Production with learning and forgetting in a competitive environment

International Journal of Production Economics 189 (2017) 52–62

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Production with learning and forgetting in a competitive environment a,⁎

b

Konstantin Kogan , Fouad El Ouardighi , Avi Herbon a b

MARK

a

Management Department, Bar-Ilan University, Ramat-Gan 52900, Israel Operations Management Department, ESSEC Business School, BP 105, 95021, Cergy Pontoise, France

A R T I C L E I N F O

A BS T RAC T

Keywords: Production Duopolistic competition Learning-by-doing Organizational forgetting Dynamic games

It has been shown that learning-by-doing enables firms to reduce marginal production costs, but that this effect weakens due to organizational forgetting. In order to assess the impact of both learning and forgetting on longterm competitiveness and a firm’s profitability, we model an experience accumulation process with depreciation and consider two competing firms that produce fully substitutable products. In this model, unit production costs decrease with the firm’s experience due to the proprietary learning process as well as the spillover of experience from the competing firm. Firms can either share or hide from each other their information about the state of their respective experience throughout the game. We found that in an equilibrium steady state, if the organizational forgetting is sufficiently large (larger than the spillover rate value), then information sharing, compared to information hiding, results both in less competitiveness and increased profits for firms. Conversely, if the organizational forgetting is small and the spillover opportunities are relatively large, then information sharing promotes both long term competitiveness and firm profits. Accordingly, firms are better off in the long term by deliberately limiting (expanding) their experience accumulation process whenever organizational forgetting is relatively large (small). A high ability of proprietary learning, however, can interfere in this relationship so that limiting the firms’ experience process will always be compatible with higher profitability.

1. Introduction Comprehensive data on all solar photovoltaic industry installations in California from 2002 to 2012 points out that 1000 additional installations by a contractor in a county reduces non-hardware marginal costs by $0.36 per watt on average; in addition to this learning by doing, 1000 installations by competitors spills over and reduces the contractor’s non-hardware costs by $0.005 per watt (Bollinger and Gillingham, 2014). Similarly, the production of the L-1011 and L1011-500 commercial aircraft in the Eighties by Lockheed Aeronautics (California) was characterized by high learning rates of approximately 35% in terms of production experience. Since commercial aircraft production is highly labor-intensive and production rates are very low, learning results primarily from a more experienced workforce. The marginal costs of producing aircraft do not, however, always decrease over time, as would be expected if production were subject to pure learning. Indeed, labor requirements were observed to increase after the strike in 1977 thereby explicitly illuminating the “forgetting effect”. By that time, the competing model, the L-1011-500 was introduced. The decision to bring out a new model significantly set back the learning process. Due to partial experience spillover, the first L-1011– 500 produced required approximately 25% more labor than the

⁎

previous model. The labor requirements for the two competing models, however, then converged over time inducing similar reduced marginal production costs (Benkard, 2000). In general, the economic and strategic implications of learning-bydoing-alone have been extensively documented (see the reviews in Arrow, 1962; Yelle, 1979; Dutton and Thomas, 1984; Cabral and Riordan, 1994; McDonald and Schrattenholzer, 2001; Fogliatto and Anzanello, 2011, Argote, 2011). However, the consequences of organizational forgetting on the competitive dynamics of industries where learning with spillovers is present remain under-investigated (Bailey, 2000; Argote et al., 1990; Benkard, 2000; Martin De Holan and Phillips, 2004). This paper seeks to fill this gap by analyzing how a firm’s competitive behavior is affected by the extent to which it accumulates experience under spillovers and organizational forgetting. To investigate this issue, we consider a duopolistic industry involving fully substitutable products, in which two firms compete on production (Cournot competition). As usual in a Cournot setup, the production decisions determine the market price and thereby the firms’ profits (Chung et al., 2012; Hu et al., 2014). Each firm’s production activity leads to accumulate experience over time through learning-bydoing. Because of its cumulative nature, experience is interpreted as a stock variable that allows for spillovers between firms (Chen et al.,

Corresponding author. E-mail addresses: [email protected] (K. Kogan), [email protected] (F. El Ouardighi), [email protected] (A. Herbon).

http://dx.doi.org/10.1016/j.ijpe.2017.04.008 Received 31 May 2016; Received in revised form 3 February 2017; Accepted 25 April 2017 Available online 27 April 2017 0925-5273/ © 2017 Elsevier B.V. All rights reserved.

