Understanding the complex relationship between R&D investment and firm growth: A chaos perspective

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Understanding the complex relationship between R&D investment and firm growth: A chaos perspective Xuchuan Yuana, , Rohit Nishantb ⁎

a b

School of Business, Singapore University of Social Sciences, Singapore 599494, Singapore Faculty of Business Administration, Université Laval, G1V 0A6, Canada

ARTICLE INFO

ABSTRACT

Keywords: Chaos Complex system Firm growth R&D investment Revenue dynamics

Time delays and feedback loops often introduce nonlinear, dynamic, chaotic, or disorderly business patterns. In this paper, we argue that a firm’s growth driven by R&D can also exhibit chaotic behavior, through a simple model, where R&D has two distinct impacts on its revenue through the demand-inducing and cost-increasing effects. On the basis of a revenue-based investment strategy with anchoring and adjustment, the firm’s growth dynamics are characterized by a two-dimensional dynamical system under either the cost-plus or profit-maximizing pricing policies. Theoretical and numerical studies are used to show that growth can drive stable, periodic, and even chaotic behaviors. The findings suggest that R&D investment has more complex impacts on growth than previously envisioned, and that managerial decisions about R&D investments can cause fluctuations and erratic growth patterns in nonlinear and complex business environments.

1. Introduction One of China’s largest solar companies recently filed for bankruptcy.1 Many of China’s state-owned shipping companies have suffered huge losses.2 One reason for the numerous failures is that Chinese companies over invested in capacity during the economic boom without planning for an eventual economic downturn (Hook, 2003). Another reason is that Chinese firms allocated substantial resources to research and development (R&D), without considering that R&D might fail to increase revenue. Indeed, a PriceWaterhouse analysis revealed that R& D investment has no significant relationship with financial performance (Viki, 2016). Although R&D investment is essential for innovation and is acknowledged as a key to long-term survival and growth, several anecdotes in various sectors show that innovation can fail to bring success. For instance, Sun Microsystems innovated and developed the programming language Java and new processor architecture, but did not perform well in the first decade of the 21st century and was eventually sold to Oracle (Hoque, 2012). Sony, once synonymous with the Walkman, and recognized for innovation, failed to outperform its nimble competitors such as LG and Samsung, with the shift toward digital storage of music (Hoque, 2012). Thus, further investigation is

needed to understand R&D’s complex impacts on performance. Failures across industry sectors show that business scenarios are fragile. Nonlinear dynamic systems characterize this business world, as the relationship between antecedents and consequences is neither linear nor immediate. Time lags often occur between business decisions and effects, so that increased investments have diminishing or even no payoffs. Evidently, decision-makers often fail to realize that their decisions will interact with external business environments and in turn influence further decision-making, constituting feedback loops (Sterman, 1989, 2004). Thus decision-making has a complex nonlinear relationship with consequences and can lead to multiple complications such as financial volatility in markets (Jurczyk, Rehberg, Eckrot, & Morgenstern, 2017) and unpredictable consumer behaviors (Langhe, Puntoni, & Larrick, 2017). The unsure impacts and feedback loops seriously threaten steady growth. Also critical but relatively unexplored is endogenous decision-making, which studies such as Baum and Wally (2003) and Luoma, Ruutu, King, and Tikkanen (2017) have demonstrated to be responsible for outcomes such as expansion, contraction, or stagnation. Against that background, we study how managerial decisions about R&D investments can influence growth patterns in firms. Our timely and important investigation provides useful insights

Corresponding author. E-mail address: [email protected] (X. Yuan). See, e.g., CNN, 21 March 2013, “Major Chinese solar company goes bankrupt: China’s Suntech Power has put its largest subsidiary into bankruptcy” by Charles Riley, available at https://money.cnn.com/2013/03/21/news/suntech-solar-bankruptcy/index.html, accessed 9 February 2019. 2 See, e.g., Reuters, 30 March 2017, “China’s COSCO Shipping reports $1.4 bln loss for 2016”, available at https://www.reuters.com/article/china-cosco-resultsidUSL5N1H522A, accessed 9 February 2019. ⁎

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https://doi.org/10.1016/j.jbusres.2019.11.043 Received 14 February 2019; Received in revised form 12 November 2019; Accepted 13 November 2019 0148-2963/ © 2019 Elsevier Inc. All rights reserved.

Please cite this article as: Xuchuan Yuan and Rohit Nishant, Journal of Business Research, https://doi.org/10.1016/j.jbusres.2019.11.043

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showing that endogenous decision-making under managerial control has effects that contrast with exogenous factors beyond managerial control. Nonlinear dynamics theory has motivated studies focusing on complexity in management systems across various operation domains such as supply chains (Hwarng & Xie, 2008; Laugesen & Mosekilde, 2006; Mosekilde & Larsen, 1988; Sterman, 1989; Wu & Zhang, 2007), organization (Anderson, 1999; Dooley & Van de Ven, 1999; Moldoveanu & Bauer, 2004), and operations (Thomas, Kevin, & Rungtusanatham, 2001, Yuan and Hwarng, 2012, 2016). Studies utilize nonlinear models and techniques to understand the factors underlying erratic business dynamics. Specifically, chaos theory and tools are widely used to study nonlinear behaviors in complex dynamic systems and to understand the complexity of management systems in various situations (Lorenz, 1993, provides an excellent introduction of chaos theory with applications in economics). R&D investment, a typical management practice, also exhibits nonlinear dynamic characteristics in which R&D investments have nonlinear outcomes such as revenue. Past studies investigating the relationship between R&D and firm growth have shown that firms are increasingly recognizing R&D investment as a key driver for growth and are investing billions annually. Leading firms in various industries specifically attribute their success to continual R&D investment and contend that innovative technologies and newly developed products are direct consequences of R&D investments. Nevertheless, R&D impacts vary. Some firms enjoy consistently and steadily increasing revenues and profits, but others experience volatile growth or even perish. Bental and Peled (1996), Matsuyama (1999), Matsuyama (2001), Walde (1999, 2002) have shown that profit-maximizing firms can derive both positive long-run growth rates and short-term fluctuations from R&D investments. Because of the unpredictable fluctuations and erratic firm performance associated with R&D, firms must understand how endogenous managerial decisions regarding R&D investments can influence growth in complex business environments. Considering that we focus on nonlinear dynamic characteristics of payoffs from R&D investment, chaos theory emerges as appropriate for considering underlying endogenous factors that have unpredictable effects on dynamic systems. By leveraging the chaos perspective, we align with several studies that have invoked chaos theory to explain erratic and disorderly growth in the business world. We develop a model showing how endogenous R&D investment decisions are related to volatile and unpredictable growth patterns. Assuming that firms are operating as monopolists, we extend earlier chaos theory work to explain how firms evolve (e.g., Feichtinger & Kopel, 1993), but distinguish R&D investments according to demandinducing effect in which innovation improves current products or develops new products, and cost-increasing effect in which R&D generates expenditures. Specifically, R&D that increases demand boosts sales quantity; while R&D that increases production costs reduces profit margins. Consequently, the two forces evoke complex, unpredictable revenue dynamics and even chaos. Motivated by industry practices, we consider that revenue-based R&D investment strategies are based on past investments and available resources. Product pricing decisions are based on two widely adopted pricing rules: cost-plus and profit-maximizing. Our study makes several contributions. First, we intuitively and realistically contrast R&D investments that increase either market demands or production costs, with contrasting impacts on firm growth, in alignment with anecdotal evidence about how R&D investments distinctly affect the labor supply.3 Second, the chaos perspective reveals that R&D investments have more nonlinear and complex relations twith

