Product markets and corporate investment: Theory and evidence

Journal of Banking & Finance 36 (2012) 439–453

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Product markets and corporate investment: Theory and evidence Evrim Akdog˘u a,⇑, Peter MacKay b a b

Graduate School of Business, Koç University, Istanbul 34450, Turkey Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

a r t i c l e

i n f o

Article history: Received 4 August 2008 Accepted 1 August 2011 Available online 6 August 2011 JEL classification: G31 Keywords: Corporate investment Externality Herding

a b s t r a c t Investment patterns often associated with agency and information problems can emerge as rational responses to product–market rivalry. We illustrate this result when industry players make simultaneous or sequential investment decisions in the face of two negative externalities. One externality arises when all competing firms invest, thus eroding the gains to investment accruing to any one firm. Another externality arises when some firms do not invest and lose out to rivals who do invest. The value of investment therefore depends on the investment’s intrinsic merits and the actions of all competitors. Our analysis can rationalize investment patterns that might appear suboptimal when such externalities are ignored. For instance, our simultaneous model can justify investment levels that might otherwise be interpreted as under- or over-investment. Our sequential model shows that value-maximizing firms might optimally herd in their investment decisions. We present evidence supporting key aspects of both the simultaneous and sequential models. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Market imperfections such as agency conflicts and information asymmetry have long figured as the usual suspects behind suboptimal corporate investment policy.1 A large literature expands on these imperfections and how financial policy can mitigate them. Consequently, policy distortions are now routinely attributed to agency or information problems. Yet, factors unrelated to these imperfections, such as product–market interactions, can lead to observationally-equivalent investment behavior. This paper proposes such alternative factors by showing how interactions between rival firms’ investment returns can rationalize a wide range of optimal investment patterns, including what might otherwise be interpreted as distortions caused by agency or information problems and herding behavior caused by managerial self-interest.

⇑ Corresponding author. Tel.: +90 212 338 1811; fax: +90 212 338 1653. E-mail addresses: [email protected] (E. Akdog˘u), [email protected] (P. MacKay). Jensen and Meckling (1976) and Jensen (1986) show that managerial self-interest leads to over-investment. However, firms under-invest if claimant interests diverge (Myers, 1977) or when project quality is unobservable (Myers and Majluf, 1984). Welch (1992) shows that noisy signals about a common value lead to rational herding where agents ignore their information and mimic first-mover actions. Scharfstein and Stein (1990) and Zwiebel (1995) show that managerial reputation concerns can cause herding in corporate investment decisions. 1

0378-4266/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2011.08.001

We model a setting where a firm’s innovative investment negatively affects other firms.2 By innovative investment we mean capital outlays that give a firm a competitive edge over its rivals via cost-reducing measures or revenue-enhancing improvements.3 Viewed in isolation, these innovations adversely affect rivals by, say, enabling one firm to gain by lowering its price or enhancing its product line. Faced with similar opportunities, rival firms form rational expectations regarding their peers’ actions which in turn affect their own innovation decisions.4 This rivalry gives rise to two negative externalities. First, duplicating a rival’s investment erodes the firm’s gains to investing compared to when the rival does not invest. Second, failure to invest when a rival invests causes the firm to fall behind and drop in value as it loses market share. We analyze how these offsetting

2 Although we have industry rivals in mind, all we need is interdependence. Thus, our framework could represent duopolists, a single firm and the rest of its industry, or even two competing industries. An example of the latter is digital photography which caused two complementary industries, cameras and film, to become strategic rivals. 3 Although we frame the model in terms of innovative investment, our modeling approach also applies to projects that change production capacity without changing the state of technology or the product space. We therefore drop the terms ‘innovative’ and ‘innovation’ from the rest of the paper to reflect this general applicability of the model. 4 In the case of digital photography, camera-makers’ innovation threatened filmmakers’ existing business. Film-makers responded by launching disposal cameras and entering the digital photography business themselves, thereby curtailing the payoff to camera-makers’ innovation.

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E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

costs of duplication and falling behind determine equilibrium investment strategies and condition optimal investment policy. We present a simultaneous model where neither firm can observe the rival’s investment decision and a sequential model where one firm can observe the first-mover’s investment. These models show how complex investment patterns can emerge from a simple set of externalities. For instance, both models show that what is typically viewed as under and over-investment can actually reflect firm-value-maximizing investment policy once set in a richer economic context.5 In standard corporate finance models, firms under-invest because claimant incentives diverge (e.g., Myers, 1977) or because information is imperfect (e.g., Myers and Majluf, 1984). In our setting, firms rationally forgo investments that appear profitable on a stand-alone basis if the cost of duplicate investment outweighs the cost of falling behind. Our model also provides an alternative to traditional explanations for over-investment such as excess perquisites consumption (e.g., Jensen and Meckling, 1976) and empire-building (e.g., Jensen, 1986). We show that firms rationally invest in projects that appear unprofitable on a stand-alone basis if the cost of falling behind outweighs the cost of duplicate investment. In our setting, competitive market forces – not incentive or information problems – lead firms to invest more or less than otherwise. Our models also offer alternative explanations for investment fads such as merger and acquisition waves, and herding behavior such as real-estate booms.6 Whereas prior studies link herding to imperfect private information (e.g., Welch, 1992) and managerial reputation (e.g., Scharfstein and Stein, 1990; Zwiebel, 1995), we show that firms can appear to coordinate investments when private information is perfect and managers have no reputation concerns. This result obtains even when herding is ruled out by construction, such as when firms act simultaneously. In particular, our simultaneous model shows that firms are more likely to make similar investment decisions when the cost of falling behind is either lower (neither firm invests) or higher (both firms invest) than the cost of duplication. In contrast, our sequential model shows that the probability that a firm will mimic first-moving rivals depends on whether the cost of falling behind is higher or lower than the cost of duplication. If the cost of falling behind is higher than the cost of duplication, the second-mover is likely to invest if the first-mover has invested and firms will thus appear to herd. If it is lower, the second-mover is unlikely to invest if the first-mover has invested and firms will appear to ‘‘anti-herd’’. We extend the sequential model to consider the possibility that rivals might differ in their costs of falling behind or in the required capital outlays. These asymmetries yield more nuanced results, where the investment rules are reflexive: Under asymmetry each firm’s investment rule depends on its own parameters (cost of falling behind, capital outlays) and those of its rival’s. We test key assumptions and predictions of our models on a sample of firms operating in concentrated industries. Using research and development as a proxy for innovative investment, we regress the interaction of own-firm and rival investment on firm value (Tobin’s q) to measure the joint effect of duplicate innovation and falling behind. We find that this net investment 5 The terms under-investment and over-investment are suitable in the context of agency and information problems, imperfections that are absent from our framework. For continuity with the literature, we use the terms only to note departures from an NPV cutoff rule that ignores externalities, without reference to the cause of such departures. 6 Martynova and Renneboog (2008) survey the literature on merger and acquisition waves. They conclude that the evidence suggests that merger waves are usually triggered by a regulatory or industry shock then extend beyond an optimal point, where hubris and managerial self-interest take over, and ‘herding-like’ behavior emerges near the end of the wave.

externality is significant and negative, which supports our assumption that externalities are present and suggests that the cost of duplication is greater than the cost of falling behind. We find that the net investment externality varies across industries in a manner consistent with our simultaneous model: When the net investment externality is high, investment is high (firms appear to over-invest) because the cost of falling behind is high. Conversely, when the net investment externality is low, investment is low (firms appear to under-invest) because the cost of duplication is high. In both these cases, the intra-industry variance of investment is low and firms appear to cluster in their decision to invest or not invest, as the model also predicts. We also find investment and externality patterns consistent with the sequential model.7 We use the correlation of current own-firm investment and lagged rival investment as a proxy for how similar or dissimilar second-mover investment is to first-mover investment. We find that when the cost of falling behind is high, second-movers are more likely to invest when first-mover rivals invest (firms appear to herd). Similarly, when the cost of duplication is high, second-movers are less likely to invest when first-mover rivals invest (firms appear to anti-herd). Finally, asymmetry across industry rivals bears on these relations in a significant way. We should point out that while our test results are consistent with our predictions, the proxies we use are quite reduced-form compared to the stylized structure underlying the analysis. Thus, the model and its associated tests should be interpreted with circumspection. We view our contribution to the literature as illustrating how the inclusion of plausible strategic interactions in the analysis of corporate investment decisions can shed new insights on observed behavior. In this sense, the model and our results should not be taken literally but do seem to fit the data well. This is not the first study to show that real-side factors can affect corporate investment. First, strategic investment is central to industrial organization (IO).8 Although our models perfect certain investment games, our contribution is not to the IO literature but to financial economics: We show that strategy and financial imperfections can produce identical investment patterns. Relatedly, the product–markets finance literature (e.g., Brander and Lewis, 1986; Maksimovic and Zechner, 1991; MacKay and Phillips, 2005; Clayton, 2009) shows that financial structure can alter firm conduct and industry equilibrium. We show how industry rivalry can affect investment policy even when finance does not. Finally, the realoptions literature (e.g., Grenadier, 2002; Shackleton et al., 2004) shows how investment is optimally accelerated or delayed in the absence or presence of competition. Our model yields similar outcomes in a one or two-period setting. We make two contributions to the literature. First, where standard models of corporate investment rely on agency and information problems to explain departures from first-best, we show that such departures might arise as an optimal response to inter-firm externalities. This is not to say that agency and information problems do not distort investment policy. All we claim is that product–market interactions must be duly considered when investigating the depth of agency and information problems and in designing policies aimed at mitigating them. Second, our analysis has implications for the coordination of corporate investment decisions. Specifically, our models predict investment patterns where firms appear to cluster (simultaneous model) or herd and anti-herd (sequential model), patterns which our empirical analysis supports. Thus, our models offer simple reasons as to why firms might rationally invest in a seemingly-coordinated fashion 7 This is possible because our models are not mutually exclusive in that they capture complementary aspects of the innovation process. Firms may well strategize both simultaneously and sequentially, as our findings suggest. 8 Tirole (1988) provides an excellent survey.