International Journal of Production Economics 189 (2017) 52–62

K. Kogan et al.

two-period models, our results suggest that the difference between production quantities of the two types of equilibrium is not cancelled out when the discounting rate is negligible. In the next section, we review the relevant literature. Section 3 develops the differential game model. Section 4 analyzes the commitment strategy, and Section 5 focuses on the contingent strategy. Section 6 compares our results. Section 7 concludes.

2015). We also assume that the stock of experience can be reduced by organizational forgetting over time. Due to the cumulative nature of experience, we use a dynamic game to determine how competitive production decisions are made under experience accumulation with forgetting in the presence of both proprietary and spillover learning. In dynamic games (e.g., Dockner et al., 2000; Jørgensen and Zaccour, 2004; Long, 2010), the mutual observability of each firm’s stock of experience over time plays an essential role in each firm’s respective strategy (Dockner et al., 2000). When each firm’s current stock of experience is observable by both parties at each period of time, a firm can update its production decisions with the current values of its own stock of experience and that of its competitor. In such instances, which are referred to as closed-loop equilibria, each firm’s production strategy is contingent upon information regarding the rival’s reaction to a change in the current stocks of experience. Conversely, if the firms’ current stock of experience cannot be observed (or can only be seen after a certain time lag) it is appropriate for each firm to optimize its production decisions throughout the game based on its sole knowledge of the initial stocks of experience given at the beginning of the game. In this case, which is referred to as open-loop equilibrium, a firm’s strategy is based on precommitment to execute a plan of production throughout the game. In reality the state of production experience is frequently and intentionally concealed by the competing firms. This extends to keeping secret main features of new products until their release. A firm may, however, discover the state of the production experience of its rivals by learning from the suppliers or customers they have in common, sending employees to trade shows and professional conferences, reverse-engineering of rival’s products and so on (von Hippel, 1988). Consequently, we consider in this paper both observable and unobservable stocks of experience as possible scenarios to determine how the observability of the rival’s current experience stocks affects competitive production decisions under experience accumulation with forgetting in the presence of both proprietary and spillover learning. To this end, we evaluate and compare production levels of both firms under open-loop and closed loop equilibria. The contribution of the present paper is manifold. In particular, the results account for learning-by-doing along with organizational forgetting over an infinite time horizon with discounting. We determine important properties of the steady-state equilibria (which do not exist under a finite planning horizon). Further, unlike previous studies, we derive a pivotal condition that compares analytically open-loop and closed-loop steady-state equilibria thereby predicting the effect of information observability on competing firms. This condition emphasizes in particular the role of organizational forgetting, experience spillovers and discounting rate. In contrast to the results obtained in

2. Literature review Our research lies at the intersection of three literatures, those concerning empirical models on determinants of learning by doing, production decisions under learning and forgetting, and game models analysis of decision rules in production problems of horizontal competition. Jarmin (1993) develops and estimates an empirical model to study the intertemporal nature of learning by doing and spillovers. He finds evidence of both proprietary and spillover learning and shows that a firm’s ability to learn from its own experience differs from its ability to learn from its rival’s experience. Benkard (2000), in addition to learning with spillovers, incorporates organizational forgetting that depreciates experience at a constant rate. Using data from the commercial aircraft industry, he suggests that the strategic effects of organizational forgetting must be taken into account when the products are labor-intensive; learning is thought to be important at the individual worker level; and the turnover is relatively high. In this paper, we integrate the influence of both spillovers and forgetting on experience accumulation. Most recent studies about the implications of forgetting in industries where learning takes place have been limited to operations management considerations. Teyarachakul et al. (2008) analyze the long-run characteristics of batch production time for a constant demand problem with learning and forgetting in production time. Teyarachakul et al. (2014) extend this result to a large class of learning and forgetting functions with some differentiability conditions. Teyarachakul et al. (2011) investigate the effect of learning and forgetting on production decisions based on the assumptions that i) the amount forgotten increases with longer interruptions in production and ii) the forgetting could be initially slow and become fast afterward. They show that these assumptions lead firms to produce smaller rather than larger quantities in the presence of learning and forgetting. Our approach differs from these studies in that we analyze how learning and forgetting affect a firm’s competitive behavior (see Table 1 for the basic features of the competition related research in learning-bydoing). Earlier analytical game-based models accounting for the effect of experience typically consider quantity-based competition over two