firm revenue than previously envisioned. Again, understanding the complex relationship is crucial for understanding how R&D investments have distinct outcomes in practice. Finally, we explore specific conditions that will evoke chaotic growth dynamics. We demonstrate that growth behavior is cyclical and even chaotic when revenue has disordered and unpredictable fluctuations. Our model shows how endogenous R&D investment decisions cause growth fluctuations. Managers can use our findings to devise strategies ensuring stable growth. In addition, we provide directions for future research. The rest of the paper is organized as follows. Section 2 briefly reviews the relevant literature. Section 3 describes the model setup and the revenue-based R&D investment strategy. Section 4 focuses on the revenue dynamics under the cost-plus pricing policy. Section 5 investigates the case with the profit-maximizing pricing policy. Key results and managerial implications are discussed in Section 6. Section 7 concludes the paper. 2. Literature review Our study builds on complexity science studies of economic and management systems, such as Arthur (1999), Anderson (1999), and Farmer (2012). The study links to literature regarding impacts of R&D investment decisions on growth and revenue and to literature that focuses on how managerial decisions impact dynamic firm behavior. Both analytical and empirical approaches have been used to study how R&D investments optimally impact growth and revenue. Baumol and Wolff (1983) analyzed how productivity growth affects relative R& D prices. Nadiri and Prucha (1991) analyzed how R&D investments increase productivity in electrical machinery industries in Japan and the United States. Falk and de Lemos (2019) analyzed small and medium-sized firms to show how R&D-sales ratios and labor productivity influence export participation and share. Glomm and Ravikumar (1994) analyzed public R&D investments for effects on macroeconomic growth. Several researchers have used analytical models to investigate R&D investment decisions in dynamic settings (Childs & Triantis, 1999; Reinganum, 1982). Bloom (2007) investigated uncertainty about future productivity and demand conditions for impacts on R&D investment decisions. Aw, Roberts, and Xu (2011) developed a dynamic structural model to investigate endogenous R&D and export decisions that affect future productivity. Studies have also used empirical approaches. Wang and Tsai (2004) used the CobbDouglas function to model productivity growth and its relationship with R&D investment. Dutta, Narasimhan, and Rajiv (2005) utilized the stochastic frontier estimation technique to estimate maximum outputs from R&D investments. Knott and Vieregger (2015) used various empirical approaches to understand optimal firm sizes for innovation. Knott and Vieregger (2019) combined both analytical and empirical approaches to show how R&D investments affect market value. Similarly, Gu (2016) considered how competition interacts with R&D investments for influencing market value. Several studies have focused on how R&D decisions affect dynamic firm behavior. Greve (2003) invoked the behavioral perspective to conclude that high performance reduces R&D intensity, which then reduces innovation. Kahneman and Tversky (1979) explained that highperforming firms decrease R&D expenditure according to whether managers tend to be risk-averse or to adopt anchoring and adjustment tendencies. The revenue-based R&D investment strategy we consider aligns with findings regarding the behavioral patterns of managers responsible for R&D investment. Bowman’s (1980, 1982) empirical investigations of managerial behavior in U.S. firms showed the “Risk Return Paradox,” in which firms realizing high returns are less likely to take risks than those realizing lower returns. Firms facing crises tend to redeem themselves by investing more in R&D and fostering innovations. Macro-economic factors such as financial crises also increase R& D’s volatility and dynamism. For example, R&D investments were hastened during the 2007 financial crisis (Coldbeck & Ozkan, 2018).

3 See, e.g., CRN, 19 February 2015, “UK software R&D boom branded a double-edged sword” by Doug Woodburn, available at https://www. channelweb.co.uk/crn-uk/news/2396138/uk-software-r-d-boom-branded-adouble-edged-sword, accessed 9 February 2019.

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Patel, Guedes, Soares, and da Conceição Gonçalves (2018) showed empirically that corporate governance causes R&D to have a less volatile relationship with performance. Some strategy studies have specifically focused on factors that influence R&D investment decisions. Hoskisson and Hitt (1988) argued that tight financial controls decrease R&D investments. Klingebiel and Rammer (2014) identified the importance of resource allocation strategies for influencing R&D and innovation. When resources are allocated to a broad range of innovation projects, product sales tend to increase. Marketing studies such as Fu and Elliott (2013) and Henard and Dacin (2010) also found that consumers express higher loyalty and purchase intentions when they perceive that firms are highly innovative. Those findings support our model assumptions that R&D investment increases demand. The anchoring strategy is also supported by recognizing effects of prior R&D failures (Khanna, Guler, & Nerkar, 2016). Some studies focus on how R&D investment relates to firm risks. Bromiley, Rau, and Zhang (2017) asserted that R&D investment is unrelated to risks, but Trigeorgis and Reuer (2015) argued that managers face dilemmas when making decisions, and that real option theory explains decision-making under uncertainty. Our study contributes to those arguments by analytically examining decision-making consequences particularly for R&D investments. Some studies have used the chaos perspective to investigate how R& D activities affect firm progress. Baumol and Wolff (1983) showed that R&D has chaotic relationships with productivity growth under specific conditions. Feichtinger and Kopel (1993) and Kopel (1996) focused on magnitude of R&D investments as related to revenue using a simple nonlinear decision-making model with R&D investment and revenue as key variables to show that R&D investment decisions can bring chaos to revenue dynamics. Whitby, Parkerb, and Tobias (2001) described a nonlinear model of duopolistic competition through R&D investment that improved product quality or technology. Simulation indicated that chaos could occur for various competing managerial policies. The interaction dynamics are solely responsible for unpredictability. However, our study diverges from those studies in that our model explicitly categorizes two distinct effects of R&D investments. In sum, some studies suggest that R&D investments convey gains but not necessarily growth. Instead, R&D investments have more complex, nonlinear, and dynamic relationships with growth than previously envisioned. However, we lack studies examining effects on firm growth or performance. Hence, to better understand how R&D investments affect revenue and growth, we developed a simple analytical model that considers two pricing policies and provides alternative explanations regarding observed growth patterns. Consequently, our research fills an important gap in the literature.

costs; for example through cost allocations, new technologies, novel designs, and advanced materials, which we call the cost-increasing effect. 5 Therefore, R&D investments expand market demand but also increase production costs. The two effects may lead to complex dynamic behavior and unexpected patterns such as periodically cyclical or even chaotic growth in revenue. For ease of reference, Table 1 lists the key parameters and notations used in our paper. For expositional simplicity and model conciseness, we assume that R&D investment has no time delay in affecting demand and production costs. To put it simply, R&D investments in each period affect demand and production costs in the same period. 6 In each period t, besides the R&D investment decision, the firm also needs to make the product price decision. To capture the impact of R&D investment on growth, the following assumptions are made based on the general functional forms:

• The demand-inducing effect.

The potential demand or the market size in each period t increases with the R&D investment decision in the same period, modeled as:

Dt > 0. It

Dt = F (It ),

• The sales quantity function.

The sales quantity of the product in each period t decreases with respect to product price but increases in terms of the potential market size:

Qt = Q Dt , pt ,

Qt > 0, Dt

Qt < 0. pt

• The cost-increasing effect.

The unit production cost in each period t increases with the amount of R&D investment in the same period, modeled as:

ct = J (It ),

ct > 0. It

• The revenue function. calculated as:

The revenue of the firm in each period t is

Rt = pt Qt = pt Q (F (It ), pt ).