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

without invoking managerial career concerns or other behavioral distortions. The paper proceeds as follows. Section 1 develops simultaneous and sequential models. Section 2 reports empirical tests and Section 3 concludes. All proofs appear in the Appendix. 2. Basic setup of the simultaneous and sequential models We begin by laying out the assumptions and framework common to our simultaneous and sequential models. Both models investigate the investment decisions of two symmetric firms, i and j, run by firm-value-maximizing managers with no reputation or career concerns. We assume these firms operate in the same industry, but this assumption is not critical so long as they are interdependent. We also assume that exit is more costly than remaining in the industry. The firms draw private values regarding an investment project, Vi and Vj, independently and uniformly distributed over [0, 1]. These realized investment values are private information, but their distribution is common knowledge.9 Adopting the investment project requires a common, sunk capital outlay of I e [0, 1].10 The investment project is not in limited supply and each firm may invest in it regardless of the other firm’s decision. This assumption sets our model apart from strategic investment models (e.g., Spence, 1977; Dixit, 1980; Fudenberg and Tirole, 1983), which more restrictively assume that investment opportunities are in limited supply.11 The firms’ only choice variable is whether to invest or not. In the simultaneous model, this choice is represented by the binary functions si(Vi) e {0, 1} and sj(Vj) e {0, 1}. If firm i(j) chooses to invest then si(Vi) = 1 (sj(Vj) = 1), otherwise si(Vi) = 0 (sj(Vj) = 0). In the sequential model, the first-mover’s decision is still represented by si(Vi). However, since the second-mover observes the firstmover’s action before making its own decision, its decision is represented by s1j ðV j Þ or s0j ðV j Þ, depending on whether the first-mover chooses si(Vi) = 1 or si(Vi) = 0. The realized investment values, Vi and Vj, are absolute values in that they reflect what each firm’s investment project is worth in isolation, i.e., they assume the rival does not invest. However, each firm’s ultimate payoff to investing depends on whether the rival firm invests. When neither firm invests, the initial state of the industry is unperturbed and each firm earns a normalized payoff of zero. Whenever at least one firm invests, the state of the industry is changed for both parties because one firm’s investment affects all competing firms: When rival j invests, firm i experiences a negative externality that depends on its own action. If firm i also invests, then the value of investment is lowered for both firms because neither firm gains the competitive edge it would have if the other firm did not invest. We allow this externality – the cost of duplicate investment – to depend on the magnitude of the rival’s realized investment value by setting the payoffs to joint investment to Vi  Vj for firm i and Vj  Vi for firm j.12 Thus, even 9 Although the firms are symmetric in terms of expected investment values, the possibility of different realized investment values can be interpreted as some firms being better suited to exploit a particular investment project. 10 We do not restrict the ex-ante expected investment payoff to be positive or negative: For I 2 [0, ½), it is positive, and for I 2 [½, 1), it is negative. Also, the sunkcost assumption is innocuous since any capital recovered is equivalent to lowering the upfront cost by the discounted amount of the salvage value. 11 Given downward sloping demand, the model can also accommodate noninnovative investment that simply expands production capacity without altering the production process or the product line. In this case, negative externalities arise because one firm’s expansion shifts the aggregate supply curve and lowers prices for all firms. 12 This is similar to Spence’s (1979) assumption that own-firm profits are decreasing in the rival’s capital stock. More recently, Khanna (1998) uses this assumption in a corporate-investment setting to solve for the optimal contract when investing together decreases the profits to each firm.

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an ex-ante profitable investment (Vi  I > 0) can earn a negative payoff if the investment turns out to be worth more in the rivals’ hands (Vi  Vj  I < 0). If firm i does not invest, however, rival j earns its absolute value of investment, Vj, and i’s failure to invest puts it at a competitive disadvantage. For simplicity, we assume the non-investing firm experiences a fixed externality of a that reflects the cost of falling behind. In short, whenever at least one firm invests the industry evolves and all firms are affected. The table below summarizes the firms’ payoffs to investment and non-investment. In each cell of the table, the first row shows firm i’s payoff and the second row shows firm j’s payoff.

In the simultaneous model, the firms do not observe each other’s type or action but form rational expectations and move accordingly. Here, we solve the game as a pure-strategy Bayesian–Nash equilibrium (BNE), where each firm maximizes expected payoff given the type-contingent strategy of the other firm where types correspond to the privately-observed realized investment values, Vi and Vj. In the sequential model, the second firm to move observes the first-mover’s action and makes an informed decision. Here, we use backward induction to solve for the equilibrium strategies and the firms’ optimal investment policies. 2.1. Simultaneous model Suppose both firms discover their private investment values at t = 1 and decide simultaneously whether to invest. We represent their investment decisions as binary functions, si(Vi) and sj(Vj), where s() = 0 means no investment and s() = 1 means investment. Conditional on knowing its private investment value Vi at t = 1, firm i’s expected payoff to investing is:

Ei ½si ¼ 1 ¼ V i  I 

Z

1

V j f ðV j ÞdV j :

ð1Þ

Vc

The integral bound Vc() represents the symmetric equilibrium investment cutoff level that guides both firms’ investment decisions: Firms invest if they draw a realized investment value above Vc and they do not invest otherwise.13 f(Vj) is the investment density function, which we assume is U[0, 1] without loss of generality. The first two terms of this equation, Vi  I, measure what firm i earns R1 when rival j does not invest. The last term, V c V j f ðV j ÞdV j , captures the negative externality that firm i experiences when rival j also invests, namely, the cost of duplicate investment. We let this externality depend on the relative attractiveness of rival j’s investment by integrating over values of Vj above the symmetric equilibrium investment cutoff level, Vj e [Vc, 1]. This is the region where rival j also innovates. Firm i also considers the expected payoff to not investing:

Ei ½si ¼ 0 ¼

Z

1

ðaÞf ðV j ÞdV j :

ð2Þ

Vc

13 In the Appendix, the proof is presented without the forced symmetry assumption on the investment cutoff (Vc). However, the unique equilibrium investment cutoff is indeed shown to be symmetric.

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Firm i still experiences a negative externality (a) when rival j invests because j’s investment puts firm i at a competitive disadvantage. To keep the model tractable, we set this cost of falling behind to a fixed, common value, a.14 This occurs when i and j draw investment values respectively below and above the investment cutoff, Vc. Firm i solves for the investment cutoff that equates the expected payoffs to investing and not investing. This is equivalent to solving for the investment cutoff that sets the net (incremental) value of investment, Ei ½si ¼ 1  Ei ½si ¼ 0  Usim i ðV i ; V c ; I; aÞ, equal to zero:

Usim i ðV i ; V c ; I; aÞ ¼ V i  I þ

Z

0.5

Vc 0.4 0.3

I

0.2 0.1

1

ða  V j Þf ðV j ÞdV j :

ð3Þ

Vc

This expression allows us to summarize the components of a firm’s investment decision. The first two terms capture what firm i earns if it invests but its rival j does not. The third term has two parts: One is the negative externality that firm i incurs when rival j also invests (Vj). Offsetting this duplication cost is the negative externality avoided, namely, the cost of falling behind (a) that firm i experiences if it fails to invest when the rival does. This term therefore enters the net expected value of investment positively. So, a  Vj is in effect the net investment externality facing firm i. Both externalities arise only if rival j innovates, i.e., if j draws an investment value above the cutoff, Vj e [Vc, 1]. Firm j solves an analogous problem. 2.1.1. Investment decisions in the simultaneous model Because the firms move simultaneously, each rival’s action is unobserved and therefore cannot act as a signal. However, because the investment density and the payoff structure are common knowledge, each firm can conjecture whether the rival will invest and set its best-response accordingly. Our first result is to derive the equilibrium investment cutoff level, Vc. Theorem 1. Define Vc as the equilibrium investment cutoff level against which firms base their investment decisions by investing if Vi > Vc or Vj > Vc. There are two symmetric pure-strategy BNE characterized as follows: (i) If a P aH, where aH ðIÞ ¼ 12 þ I, firms always invest, regardless of their private investment values. That is, if a P aH then the equilibrium investment cutoff is Vc = 0. (ii) If a < aH then the equilibrium ffi investment cutoff is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V c ða; IÞ ¼ ða  1Þ þ a2  4a þ 2I þ 2.

0.0

0.0

0.2

0.4

0.6

αL

αH

0.8

1.0

α

Fig. 1. Investment in the simultaneous-move model. Simultaneous-model investment cutoffs as a function of the cost of falling behind (a) and a fixed-capital outlay of I = 0.2. The straight horizontal line (at I = 0.2) represents the investment cutoff in the absence of negative investment externalities. The dashed line shows investment cutoffs (Vc) in the presence of investment externalities: Firms are less likely to invest relative to the no-externality case when a is low (a < aL), more likely to invest when aL 6 a 6 aH, and always invest if a is high (a > aH).

decreases in I both lower the investment cutoff level compared to when externalities are absent. A higher a (or lower I) causes the relative payoff to not investing to become more negative in expectation. In this range, the downside of not investing when the rival invests weighs more than the cost of duplicating the rival’s investment. This lowers the equilibrium investment cutoff level and prompts firms to invest more often. In fact, when a is high enough (a P aH), falling behind is never a viable option. In this range, investing is always preferable and firms innovate for all realized investment values, i.e., the cutoff level is Vc = 0.15 Not surprisingly, a lower a (or higher I) has the opposite effect. When a is low, the penalty to not investing when a rival invests is low because the cost of falling behind is low. Since firms are not sure to capture the full value of their investment because of the possibility of duplication, a low cost of falling behind causes them to invest more discriminately, investing only when they draw a relatively high investment value. Accordingly, the equilibrium cutoff is higher and investment occurs less often. Our next result formalizes these ideas. 1 Corollary 1. Define aL ðIÞ ¼ 1þI 2 and aH ðIÞ ¼ 2 þ I. When a < aL, the equilibrium cutoff level is greater than the capital outlay, Vc > I, and when aL 6 a < aH, the equilibrium cutoff level is lower than the capital outlay, Vc < I.

The economic interpretation is as follows. In a benchmark model where firms make investment decisions in isolation, firms either earn their net-of-cost realized investment value, Vi  I, by investing or normalized profits of zero by not investing. In our model, two additional considerations enter the investment decision. On the one hand, the expected payoff to investing is lower because of the positive probability the opponent will duplicate the investment, eroding the gains to investment for itself and its rival. On the other hand, the expected payoff to not investing is also lower – in fact, it is negative – because of the positive probability the opponent will invest and impose an externality of –a on the rival by causing it to fall behind. Fig. 1 shows how the equilibrium investment cutoff level (Vc) balances these two externalities over a range of a (the cost of falling behind) and a given capital outlay (I = 0.2). Increases in a and

In equilibrium, there is a set of investment values (I < Vi, Vj < Vc), where firms rationally forego investments that appear profitable on a stand-alone basis. These are marginal projects in the sense that perceived profits turn to losses if rivals also innovate. Thus, what might look like under-investment can emerge as optimal investment once we account for externalities. For instance, in Zwiebel (1995) under-investment is attributed to managers’ reputational concerns. In standard corporate finance models, firms under-invest because claimant incentives diverge (e.g., Myers, 1977) or because information is imperfect (e.g., Myers and Majluf, 1984). In our setting, ‘‘under-investment’’ arises as a result of competitive forces in the product markets rather than as a distortion caused by shareholder–bondholder conflicts or information problems. Similarly, our model provides an alternative explanation for ‘‘over-investment’’ to the excess perquisites consumption

14 In addition to simplifying the analysis, this assumption also has economic content: a could reflect industry-specific components of the payoff structure and depend on industry concentration and the characteristics of each industry (i.e., the availability and value of investment opportunities).

