickson and Whited (2000, 2002) that yields consistency in the presence of measurement error in . These estimators are based on taking expectations of polynomials derived from the classical error-in-variables model. We estimate these models using the GAUSS code available on Toni Whited’s web site. When we regress a firm’s investment on its own −1 , the coefficients in the first row of table B1 are positive, statistically significant, and of similar size whether we include fixed effects for firms, years, and four-digit industries or not. The GMM estimates, which use second-order moments, are larger but not extremely so. The results in the second row indicate that the negative response of firm-level investment to aggregate −1 is also robust to the inclusion of fixed effects firms and four-digit industries, but that the effects are much stronger for the GMM estimations using fourth-order moments. This might be expected given that our measure of aggregate does not have the extent of skewness in logs under which the Erickson and Whited estimator performs most effectively (Erickson and Whited, 2012). It is worth noting, however, that the GMM results are consistent with the qualitative implications of our theoretical model.
36
Table B1 Robustness of Investment Regressions in Figure 1, 1962-2006 Dependent variable: log of firm investment as percent of capital (I /K−1 ) log(−1 )
0.518 (87.8)
1.044 (12.9)
0.698 (77.5)
0.603 (94.2)
0.532 (79.1)
0.599 (83.6)
log(−1 )
-0.251 (15.7)
-3.871 (11.8)
-0.526 (29.0)
-0.422 (26.7)
firm effects
no
no
yes
no
no
no
no
no
yes
no
year effects
no
no
no
yes
no
yes
n.a.
n.a.
n.a.
n.a.
4-digit SIC
no
no
no
no
yes
yes
no
no
no
yes
estimation
OLS
gmm
OLS
OLS
OLS
OLS
OLS
gmm
OLS
OLS
2
.084
.169
.415
.134
.127
.180
.002
.027
.363
.062
Note. — T-statistics based on robust standard errors are in parentheses. The independent variables Q −1 are firm-specific Tobin’s Qs measured at the end of the previous year, and the Q −1 are annual value-weighted means of Tobin’s Qs for all Compustat firms. There are 141,956 observations included in each regression.
Different signs of the two slopes in Figure 1 The following derives the statistical conditions needed in the i.i.d. case for there to arise a positive slope in panel A of figure 1 and a negative slope in panel B. Suppose that investment () and satisfy the following two structural equations = − +
and
= +
where j,t and j,t have zero mean and where and are i.i.d. Then in the two univariate regressions = 0 + 1 + 1
and
= 0 + 1 +2
OLS estimates have the following properties: (ˆ1 ) = −
2 2 + 2 37
³ ´ and ˆ1 = −
(I /K)i,t
Pooled relation in Panel A of Fig 1
low-Q year
high-Q year
QL
QH
Qi,t-1
Firm Q Figure 10: Schematic view of panel A of Fig. 1 2
Then if , the aggregate effect is negative. But (ˆ1 ) 0 ⇔ 2 + 2 , and ¡ ¢ this too will hold if 2 2 + 2 is sufficiently small. In our Compustat data, this ratio is 0.325. As a first pass assume that the OLS estimates ³ ´ in figure 1 coincide with their mean values, in which case (ˆ 1 ) = 0518 and ˆ = −0251, and we have = + 0251 leaving us with the equation 0518 = − ( + 0251) 0325, implying that = 0888 and = 0888 + 0251 = 1139. Graphically we show the pooled panel relation as the dashed red line in figure 10, and it depicts two cross section relations — one for a low aggregate Q year with an aggregate equal to L , and one for a high-Q year with an aggregate equal to H Then figure 11 shows the negative aggregate relation that emerges and that is plotted in panel B of figure 1.
Appendix C: Proofs Proof of Proposition 1: Let satisfy the equation µ Z µ =
1 + 1−1 + + (1 − )
¶
38
¶1 [ + (1 − ) ] ()
(40)
(I /K)i,t
Aggregate relation plotted in Panel B of Fig. 1
QL
QH
Qi,t-1
Aggregate Q Figure 11: Schematic view of panel B of Fig. 1 and let satisfy the equation !−1 Ã R 0 − 0 R 0 − 0 ( ) (1 − + [0 ]) (0 ) ( ) (0 ) R = 1 − R − 0 ( + [1 − ] 0 ) (0 ) − ( 0 + [1 − ] 0 ) (0 )
(41)
First, suppose that a solution for exists (its existence will be shown at the end of 0 0 0 this proof). Then and are defined, and is defined in (41). Since = , (9) implies µ ¶− Z − − = (0 ) [ 0 + (1 − ) 0 ] (0 ) (42) 0 Therefore
0 = Now
µ Z ¶−1 0 − 0 0 0 ( ) [ + (1 − ) ] ( )
= = 1 0 1 − + ( + − )
Therefore 1
1
´ ³ 1 1 − + [ + ] −
1
1 ´ ³ 1 1 − 1 − +
1
µ Z ¶−1 0 − 0 0 0 ( ) [ + (1 − ) ] ( ) = 39
i.e., ¶1 µ ¶ µ Z 1 1 1 0 − 0 0 0 ( ) [ + (1 − ) ] ( ) 1 − + ( + ) = + i.e., = 1
+
³ R
1 − + 1 ( + )
+ + (1 − ) ´1 = ¢ ¡ R 1−1 0 )− [ 0 + (1 − ) 0 ] (0 ) 1 − 0 0 0 0 1 + ( ( ) [ + (1 − ) ] ( )
i.e., (12). Then = 1 ( + − ) =
1
³ +−
++(1−) 1+ 1−1
´ , i.e., (13).