• The R&D investment decision.

The amount of R&D investment in period t + 1 increases in the revenue and the decision in the previous period t, modeled as:

3. Model setup

It + 1 = G It , Rt ,

We consider a monopolistic firm that invests in R&D in each period, e.g., year, denoted by t.4 The resources allocated in R&D are devoted to develop new products and/or upgrade current products. There are two potential effects of the R&D investment on operations. First, newly developed or upgraded products attract potential demand, as consumers prefer products that feature novel designs, innovative technologies, new functions, and/or improved quality (Klingebiel and Rammer, 2014), which we call the demand-inducing effect. However, R&D investments have indirect expenditures that increase manufacturing

It + 1 It

0,

It + 1 Rt

0.

Substituting the function for the amount of R&D investment into the revenue function, the system dynamics in terms of the R&D investment and revenue are summarized as: 5

Even if the firm focuses on developing new production technologies through R&D and reduces unit variable production costs, the fixed cost per unit still increases because R&D investment is a fixed expenditure. Our model can still be applied if the R&D investment increases the overall unit production cost. 6 For the demand-inducing and the cost-increasing effects, we do not distinguish whether products are newly developed or upgraded versions of current products. Nevertheless, our model generalizes to different product types and where the demand-inducing and cost-increasing effects have time lags. Under such situations, the dimension of the dynamical model increases, but the key results and the managerial implications hold qualitatively.

4 We use the discrete time dynamic model to show that R&D investment decisions are part of firms’ resource planning and allocation, depending on the available resources, especially revenues accumulated from past periods. Besides, R&D decisions often have delayed impacts on firm performance. The literature has widely incorporated the delay effect in economic modeling. Rotemberg (2003) argued that in modeling cyclical productivity dynamics, we must consider that significant lags occur in the diffusion of new technologies.

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typically considered a long-term strategy for allocating resources often according to previous decisions and adjusted according to market conditions, the firm’s financial status, and other factors. Motivated by industry practices and the anchoring and adjustment nature of human decision-making, we assume that, in each period, the manager adopts revenue-based R&D investment strategies and allocates resources according to past decisions and revenues generated in past periods. 11 Revenue-based investment strategies have been widely observed and documented in practice, as firms often tie R&D investment to the available resources, especially internal sources of funds, such as revenues. To model the revenue-based R&D investment strategy with anchoring and adjustment, we assume the R&D investment decision in period t + 1 is given by the exponential smoothing rule:

Table 1 Key parameters and notations. Parameters

Notations

Rt It Dt ct pt Qt >0

The The The The The The The The

(1, ¯) [0, 1]

revenue in period t R&D investment decision in period t potential demand in period t unit production cost in period t selling price in period t sales quantity in period t marginal expenditure factor in the R&D investment strategy markup factor in the cost-plus pricing policy

The anchoring factor in the R&D investment strategy

It + 1=G (It , Rt ), Rt + 1=pt + 1 Q (F (G (It , Rt )), pt + 1 ),

(1)

It + 1 = It + (1

where the model has no exogenous randomness or uncertain factors. The model (1) is a typical deterministic dynamic system. For analytical tractability, rather than use the general functional forms, we use concrete equations widely adopted in economics and management studies to analyze system dynamics. Specifically, potential demand, production cost, and sales quantity are assumed linear in terms of R&D investment or product price. These functions are:

Dt =D0 + d (It I0), d > 0, ct =c0 + c (It I0), c > 0,

Qt = max(Dt

) Rt ,

(3)

[0, 1] captures the smoothing propensity in decision-making, where (0, 1) is the fixed protermed the anchoring factor. The parameter portion of the revenue in the previous period that can be devoted to R& D. We term as the marginal expenditure factor. As R&D investment is often considered a long-term operational strategy, we assume and are set at the beginning of the horizon and kept as constant thereafter.12 By (3), the R&D investment in period t + 1 anchors at the decision in period t and is adjusted by the available resources: the previous period’s revenue. The anchoring and adjustment characteristic makes it possible that in certain periods, the R&D investment is more than the revenue generated in the previous period. The firm may have to use its internal capital savings or external funding sources to finance its R&D activities and sustain operations (Lewis & Tan, 2016).13 By (3), if = 1, then the firm invests the same amount of resources in R&D in every period such that It = I0 for t = 1, 2, , where I0 is the initial amount committed into R&D. While if = 0 , then the firm adjusts the R&D investment decision by the revenue in the previous period, without anchoring on previous decisions. Under this situation, the model (1) can be reduced to a one-dimensional dynamic system and we can theoretically prove that the revenue dynamics exhibit stability, periodic cycles, and chaos (e.g., Yuan & Hwarng, 2012, 2016). For (0, 1) . As a long-term conciseness, we focus on the case with at the beginning of the time strategy, the manager decides and horizon: t = 0 . The two parameters are kept as constant in each period. Given the R&D investment strategy, motivated by industry practices, in Sections 4 and 5, we consider two pricing policies separately, namely the cost-plus policy and the profit-maximizing policy. To keep the notations short, we assume the initial R&D investment and the revenue are both zero; that is, I0 = 0 and R 0 = 0 , so that the demand and the cost functions are simplified as Dt = D0 + dIt , ct = c0 + cIt , respectively. By (2) and (3), the dynamical system (1) is reduced to a two-dimensional

(2)

kpt , 0), k > 0,

where the demand-amplifying slope, d, measures the marginal effect of the R&D investment on the demand,7 and the cost-increasing slope, c, measures the marginal effect of the investment on the production cost.8 The demand slope k captures customers’ price sensitivity, which also measures consumers’ demand elasticity. 3.1. R&D investment strategy R&D investment and product price are two key operational decisions that determine the dimension of the dynamical system (1). In this section, we discuss how managers make R&D investment decisions. Past decisions or choices often influence managerial decisions.9 The anchoring-and-adjustment policy is frequently followed in both practice and research (Sterman, 1989).10 In practice, R&D investment is 7 We thank the referee for pointing out the economic interpretation of the potential demand function, where R&D investment often has diminishing marginal effect on demand with respect to the amount of devoted resources. We assume that the potential demand linearly increases in the R&D investment amount mainly for analytical tractability. In the simulation, we also tried the concave demand function in terms of the R&D investment, and found qualitatively the same results. That is, stability, periodic cycles, and chaos are observed in the revenue dynamics. 8 We thank the referee for indicating the relationship between the unit production cost and the R&D investment. We assume It includes mainly the direct cost related to R&D activities. Intuitively, for the production cost function, each unit of the sold product bears a direct R&D cost of It / Qt . For analytical simplicity, we assume the unit production cost increases in the R&D investment because of the cost increasing effect. We also consider the other cost increments in R&D activities, especially the indirect costs related to R&D. 9 Firms often implement medium or long-term strategies that lead to correlated operational decisions. But managers’ behaviors such as inertia in decisionmaking also influence how decisions are made in practice (Patel & Chrisman, 2014). 10 Lach and Rob (1996) developed a model of industry evolution where endogenous innovations drive the dynamics, followed by subsequent embodiments in physical capital. Their model stresses the causal relationship between past R&D expenditures and current investments in machinery and equipment. The causality pattern is supported by the data and explains the observed higher