15 Exit is another option, one which we ignore in the base model but tackle later in the ‘extensions’ section (Section 2.3.1). For now, we assume that exit is more costly than remaining in the industry or, equivalently, that the continued operation of existing assets is worth at least a.

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

(e.g., Jensen and Meckling, 1976) and empire-building (e.g., Jensen, 1986) hypotheses central to corporate finance. Indeed, we show that there is a set of investment values (Vc < Vi, Vj < I) where firms rationally invest in investments that appear unprofitable on a stand-alone basis. In these situations, the cost of falling behind outweighs the cost of duplicate investment. Thus, in our setting, competitive forces – not misaligned incentives between managers and shareholders – are what cause firms to step up investment.16 Building on Brander and Lewis (1986), where firms use debt strategically to commit to an aggressive investment policy (‘‘over-investment’’) and deter rivals from investing (‘‘underinvestment’’), Clayton (2009) shows that the indebted firm might invest less aggressively (under-investment) if its investment decision precedes its financing decision. Our model shows how similar outcomes, i.e., the appearance of over-investment or under-investment, can result from payoff interactions, without reference to strategic debt interactions in the product market. Another implication of the model is that firms appear to cluster in their investment decisions even when the conditions needed for herding are absent. Contrary to herding models, in our model firms (a) act simultaneously rather than sequentially, (b) do not observe each other’s actions, and (c) do not ignore their private information. We find that the existence of negative investment externalities means we are more likely to observe similarity in investment decisions that might be mistaken for herding behavior. The following corollary formalizes this result. Corollary 2. When firms facing negative investment externalities move simultaneously the probability that they act similarly (both firms invest or no firm invests) is always greater than the probability that they act differently (one invests while the other does not invest). The probability that both firms invest is (1  Vc)2 and the probability that neither firm invests is V 2c . Although the probability that just one firm invests is always positive, namely, 2ðV c  V 2c Þ, the probability that both firms act the same way is always higher. The reason is as follows. As a becomes large, Vc approaches zero and the probability that both firms invest goes to one. Conversely, as a becomes small, Vc gets large and the probability that neither firm invests goes to one. In both of these cases, the probability that only one firm invests is smaller than the probability that both firms take the same action. Thus, what might appear as concerted action can in fact arise as the outcome of a rational, non-cooperative, competitive environment where no firm conditions on the observed actions of other firms. 2.2. Sequential model Our simultaneous model establishes that negative investment externalities can lead to seemingly-coordinated investment decisions even when firms do not observe rivals’ actions. Since the herding literature concerns itself with how first-mover actions affect second-mover decisions, we now present a sequential-move game that links our analysis to previous work. We find that negative investment externalities can cause firms to mimic or counter rival investment decisions (‘‘herd’’ or ‘‘anti-herd’’) even when managers are fully aligned with shareholders. In the sequential model, both firms discover their private investment types at t = 1 but only one of them (firm i) can invest at that time by setting si(Vi) e {0, 1}. The second firm (j) only 16 Other studies, for example Narayanan (1985) and Bebchuk and Stole (1993), suggest that managers concerned with reputation may seek to increase the stock price in the short-run at the expense of long-run shareholder value. In addition, Grundy and Li (2010) show that executives design their investment strategies based on the current investor sentiment and also their own interest in the company. Underand over-investment could therefore also reflect such incentive distortions.

443

decides whether to innovate at t = 2, after observing the first mover’s action. Notationally, if j observes that i chose to invest, then j’s investment decision is given by s1j ðV j Þ 2 f0; 1g. If j observes that i did not invest, then j’s investment decision is given by s0j ðV j Þ 2 f0; 1g. In what follows, Vci represents the first mover’s equilibrium investment cutoff function. The second mover has two such cutoff functions, depending on whether the first mover invests ðV 1cj Þ or not ðV 0cj Þ. The first mover’s expected payoffs to investing and not investing are:

Ei ½si ¼ 1 ¼ V i  I 

Z

1

V 1cj

ðV j Þf ðV j ÞdV j ;

ð4Þ

and

Ei ½si ¼ 0 ¼

Z

1

V 0cj

ðaÞf ðV j ÞdV j :

ð5Þ

Similar to the simultaneous model, the first two terms of Eq. (4), Vi  I, is what the first-mover earns if it invests and the secondR1 mover does not. The third term, V 1 ðV j Þf ðV j ÞdV j , captures the negcj ative externality the first-mover incurs when the second-mover also invests, i.e., when V j > V 1cj . Eq. (5) reflects the negative externality the first-mover experiences (a) when it does not invest but the second-mover does, i.e., when V j > V 0cj . Collecting all these components, the net expected payoff to investment for the firstmover (firm i) becomes: 1 0 Useq i ðV i ; a; I; V cj ; V cj Þ ¼ Ei ½si ¼ 1  Ei ½si ¼ 0

¼ Vi  I 

Z

1

V 1cj

ðV j Þf ðV j ÞdV j þ

Z

1

V 0cj

ðaÞf ðV j ÞdV j : ð6Þ

The payoffs facing the second-mover depend on the firstmover’s decision. If the first mover has invested (si = 1), the second-mover’s net expected payoff to investment is: 1 1 Useq j ðV j ; a; I; V ci Þ ¼ Ej ½sj ¼ 1  Ej ½sj ¼ 0

¼ Vj  I 

Z

V ci

ðV i ÞgðV i jsi ÞdV i þ a:

ð7Þ

0

Here, g(Vi|si) represents the conditional density of the first-mover’s investment value given that the second-mover has observed the first-mover invest. The last term in Eq. (7) reflects the fact that the second-mover (j) avoids a certain payoff of a by investing when firm i does. If the first-mover does not invest (si = 0), the second-mover’s payoffs are similar to those it faces in isolation: Investing ðs0j ¼ 1Þ ensures the absolute value of investment, Vj  I, and not investing ðs0j ¼ 0Þ preserves the industry status quo and earns normalized profits of zero. 2.2.1. Investment decisions in the sequential model The sequential model produces four different pure-strategy equilibria that depend on the magnitude of the cost of falling behind (a). Common to these equilibria, we find that when the first-mover does not invest the second mover’s investment decision is perfectly efficient in the sense that the second mover’s investment cutoff function coincides with the benchmark case of no externalities ðV 0cj ¼ IÞ. A richer set of outcomes arises when the first-mover does invest and the second-mover’s decision again depends on the relative costs of duplication and falling behind. In this case, the first-mover’s decision reflects the second-mover’s anticipated reaction to the first-period investment decision.17 17 Few studies in the herding literature examine the effect of the second-mover’s decisions on the first-mover’s actions. Exceptions include Gül and Lundholm (1995) and Choi (1997).

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

444

1.0

AntiHerd

Herd

0.8

Vcj1 V cj0 = I

0.6

I

0.4

Vci 0.2 0.0

0.0

αL

0.2

0.4

1.2

αH

1.6

2.0

α

αHH

Fig. 2. Investment in the sequential-move model. Sequential-model investment cutoffs as a function of the cost of falling behind (a) and a fixed-capital outlay of I = 0.5. The dotted line depicts the first-mover’s equilibrium investment cutoff function. The dashed (solid) line shows the second-mover’s investment cutoffs when the first-mover has (not) invested. When a is high (a > aH), the first-mover is more likely to invest relative to the no-externality case and the second-mover always invests: Firms appear to herd. When a is low (a < aL), the first-mover is still more likely to invest relative to the no-externality case but the second-mover never invests: Firms appear to anti-herd.

These findings are discussed in the following theorems and illustrated in Fig. 2. 6Iþ3 Theorem 2. Define aL(I) = 3I1 3I and aH(I) = 62I. When aL < a 6 aH the investment cutoff functions are given by:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ci ða; IÞ ¼ 5 þ 2a  2I þ 2 7  6a þ 2Ia; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 1cj ða; IÞ ¼ 2 þ 7  6a þ 6I þ 2Ia; and V 0cj ¼ I: Theorem 2 shows that just as in the simultaneous model, for intermediate values of a, investment cutoffs in the sequential model balance the capital outlay (I) and the cost of falling behind (a). Outside this range, when the first firm invests, the second firm either always invests or never invests. Recall that the investment values (Vi and Vj) are distributed over the [0, 1] interval. Therefore, an equilibrium cutoff of zero means that firms will always invest because any investment value they draw will equal or exceed this cutoff. Conversely, an equilibrium cutoff of one means that firms never invest because they can never draw an investment value above one. These polar cases therefore offer an alternative explanation for patterns that the herding literature ascribes to informational cascades or managerial career concerns. Theorems 3 and 4 formalize these results which we parenthetically term ‘‘herding’’ and ‘‘anti-herding’’. Theorem 3 (Herding). 6Iþ3 1þ2I (i) Define aH ðIÞ ¼ 62I and aHH ðIÞ ¼ 2ð1IÞ . When aH(I) < a < aHH(I) the equilibrium investment cutoff levels are: V ci ¼ 12 þ I  aþ aI, V 1cj ¼ 0, and V 0cj ¼ I. 1þ2I (ii) Define aHH ðIÞ ¼ 2ð1IÞ . When a P aHH(I) the equilibrium investment cutoff levels are given by: Vci = 0, V 1cj ¼ 0, and V 0cj ¼ I.