Table 1 Basic features of research on dynamic games with learning-by-doing. Research

Learning

Spillover

Forgetting

Time Horizon

Equilibrium type

Analytical comparison of FNE and OLNE

Spence (1981)

+

–

–

Finite horizon with discounting

+

Fudenberg and Tirole (1983)

+

–

–

Infinite horizon with discounting

FNE and OLNE for two periods OLNE

Jørgensen and Zaccour (2000) Stokey (1986)

+

+

–

+

–

FNE and OLNE for two periods FNE

–

Miravete (2003) Besanko et al. (2010)

Fixed cost learning Finite stock of knowledge + +

Complete spillover – –

Finite, discrete-time with discounting Infinite horizon with discounting

– +

FNE and OLNE FNE

+ –

+ +

– +

Infinite horizon with discounting Discrete-time, infinite horizon with discounting Finite horizon without discounting Infinite horizon with discounting

FNE and OLNE FNE and OLNE

– +

Kogan et al. (2016) This paper

53

–

+

International Journal of Production Economics 189 (2017) 52–62

K. Kogan et al.

Ci (Xi (t ),Xj (t )) = c − γXi (t ) − εXj (t )

periods, especially when analyzing contingent equilibria (e.g., Spence, 1981; Fudenberg and Tirole, 1983; Jørgensen and Zaccour, 2000). In these models, output is typically higher under contingent rather than precommitment strategy. In a more general model, however, Stokey (1986) develops a differential game with complete spillovers and shows that the contingent production strategy in this scenario is higher than that of the static Cournot-Nash equilibrium with no learning. To overcome tractability issues, Miravete (2003) assumes no spillovers and only fixed cost reduction due to accumulated output while the unit cost remains constant. Besanko et al. (2010) also assume no spillovers, but account for the effect of learning-by-doing on the marginal cost by considering two firms characterized by a finite stock of discrete “knowhow” states and logit demands. Due to a tractability problem, a numerical method is employed to compute subgame perfect equilibria of price competition that includes organizational forgetting. The numerical study suggests that organizational forgetting intensifies pricing competition. Kogan et al. (2016) focus on stochastic learning conjectures without forgetting and discounting future profits. Assuming a steady state cannot exist, they analyze transient solutions over a finite time horizon. Their numerical simulations show the relationship between open-loop and closed-loop transient equilibrium quantities may possibly be altered. Unlike Kogan et al. (2016), we provide an analytical comparison of open-loop and closed-loop steady-state equilibria. An unexamined question is the extent to which the observability of the rival’s current experience stocks affects competitive production decisions under experience accumulation with forgetting in the presence of both proprietary and spillover learning. We follow the approach of comparing decision rules involved by different informational structures, but with the innovation that we apply the comparison in the context of learning with forgetting and spillovers. Doing so, we evaluate the impact of observability of each other’s experience level on the production decisions in a quantity-based competition duopoly market.

(4)

i , j = 1,2,i ≠ j , where c = ci , γ = γi , ε = εi ,i = 1,2, i.e., we assume symmetric conditions. Eq. (4) implies that the utility of learning based on cumulative experience Xi declines due to the declining efficiency of experience accumulation implied by the forgetting effect in (3). The presence of the forgetting effect generalizes the linearity assumption based on the cumulative production Yi , which is widely used in the literature (e.g., Jørgensen and Zaccour, 2000; Fudenberg and Tirole, 1983). Given the discounting rate δ > 0 of future profits and assuming α − (qi (t ) + qj (t )) − c + γXi (t ) + εXj (t ) ≥ 0 , each firm maximizes its cumulative profit (the time index t is omitted for convenience): ∞

Max qi

∫

e−δ t [α −(qi + qj ) − c + γXi + εXj ] qi dt (5)

0

subject to constraint (3), i , j =1,2,i ≠ j . Depending on whether the stocks of experience are mutually observable or not, the firms identify which strategy, either open-loop Nash equilibrium (OLNE) or closed-loop Nash equilibrium (CLNE), they will apply throughout the game. We first characterize the open-loop equilibrium and then analyze the closed-loop Nash equilibrium. 4. Nash equilibrium under mutually unobservable stocks of experience This section will derive open-loop production strategies of the firms, i.e., each firm precommits to execute a production plan given the competitor’s precommitment. The current value Hamiltonians for the differential game (3) and (5) are (e.g., Grass et al., 2008):