(footnote continued) volatility of physical investments relative to that of R&D expenditures. 11 Another important reason that previous decisions influence current R&D investment decisions is that ongoing investments in specific R&D projects are usually long term. 12 In practice, firms may update/adjust R&D investment policies occasionally, which means and may change accordingly over time. Our focus is the complexity of the revenue dynamics under R&D investment in a deterministic setting, so we refrain from the confounding effect of varied parameters by keeping and constant in the analysis and simulation. 13 We thank the reviewer for highlighting the financial aspect of firms in terms of the source of funding for R&D investment. In Appendix B, we also develop alternative models where the firm uses a portion of the net profit in each period to finance R&D investment based on the exponential smoothing rule. The models with the net profits as the source of funding for R&D have similar structure to the corresponding models with revenue as the funding source, and the dynamical behaviors in terms of the net profit are also qualitatively the same as the revenue dynamics. 4

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model in terms of the revenue and the R&D investment decision. We first analyze the stability of the equilibrium point in the dynamical system and then resort to numerical simulations to investigate the impact of various factors on the system dynamics. By using the simulation approach, a wide range of parameter values can be examined to simulate various managerial decisions about R&D investments, particularly the anchoring and the marginal expenditure factors.14

According to the dynamical system theory (e.g., Devaney, 1989 & Lorenz, 1993), the following scenarios occur in terms of the stability of the equilibrium point and the long-run behavior of the dynamical system:

• The system is convergent and the equilibrium point is a stable node if r < 1; The • system is divergent in one direction while convergent in the other 2

direction, and the equilibrium point is a saddle point if r2 > 1.

4. The cost-plus pricing policy

Therefore, the modulus of the second eigenvalue r2 plays a critical role in the stability of the equilibrium points. We first solve the equilibrium points (if any) to the dynamical system (5). Denote R as the solution to the equation H (R, , ) = 0 , where

In this section, we consider that the firm adopts the cost-plus pricing policy, widely practiced in various industries (e.g., Shubik, 1962), where the product price is determined by adding a specific markup to the production cost. The product price under the cost-plus pricing policy is:

pt = ct , 1 <

< ¯,

H (R, , ) = (c0 + c R)[Q0 + (d

(4)

Proposition 1. Under the case with d k c = 0 , if c Q0 1, both the revenue and the R&D investment increase over time; if c Q0 < 1, the equilibrium point (I , R ) is solved as:

k c ) It + 1, 0).

R =

After substituting Eq. (3) into the above revenue function, the dynamical system is reduced to a two-dimensional model in terms of the R& D investment decision and the revenue as below:

It + 1= It + (1 + (1

) Rt )]max(Q0 + (d

(5)

R =

where Q0 = D0 k c0 is the initial sales quantity with I0 = 0 and p0 = c0 . To study the dynamical characteristics of the two-dimensional system, we first solve the equilibrium points of (5), denoted as (I , R ) , where I 0 and R 0 . If the equilibrium point exists, then:

R = (c0 + c R )[Q0 + (d

(1 ( , )

(1

k c ) R ], and I = R .

) )

( , )

,

1

c0 Q0 . c Q0

If c Q0 > 1, R < 0 , and the system has no equilibrium points, and both the revenue and R&D investment grow steadily in t. If 0 < c Q0 < 1, R > 0 and a unique equilibrium point, (I , R ) , exists, where R = R > 0 and I = R . At the equilibrium point (I , R ) , the ) c Q0 (0, 1) , second eigenvalue is calculated as r2 = + (1 indicating that the equilibrium point is a stable node, and the system will converge to (I , R ) . □

Thus, the equilibrium point is independent of the anchoring factor . However, the stability of the equilibrium point (I , R ) indeed depends on , analyzed below. To simplify the analysis, we focus on the dynamical system (5) without the nonnegative constraint on the revenue function. Denote ( , ) = c0 (d k c ) + cQ0 + 2c (d k c )( It + (1 ) Rt ) . The Jacobian of the dynamical system (5) at any point (It , Rt ) can be calculated as:

J=

c 0 Q0 , and I = R , c Q0

Proof. If c Q0 = 1, H (R, , ) = 0 has no solutions, and both the 1, the revenue and R&D investment grow steadily in t. If c Q0 solution to H (R, , ) = 0 is calculated as:

k c )( It

) Rt ), 0),

1

which is a stable node, and the system dynamics converge to (I , R ) .

) Rt ,

Rt + 1= [c0 + c ( It + (1

(7)

R,

and if R 0 , then the equilibrium point satisfies R = R and I = R ; otherwise, the equilibrium point of the dynamical system (5) does not exist. If d k c = 0 , Eq. (7) is a linear function in terms of R. Starting with the initial point {(I0 , R 0 ) = (0, 0)} , the revenue dynamics of (5) are depicted as below:

1 measures the markup where is termed as the markup factor and ratio. Price is also a long-term operational decision, so we assume is a constant in each period, which is determined at the beginning of the time horizon in the model: t = 0 . The markup factor is bounded above by ¯ > 1. Substituting Eqs. (4) and (2) with the initial point {(I0 , R 0 ) = (0, 0)} into the revenue function in the dynamical system (1), the revenue in period t + 1 is calculated as:

Rt + 1 = pt + 1 Qt + 1 = (c0 + cIt + 1)max(Q0 + (d

k c ) R]

Therefore, for the simplest case, the revenue dynamics under the cost-plus pricing policy either increase period-by-period or converge to a stable node. In terms of the growth pattern, the firm may grow steadily or achieve a constant revenue in each period in the long run. 0, In the following section, we consider the cases with d k c where H (R, , ) = 0 is a quadratic function in terms of R, and the equation H (R, , ) = 0 has at most two solutions, denoted as R1 and R2 , respectively. Denote A ( ) = c (d k c) 2, B ( ) = [c0 (d k c ) + cQ0] 1, and C ( ) = c0 Q0 > 0 . Eq. (7) is simplified as H (R, , ) = A ( ) R2 + B ( ) R + C ( ) , and the discriminant is calculated as ( ) = B ( )2 4A ( ) C ( ) . Clearly, if ( ) < 0 , then H (R, , ) = 0 has no solutions; if ( ) > 0 , then:

(6)

) ( , ) and with the trace and the determinant as trJ = + (1 detJ = 0 , respectively. The discriminant of the Jacobian is denoted as = (trJ ) 2 4detJ = [ + (1 ) ( , )]2 . The two eigenvalues are ) ( , ) , which are both real numbers thus r1 = 0 and r2 = + (1 at the equilibrium points. The stability of the equilibrium point depends on the modulus of the two eigenvalues, denoted as ri , i = 1, 2 .

R1 = If

B( )

( )

2A ( )

, and R2 =

B( ) + 2A ( )

( ) = 0 , one unique solution exists: R =

( )

.

B( ) . 2A ( )

Depending on

( )

and A ( ) , the following result in terms of the system dynamics (5) holds. The proof is provided in Appendix A.

14

We can also use real-world data to estimate the model parameters and then perform a simulation. However, other confounding factors can endogenously determine R&D investment and pricing decisions, leading to biased estimation. Besides, we lack wide ranges of parameter values to capture real of decisionmaking scenarios.