In (10), we substitute [1 − + (0 )] 0 for (), which leads to R 0 − 0 ( ) ( + [1 − + (0 )] 0 ) (0 ) () R = ≡ (1 − + []) − ( 0 + [1 − ] 0 ) (0 )
and thence to !−1 Ã R 0 − 0 R 0 − 0 ( ) (1 − + [0 ]) (0 ) ( ) (0 ) R = 1 − R − 0 ( + [1 − ] 0 ) (0 ) − ( 0 + [1 − ] 0 ) (0 )
which, together with (12), leads to (41). Existence of D.–Next, we show that exists when (11) holds. Divide both sides of (40) by to get µ ¶ ¶1 µ Z 1 + 1−1 − [ + (1 − ) ] 1 = + + (1 − ) ¶ µ Z µ ¶1 −1 + 1−1 = [ + (1 − ) ] + + (1 − )
Since −1 ranges from zero to infinity as ranges over the positive line, and since 0, a necessary and sufficient condition for a solution for to exist is that µ Z µ ¶ ¶1 1−1 [ + (1 − ) ] ≤ 1 + + (1 − ) This is equivalent to 1≥
1
Z
−1
+ (1 − ) ( + () + (1 − ) )
Since and are positive, for (43) to hold it suffices that ¶1− Z Z µ 1 1 −1 + (1 − ) 1−+ = 1 ( + (1 − ) ) 40
(43)
i.e., (11). Derivation of (27).– 1 (1 + )− = − = −1 (1 + ) − 1 1 − (1 + ) so that
= −1 (1 + ) − 1 ⇒ 1 + = −1 (1 + )
which leads to (27). Proof of Proposition 2: A Taylor’s expansion around = 0 yields ¸¶−1+1 µ ∙ + − ≈ 1+ = (1 + )1− ∗
Proof of Proposition 3: When projects with ∈ [0 max ] are implemented, the number of new machines (per unit of ) built is =1−
µ
max
¶−1
(44)
Now, when the marginal project just breaks even, max = ˆ, and we have (33). Solving (44) for the marginal project yields max = (1 − )−
(45)
Letting () denote the unnormalized C.D.F. defined in (30), the cost of per unit of is µ ¶−1 Z Z max 1 max () = cost = changing variables to = , cost =
à ¯max ∙ ¸1−1 ! ¯ ¯ −1 = = 1 − max ¯ 1 1 − 1 1 1− ! à 1 − (1 − )1− (using (45)). = 1− Z
max
1−1
Since the cost is in units of today’s goods, this implies (32). 41
References [1] Abel, Andrew B., and Janice C. Eberly. 2011. “How Q and Cash Flow Affect Investment without Frictions: An Analytic Explanation.” Review of Economic Studies 78 (4): 1179—1200. [2] Acemoglu, Daron, Gino Gancia, and Fabrizio Zilibotti. 2012. “Competing Engines of Growth: Innovation and Standardization.” Journal of Economic Theory 147 (2): 570—601. [3] Acemoglu, Daron, Ufuk Akcigit, Nicholas Bloom, and William R. Kerr. 2013. “Innovation, Reallocation and Growth.” Working Paper no. 18993 (April), NBER, Cambridge, MA. [4] Araujo, Luis, and Raoul Minetti. 2012. “Credit Crunches, Asset Prices, and Technological Change.” Working Paper no. 1204, CASMEF. [5] Arrow, Kenneth. 1962. “The Economic Implications of Learning by Doing.” Review of Economic Studies 29 (3): 155—73 [6] Asker, John, Joan Farre-Mensa, and Alexander Ljungqvist. 2011. “Comparing the Investment Behavior of Publicly Traded and Privately Held Firms.” Manuscript (July). [7] Atkeson, Andrew, Aubhik Khan, and Lee Ohanian. 1996. “Are Data on Industry Evolution and Gross Job Turnover Relevant for Macroeconomics?” CarnegieRochester Conference Series on Public Policy 44: 216-250 [8] Axtell, Robert. 201. “Zipf Distribution of U.S. Firm Sizes.” Science 293 (5536): 1818-20. [9] Baker, Malcolm, Jeremy Stein, and Jeffrey Wurgler. 2003. “When Does the Market Matter? Stock Prices and the Investment of Equity-Dependent Firms.” Quarterly Journal of Economics 118 (3): 969—1005. [10] Barro, Robert J. (2006). “Rare Disasters and Asset Markets in the Twentieth Century.” Quarterly Journal of Economics, 121 (3): 823—66. [11] Becker, Bo, and Victoria Ivashina. 2011. “Cyclicality of Credit Supply: Firm Level Evidence.” Working Paper no. 17392 (September), NBER, Cambridge, MA. [12] Bilbiie, Florin, Fabio Ghironi, and Marc Melitz. 2012. “Endogenous Entry, Product Variety, and Business Cycles.” Journal of Political Economy 120 (2): 304—45 [13] Campbell, Jeffrey. 1998. “Entry, Exit, Embodied Technology, and Business Cycles.” Review of Economic Dynamics 1 (2): 371—408. 42