Hi = [α − (qi + qj ) − c + γXi + εXj ] qi + ψi (qi − ρXi ) + ψi j (qj − ρXj )

(6)

j

i , j =1,2, i ≠ j , where the co-state variables ψi (t ) and ψi (t ) satisfy the following conditions:

3. The model Consider two firms i = 1,2 producing substitutable goods. At each period of time t over an infinite planning horizon, the firms choose a quantity qi (t )≥0 to be produced per time unit (output). The price P (Q (t )) ≥ 0 of the products at time t depends negatively on the total output at time t , Q (t ) = ∑i qi (t ):

P (Q (t )) = α − β (q1 (t ) + q2 (t ))

ψi̇ = ρψi −

∂Hi = (δ + ρ) ψi − γq ∂Xi i

ψi̇ j = ρψi j −

(7a)

∂Hi = (δ + ρ) ψi j − εq ∂Xj i

(7b) j

under the transversality conditions: lim e−δt ψi (t )=0 and lim e−δt ψi (t )=0 .

(1)

t →∞

where α>0 is the potential demand and β = 1. The cumulative output Yi (t ) of firm i, given by:

∂Hi = α − 2qi − qj − c + γXi + εXj + ψi = 0 ∂qi

Yi̇ (t ) = qi (t )

i , j = 1, 2,

(2)

where Yi̇ (t ) denotes the rate of change of firm i ’s cumulative output Yi (t ), affects the production experience level, Xi (t ) >0 , i = 1,2 . As in Benkard (2000), we distinguish between cumulative production Yi (t ) and experience Xi (t ), which evolves according to:

Xi̇ (t ) = qi (t ) − ρXi (t ),

Xi (0)≥0

t →∞

The equilibrium production quantities are found from:

qi =

i≠j , that is:

α − qj − c + γXi + εXj + ψi 2

(8)

Since the initial experience of the firms is identical, the symmetric equilibrium production quantities are:

(3)

α − c + γXi + εXj + ψi ⎧ ⎪ , ifα − c + γXi + εXj + ψi >0 3 qi = ⎨ ⎪ 0, ifα − c + γXi + εXj + ψi≤0 ⎩

We assume the initial experience of the firms is identical, X1 (0) = X2 (0) to consider symmetric equilibria. Eq. (3) implies that the greater the experience the more difficult it is to further accumulate it, because of the forgetting effect, ρXi (t ), where ρ > 0 and constant. Consequently, there can be saturation in accumulating experience and accordingly in the firm’s ability to cut its unit production cost. Note that, as in Teyarachakul et al. (2011), the amount forgotten also increases with longer interruptions in production, i.e., qi (t )=0 . Learning from its own production experience and its competitor’s experience spillovers reduces the firm’s initial unit production cost, ci >0 , at the marginal rates γi >0 , εi≥0 , respectively, that is:

(9)

i , j = 1,2, i ≠ j . Letα − c + γXi + εXj + ψi≤0 , then qi = 0 which, when substituting into (7) and accounting for (3), results for the symmetric case in two state/co-state equations:

ψi̇ = (δ+ρ) ψi

(10)

Xi̇ = − ρXi

(11)

i = 1,2 . That is, ψi = Ae(δ + ρ) t , which does not satisfy the transversality lim e−δt ψi (t ) = 0 , A = 0, unless but then condition, t →∞

54

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K. Kogan et al.

⎛ δ + 2ρ ⎞ We next assume that 3ρ > γ ⎜ δ + ρ ⎟ + ε to study the steady state. ⎠ ⎝ Substituting qi from (9) into (7) and accounting for (3), we get the following canonical system:

α − c + γXi + εXj + ψi≤0 does not hold. Accordingly, idling cannot be optimal at a terminal interval of time. We next assume that the steady state marginal cost C = c S, OLNE is greater than or equal to some given minimal marginal cost cm that the firm may achieve in the long run. If this assumption does not hold, then the firms eventually lose their ability to learn as a means of reducing their marginal costs, as will be discussed below. To find a steady state, we set Xi̇ =0 , and thereby qi=ρXi , i=1,2 . Our results are then summarized as follows.