Proposition 2. Under d

• If 5

( ) < 0 and d

k c

0 , the system dynamics (5) are:

k c > 0 , both the revenue and R&D investment

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increase over time. ( ) = 0 and d k c > 0 , the following cases hold: – if B ( ) > 0 , both the revenue and R&D investment increase over time; – if B ( ) 0 , one equilibrium point exists: (I , R ) , which is unstable and R = R > 0 and I = R . If ( ) > 0 , the following cases hold: – If d k c > 0 and B ( ) < 0 , the dynamical system has two equilibrium points, denoted as (I1 , R1 ) and (I2 , R2 ), where R1 = R1 > 0, I1 = R1 , R2 = R2 > 0 , and I2 = R2 . The equilibrium ) ( ) 2, (I1 , R1 ) is a point (I2 , R2 ) is a saddle point. If (1 saddle point; otherwise, (I1 , R1 ) is a stable node. – If d k c > 0 and B ( ) > 0 , the dynamical system has no equilibrium points and both the revenue and R&D investment increase over time. – If d k c < 0 , the dynamical system has one equilibrium point R = R1 > 0 I = R. (I , R ) , where and If (1 ) ( ) 2, (I , R ) is a saddle point; otherwise, (I , R ) is a stable node.

with complex dynamical systems in business. The impact of We first consider the impact of the marginal expenditure factor on the system dynamics. In the numerical example, the anchoring factor is fixed at = 0.1, and the base demand is set as Q0 = 10 . The left panel in Fig. 1 is the bifurcation plot of the revenue with respect to . The figure reveals that the system exhibits stability with one stable equilibrium point, cyclical behavior with periodic points, and eventually chaos with the increment of . To further verify whether the dynamic behavior shown in the bifurcation plot exhibits chaos or not, we numerically calculate the Lyapunov exponent with respect to using the method in Eckmann and Ruelle (1985). Negative and positive Lyapunov exponents indicate a stable and chaotic system, respectively; if the Lyapunov exponent is zero, the system exhibits periodic cycles. The right panel in Fig. 1 plots the Lyapunov exponent of [0.18, 0.2], indicating that the region is the region with respect to indeed chaos when is beyond 0.19. The impact of The anchoring factor also impacts the system dynamics. In the numerical example, the marginal expenditure factor is fixed at = 0.2 , and the base demand is set as Q0 = 10 to investigate the impact of on the system dynamics. The left panel in Fig. 2 is the bifurcation plot of the revenue with respect to . The figure shows that the system exhibits stability with one stable equilibrium point, cyclical behavior with periodic points, and eventually chaotic behavior with the decrement of . The right panel in Fig. 2 plots the Lyapunov exponent of [0, 0.4], indicating that the region is indeed chaos the region with when is lower than 0.15. To visually depict the dynamic patterns of the model, Fig. 3 shows the return map of the periodic cycle under = 0.18 and = 0.1 (left panel), and the strange attractor under = 0.2 and = 0.1 (right panel). To plot the return map, we ran the dynamic system (8) for 10000 time units to reach equilibrium and plotted the last 1000 data points. The strange attractor also demonstrates chaos in the dynamic system. Therefore, given all the other parameters fixed, the proportion of revenue devoted to R&D ( ) and the anchoring and adjustment in the decision-making behavior ( ) jointly affect the revenue dynamics, which may converge to a stable node, oscillate periodically, or even vary disorderly into the chaotic state. Figs. 1 and 2 show that when more revenue is devoted to R&D ( becomes larger), or when the R&D investment decision is less anchored on the previous decision ( becomes smaller), the revenue dynamics could change from stable to periodic or even chaotic. By the dynamical system (8), if decreases ) becomes larger: the R&D and increases simultaneously, then (1 investment decision is adjusted more by a higher proportion of revenue. Thus, if the manager adjusts the R&D investment decision by relying more on the available resources such as the revenue, the firm shows a more fluctuating or even chaotic growth path.

• If •

Proposition 2 shows that the two-dimensional revenue dynamical system (5) under the cost-plus pricing policy may exhibit various patterns such as monotonically increasing, converging to a stable node, and periodically oscillating. The existence of the saddle point under d k c < 0 and ( ) > 0 indicates that the dynamical system may exhibit more complex behavioral patterns. 4.1. Numerical example In this section, we conduct numerical studies to investigate the dynamical behavior of the two-dimensional system (5). The model includes three decisions: the anchoring factor , the marginal expenditure factor , and the markup factor . We focus on the impact of the marginal expenditure factor and the anchoring factor on the system dynamics.15 In the following numerical example, to show how endogenous decisions about R&D investment impact revenue dynamics, we use the following parameter values:

• The demand slope k = 1; • The demand-amplifying slope d = 1; • The cost-increasing slope c = 1; • The markup factor = 2; • The initial unit production cost is normalized as c

0

= 0.

Based on the above parameter values, the two-dimensional dynamical system (5) is reduced to:

It + 1= It + (1 ) Rt , Rt + 1= ( It + (1 ) Rt )max(D0

( It + (1

) Rt ), 0),

5. The profit-maximizing pricing policy

(8)

In this section, we consider that the firm adopts the profit-maximizing pricing policy: the price is charged to maximize the profit in each period. Specifically, the product price is determined by maximizing the profit in each period as below:

which is a typical piecewise-smooth map, where border-collision and period-doubling bifurcations are typically observed in the dynamical system, and may also lead to chaos. Many techniques are available for analyzing the piecewise smooth maps (e.g., Bernardo, Budd, Champneys, & Kowalczyk, 2008). For concise discussion, we keep the mathematical analysis simple and parsimonious and strongly focus on the managerial insights from the model. Therefore, in the following numerical simulation, we resort to the bifurcation plot, Lyapunov exponent, and the return map to characterize the behavioral patterns of the revenue dynamics under R&D investment. The three approaches are widely used in the economics and management literature for dealing

pt = argmax p p

ct

Dt

kp ,

which is given by

pt =

D + kc0 + (d + kc ) It Dt + kct = 0 . 2k 2k

(9)

The sales quantity in each period is calculated as:

Qt = max

15

As a managerial decision, the markup factor in the cost-plus pricing policy will also affect the system dynamics. However, for the conciseness, we will not focus on the impact of on the revenue dynamics.

Dt

kct 2

, 0 = max

D0

kc0 + (d 2k

kc ) It

, 0 .

(10)

Substituting Eqs. (9), (10), and (2) with the initial point 6

Journal of Business Research xxx (xxxx) xxx–xxx

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Fig. 1. Left panel: Bifurcation plot with respect to . Right panel: Lyapunov exponent with respect to . The figure is drawn based on the dynamical system (8), starting with = 0.1, and incremented by 0.0001. Parameter values are: k = 1, d = 1, c = 1, c0 = 0, = 2, = 0.1, Q0 = 10 .

Fig. 2. Left panel: Bifurcation plot with respect to . Right panel: Lyapunov exponent with respect to . The figures are based on the dynamical system (8), starting with = 0 , and incremented by 0.0001. Parameter values are: k = 1, d = 1, c = 1, c0 = 0, = 2, = 0.2, Q0 = 10 .

Fig. 3. Left panel: An illustration of the periodic cycle with period 8 in the periodic region. Right panel: An illustration of the strange attractor in the chaotic region. The figures are based on the dynamical system (8). Parameter values are: k = 1, d = 1, c = 1, c0 = 0, = 2, Q0 = 10 .