[14] Carroll, Glenn, and T. Michael Hannan. 2000. The Demography of Organizations and Industries. Princeton: Princeton Univ. Press. [15] Chari, V. V., and Hugo Hopenhayn. 1991. “Vintage Human Capital, Growth, and the Diffusion of New Technology.” Journal of Political Economy 99 (6): 1142—65. [16] Chesbrough, Henry W. 2002. “Making Sense of Corporate Venture Capital.” Harvard Business Review 80 (3): 90—99. [17] Chirinko, Robert and Huntley Schaller. 1995. “Why Does Liquidity Matter in Investment Equations?” Journal of Money, Credit and Banking 27 (2): 527—48. [18] Eeckhout, Jan, and Boyan Jovanovic. 2002. “Knowledge Spillovers and Inequality” American Economic Review 92 (5): 1290-1307. [19] Erickson, Timothy, and Toni M. Whited. 2000. “Measurement Error and the Relationship between Investment and Q.” Journal of Political Economy, 108 (5): 1027—57. [20] Erickson, Timothy, and Toni M. Whited. 2002. “Two-Step GMM Estimation of the Errors-in-Variables Model using Higher-Order Moments.” Econometric Theory 18 (3): 776—99. [21] Erickson, Timothy, and Toni M. Whited. 2012. “Treating Measurement Error in Tobin’s Q.” Review of Financial Studies, 25 (4): 1286—1329. [22] Fazzari, Steven, Glenn Hubbard, and Bruce Petersen. 1988. “Financing Constraints and Corporate Investment,” Brookings Papers on Economic Activity: 141—206. [23] Feuerverger, Andrey and Peter Hall. 1999. “Estimating a Tail Exponent by Modelling Departure from a Pareto Distribution.” Annals of Statistics 27 (2): 760-81. [24] Fisher, Jonas. 2006. “The Dynamic Effects of Neutral and Investment-Specific Technology Shocks.” Journal of Political Economy 114 (3): 413—51. [25] Gilchrist, Simon, and John Williams. 2000. “Putty—Clay and Investment: A Business Cycle Analysis.” Journal of Political Economy 108 (5), 928—60. [26] Goldsmith, Raymond W. 1955. A Study of Savings in the United States. Princeton, NJ: Princeton Univ. Press. [27] Gompers, Paul. 1994. “The Rise and Fall of Venture Capital.” Business and Economic History 23 (1): 1—26.
43
[28] Gompers, Paul. 2002. “Corporations and the Financing of Innovation: The Corporate Venturing Experience.” Federal Reserve Bank of Atlanta Economic Review (Q4): 1—17. [29] Gompers, Paul, Anna Kovner, Josh Lerner, and David Scharfstein. 2008. “Venture Capital Investment Cycles: The Impact of Public Markets.” Journal of Financial Economics 87 (1): 1—23. [30] Greenwood, Jeremy, Zvi Hercowitz, and Per Krusell. 1997. “Long-Run Implications of Investment-Specific Technological Change.” American Economic Review 87 (3): 342—62. [31] Greenwood, Jeremy, Zvi Hercowitz, and Per Krusell. 2000. “The Role of Investment-Specific Technological Change in the Business Cycle.” European Economic Review 44 (1): 91—115. [32] Grossman, Gene, and Carl Shapiro. 1982. “A Theory of Factor Mobility.” Journal of Political Economy 90 (5): 1054—69. [33] Hall, Robert E. 1978. “Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence.” Journal of Political Economy, 86 (6), 971—87. [34] Hall, Robert E. 2001. “The Stock Market and Capital Accumulation.” American Economic Review, 91 (5): 1185—1202. [35] Hayashi, Fumio. 1982. “Tobin’s Marginal q and Average q: A neoclassical Interpretation.” Econometrica 50 (1): 213—24. [36] Jovanovic, Boyan, and Yaw Nyarko. 1996. “Learning by Doing and the Choice of Technology.” Econometrica 64(6): 1299—1310. [37] Jovanovic, Boyan. 2009. “Investment Options and the Business Cycle.” Journal of Economic Theory 144 (6), 2247—65. [38] Jovanovic, Boyan, and Peter L. Rousseau. 2001a. “Why Wait? A Century of Life Before IPO.” American Economic Review (Papers and Proceedings) 91 (2): 336—41. [39] Jovanovic, Boyan, and Peter L. Rousseau. 2001b. “Vintage Organization Capital.” Working Paper no. 8166, NBER, Cambridge, MA. [40] Jovanovic, Boyan, and Peter L. Rousseau. 2005. “General Purpose Technologies.” In Handbook of Economic Growth, vol. 1b, edited by Philippe Aghion and Steven N. Durlauf. Amsterdam: Elsevier North Holland.