⎛ γ (α − c + γXi + εXj ) γ⎞ ψi̇ = ⎜δ + ρ − ⎟ ψi − , ⎝ 3⎠ 3

Xi̇ =

⎛ δ + 2ρ ⎞ Proposition 1. If3ρ > γ ⎜ δ + ρ ⎟ +ε and c S, OLNE ≥c m , the game has a ⎠ ⎝ unique symmetric OLNE steady state characterized by positive experience, production and shadow price: (δ + ρ)(α − c ) (δ + ρ)(3ρ − γ − ε ) − γρ

(12a)

ρ (δ + ρ)(α − c ) = (δ + ρ)(3ρ − γ − ε ) − γρ

(12b)

XiS, OLNE = qiS, OLNE

ψiS, OLNE =

− (ρ −

γ )X 3 i

(14)

⎡ γ ⎢ δ+ρ− 3 0 ⎢ ⎢ 0 δ+ρ− J=⎢ 1 ⎢ 0 ⎢ 3 1 ⎢0 ⎣ 3

− γ 3

γ2 3 γε

−3 γ 3 ε 3

⎤ ⎥ ⎥ γ2 − 3 ⎥ ⎥ ε − ρ3 ⎥ ⎥ γ − ρ ⎥⎦ 3

−

γε 3

(15)

The determinant of the Jacobian matrix is:

γρ (α − c ) (δ + ρ)(3ρ − γ − ε ) − γρ

(12c)

Δ(J ) =

Corollary 1. The greater the proprietary learning abilityγ and the spillover effect ε , the greater the firms’ open-loop steady state production and experience. Moreover, the marginal effect of ρ proprietary learning is 1 + δ + ρ times stronger than that due to spillover. ■ By setting γ and ε to zero, we obtain the classical Cournot-Nash α−c static game equilibrium, qiCN = 3 < qiS, OLNE . Accordingly, both proprietary learning and spillovers induce higher production of the firms compared to the myopic Cournot-Nash production. Importantly, assuming again c S, OLNE ≥c m , we observe that even if we substitute c (γ + ε )(δ + ρ)(α − c ) with c S, OLNE = c − (γ +ε ) XiS, OLNE = c − (δ + ρ)(3ρ − γ − ε) − γρ , we find that

< qiS, OLNE =

ρ (δ + ρ)(α − c ) , (δ + ρ)(3ρ − γ − ε ) − γρ

[δ (3ρ − γ − ε ) + ρ (3ρ −2γ − ε )][δ (3ρ − γ + ε ) + ρ (3ρ −2γ + ε )] 9

from which a sufficient condition for a positive value of Δ(J ) is δ (3ρ − γ − ε ) + ρ (3ρ − 2γ − ε ) ≥ 0 , which is fulfilled when accounting for the steady state conditions of Proposition 1. On the other hand, using Dockner’s formula, the sum of the principal minors of J of order 2 minus the squared discounting rate,

Proof. See the Appendix.■ From equations (12) we readily conclude:

[3ρ (δ + ρ) − γρ](α − c ) 3[(δ + ρ)(3ρ − γ − ε ) − γρ]

3

i , j = 1,2, i ≠ j . To study the stability of the steady state, we construct the Jacobian matrix associated with the canonical system (13)-(14):

i = 1, 2.

qiCN =

α − c + εXj + ψi

(13)

⎡ ⎞ ⎛γ γρ ⎤ ⎥ K = 2 ⎢ (δ + ρ ) ⎜ − ρ ⎟ + ⎠ ⎝3 3⎦ ⎣ γ (δ + 2ρ)

is negative if 3ρ > δ + ρ . Given that Δ(J ) >0 and K < 0 , the Jacobian matrix has two eigenvalues with negative real parts and two with positive real parts. To determine whether the optimal path is monotonic or follows transient oscillations, we compute:

Ω = K2 − 4Δ(J ) 2 ⎧ ⎞ ⎛γ ⎪⎡ γρ ⎤ ⎥ ⎢ (δ + ρ ) ⎜ − ρ ⎟ + = 4⎨ ⎪ ⎠ ⎝3 3⎦ ⎩⎣

as stated below.