{(I0 , R 0 ) = (0, 0)} into the revenue function in the dynamical system (1), the revenue of the firm in period t + 1 is calculated as:

dynamical system (1) is reduced to a two-dimensional dynamical model in terms of the R&D investment decision and the revenue as below:

Rt + 1=pt + 1 Qt + 1,

It + 1 = It + (1

(

D + kc + (d + kc ) It + 1 D = 0 0 2k max 0

=max

(

1 [Q0 4k 2

+ 2(dD0

kc0 + (d 2k

k2cc

0 ) It + 1

kc ) It + 1

+

)

, 0 ,

(d 2

k2c 2) It2+ 1],

(11)

) Rt ,

Rt + 1

)

0 ,

= max

where we assume Q0 = D0 kc0 > 0 . After substituting Eq. (3) into the above revenue function, the

1 [Q02 + 2(dD0 4k 2

( It + (1 7

) Rt ) 2], 0 .

k2cc0)( It + (1

) R t ) + (d 2

k 2c 2)

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To study the dynamics of the above two-dimensional system, we first solve the equilibrium points of (11), denoted as (I , R ) , where I 0 and R 0 . If the equilibrium point exists, then:

R =

1 [Q02 + 2(dD0 4k 2

k2cc0) R + (d2

R1 =

(1 2k2

( )

(1

) )

( )

2k 2

Proposition 4. Under d

•

)

with the trace and the determinant as trJ = + 2k2 ( ) and detJ = 0 , respectively. The discriminant of the Jacobian is denoted as

= (trJ ) 2

4detJ =

(1

+

(1

)

2k2

)

2

( ) . The two eigenvalues are thus

r1 = 0 and r2 = + 2k2 ( ) . Therefore, the magnitude of r2 plays an important role in the stability of the equilibrium points. We first consider the equilibrium points of the dynamical system (11). Denote R as the solution to the equation H (R, ) = 0 , where H (R, ) = Q02 + [2(dD0

k2cc0)

4k 2] R + (d 2

k 2c 2 ) 2R2 ,

B+ 2A

,

kc , the system dynamics (11) are as follows:

• If < 0 and d > kc , both the revenue and R&D investment increase over time. • If = 0 and d > kc , the following cases hold:

(12) (1

, and R2 = B( )

k2c 2)( R )2], and I = R .

,

2A

and if = 0 , then R = 2A ( ) is the one unique solution. Depending on and A, the following result in terms of the system dynamics (11) holds. The proof is in Appendix A.

Thus, the equilibrium point is independent of the anchoring factor . However, the stability of the equilibrium point (I , R ) indeed depends on , which is analyzed in the following section. To simplify the analysis, we focus on the dynamical system (11) without the nonnegative constraint on the revenue function. Denote ( ) = dD0 k 2cc0 + (d 2 k 2c 2)( It + (1 ) Rt ) . The Jacobian of the dynamical system (11) at any point (It , Rt ) can be calculated as:

J=

B

(13)

– if B > 0 , both the revenue and R&D investment increase over time; – if B 0 , the dynamical system has one equilibrium point (I , R ) which is unstable with R = R > 0 and I = R . If > 0 , the following cases hold: – If d kc > 0 and B < 0 , the dynamical system has two equilibrium (I2 , R2 ) , (I1 , R1 ) points, denoted as and where R1 = R1 > 0, I1 = R1 , R2 = R2 > 0 , and I2 = R2 . (I2 , R2 ) is a ) ( ) > 8k 2, (I1 , R1 ) is a saddle point; saddle point. If (1 otherwise, (I1 , R1 ) is a stable node. – If d kc > 0 and B > 0 , the dynamical system has no equilibrium points and both the revenue and R&D investment increase over time. – If d kc < 0 , the dynamical system has one equilibrium point (I , R ) , R = R1 > 0 I = R. where and If (1 ) ( ) > 8k 2, (I , R ) is a saddle point; otherwise, (I , R ) is a stable node.

and if R 0 , then the equilibrium point satisfies R = R and I = R ; otherwise, the equilibrium point of the dynamical system (11) does not exist. For the case with d = kc, H (R, ) is a linear function in terms of R. Starting with the initial point (I0 , R0 ) = (0, 0) , the revenue dynamics in (11) are depicted as below:

By Proposition 4, similar to the two-dimensional revenue dynamical system (5) under the cost-plus pricing policy, the revenue dynamics of the system (11) under the profit-maximizing pricing policy may exhibit various patterns such as monotonically increasing, converging to a stable node, and periodically oscillating. The existence of the saddle point under d k c < 0 and > 0 indicates that the dynamical system will exhibit more complex behavioral patterns such as chaos.

Proposition 3. Under d = kc , if dD0 k 2cc0 2k2 , both the revenue and R&D investment increase over time; if dD0 k 2cc0 < 2k 2 , then the equilibrium point (I , R ) is solved as:

5.1. Numerical example

R =

Q02 2(dD0

4k 2

k 2cc0)

In this section, we conduct numerical studies to investigate the revenue dynamics of the system (11). We exclude the period-2 cycle because of the non-negativity constraint on the revenue in (11). In the following numerical example, we use the following parameter values:

, and I = R ,

which is a stable node, and the system dynamics converge to (I , R ) . Proof. If d 2

R =

4k 2

k 2c 2 = 0 , the solution to H (R, ) = 0 is solved as:

Q02 2(dD0

k 2cc0)

• The demand slope k = 1; • The demand-amplifying slope d = 0.5; • The cost-increasing slope c = 1; • The initial unit production cost is normalized as c

.

Since D0 kc0 > 0, dD0 k2cc0 > 0 . If dD0 k 2cc0 > 2k 2 , then R < 0 , the dynamical system has no equilibrium points, and both the revenue and R&D investment increase in t. If dD0 k 2cc0 < 2k 2 , then R > 0 , the equilibrium point satisfies R = R and I = R . At (I , R ) , the second eigenvalue is calculated as r2 =

+

(1

) 2k2

dD0

0

= 0.

Based on the above parameter values, the two-dimensional dynamical system (11) is reduced to:

It + 1 = It + (1

k 2cc0 < 1. Therefore,

(14)

) Rt ,

1 D02 + D0 4k 2

3 ( It + (1 4

(I , R ) is a stable node, and the system will converge to (I , R ) . □

Rt+ 1 = max

Therefore, for the simplest case, the system dynamics under the profit-maximizing pricing policy either increase period-by-period or converge to a stable node. In terms of the growth pattern, the firm may grow steadily or achieve a constant revenue in each period in the long run. kc , where In the following section, we consider the cases with d H (R, ) is a quadratic function in terms of R, and the equation H (R, ) = 0 has at most two solutions, denoted as R1 and R2 , respec4k 2 , and tively. Denote A = (d 2 k 2c 2) 2, B = 2(dD0 k 2cc0) C = Q02 > 0 . Eq. (13) is simplified as H (R, ) = AR2 + BR + C . The = B2 4AC . Clearly, if < 0 , then discriminant is calculated as H (R, ) = 0 has no solutions; if > 0 , then:

The impact of We first consider the impact of the marginal expenditure factor on the system dynamics. In the numerical example, the anchoring factor is fixed at = 0.02 , and the base demand is set as Q0 = 20 . The left panel in Fig. 4 is the bifurcation plot of the revenue with respect to , where the system changes from stable to periodic, and eventually chaotic with the increment of . The right panel in Fig. 4 [0.28, 0.303]. The plots the Lyapunov exponent in the region with positive Lyapunov exponent indicates that the system indeed becomes chaotic in the region when is beyond a certain level, say, 0.295. The impact of The anchoring factor also impacts the system dynamics. In the numerical example, the marginal expenditure factor is 8

It + 1

Rt

) Rt )2 , 0 .