This result can be understood as follows. For high values of a (a > aH), it is too costly for the second firm to forgo investing if the first firm has already invested: The cost of falling behind is just too high. This forces the second firm to mimic the first firm’s investment decision regardless of the investment value it draws (Vj). This result thus appears to correspond to ‘‘herding’’ (or ‘‘cascading’’), as commonly understood in the literature, in the sense that the second firm ignores its private information once its learns of the first firm’s action. However, in our model this result does not arise because the second firm learns something new about its own

investment value by observing the first firm’s action.18 Rather, the second firm’s payoffs are altered by the first firm’s action such that mimicking the first firm’s decision to invest is always a dominant strategy, i.e., V 1cj ¼ 0 if a > aH. In the simultaneous model, a low a causes under-investment (Vc > I) for all a < aL. In the sequential model, an extra equilibrium appears when a gets very low. For very small a (a < aL), the second firm has little incentive to invest once the first firm has invested. In fact, when the cost of falling behind is small enough it is never rational for the second firm to follow suit. This is because the second firm only captures the incremental value of investing, Vj  Vi  I, whose conditional expected value is more negative than a. Firms thus appear to engage in a form of ‘‘anti-herding’’. As before, this outcome arises solely through the effect the first firm’s action has on the second firm’s payoff. We formalize this idea as follows: Theorem 4 (Anti-herding). Define aL ðIÞ ¼ 3I1 3I . When a 6 aL(I) the equilibrium investment cutoff levels are given by: Vci(a, I) = I  a + aI, V 1cj ¼ 1, and V 0cj ¼ I. Another implication of this result is that the anticipation of the second firm’s investment decision causes the first firm to invest more than it would absent the negative externalities (i.e., Vci = I  a + aI < I, for all I, a e [0, 1]). The intuition behind this result is as follows. Given that the second firm never invests once the first firm has, the first firm is able to capture the absolute value of its investment whenever it does invest. If this were the only consideration, we would expect the first firm to ignore the negative externalities and set Vci = I. However, the first firm still has to consider the possibility that it will fall behind if the second firm invests once the first firm has decided not to invest. In equilibrium, the first firm acts pre-emptively by investing more aggressively, i.e., it uses an investment cutoff below the capital outlay (Vci < I). 2.3. Extensions We now extend the base sequential model to consider factors such as market entry and exit and firm heterogeneity, namely, differences in capital outlays and the cost of falling behind. 2.3.1. Market entry and exit The model assumes two firms operating in a closed market. It therefore seems natural to ask how our results might change in the face of market entry and exit. There are two ways to approach the question within the context of our model. One way is to consider investment as a discrete entry/exit decision. The second way is to ask how the presence (or absence) of a hypothetical third rival would affect our findings. We review each of these approaches in turn. Referring to Fig. 2, consider the cutoff function for the secondmover when the first-mover has invested (dashed line). If the cost of falling behind is very low (a < aL), the second-mover never invests (‘‘enters’’) and thus concedes the industry to the first-mover (anti-herding). If the cost of falling behind is very high (a > aH), the second-mover always invests (‘‘enters’’) and thus contests the industry (herding). For intermediate costs of falling behind (aL < a > aH) the second-mover may or not enter, depending on its realized investment value (Vcj). Mindful of the second-mover’s strategies, the first-mover is less likely to refrain from investing 18 Gül and Lundholm (1995) and Zhang (1997) show that in a sequential model, a herdlike ‘‘clustering’’ of actions arises if agents are allowed to choose not only between two possible actions but also when to take that action. In our model, ‘‘herding’’ occurs even though firms only decide whether to invest at all at a set point in time. In a different setting, Park (2011) also focuses on ‘‘concurrent’’ herding – as opposed to the sequential herding usually linked to informational cascades – in order to show that asymmetry in herding can lead to asymmetric volatility in foreign exchange markets.

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

(‘‘exiting’’) than when it faces potential entry. The first-mover acts pre-emptively and deters second-mover entry. Analogous results arise in the capital outlay space (see Figs. 3 and 4), where prior investment (‘‘entry’’) by the first-mover deters second-mover entry relative to the no-externality case. We can also tackle the question by projecting how a third rival would affect our results. In the base model, when the cost of falling behind dominates the cost of duplication, anticipated (first-mover) or observed (second-mover) investment by one firm raises its rival’s propensity to invest to avoid falling behind. Introducing a third rival increases the probability that at least one competitor invests, which compels the two other firms to invest more often. The reverse logic applies when the cost of duplication dominates the cost of falling behind: Due to the greater risk of duplication, introducing a third rival decreases the probability that at least one competitor invests, which compels the two other firms to invest less often. In short, increasing (decreasing) competition tends to accentuate (attenuate) the results derived in the base two-firm model. Depending on the relative costs of falling behind and duplication, the propensity to invest could rise or fall.

445

2.3.2. Asymmetric costs of falling behind In this section, we allow firms to differ in their costs of falling behind by considering asymmetric a’s (ai and aj rather than a common a). Theorem 5 (below) lists the investment cutoff functions for all ranges of a for both firms. Not surprisingly, the asymmetric-a model is a generalized version of the symmetric-a model. The main difference is that in the base model the a cutoffs (i.e., aL, aH) for both firms are functions of the symmetric cost of falling behind (a). In the asymmetric case, these cutoffs depend on the common capital outlay (I) and on each firm’s cost of falling behind (ai, aj). Theorem 5. Define

aLi ¼ aHi ¼

2  2aj þ I2  2Iaj þ a2j 2ðI  1Þ

and

3  4I  4aj þ 4I2  8Iaj þ 4a2j 8ðI  1Þ 1 2

3 2

1 2

1 2

; 3 4

3 2

1 2

1 2

aLj ¼  þ I þ ai I  ai and aHj ¼ þ I þ ai I  ai :

(i) When aLi < ai 6 aHi and aLj < aj 6 aHj the investment cutoff functions are given by:

1.0

Vcj

0.8

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ci ðai ; aj ; IÞ ¼ 5 þ 2aj  2I þ 2 7 þ 6I  4aj  2ai þ 2Iai ;

Vc = I

Vci

0.6

V 1cj ðai ; aj ; IÞ ¼ 2 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 þ 6I  4aj  2ai þ 2Iai ; and V 0cj ¼ I:

0.4 0.2 0.0

0.0

0.2

0.4

0.6

0.8

1.0

Ι

Fig. 3. High cost of falling behind. Equilibrium investment cutoffs as a function of the fixed-capital outlay (I) and the cost of falling behind (a = 0.5). The solid line (Vc = I), shows the no-externalities case. The dotted line shows the first-mover’s investment cutoff function (Vci). The dashed (solid) line shows the second-mover’s investment cutoff function when the first-mover has (not) invested. Whenever the investment cutoff function lies above (below) the solid line, the propensity to invest is lower (higher) than in the no-externalities case (aka under-investment and overinvestment). When the cutoff function equals 1 the firm invests with probability 1; when the cutoff function equals 0 the firm invests with probability 0.

1.0

Vcj

0.8

Vc = I

0.6

Vci

0.4 0.2 0.0

0.0

0.2

0.4

0.6

0.8

1.0

Ι

Fig. 4. Low cost of falling behind. Equilibrium investment cutoffs as a function of the fixed-capital outlay (I) and the cost of falling behind (a = 0.5). The solid line (Vc = I), shows the no-externalities case. The dotted line shows the first-mover’s investment cutoff function (Vci). The dashed (solid) line shows the second-mover’s investment cutoff function when the first-mover has (not) invested. Whenever the investment cutoff function lies above (below) the solid line, the propensity to invest is lower (higher) than in the no-externalities case (aka under-investment and overinvestment). When the cutoff function equals 1 the firm invests with probability 1; when the cutoff function equals 0 the firm invests with probability 0.

(ii) (Herding). When aLi < ai < aHi and aj > aHj the equilibrium investment cutoff levels are: V ci ¼ 12 þ I  ai þ ai I, V 1cj ¼ 0, and V 0cj ¼ I. (iii) (Anti-herding). When aLi < ai 6 aHi and aj 6 aLj the equilibrium investment cutoff levels are given by:

V ci ða; IÞ ¼ I  ai þ ai I;

V 1cj ¼ 1; and V 0cj ¼ I:

(iv) Whenai > aHi and aLj < aj < aHj the investment cutoff functions are given by:

V ci ¼ 0;

V 1cj ¼

1 þ I  aj ; and V 0cj ¼ I: 2

Figs. 3 and 4 show investment cutoffs as a function of the capital outlay when the cost of falling behind is the same for both firms (a = ai = aj). As one would expect, when a is low (see Fig. 4, (a = 0.1) the propensity to invest is low (relative to the no-externalities case) because the risk of duplication is high compared to the cost of falling behind. This effect is strongest for the first-mover but also holds for the second-mover. The effect is much attenuated when a is high (see Fig. 3, a = 0.5) because the cost of falling behind looms relatively large. Figs. 5 and 6 illustrate the asymmetric case (ai – aj). In Fig. 5, the first-mover’s cost of falling behind is low (ai = 0.1) and the second-mover’s cost is high (ai = 0.5); Fig. 6 shows the opposite (ai is high and aj is low). Comparing these cases to the symmetric-a cases depicted in Figs. 3 and 4 shows the differential impact asymmetric ai has on the first-mover compared to the second-mover. Corollary 3 states these contrasts more formally. Corollary 3 (First-mover, i). When ai (own cost) increases, i’s propensity to invest increases. When aj (rival’s cost) increases, i’s propensity to invest decreases. (Second-mover, j): When either aj (own cost) or ai (rival’s cost) increases, j’s propensity to invest increases.

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

446

This analysis shows the dramatic effect that any change in the cost of falling behind has on the second-mover’s investment policy, whether a change in its own cost of falling behind (aj) or that of its rival, the first-mover (ai). Given the sensitivity of the secondmover’s investment policy to the cost of falling behind, the firstmover’s investment policy is understandably also affected: Since an increase in its own cost of falling behind (aj) increases the likelihood that the second-mover will duplicate any positive investment made by the first-mover (firm i), firm i then lowers its propensity to invest in the first place to avoid the expected cost of duplication. 2.3.3. Asymmetric capital outlays In this section, we allow firms to differ in their capital outlays by considering asymmetric I’s (Ii and Ij rather than a common I). Theorem 6 (below) lists the investment cutoff functions for all ranges of a for both firms. Not surprisingly, the asymmetric-I model is a generalized version of the symmetric-I model. The main difference is that in the base model the a cutoffs (i.e., aL, aH) for both firms are functions of the symmetric capital outlay (I). In the asymmetric case, these cutoffs depend on the common cost of falling behind (a) and on each firm’s capital outlay (Ii, Ij).