⎛ δ + 2ρ ⎞ Proposition 2. Let3ρ > γ ⎜ δ + ρ ⎟ + εandc S, OLNE ≥c m . If γ > 0 , then the ⎠ ⎝ static Cournot-Nash equilibrium production is lower than its dynamic, steady-state counterpart for any marginal cost in the region of S , OLNE [c S, OLNE ,c], that is, qiCN C ∈[c S, OLNE , c]
S , OLNE

then qiCN C = c S, OLNE = q , i = 1,2 . ■ i Proposition 2 shows that if the learning ability along with spillovers is disregarded, the resulting static Cournot-Nash outcome based on any unit production cost ranging from the initial cost up to the steady state one is lower than its dynamic counterpart. This outcome is different from those obtained for similar Cournot competition-based differential games without learning. In particular, it has been found that the static Cournot-Nash equilibrium production level is either greater than that in the precommitment steady-state equilibrium when the prices are sticky (Fershtman and Kamien, 1987) or equal to it under costly dynamics of output adjustments (Dockner, 1992). Our result emanates from the equilibrium output equation defined in (9). The classical principle, “marginal revenue equals marginal cost”, underlies the equilibrium equation with the co-state variable (shadow price) being the only term that distinguishes the static Cournot-Nash equilibrium from the dynamic one with potential for further learning. This term is positive at the steady state (see Proposition 1) and plays a strategic role by introducing learning opportunity - marginal revenue from increasing experience by one more unit thereby reducing the unit production cost.

−

⎫ [δ (3ρ − γ − ε ) + ρ (3ρ −2γ − ε )][δ (3ρ − γ + ε ) + ρ (3ρ −2γ + ε )] ⎪ ⎬ ⎪ 9 ⎭

=

4ε 2 (δ + ρ)2 , 9

which is clearly positive. This implies that the path converging to the saddle-point is monotonic near the steady state. Consequently, we get the following result.

⎛ δ + 2ρ ⎞ Proposition 3. If3ρ > γ ⎜ δ + ρ ⎟ + ε and c S, OLNE ≥c m , the unique ⎠ ⎝ symmetric OLNE steady state exhibits a (local) two-dimensional stable manifold and the convergence to the steady state is monotonic. ■ Fig. 1 depicts the phase-portrait diagram for symmetric conditions. It shows that when the initial production experience is small (large), the optimal strategy consists of setting the initial production quantity at a relatively low (large) level and then to increase (decrease) it until the steady state is reached. In other words, the industry output should progressively increase (decrease) with the experience accumulated collectively. This result implies that it is not optimal for a firm with too little experience to set its production quantity at a high initial level in order to quickly outperform its competitor. Conversely, a firm with considerable initial experience should rely on organizational forgetting 55

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K. Kogan et al.

=0

qï =

1 [(ε + 3δ ) q i̇ −ρ (δ + ρ)(α −c )] 3

(18)

In this case, the equilibrium output and co-state grow linearly as:

=0

qiOLNE =

ψi =

ρ (δ + ρ)(α − c ) t +G (ε+3δ )

(19a)

γ [A1 G (δ + ρ) + B1 + B1 (δ + ρ) t ] B1 (δ + ρ)2

where

G

is

the

integration

(19b) constant,

A1 = A

ρ

ε = 3 + δ and γ [A1 G (δ + ρ) + B1]

B1 = 3 (δ+ ρ)(α−c ). The constant is found from ψi (0) = and G = qiOLNE (0) =

C = c − γXi + εXj = c m

(16)

qi =

XiOLNE (t ) =

2β (Xi (0) − XiS, OLNE ) A + 2ρ −

A2

A2 +4B ) t

1

e 2 (A −

+ Ee−ρt +

+ 4B

i = 1, 2 , where E = X (0) −

Xi̇ = 0 ρ (c − c m )

2β (Xi (0) − XiS, OLNE ) A +2ρ − A2 +4B

−

qiS, OLNE ρ

qiS, OLNE

Proposition 5. Let the constraint C ≥ c m be binding for t ≥ t *. If ρ (c − c m ) α − cm < 3 , then the static Cournot-Nash equilibrium output based γ+ε

ρ

on c m is higher than its dynamic counterpart qiOLNE < qiCN

.