Journal of Business Research xxx (xxxx) xxx–xxx

X. Yuan and R. Nishant

Fig. 4. Left panel: Bifurcation plot with respect to . Right panel: Lyapunov exponent with respect to . The figures are based on the two-dimensional dynamical system (14), starting with = 0.1, and incremented by 0.0001. Parameter values are: k = 1, d = 0.5, c = 1, c0 = 0, = 0.02, Q0 = 10 .

fixed at = 0.3, and the base demand is set as Q0 = 20 to investigate the impact of on the system dynamics. The left panel in Fig. 5 is the bifurcation plot of the revenue with respect to , by which we see that the system exhibits stability, periodic cycles, and chaos with the decrement of . The right panel in Fig. 5 plots the Lyapunov exponent of [0, 0.4], where the positive Lyapunov exponent the region with indicates that the system dynamics become chaotic in the region when is lower than a certain level, say, 0.03. To further visualize the dynamic system patterns, Fig. 6 shows the return map of the periodic cycle under = 0.26 and = 0.05 (left panel), and the strange attractor = 0.3 and = 0.02 (right panel). The existence of the stranger attractor further confirms that the dynamics of the two-dimensional system (11) are chaotic under certain regions of and . Therefore, given all the other parameters fixed, the results are similar to those under the cost-plus pricing policy, with the profit-maximizing pricing policy, the proportion of revenue devoted to R&D ( ) and the anchoring and adjustment in the decision-making behavior ( ) jointly affecting the dynamics of the revenue process. When greater revenue is devoted to R&D ( becomes larger), or the R&D investment decision is less anchored on the previous decision ( becomes smaller), the revenue dynamics could change from stable to periodic or even chaotic. If the R&D investment decision is adjusted more by the revenue, growth dynamics could be more fluctuating or even chaotic.

specific dynamic phenomenon similar to macroeconomic cycles, with intermittent contraction and expansion that can be chaotic. A lack of order and predictability in growth over time can reduce the effectiveness of managerial judgments. Undoubtedly, exogenous factors contribute to fluctuations in growth. Managerial decisions, such as about R &D investments, also contribute to the dynamics and fluctuations even in the absence of exogenous factors. Understanding how endogenous decision-making affects growth is important because managers control whether firms undertake the actions necessary to ensure stable growth. We offer interesting insights by investigating growth dynamics through a simple parsimonious R&D investment model. By distinguishing different consequences of R&D investment, we demonstrate that managerial decision-making is crucial to dynamic firm performance. Our findings suggest that R&D investment can evoke various behavioral patterns regarding revenue dynamics. Specifically, the marginal expenditure factor is important: when the proportion of revenue invested into R&D increases, revenue dynamics may move from a stable state to periodic cycles or even chaos, implying that managers can achieve stable growth by allocating a smaller proportion of revenue into R&D. Empirical studies such as Artz, Norman, Hatfield, and Cardinal (2010) support the findings in showing that R&D investment has a U-shaped relationship with product announcements, but product announcements are positively associated with sales growth. The anchoring factor also significantly impacts revenue dynamics. A high anchoring factor indicates that R&D investments are based more on previous decisions. Our results indicate that a decreased anchoring factor may change revenue dynamics from stability to periodic cycles or even chaos, implying that stable growth requires managers to rely more on previous investment decisions and put fewer revenue resources into

6. Discussion Researchers and practitioners are increasingly considering dynamic business phenomena. In this study, we focus on growth paths of firms, a

Fig. 5. Left panel: Bifurcation plot with respect to . Right panel: Lyapunov exponent with respect to . The figures are based on the two-dimensional dynamical system (14), starting with = 0 , and incremented by 0.0001. Parameter values are: k = 1, d = 1, c = 1, c0 = 0, = 0.3, Q0 = 10 . 9

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Fig. 6. Left panel: An illustration of the periodic cycle with period 4 in the periodic region. Right panel: An illustration of the strange attractor in the chaotic region. The figures are based on the two-dimensional dynamical system (14). Parameter values are: k = 1, d = 1, c = 1, c0 = 0, = 2, Q0 = 10 .

R&D. The finding aligns with Govindarajan, Rajgopal, Srivastava, and Enache (2019), who argued that managers should not treat R&D as discretionary. The finding also conforms to practice in that managers often rely on previous decisions when allocating resources. To manage volatility and to achieve smooth operations and growth, managers should combine past experiences with new, more varied information, especially where they place higher weight on previous decisions. If the revenue dynamics exhibit steadily increasing or stable behavior, firms can grow continuously and sustainably. When a firm’s revenue periodically oscillates, growth may go up and down. However, if the dynamic behavior resembles chaos, growth becomes unpredictable and sensitive to initial conditions. In other words, routine managerial tasks and past experiences no longer predict the future; further actions might prove useless, perhaps causing a firm’s sudden decline or death (Feichtinger & Kopel, 1993). To “heal” or “recover”, managers must know which direction to follow. They must understand the system’s global performance before making corrective decisions. Identifying whether the system carries potential chaos is an important management strategy. Chaos could destroy any attempts to forecast and plan by suddenly changing a system into a completely different state without changes to underlying structures or decisionmaking, a common situation when managers follow usual strategies and practices but find their firms showing totally different performance, which may offer one explanation for the “sudden death” of firms that were in healthy financial conditions before they abruptly experienced trouble. Managers often blame external environments for causing failure when their decisions are actually the underlying cause. As chaos is also sensitive to initial conditions, a slight change of decision may bring cause a chaotic system to become stable. Thus, managers particularly need to know the chaotic regimes in management systems (Feichtinger & Kopel, 1993) to know which directions to take for escaping or preventing crises. Our findings complement studies grounded in complexity theory that deals with relevant topics such as innovation policies. For instance, Frenken (2017) argued that innovation is too complex to be solely explained by traditional paradigms such as market or system failures, and that we need a complexity perspective that views innovation outcomes as constituents of networks. Our study contributes to this view by showing that the innovation domain subsuming R&D investment indeed exhibits complex characteristics. Our study also contributes to the growing, nascent stream of research on the role of internal stakeholders such as R&D investment directors. In particular, our study echoes studies such as Shaikh, O’Brien, and Peters (2018), which argued that besides exogenous factors influencing R&D, internal stakeholders such

as directors play a key role in preserving R&D investments and its relationship with financial resources at disposal. Our study suggests that managerial decision-making is also critical if R&D is to benefit firms financially. 7. Conclusion We developed a simple and parsimonious model that considers R&D investment as driving growth in firms. The model shows that management decisions can cause instability or even chaos in systems, even in the absence of exogenous factors. We provide a different perspective to R&D research, particularly to empirical studies that often control for exogenous factors and test linear or curvilinear relationships. Our model is stylized and simple, but it captures many practical aspects of R &D investment and reveals pitfalls that intuition may overlook (Feichtinger & Kopel, 1993). The results show that a minute decision change could cause system fluctuations and disorder, even though the original intention was to eliminate variability and achieve steady growth. Hence, managers can manage operations more effectively and achieve their objectives by understanding chaos and nonlinearity dynamics in the vibrant business environment. Our study provides a starting point to develop a more nuanced understanding of the relationship between managerial decisions about R&D investments and firm growth. Further investigation is needed for considering many potential directions, both academically and practically relevant. Future research could improve the current model by relaxing assumptions, incorporating more factors into R&D investment decisions, using empirical data to estimate the relationship between R& D investment and revenue, and considering the impact of other factors such as interest rates on resource allocations. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This research was supported by the National Natural Science Foundation of China (Grant No. 71502044), China Postdoctoral Science Foundation (Grant No. 2015M570300), and the research grant from Harbin Institute of Technology and Université Laval (Faculty of Business Administration).