(iii) (Herding). When a P aHH(I) the equilibrium investment cutoff levels are given by:

V ci ¼ 0;

V 1cj ¼ 0; and V 0cj ¼ Ij :

(iii) (Anti-herding). When a 6 aL(I) the equilibrium investment cutoff levels are given by:

V ci ða; IÞ ¼ Ii  ai þ ai Ij ;

V 1cj ¼ 1; and V 0cj ¼ Ij :

Figs. 7 and 8 show investment cutoffs as a function of the cost of falling behind (a) when the capital outlay is the same for both firms (I = Ii = Ij). As we would expect, we observe that, for all values of a, the propensity to invest decreases as I increases, i.e., the cutoff functions in Fig. 7 (I = 0.1) never lie above the cutoff functions in Fig. 8 (I = 0.5). This happens for the simple reason that increasing the capital outlay deters investment in general. Figs. 9 and 10 illustrate the asymmetric case (Ii – Ij). In Fig. 9, the first-mover’s capital outlay is low (Ii = 0.1) and the secondmover’s outlay is high (Ij = 0.5); Fig. 10 shows the opposite (Ii is high and Ij is low). Comparing these cases to the symmetric-I cases

Vcj

Theorem 6. Define 1.0

2Ij  1 þ Ii 4Ij þ 3 þ 2Ii ; aH ¼  and 3 þ Ij 2ð3 þ Ij Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ 2Ij  þ 3I2j  2Ij  2 þ 2Ii : 2

aL ¼  aHH

0.8

Vc = I

0.6

Vci

0.4

(i) When aL < a 6 aH the investment cutoff functions are given by:

0.2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ci ða; Ii ; Ij Þ ¼ 5 þ 2a  2Ij þ 2 7 þ 4Ij  6a þ 2Ii þ 2aIj ;

0.0

V 1cj ða; Ii ; Ij Þ ¼ 2 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 þ 4Ij  6a þ 2Ii þ 2aIj ; and V 0cj ¼ Ij :3:

(ii) (Herding). When aH(I) < a < aHH(I) the equilibrium investment cutoff levels are:

V ci ¼

1 þ Ii  ai þ ai Ij ; 2

V 1cj ¼ 0; and V 0cj ¼ Ij :

0.0

0.2

0.4

0.6

0.8

1.0

Ι

Fig. 6. Asymmetric alphas (ai = 0.5 aj = 0.1). Equilibrium investment cutoffs as a function of the fixed-capital outlay (0 < I < 1) and the cost of falling behind. The solid line (Vc = I), shows the no-externalities case. The dotted line shows the firstmover’s investment cutoff function (Vci). The dashed (solid) line shows the secondmover’s investment cutoff function when the first-mover has (not) invested. Whenever the investment cutoff function lies above (below) the solid line, the propensity to invest is lower (higher) than in the no-externalities case. When the cutoff function equals 1 the firm invests with probability 1; when the cutoff function equals 0 the firm invests with probability 0.

Vcj 1.0

1.0

Vci 0.8

0.8

Vc = I

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Ι

Fig. 5. Asymmetric alphas (ai = 0.1 aj = 0.5). Equilibrium investment cutoffs as a function of the fixed-capital outlay (0 < I < 1) and the cost of falling behind. The solid line (Vc = I), shows the no-externalities case. The dotted line shows the firstmover’s investment cutoff function (Vci). The dashed (solid) line shows the secondmover’s investment cutoff function when the first-mover has (not) invested. Whenever the investment cutoff function lies above (below) the solid line, the propensity to invest is lower (higher) than in the no-externalities case. When the cutoff function equals 1 the firm invests with probability 1; when the cutoff function equals 0 the firm invests with probability 0.

Vcj1

V cj0 = I

Vci 0.0

I 0.4

0.8

1.2

1.6

2.0

α

Fig. 7. Low fixed-capital outlay (I = 0.1). Equilibrium investment cutoffs as a function of the fixed-capital outlay and the cost of falling behind (0 < a < 2). The solid line (Vc = I), shows the no-externalities case. The dotted line shows the firstmover’s investment cutoff function (Vci). The dashed (solid) line shows the secondmover’s investment cutoff function when the first-mover has (not) invested. Whenever the investment cutoff function lies above (below) the solid line, the propensity to invest is lower (higher) than in the no-externalities case. When the cutoff function equals 1 the firm invests with probability 1; when the cutoff function equals 0 the firm invests with probability 0.

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

in Figs. 7 and 8 shows the differential impact asymmetric I has on the first-mover compared to the second-mover. Corollary 4 states these contrasts more formally. Corollary 4 ( [First-mover, i]). When Ii (own outlay) increases, i’s propensity to invest decreases. When Ij (rival’s outlay) increases, i’s propensity to invest increases. [Second-mover, j]: When either Ij (own outlay) or Ii (rival’s outlay) increases, j’s propensity to invest decreases. For the first-mover, an increase in own capital outlay lowers its propensity to invest because, ceteris paribus, the realized investment value (Vi) is less likely to exceed Ii. However, an increase in the rival’s capital outlay has the opposite effect: Knowing the second-mover’s best response, the first-mover invests more aggressively in the first place, a form of deterrence. The reasoning changes for the second-mover because its decision is conditioned on the first-mover’s: If the first-mover has invested despite a high investment hurdle, the second-mover infers that the first-mover has drawn a high investment value (Vi). This translates to a high cost of duplication for the second-mover, who therefore lowers its propensity to invest.

0.8

V cj0 = I

0.6

I Vci 0.2

0.0

0.4

0.8

1.2

1.6

2.0

α

Fig. 8. High fixed-capital outlay (I = 0.5). Equilibrium investment cutoffs as a function of the fixed-capital outlay and the cost of falling behind (0 < a < 2). The solid line (Vc = I), shows the no-externalities case. The dotted line shows the firstmover’s investment cutoff function (Vci). The dashed (solid) line shows the secondmover’s investment cutoff function when the first-mover has (not) invested. Whenever the investment cutoff function lies above (below) the solid line, the propensity to invest is lower (higher) than in the no-externalities case. When the cutoff function equals 1 the firm invests with probability 1; when the cutoff function equals 0 the firm invests with probability 0.

1.0

Vcj1

0.8

Vcj0 = I j

0.6

Ij 0.4

Vci1

0.2 0.0

0.0

0.4

Vc0i = I i 0.8

1.2

1.6

Vcj1

0.8

Vcj0 = I j

0.6

Ij 0.4

1 ci

V

Vc0i = I i

0.2 0.0

0.0

0.4

0.8

1.2

1.6

Ii 2.0

α

Fig. 10. Asymmetric outlays (Ii = 0.5 Ij = 0.1). Equilibrium investment cutoffs as a function of the fixed-capital outlay (0 < I < 1) and the cost of falling behind. The solid line (Vc = I), shows the no-externalities case. The dotted line shows the firstmover’s investment cutoff function (Vci). The dashed (solid) line shows the secondmover’s investment cutoff function when the first-mover has (not) invested. Whenever the investment cutoff function lies above (below) the solid line, the propensity to invest is lower (higher) than in the no-externalities case. When the cutoff function equals 1 the firm invests with probability 1; when the cutoff function equals 0 the firm invests with probability 0.

3. Empirical analysis

0.4

0.0

1.0

In sum, considering asymmetric costs of falling behind or capital outlays imparts more realism to the analysis and yields a richer set of results, where the investment policies are reflexive and more nuanced than in the symmetric case. We now turn to our empirical tests.

Vcj1

1.0

447

Ii 2.0

α

Fig. 9. Asymmetric outlays (Ii = 0.1 Ij = 0.5). Equilibrium investment cutoffs as a function of the fixed-capital outlay (0 < I < 1) and the cost of falling behind. The solid line (Vc = I), shows the no-externalities case. The dotted line shows the firstmover’s investment cutoff function (Vci). The dashed (solid) line shows the secondmover’s investment cutoff function when the first-mover has (not) invested. Whenever the investment cutoff function lies above (below) the solid line, the propensity to invest is lower (higher) than in the no-externalities case. When the cutoff function equals 1 the firm invests with probability 1; when the cutoff function equals 0 the firm invests with probability 0.

This section tests the ability of our model to explain observed firm investment behavior. We describe our research design, data sources, sampling methods, and our empirical results. 3.1. Research design Our research design follows from the implications of Corollary 2 (simultaneous model) and Theorems 3 and 4 (sequential model). The path from theory to empirics involves several preliminaries that underpin our proposed research design. Briefly, we must find a way to capture the investment externalities, measure their importance, and link their relative magnitude across industries to predictions of the models. We now address each of these points in turn. Our first task, capturing investment externalities, is complicated by the fact that we model two distinct externalities, neither of which is directly observable. We therefore collapse the cost of falling behind (a) and the duplication cost (Vj) into a single net investment externality (a  Vj): When firm i invests along with its rival it incurs the duplication cost but avoids the cost of falling behind. So long as these externalities are unrelated, a and a  Vj move in the same direction and the implications of our models, which are stated in terms of the cost of falling behind, also hold in terms of the net investment externality.19 The second question is how to detect the net investment externality empirically. To this end, we regress the interaction of ownfirm investment and rival investment on own-firm Tobin’s q (our proxy for firm value). This interaction tells us whether investment by rival firms conditions the effect own-firm investment has on firm value, such as through a net investment externality. The coefficient on this interaction indicates whether the net investment externality is negative, positive, or possibly nil if the two 19 More precisely, this assumption only fails if the externalities are negatively related and the duplication cost falls quicker than the cost of falling behind rises.

448

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

externalities are of equal magnitude. This interaction helps to establish the applicability of the model by testing one of its assumptions but it does not tell us whether the predictions of the models are supported. The next step does. Third, we wish to test if the variation in net investment externality we observe across industries can be reconciled with our simultaneous or sequential models. To structure this test we turn to Corollary 2 of the simultaneous model and Theorems 3 and 4 of the sequential model. Corollary 2 states that firms tend to behave similarly when the cost of falling behind (a is either low, when no firm invests because duplicate investment is too costly, or high, when all firms invest because falling behind is too costly. This clustering in investment does not obtain for intermediate levels of a or, ceteris paribus, intermediate levels of a  Vj. The empirical implication is that we should observe low variance in investment rates in industries with either low or high levels of the net investment externality and high variance in investment rates in industries characterized by intermediate levels of the net investment externality. Another implication of the simultaneous model is that firms tend to invest less relative to the no-externality case if the cost of falling behind is low, more if it is high, and normally for intermediate levels. The empirical implication is that we should observe mean investment rates increasing in the level of the net investment externality, a  Vj. The sequential model has the following implications relative to the no-externality case: When the cost of falling behind is low the second-mover invests less if the first-mover has invested; when it is high the second-mover invests more if the first-mover has invested. Theorems 3 and 4 further show that firms tend to invest differently when the cost of falling behind is low (anti-herding) but tend to invest similarly when it is high (herding). The empirical implication is that the correlation between current own-firm investment and lagged rival investment should increase in the level of the net investment externality, a  Vj. Table 1 summarizes the various empirical implications of the simultaneous and sequential models. We test these implications using two interaction variables. First, the simultaneous-model measure takes on a value of minus one (1) if industry–year investment has both low variance and low mean compared to other industries in a given year, zero (0) if industry–year investment variance is high (low and high mean investment both included), and plus one (+1) if the industry–year investment has low variance but high mean. Corollary 2 predicts that sorting industries in this manner should rank them from low to high net investment externality. Thus, we use a three-way interaction between this measure, own-firm investment, and mean rival investment to test that prediction. A positive sign on this interaction would support the simultaneous model.20 Second, we use the correlation between current own-firm investment and lagged mean rival investment to test the sequential model. According to Theorems 3 and 4, this measure should rank industries from low to high net investment externality. Thus, we expect a positive sign on the interaction between this measure, own-firm investment, and mean rival investment. Note that the simultaneous and sequential models might both be operative since they are not mutually exclusive: Firms may well strategize both concurrently and sequentially. Finally, to reflect the possibly of firm asymmetry considered in our sequential-model extensions, we further interact the interaction variable used in the sequential model with two measures of 20 An alternative would be to compare estimates of the net investment externality across sub-samples. However, using interactions preserves degrees of freedom and thus maximizes the power of our tests. This is important because our sample is small (2541 firm-years) and a sub-sample approach would stretch statistical power.