44
[41] Jovanovic, Boyan, and Balazs Szentes. 2013. “On the Market for Venture Capital.” Journal of Political Economy 121 (3): 493—527. [42] Justiniano, Alejandro, and Giorgio Primiceri. 2008. “The Time Varying Volatility of Macroeconomic Fluctuations.” American Economic Review 98 (3): 604—41. [43] Justiniano, Alejandro, Giorgio Primiceri, and Andrea Tambalotti. 2011. “Investment Shocks and the Relative Price of Investment.” Review of Economic Dynamics 14 (1): 101—21. [44] Kaplan, Steven, and Antoinette Schoar. 2005. “Private Equity Performance: Returns, Persistence and Capital Flows.” Journal of Finance 60 (4): 1791—1823. [45] Kendrick, John. 1961. Productivity Trends in the United States. Princeton NJ: Princeton Univ. Press. [46] Kortum, Samuel, and Josh Lerner. 2000. “Assessing the Contribution of Venture Capital to Innovation.” Rand Journal of Economics 31 (4): 674—92. [47] Lucas, Robert E., Jr. 1978. “Asset Prices in an Exchange Economy.” Econometrica 46 (6): 1429—55. [48] Luttmer, Erzo. 2007. “Selection, Growth, and the Size Distribution of Firms,” Quarterly Journal of Economics (2007) 122 (3): 1103—1144. [49] Martimort, David. 1999. “The Life Cycle of Regulatory Agencies: Dynamic Capture and Transaction Costs.” Review of Economic Studies 66 (4): 929—47. [50] National Venture Capital Association. 2001. National Venture Capital Association Yearbook. New York: Thomson Reuters. [51] Papanikolaou, Dimitris. 2011. “Investment Shocks and Asset Prices.” Journal of Political Economy 119 (4): 639—85. [52] Prescott, Edward C. 1980. “Comments on the Current State of the Theory of Aggregate Investment Behavior.” Carnegie-Rochester Conference Series on Public Policy 12 (1): 93—101. [53] Prusa, Thomas J., and James Schmitz. 1991. “Are New Firms an Important Source of Innovation?: Evidence from the PC Software Industry.” Economics Letters 35 (3), 339—42. [54] Prusa, Thomas J., and James Schmitz. 1994. “Can Companies Maintain their Initial Innovation Thrust? A Study of the PC Software Industry.” Review of Economics and Statistics 76 (3): 523—40.
45
[55] Reinganum, Jennifer. 1983. “Uncertain Innovation and the Persistence of Monopoly.” American Economic Review 73 (4): 741—48. [56] Sargent, Thomas J. 1980. “Tobin’s Q and the Rate of Investment in General Equilibrium.” Carnegie-Rochester Conference Series on Public Policy 12 (1): 107—54. [57] Solow, Robert. “Investment and technological progress,” in: K. Arrow, S. Karlin, P. Suppe (Eds.), Mathematical Methods in the Social Sciences, Stanford University Press, Stanford, CA, 1960. [58] Sorensen, Morten, Neng Wang and Jinquang Yang. “Valuing private equity.” July 2013. [59] U.S. Department of Commerce, Bureau of Economic Analysis. 2006. “National Income and Product Accounts.” Washington, DC. [60] Wright, Stephen H. 2004. “Measures of Stock Market Value and Returns for the U.S. Nonfinancial Corporate Sector, 1900-2002.” Review of Income and Wealth, 50 (4): 561—84. [61] Yorukoglu, Mehmet. 1998. “The Information Technology Productivity Paradox.” Review of Economic Dynamics 1 (2): 551—92.
46