10

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Appendix A. Proofs A.1. Proof of Proposition 2 Proof. Since d

0, H (R, , ) is a quadratic function in terms of R. Depending on A ( ) and

k c

( ) , we have the following cases:

• If ( ) < 0, H (R, , ) = 0 has no solutions. The equilibrium point of the dynamical system (5) does not exist. The case holds if d k c > 0. Both the R&D investment and revenue increase over time. If • ( ) = 0, one solution exists: R = to H (R, , ) = 0. The case holds if d k c > 0. Thus, if B ( ) > 0, R < 0 indicating the dynamical B( ) 2A ( )

•

system has no equilibrium points; otherwise, the equilibrium point (I , R ) satisfies R = R > 0 and I = R , and at the equilibrium point, the second eigenvalue is calculated as r2 = 1. The equilibrium point is an unstable node, since A ( ) > 0 . If ( ) > 0 , two solutions exist: R1 and R2 , to H (R, , ) = 0 , leaving the following cases: – If d k c > 0 , i.e., A ( ) > 0 . If B ( ) < 0, R1 > 0 and R2 > 0 . The dynamical system has two equilibrium points, denoted as (I1 , R1 ) and (I2 , R2 ), where R1 = R1 > 0, I1 = R1 , R2 = R2 > 0 , and I2 = R2 . The second eigenvalue r2 at each equilibrium point can be calculated as:

r2 = 1

(1

)

( ) , r2 = 1 + (1

)

( ) > 1.

) ( ) 2, (I1 , R1 ) is a saddle point; otherwise, (I1 , R1 ) is a stable node. Thus, (I2 , R2 ) is always a saddle point. If (1 – If d k c > 0 , i.e., A ( ) > 0 . If B ( ) > 0, R1 < 0 and R2 < 0 . The dynamical system has no equilibrium points. Since A ( ) > 0 , the dynamical system will grow in t. – If d k c < 0 , i.e., A ( ) < 0 . If B ( ) < 0, R1 > 0 and R2 < 0 . The dynamical system has one equilibrium point (I1 , R1 ) , where R1 = R1 > 0 and I1 = R1 . At (I1 , R1 ) , the eigenvalue of r2 can be calculated as: r2 = 1

(1

)

( ).

) ( ) 2, (I1 , R1 ) is a saddle point; otherwise, (I1 , R1 ) is a stable node. If (1 – If d k c < 0 , i.e., A ( ) < 0 . If B ( ) > 0, R1 > 0 and R2 < 0 . The dynamical system has one equilibrium point (I , R ) , where R = R1 > 0 and I = R . At (R , I ) , the eigenvalue of r2 can be calculated as: r2 = 1

(1

If (1

)

)

( ). 2, (I , R ) is a saddle point; otherwise, (I , R ) is a stable node. □

( )

A.2. Proof of Proposition 4 Proof. Under the case d

kc , depending on A and

, we have the following cases:

• If < 0, the equation has no solutions. The equilibrium point does not exist. The case holds if d > kc . Both the revenue and the R&D investment increase over time. If • = 0, one solution exists: R to H (R, ) = 0. The case holds if d > kc . Thus, if B > 0, R < 0, and the dynamical system has no equilibrium •

points; while if B 0 , the equilibrium point (I , R ) satisfies R = R and I = R . The second eigenvalue at the equilibrium point is calculated as r2 = 1. The equilibrium point is unstable. If > 0 , two solutions exist, denoted as R1 and R2 . Then, we have the following cases: –If d kc > 0 and B < 0, R1 > 0 and R2 > 0 . The dynamical system has two equilibrium points, denoted as (I1 , R1 ) and (I2 , R2 ), where R1 = R1 > 0, I1 = R1 , R2 = R2 > 0 , and I2 = R2 . The eigenvalue of r2 at the two equilibrium points can be calculated as:

r2 = 1

(1

) 4k 2

, r2 = 1 +

(1

) 4k 2

. (1

)

> 2 , the first equilibrium point is a saddle point; otherwise, (I1 , R1 ) is a Thus, the second equilibrium point is always a saddle point. If 4k2 stable node. – If d kc > 0 and B > 0, R1 < 0 and R2 < 0 . The dynamical system has no equilibrium points and since A > 0 , both the revenue and R&D investment increase over time. – If d kc < 0 and B < 0, R1 > 0 and R2 < 0 . The dynamical system has one equilibrium point (I , R ) , where R = R1 > 0 and I = R . At (1 ) (1 ) > 2 , the equilibrium point is a saddle point; otherwise, (I1 , R1 ) is (R , I ) , the eigenvalue of r2 can be calculated as r2 = 1 . If 4k2 4k 2 a stable node. – If d kc < 0 and B > 0, R1 > 0 and R2 < 0 . The dynamical system has one equilibrium point (I , R ) , where R = R1 > 0 and I = R . At (1 ) (1 ) > 2 , the equilibrium point is a saddle point; otherwise, (I , R ) is (I , R ) , the eigenvalue of r2 can be calculated as r2 = 1 . If 4k2 4k 2 a stable node. □ Appendix B. The profit-based R&D investment strategy In this section, we develop the model where the net profit in each period is source of funding for R&D investment. The firm also adopts the exponential smoothing rule in determining the R&D investment in each period because of the anchoring and adjustment in decision-making. Denote the net profit as t in each period t, which is calculated as: t

= (pt

ct ) Qt

(15)

It ,

that is, the profit from product sales after the R&D investment. 11

Journal of Business Research xxx (xxxx) xxx–xxx

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Under the anchoring and adjustment policy, the R&D investment amount in period t + 1 is given by

It + 1 = It + (1

)

(16)

t.

Under the cost-plus pricing policy, substituting the price in (4), the net profit in period t + 1 is calculated as: t + 1=max{(pt + 1

=max{(

ct + 1) Qt + 1

It + 1, 0} = max{(

1) c0 Q0 + [(

1)(c0 (d

1)(c0 + cIt + 1)(Q0 + (d

k c ) + cQ0)

1] It + 1 + (

k c ) It + 1)

1) c (d

It + 1, 0}

k c ) It2+ 1, 0},

(17)

where Q0 = D0 k c0 with I0 = 0 and p0 = c0 . The dynamical system under the cost-plus pricing policy with the net profit as the source of funding is summarized as:

It + 1= It + (1 t + 1=max{(

+(

) t, 1) c0 Q0 + [(

1) c (d

1)(c0 (d

k c )[ It + (1

2 t] ,

)

k c ) + cQ0)

1][ It + (1

)

(18)

t]

0}.

The structure of the above dynamical system (18) is similar to the model in Section 4. Following the analysis in Section 4, we can prove the existence of conditions where the dynamical system can exhibit stability, periodic cycles, and even chaos in terms of the net profit. Under the profit-maximizing pricing policy, substituting the optimal price in (9) into the profit function, we can calculate the net profit as: t + 1=max{(pt + 1

=max = max

{

D0

ct + 1) Qt + 1

kc0 + (d 2k

1 [D0 4k 2

kc ) It + 1

It + 1, 0}, max

kc0 + (d

(

D0

kc0 + (d 2k

kc ) It + 1]2

kc ) It + 1

, 0

)

}

It + 1, 0 ,

(19)

It + 1, 0 .

The dynamical system under the profit-maximizing pricing policy with the net profit as the source of funding is summarized as:

It + 1= It + (1 1 t + 1=max 4k 2

) D0

t,

kc0 + d

kc [ It + (1

)

2 t]

It + 1

t

, 0 .

(20)

The structure of the dynamical system (20) is similar to the model in Section 5. Following the analysis in Section 5, we can prove conditions exist in which the dynamical system can exhibit stability, periodic cycles, and even chaos in terms of the net profit.

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