Table 1 Empirical predictions. Summarized below are the empirical implications of our simultaneous and sequential models. In the absence of competition, the value of firm i’s innovation is Vi. In the presence of competition, i weighs the cost of falling behind (a) if it does not innovate when its rival innovates, and the cost of duplication (Vj) if both it and its rival innovate. When both firms innovate, a is avoided so the net innovation externality is given by (a  Vj). Mean investment is the average investment of all firms included in our sample where investment is proxied by the R&D rate. Investment variance is the level of deviation from the investment mean within an industry. Investment correlation is the correlation between own-firm investment and lagged rival investment. Cost of falling behind (a)

Net externality (a  Vj)

Mean investment

Investment variance

Panel A – Simultaneous model Low Low Medium Medium High High

Low Medium High

Low High Low

Cost of falling behind (a)

Mean investment

Investment correlation

Low Medium High

High Medium Low

Net externality (a  Vj)

Panel B – Sequential model Low Low Medium Medium High High

intra-industry heterogeneity. First, we use within-industry variance in investment rates as a proxy for asymmetry in the cost of falling behind (asymmetric alpha, a). Second, we use within-industry variance in firm size (log of assets) as a proxy for asymmetry in the capital needed to make the investment (asymmetric capital outlay, I). These two interaction variables act as modifiers on the basic (symmetric) sequential model in the sense that significant regression coefficients on these variables will indicate the applicability of the sequential-model extensions. 3.2. Data sources, sample selection, and variable construction We use manufacturing firms in the merged COMPUSTAT–CRSP database produced by Wharton Research Data Services. We use COMPUSTAT for the financial accounting and operating variables. We use CRSP for historical industry classifications because COMPUSTAT only reports current industry classifications. We merge in COMPUSTAT’s business-segment files to measure diversification levels. Using these segment files limits the sample to 1977–2003. Given the nature of our model, we limit our sample to industries that are likely to exhibit strategic interaction, namely, concentrated industries. We use the US Census of Manufacturers’ Herfindahl–Hirschman Index (HHI) to measure industry concentration and the US Department of Justice’s cutoff (HHI of 1800 and up) to identify concentrated industries. Our proxy for firm value is Tobin’s q which we measure by dividing the market value of equity plus the book value of debt and preferred stock minus deferred taxes by the book value of assets. Because we are primarily interested in innovative investment (versus capacity expansion), our proxy for the investment rate is research and development expense divided by the lagged book value of assets.21 We measure rival investment as the mean investment rate for all firms in a given firm’s industry–year, own-firm excluded. Our regressions control for known determinants of Tobin’s q. These are: Profitability (the ratio of earnings before extraordinary items and depreciation (minus dividends) to total assets), financial leverage (total debt divided by total assets), firm size (natural logarithm of total assets), cash flow volatility (standard deviation of cash flow divided by total assets using up to ten (minimum four) 21 Coverage for research and development expense (COMPUSTAT data item # 46) is low (over half are missing). To preserve the sample, we therefore set investment to zero in cases where these data are missing.

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449

Table 2 Summary statistics. Reported are summary statistics for firms in concentrated manufacturing industries, 1977–2003. We use the Census of Manufacturers’ Herfindahl–Hirschman Index (HHI) to measure industry concentration and the Department of Justice’s cutoff (HHI of 1800 and up) to identify concentrated industries. Our proxy for investment is research and development expense divided by the book value of assets (the investment rate). Our proxy for Tobin’s q is the market value of equity plus book value of debt and preferred stock minus deferred taxes, divided by the book value of assets. Mean rival investment and Tobin’s q are the mean values of these variables for all firms in a given firm’s four-digit SIC industry–year, own-firm excluded. Our proxies for financial status are: cash flow (earnings before extraordinary item and depreciation minus dividends), financial leverage (total debt divided by total assets), and firm size (log of total assets). Our proxies for risk are: cash flow volatility (standard deviation of cash flow divided by assets), diversification (one minus the Herfindahl of sales across the firm’s four-digit SIC industries), and the capital-labor ratio (net property, plant, and equipment per employee).

Own-firm investment Mean rival investment Own-firm Tobin’s q Mean rival Tobin’s q Cash flow/assets Financial leverage Size (log of assets) Cash flow volatility Diversification Capital-labor ratio

Mean

Median

Std. dev.

Minimum

Maximum

N

0.038 0.037 1.435 1.425 0.050 0.246 4.912 0.075 0.159 0.036

0.019 0.029 1.188 1.267 0.073 0.231 4.540 0.055 0.000 0.020

0.054 0.032 0.834 0.622 0.118 0.170 2.258 0.067 0.242 0.049

0.000 0.000 0.407 0.415 1.174 0.000 0.272 0.005 0.000 0.000

0.516 0.220 7.682 8.215 0.256 0.782 9.721 0.505 0.763 0.429

2541 2541 2541 2541 2541 2541 2541 2541 2541 2541

annual observations), diversification (one minus the Herfindahl of sales across a firm’s four-digit SIC industries (computed from segment data)), and operating leverage (fixed-capital stock (net property, plant, and equipment, in $ millions) divided by the number of employees). Finally, our regressions control for industry and year fixed effects by demeaning all firm-level variables. We delete observations with negative sales or assets, and those with a CRSP permanent number equal to zero. We delete influential observations as follows. We drop observations where Tobin’s q is over ten, asset-normalized cash flow is less (more) than negative (positive) two, and financial leverage lies outside the [0, 1] interval. Finally, we exclude observations above the ninety-ninth percentile of all variables. Cash flow can take on negative values, so we also drop observations below the first cash-flow percentile. We mitigate endogeneity bias by using twice-lagged values of the variables as instruments in our regressions. We therefore require that each firm-year have at least 2 years of lagged data.22 The final sample consists of 2541 firm-years, summarized in Table 2. Our regressions are estimated using a system Generalized Method of Moments (GMM) procedure where the dependent variable is Tobin’s q. Following Arellano and Bover (1995) and Blundell and Bond (1998), two equations are estimated simultaneously: (1) First-differences instrumented by the twice-lagged levels of all the variables (and their squared values), and (2) Levels instrumented by the once-lagged first-differences of all the variables (and their squared values).23 The Hansen’s J-statistics shown at the bottom of the table test the joint hypothesis that the instruments are exogenous (uncorrelated with the residuals) and the model is well-specified.24 Newey–West standard errors correct the for heteroscedasticity and first-order autocorrelation.

22 Because the sample is highly unbalanced, with some firms appearing only a few years and others remaining throughout the entire panel, this number of lags reflects a trade-off between reducing endogeneity bias, losing observations and statistical power, and introducing large-firm and survivor biases. 23 Only coefficients for the first-differences equation are reported as this regression accounts for firm fixed effects whereas the levels regression does not. As Blundell and Bond (1998) show, adding the levels equation yields system GMM estimates with smaller finite sample bias and greater precision than the stand-alone first-difference equation GMM estimates when the explanatory variables are persistent, which is certainly the case here. 24 As commonly happens, the Hansen’s J-statistics reject the over-identifying restrictions. This could mean that the model is not fully specified or that the instruments are correlated with the residuals. However, J-tests hold asymptotically and are known to over-reject in finite samples (Ferson and Foerster, 1994) and Leamer (1983) shows that large-sample specification tests are sensitive to even small departures from the ‘‘true’’ model.

3.3. Empirical results Table 3 shows results for four standardized regressions.25 Model 1 serves as a benchmark for Models 2–7 where we add the simultaneous-model and sequential-model measures. Turning to Model 1, we first find that own-firm investment has a strong positive effect on firm value. This result is comforting in that the average sample firm makes value-increasing investments. Of more direct interest, we find that the sensitivity of Tobin’s q to the interaction of own-firm and mean rival investment – our proxy for the net investment externality – is significant and negative. The fact that this interaction is significant supports our assumption that externalities are present; the fact that it is negative suggests that duplication is more costly than falling behind. Although encouraging, these results only show that the base conditions for our model are met. Our main predictions are tested in Models 2–7 where we add the simultaneous-model and sequential-model measures. Turning to Model 2, we find that the interaction of the simultaneous-model measure with the own-rival investment interaction is significantly positive, as predicted. This result suggests that, consistent with patterns predicted by the simultaneous model, the net investment externality is greater (less negative) in industries with low investment variance and high mean investment, and lower (more negative) in industries with low investment variance and low mean investment. This finding supports our prediction that firms are most (least) likely to invest  and to do so symmetrically (asymmetrically)  when the cost of falling behind is higher (lower) than the cost of duplication. Model 3 shows that the interaction of the sequential-model measure with the own-rival investment interaction is significantly positive, as predicted. This suggests that, consistent with the sequential model, the net investment externality is greater (less negative) in industries where second-movers tend to mimic firstmovers (‘‘herding’’) and lower (more negative) in industries where second-movers tend not to mimic first-movers (‘‘anti-herding’’).26 25 We present standardized regressions to facilitate interpretation and comparisons within and between models. We obtain standardized regressions by first subtracting the sample mean from each observation and then dividing this result by the sample standard deviation. The resulting regression coefficients show how the dependent variable responds (in standard-deviation units) to a one-standard deviation change in each regressor, holding all other regressors constant. This transformation also explains why no intercept is included in the regressions. 26 Prior studies document herding by institutional investors (e.g., Nofsinger and Sias, 1999; Chang et al., 2000; Villatoro, 2009) and security analysts recommendations (e.g., Graham, 1999; Welch, 2000). We find evidence of similar patterns in corporate investment decisions.

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450

within-industry variance in firm size as a proxy for asymmetry in required capital outlays (ORS  size asymmetry). Although Model 5 shows no significant role for the ORS  investment asymmetry interaction, this variable turns statistically significant in Model 7, where it included along with the ORS  size asymmetry interaction. The latter interaction is also statistically significant when the ORS  investment asymmetry interaction is excluded (Model 6). These results attest to the importance of considering intra-industry asymmetries, both in theory and empirically. What can we say about the economic importance of these results? First, the coefficient on the simultaneous-model measure interaction is about half the magnitude of the coefficient for the own-rival investment interaction. In the case of the sequentialmodel measure interaction, the coefficients are of similar magnitudes. Thus, our measures are quite successful in picking

Although our tests appear to support both the simultaneous and sequential models, it is unclear whether both models are operative or if the two model measures are simply correlated. Model 4 addresses this question by adding the simultaneous-model and sequential-model measures together. Both measures continue to be statistically significant, indicating that both models are operative. This means that firms not only condition on observed rival investment (sequential-model), but also conjecture as to their rivals’ current decisions (simultaneous-model). Models 5–7 add interaction variables meant to reflect the possibly of firm asymmetry considered in our sequential-model extensions. Model 5 interacts own-rival investment with the sequential-model measure against within-industry variance in investment rates as a proxy for asymmetry in the cost of falling behind (ORS  investment asymmetry). Model 6 interacts own-rival investment with the sequential-model measure against

Table 3 Multivariate GMM regressions. Standardized system GMM regressions for firms in concentrated manufacturing industries (HHI of 1800 and up). Following Arellano and Bover (1995) and Blundell and Bond (1998), two equations are estimated in each model: (1) (reported coefficients): First-differences instrumented by the twice-lagged levels of all the variables (and their squared values), and (2) (coefficients not reported): Levels instrumented by the once-lagged first-differences of all the variables (and their squared values). The dependent variable is Tobin’s q (market value of equity plus the book value of debt and preferred stock minus deferred taxes, divided by the book value of assets). Our proxy for investment is research and development expense divided by the lagged book assets (the investment rate). Mean rival investment is the mean investment rate for all firms in a given firm’s industry–year, own-firm excluded. The interaction of own-firm and mean rival investment shows how firm value is affected when both firms innovate. This is our proxy for the net innovation externality. The simultaneous-model measure takes on a value of minus one (1) if industry–year investment has both low variance and low mean relative to other industries in a given year, zero (0) if industry–year investment variance is high (low and high mean investment both included), and plus one (+1) if the industry– year investment has low variance but high mean. The sequential-model measure is the correlation between own current investment and lagged mean rival investment. Investment asymmetry (intra-industry variance of investment rates) is a proxy for when the cost of falling behind (alpha, a) varies across industry rivals. Size asymmetry (intraindustry variance of log of assets) is a proxy for when the capital outlay (I) varies across industry rivals. The control variables are: cash flow (earnings before extraordinary item and depreciation minus dividends), financial leverage (total debt divided by total assets), and firm size (log of total assets), cash flow volatility (standard deviation of cash flow divided by assets), diversification (one minus the Herfindahl of sales across the firm’s four-digit SIC industries), and the capital-labor ratio (net property, plant, and equipment per employee). We also control for industry and year fixed effects by demeaning all firm-level variables. Standard errors in parentheses are corrected for heteroscedasticity and firstorder autocorrelation using the Newey–West procedure. The Wald statistic tests the null hypothesis that none of the regression coefficients is different than zero. The Hansen Jstatistic tests whether the over-identifying restrictions are rejected. The incremental Hansen J-statistic tests whether the variables added in Models 2–7 improve the specification compared to Model 1. The last column (FP R2) reports the adjusted R-square of the endogenous variables regressed on the instruments.

Own-firm investment Mean rival investment Own  rival investment

Model 1

Model 2

Model 3

Model 4

Model 5

Model 6

Model 7

FP R2

0.13a (0.02) 0.00 (0.02) 0.02a (0.00)

0.13a (0.02) 0.00 (0.02) 0.03a (0.00) 0.02a (0.00)

0.12a (0.02) 0.00 (0.02) 0.04a (0.00)

0.11a (0.02) 0.01 (0.02) 0.05a (0.01) 0.02a (0.00) 0.03a (0.00)

0.11a (0.02) 0.01 (0.02) 0.05a (0.01) 0.02a (0.00) 0.04a (0.01) 0.01 (0.00)

0.11a (0.02) 0.01 (0.02) 0.05a (0.01) 0.02a (0.00) 0.01a (0.00)

0.12a (0.02) 0.01 (0.02) 0.04a (0.01) 0.01a (0.00) 0.02a (0.00) 0.03a (0.00) 0.06a (0.01) 0.09a (0.03) 0.13b (0.05) 0.08a (0.02) 0.02a (0.00) 0.19a (0.03) 0.02 (0.01) 0.13a (0.02)

0.60

2350 518a 611a 0.0001 10.74a

Own  rival investment  simultaneous-model measure

0.03a (0.00)

ORS: own  rival investment  sequential-model measure ORS  investment asymmetry

a

ORS  size asymmetry Mean rival Tobin’s q Cash flow/assets Financial leverage Size (log of assets) Cash flow volatility Diversification Capital-Labor Ratio Degrees of freedom Wald statistic (H0: all bi = 0) Hansen J-statistic Hansen J-statistic p-value Incremental Hansen J-statistic a b c

Statistical significance at the 1% confidence level. Statistical significance at the 5% confidence level. Statistical significance at the 10% confidence level.

0.11 (0.03) 0.11b (0.05) 0.09a (0.02) 0.01a (0.00) 0.18a (0.03) 0.02c (0.01) 0.12a (0.02)

0.10 (0.03) 0.14a (0.05) 0.09a (0.02) 0.01a (0.00) 0.19a (0.03) 0.02c (0.01) 0.12a (0.02)

0.10 (0.03) 0.13b (0.05) 0.08a (0.02) 0.02a (0.00) 0.20a (0.03) 0.01 (0.01) 0.12a (0.02)

0.09 (0.03) 0.13b (0.05) 0.08a (0.02) 0.02a (0.00) 0.20a (0.03) 0.02 (0.01) 0.12a (0.02)

0.09 (0.03) 0.13b (0.05) 0.08a (0.02) 0.01a (0.00) 0.20a (0.03) 0.02 (0.01) 0.12a (0.02)

0.04 (0.01) 0.09a (0.03) 0.13b (0.05) 0.08a (0.02) 0.02a (0.00) 0.19a (0.03) 0.02 (0.01) 0.13a (0.02)

2358 627a 600a 0.0001

2356 591a 601a 0.0001 0.56

2356 463a 590a 0.0001 10.10a

2354 453a 596a 0.0001 4.55

2352 453a 596a 0.0001 4.60

2352 468a 602a 0.0001 1.61

a

a

a

a

a

0.66 0.50 0.51 0.37 0.58 0.35 0.69 0.57 0.50 0.96 0.85 0.73 0.71

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

up variation in the net investment externality, and in a manner consistent with the predictions of the models. Second, the incremental Hansen’s J-statistics reported at the bottom of the table indicate that adding the simultaneous and sequential-model measures significantly improves model fit.

4. Conclusion We examine how investment policy is affected by two negative investment externalities. First, duplicating a rival’s innovation erodes the gains to investing compared to when the rival does not invest. Second, failing to invest when a rival does invest causes the firm to fall behind. We analyze how the interplay of these negative externalities affects equilibrium investment behavior and alters optimal investment policy compared to when such externalities are absent. We present a simultaneous model where neither firm can observe the rival’s investment decision and a sequential model where one firm can observe the first-mover’s investment. Both models show that what might otherwise appear as under- and over-investment can actually reflect firm-value-maximizing investment policy when framed in a strategic competitive setting. We also offer alternative explanations for coordination of corporate investment decisions. Our simultaneous model shows that negative investment externalities increase the probability that firms will rationally make similar investment decisions (both invest or neither invest). Our sequential model shows that the probability that a firm will mimic the actions of first-moving rivals depends on the relative size of the negative externalities the follower experiences if it does or does not mimic the leader. If the cost of falling behind is high firms will appear to herd. Conversely, if the cost of duplicate investment is high firms will appear to anti-herd. Thus, our models offer simple reasons as to why firms might rationally act in a seemingly coordinated fashion without invoking managerial career concerns or other behavioral distortions. To reflect the heterogeneity observed across firms in reality, we extend the sequential model to accommodate asymmetry in the cost of falling behind and capital outlays. This imparts more realism to the analysis and yields a richer set of results, where the investment policies are reflexive and more nuanced than in the symmetric case. We also show, within the context of our model, how the main results can be interpreted to describe entry or exit into the market. Finally, we test the models empirically and support key aspects of both the simultaneous and sequential models. Our analysis illustrates how a realistic competitive setting is sufficient to generate a wealth of investment policies and patterns that differ from standalone firm outcomes. This work has important implications for future empirical and theoretical research. While unchecked agency and information problems assuredly distort investment policy, our analysis shows that product–market interactions can yield observationally-equivalent investment patterns. Thus, factors such as the negative investment externalities we examine must be duly considered when assessing these problems empirically and in designing policies aimed at mitigating them.

451

from the comments of an anonymous referee. Remaining errors are ours. Appendix A Proof of Theorem 1. Let Vcj (Vci) be the investment cutoff, above which j (i) chooses to innovate and below which it does not.27 (Note that this allows for the possibility of asymmetric equilibria.) We present a proof in two parts:

(i) We start by assuming that Vcj = 0 (or Vcj 6 0) and solve for i’s best-response (Vci); we then identify the region in which our initial conjecture is correct. If Vcj = 0, firm i’s payoff from investing becomes:

Usim i ðV i ; V c ; I; aÞ ¼ V i  I þ

Z

1

ða  V j Þf ðV j ÞdV j :

This means that firm i’s best response is:

V ci ðI; aÞ ¼

1 þ I  a: 2

ð9Þ

Firm j runs through the same analysis (assumes that Vci = 0) and finds the same best-response function. For our initial conjecture to be correct, we now restrict both cutoff functions (best-response functions) to be less than or equal to zero. When a > 12 þ I  aH , the conditions Vci, Vcj 6 0 are satisfied and Vci = Vcj  Vc = 0 is the symmetric equilibrium. h (ii) We now assume the equilibrium cutoff(s) fall somewhere in the intermediate range of 0 6 Vci, Vcj 6 1. We solve for the equilibrium and then show the region in which our initial assumption is satisfied. Given its beliefs about Vcj, firm i chooses its own cutoff, Vci, by solving for the investment value that sets its net expected payoff of innovation equal to zero:

Ui ðV i ; V c ; I; aÞ ¼ V i  I þ

Z

1

ða  V j Þf ðV j ÞdV j :

This gives us,

V ci ðV cj ; a; IÞ ¼ 

V 2cj 1 þ aV cj þ I  a þ : 2 2

ð11Þ

By symmetry, the second firm has an analogous best-response function,

V cj ðV ci ; a; IÞ ¼ 

V 2ci 1 þ aV ci þ I  a þ : 2 2

ð12Þ

The symmetric equilibrium cutoff falls where the two best-response functions meet, namely, V ci ¼ V cj  V c . Algebra shows that this symmetric equilibrium cutoff function is given by:

V c ða; IÞ ¼ ða  1Þ þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2  4a þ 2I þ 2:

ð13Þ

Next, we prove that beliefs are in fact correct in equilibrium, in other words, that the condition 0 6 Vc 6 1 (or 0 6 Vci, Vcj 6 1) is satisfied. For Vc P 0,

1 2

We thank the editor (Ike Mathur), Sudipto Dasgupta, Gilles Hillary, David Mauer, Todd Milbourn, Rex Thompson, Mike Vetsuypens, Mungo Wilson, and participants at the meetings of the Midwestern Finance Association and the Financial Management Association for useful comments. The paper benefited largely

ð10Þ

V cj

a2  4a þ 2I þ 2 > 1  2a þ a2 ) a < þ I  aH : Acknowledgements

ð8Þ

0

ð14Þ

For Vc 6 1,

27 Both parts of the proof assume that the cutoff rule applies. In other words, firm i believes that sj(Vj) = 1 implies Vj > Vcj and sj(Vj) = 0 implies Vj > Vcj. In equilibrium, these beliefs are correct since the net expected payoff of investment is increasing in the realized investment values for both firms.

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

452

a2  4a þ 2I þ 2 < 4  4a þ a2 ) I < 1;

ð15Þ

which is always true. Therefore, Vc(a, I) as shown above is the symmetric equilibrium in the region where a < 12 þ I  aH . h

Proof of Corollary 1. Note that a < 12 þ I  aH implies that Vc P 0 (see proof of Theorem 1(ii)). For Vc 6 I, we need the following inequality to hold:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ IÞ ða  1Þ þ a2  4a þ 2I þ 2 6 I ) a P  aL : 2

ð16Þ

ð1 þ IÞ 1 6 a 6 þ I: 2 2 Similarly, the condition Vc > I is satisfied when

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ IÞ : a2  4a þ 2I þ 2 > I ) a < 2

ð1 þ V ci Þ þ I  a: 2

ð17Þ

As a < ð1þIÞ  aL implies that a < 12 þ I  aH , the condition Vc > 0 is 2 already satisfied when we impose the binding constraint a < aL. Also, as we show in the proof of Theorem 1(ii), the condition Vc 6 1 is true for all I 6 1. Hence,

1þI 1 < a < þ I  aH and V c < I when aL  2 2 1þI V c > I when aL <  aL :  2 Proof of Corollary 2. In the simultaneous model, both firms innovate when their respective investment values (Vi and Vj) are higher than the symmetric equilibrium cutoff, Vc. Since Vi and Vj are the independently and uniformly distributed, the probability that both firms innovate is (1  Vc)2. Similarly, neither firm innovates when both investment values are smaller than this symmetric cutoff; hence this outcome occurs with probability V 2c . Finally, the probability that only one firm invests and the other does not is equal to ð1  V 2c  ð1  V c Þ2 Þ ¼ 2ðV c  V 2c Þ: h Proof of Theorem 2. We use backward induction to solve for the equilibrium strategies in the sequential model. We start with the second firm’s response to its beliefs about the first-mover’s actions and then solve for the first firm’s best-response to the second mover’s strategy.

Upon observing that the first firm has not invested, the second firm behaves as if there are no innovation externalites: Its investment cutoff is V 0cj ¼ I. First firm: The first firm’s net expected payoff given the second firm’s strategy is:

Upon observing that the first firm has invested, the second firm forms beliefs about the first-mover’s investment value given this new information: It is greater than some cutoff value of the first firm, Vci where 0 < Vci < 1. Given these beliefs, (Vi > Vci), the second firm’s net expected payoff to investing is: 1 1 Useq j ða; I; V ci Þ ¼ Ej ½sj ¼ 1  Ej ½sj ¼ 0

¼ Vj  I 

ðV i ÞgðV i jsi ÞdV i þ a;

1

V 1cj

ðV j Þf ðV j ÞdV j þ

Z

1 V 0cj

ðaÞf ðV j ÞdV j :

ð20Þ

Firm i’s beliefs about rival j’s cutoff functions are that ci Þ V 1cj ¼ ð1þV þ I  a, where 0 < V 1cj < 1 and V 0cj ¼ I. Incorporating 2 these beliefs into its payoff function, firm i’s best-response cutoff function becomes:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ci ðÞ ¼ 5  2I þ 2a þ 2 7 þ 6I  6a þ 2aI  h1 ða; IÞ: Substituting this back into

V 1cj ðÞ ¼ 2 þ

ð21Þ

V 1cj ðÞ,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 þ 6I  6a þ 2aI  h2 ða; IÞ:

ð22Þ

Now, we have to identify the regions where the conditions 0 < V ci ; V 1cj < 1 are satisfied. First, note that the second firm’s reaction function ðV 1cj Þ reaches both zero and 1 before the first firm’s function (Vci) does. To guarantee that both these functions are within the specified range, we can simply restrict V 1cj to be between 6Iþ3 zero and 1. Then, 0 6 V 1cj 6 1 is true when aL  3I1 < a < 62I  aH ; 3I which also guarantees that 0 6 Vci 6 1. Therefore, for intermediate values of a where aL  3I1 3I < 1 1 a < 6Iþ3 62I  aH and 0 6 V c1 ; V c2 6 1, V ci ¼ h1 ða; IÞ; V cj ¼ h2 ða; IÞ and V 0cj ¼ I is the pure-strategy equilibrium. h Proof of Theorem 3 (i) Here, we start with the assumption that V 1cj ¼ 0 (or V 1cj 6 0), solve for the first-mover’s response, and then check that our assumption is actually true in equilibrium. The assumption of V 1cj ¼ 0 and V 0cj ¼ I means the expected payoff of firm i is: 1 0 Useq i ða; I; V cj ; V cj Þ ¼ V i  I 

Z

1

ðV j Þf ðV j ÞdV j þ

0

Z

1

ðaÞf ðV j ÞdV j :

I

ð23Þ

V ci ðI; aÞ ¼

V ci

Z

The cutoff we are solving for is the investment value, Vci, that sets the above equation equal to zero. In this case,

Second firm: (CASE 1: Upon observing si(Vi) = 1)

Z

ð19Þ

Second firm: (CASE 2: Upon observing si(Vi) = 0)

Ui ða; I; V 1cj ; V 0cj Þ ¼ V i  I 

This means that the condition 0 6 V c 6 I is satisfied when

ða  1Þ þ

V 1cj ¼

ð18Þ

where g(Vi|si) represents the conditional density of the first-mover’s investment value given that the second-mover has observed the first-mover innovate. Firm j invests for realized investment values that are greater than some investment cutoff ðV 1cj Þ that sets the above expression to zero, and does not invest otherwise. Accordingly, the cutoff becomes:

ð24Þ

Now, we have to make sure that the second firm’s cutoff function is in fact always less than zero. Given its beliefs about the first firm’s investment value upon observing that firm i has invested (Vi > Vci and 0 6 Vci 6 1), firm j invests when its own value is greater than V 1cj where,

V 1cj ðI; aÞ ¼

0

1 þ I  a þ aI: 2

ð1 þ V ci Þ þ I  a: 2

ð25Þ

Given that V ci ¼ 12 þ I  a þ aI we substitute to get,

  1 1 1 þ þ I  a þ aI þ I  a 2 2 2 3 3 3 1 ¼ þ I  a þ aI: 4 2 2 2

V 1cj ðI; aÞ ¼

V 1cj 6 0 holds when,

ð26Þ

E. Akdog˘u, P. MacKay / Journal of Banking & Finance 36 (2012) 439–453

3 3 3 1 6I þ 3 þ I  a þ aI 6 0 ) a >  aH ðIÞ: 4 2 2 2 6  2I

ð27Þ

This assumes that Vci P 0. This condition is true when,

1 1 þ 2I þ I  a þ aI P 0 ) a 6  aHH ðIÞ: 2 2ð1  IÞ

ð28Þ

Therefore, in the region aH(I) < a < aHH(I) the equilibrium investment cutoff levels are V ci ¼ 12 þ I  a þ aI; V 1cj ¼ 0 and V 0cj ¼ I. (ii) Now, we check the consequence of Vci = 0 which occurs for 1þ2I a > 2ð1IÞ  aHH ðIÞ. This will cause j to have a different bestresponse function:

V 1cj ¼

1 þ V ci 1 þ I  a ¼ þ I  a: 2 2

ð29Þ

Then we check if V 1cj is ever positive when Vci = 0. V 1cj is greater than zero if a < 12 þ I and less than zero otherwise. If 1þ2I > 12 þ I, V 1cj is never positive when Vci 6 0. It is clear that 2ð1IÞ 1þ2I this always holds true as 2ð1IÞ > 1þ2I for all 1 > I P 0. Hence, 2 the only equilibrium for a > aHH is V ci ¼ V 1cj ¼ 0 and V 0cj ¼ I. h Proof of Theorem 4. The proof is similar to that of Theorem 3, except that now we start by assuming that V 1cj P 1 and solve for the first firm’s best-response. Then, we check whether V 1cj is in fact greater than 1 in the specified regions.When V 1cj ¼ 1, the first firm’s payoff is: 1 0 Useq i ða; I; V cj ; V cj Þ ¼ V i  I þ

Z

1

ðaÞf ðV j ÞdV j :

ð30Þ

I

As a result, the first firm’s investment cutoff becomes

V ci ¼ I  a þ aI:

ð31Þ V 1cj

1þV ci 2

Recall (from the proof of Theorem 2) that ¼ þ I  a. This also assumes that 0 6 V ci 6 1 which we check below. Substituting Vci from the above expression, we get

V 1cj ¼

1 1 1 3 3 1 þ ðI  a þ aIÞ þ I  a ¼ þ I  a þ aI: 2 2 2 2 2 2

ð32Þ

Now, we check what conditions satisfy the inequality V 1cj P 1:

1 3 3 1 3I  1 3  I 3I  1 þ I  a þ aI > 1 ) > )a<  aL ðIÞ; 2 2 2 2 2 2 3I ð33Þ which is also a sufficient condition for Vci P 0. Therefore, the strategies V ci ¼ I  a þ aI; V 1cj ¼ 1 and V 0cj ¼ I constitute an equilibrium when a < aL(I). h Proof of Theorem 5. Similar to the proof of Theorem 2 where the common externality, a, is replaced with ai for firm i and aj for firm j (ai – aj). h Proof of Corollary 3. Follows Theorem 5. h

@V ci @ ai

ci < 0; @V > 0; @ aj

@V 1cj @ ai

< 0;

@V 1cj @ aj

< 0 from

Proof of Theorem 6. Similar to the proof of Theorem 2 where the common capital outlay, I, is replaced with Ii for firm i and Ij for firm j (Ii – Ij). h Proof of Corollary 4. Theorem 6. h

@V ci @Ii

ci < 0, @V > 0, @Ij

@V 1cj @Ii

< 0,

@V 1cj @Ij

< 0. Follows from

453

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