ρ (c − c m )

1 ⎛⎜A − A2 +4B ⎞⎟ t ⎠

− qic

α − cm

C=cm

;

otherwise, if γ + ε ≥ 3 , then qiOLNE ≥qiCN C = c m for t ≥ t *. ■ To sum up our observation, the greater the proprietary learning rate γ and/or the rate of learning from the competitor’s experience spillovers ε , the higher the firms’ output and experience at the steady state and the higher the likelihood that the steady state condition ⎛ δ + 2ρ ⎞ 3ρ > γ ⎜ δ + ρ ⎟ +ε will be eventually violated. Specifically, if the rate of ⎠ ⎝ proprietary learning and learning from the competitor is three times faster than the forgetting rate, the steady state condition will not hold. As a result, any increase in output and subsequent decrease in product price will be fully offset by a swift marginal cost reduction. Accordingly, the firms will gain greater profits by increasing (rather than steady) outputs and the marginal production cost will eventually be reduced to the minimally possible one, c m .

Proof. See the Appendix. ■ It is straightforward to verify that given the steady state condition from Lemma 1, ψi (defined by (A5)-(A7)) is positive for any t ≥ 0 and Xi (0) ≥ 0 rather than just at the steady state and that the denominator in (A7) is always positive. Consequently, if the initial experience is lower than that at the steady state Xi (0) 0 and conditions of Proposition 4 do not ⎛ δ + 2ρ ⎞ hold, 3ρ < γ ⎜ δ + ρ ⎟ + ε , i.e., B < 0 and A2 + 4B > 0 , then: ⎠ ⎝

qiOLNE (t ) = β (Xi (0) + Xic ) e 2 ⎝

α − cm + ψ

i i , j = 1,2, i ≠ j . That is, qi = ρXi = γ + ε = . Consequently, if 3 the constraint is binding, depending on value of ψi , the static Cournot game does not necessarily imply lower competitiveness than the dynamic one.

+ qiS, OLNE

A2 +4B ) t

1 (α − c + γXi + εXj + ψi ) 3

dψi = 0

⎛ δ + 2ρ ⎞ Proposition 4. If3ρ > γ ⎜ δ + ρ ⎟ + ε and C ≥ c m ,the unique symmetric ⎠ ⎝ OLNE outputs and experiences are given by: 1

(20)

will be These solutions imply that the requirement of C ≥ eventually violated and the model will no longer reflect the reality of non-negative marginal costs. In such a case, one needs to introduce a state constraint, C ≥ c m , into the model. When the constraint is binding, i.e., C = c m ,the co-states will have negative jump so that it is optimal to maintain constant output with a constant marginal cost. In this case the firms are forced into a steady state behavior, determined by:

i = 1,2 . Next differentiating (16) one more time we arrive at the following result.

qiOLNE (t ) = β (Xi (0) − XiS, OLNE ) e 2 (A −

, i.e.,

cm

to reach the steady state. We next characterize a closed-form solution of the equilibrium time path of the canonical system (13)-(14) that converges asymptotically to the steady state for any given initial states of experience. Differentiating the optimal control in (9) over an interval of time and accounting for (γ +ε )(qi −ρXi ) + (δ + ρ) ψi − γqi (7) and (3), we have q̇i = , which, when employing 3 (9) again, leads to:

[3(δ + ρ) + ε] qi − (δ + 2ρ)(γ +ε ) Xi − (δ + ρ)(α − c ) 3

3B1 (δ + ρ)2

B1 (δ + ρ)2

ρ (α − c ){(δ + ρ)2 [α − c + (γ + ε ) Xi (0)] + γ} G= [3ρ (δ + ρ)2 (α − c ) − γ (ε+3δ )]

Fig. 1. Phase-portrait diagram in the state-costate space.

qi̇ =

B1 (δ + ρ)2 [α − c +(γ + ε ) Xi (0)] + γ [A1 G (δ + ρ) + B1]

5. Nash equilibrium under mutually observable stocks of experience We now assume that the firms’ production experience is common knowledge throughout the game, i.e., each firm can observe its own production experience and that of the competitor at any time so that both firms make contingent production decisions throughout the planning horizon. Here also, we use the maximum principle, but now account for the fact that the firms’ policy depends on states X1 and X2 . This does not impact the optimality condition (9) and only changes the co-state

(17)

where qic = − qiS, OLNE >0andXic = − XiS, OLNE . That is, the output increasingly grows rather than tending to a steady state. Similar growth is ⎛ δ + 2ρ ⎞ observed when 3ρ = γ ⎜ δ + ρ ⎟ + ε , and (A4) takes the form: ⎠ ⎝ 56

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equations (7) to:

ψi̇ = (δ + ρ) ψi − γqi + j

j

∂qj ∂Xi

ψi̇ = (δ + ρ) ψi − εqi +

qi −

∂qj ∂Xj

∂qj ∂Xi

qi −

ψi j ,

∂qj ∂Xj

On the other hand, assuming that the steady state conditions of Proposition 1 hold and either a > δ + ρ < ε , or a < δ + ρ > ε , then the steady state conditions of Proposition 6 hold and:

(21a)

qiS, OLNE >qiS, FNE ,

ψi .

ciS, OLNE >ciS, FNE andqiCN

(21b)

Then, considering the linear strategies:

qi = r + aXi + bXj .

for a steady state, ψ̇i = 0 and Xi̇ = 0 , we obtain the following feedback Nash equilibrium (FNE). ρ ⎡ γ δ+ρ⎢ ⎣

+

b (δ + ρ − ε ) ⎤ a−δ−ρ ⎥ ⎦

i , j = 1, 2, and c S, FNE ≥c m there

ρ (a − δ − ρ)(δ + ρ)(α − c ) (a − δ − ρ)[(δ + ρ)(3ρ − γ − ε ) − ρ (γ − b )] − ρb (a − ε ) (23b)

ρ (α − c )[(γ − b )(a − δ − ρ) + b (a − ε )] . (a − δ − ρ)[(δ + ρ)(3ρ − γ − ε ) − ρ (γ − b )] − ρb (a − ε )

(24)

(26)

b (a − ε ) ⎤ andc S, FNE ≥c m . a − δ − ρ⎥ ⎦

(27)

ρ+δ=ε

b (a − ε )

of γ − b + a − δ − ρ . If it is negative, then the experience shadow price is negative and therefore the dynamic equilibrium output will be smaller than that of the static Cournot game This implies a converse case to that of Proposition 2.

−b+

i ≠ j . The Jacobian matrix for the Eq. (24) is symmetric:

Theorem 1. Let ρ > a + b and the steady state conditions of Propositions 1 and 6 hold. If ρ + δ < ε , then the OLNE steady state output is lower than the FNE steady state output, that is, qiS, OLNE ε , then qiS, OLNE >qiS, FNE . ■ Theorem 1 identifies a pivotal steady state condition of the dynamic Cournot competition. Specifically, given the discounting rate value, δ , when the forgetting rate, ρ , is fully offset (to the extent of the discounting rate δ ) by the experience spillover from the competitor, ε , i.e.,

Assume now that the conditions of Proposition 6 hold. Then, qiS, FNE >0 , XiS, FNE >0 , while the sign of ψiS, FNE is determined by the sign

ρ ⎡ γ δ+ρ⎢ ⎣

(25)

Based on Propositions 6 and 8 along with the stability requirements, we conclude:

Proof. See the Appendix. ■ Note that if the condition c S, FNE ≥c m does not hold, then as for the OLNE case with binding state constraint, the firms are forced to a c −c m steady state behavior with Xi = γ + ε . Therefore, the closed-loop production rates are:

Proposition 7. Let3ρ > γ + ε +

. C = c S, FNE

6. Comparison of equilibria (23c)

⎧ r + aXi + bXj , ifc − γXi − εXj >c m qi = ⎨ ρXi , ifc − γXi − εXj = c m ⎩

CN

The trace and determinant of (26) are Tr (J )=2(a − ρ) and Δ(J ) = (a−ρ − b )(a−ρ + b ), respectively. A globally asymptotically stable steady state requires that both Tr (J ) <0 and Δ(J ) >0 , which hold if ρ > a + b . Under this condition, given (Tr (J ))2 −4Δ(J )=4b 2 >0 , the convergence to the globally, asymptotically stable steady state is monotonic (Fig. 2).

(23a)

ψiS, FNE =

i

⎡ a−ρ b ⎤ J=⎢ ⎥ ⎣ b a−ρ ⎦

(a − δ − ρ)(δ + ρ)(α − c ) = (a − δ − ρ)[(δ + ρ)(3ρ − γ − ε ) − ρ (γ − b )] − ρb (a − ε )

qiS, FNE =

>q

Xi̇ (t ) = r + bXj + (a−ρ) Xi

exists a unique symmetric FNE steady state which is characterized by:

XiS, FNE

C = c S, FNE

We now study the stability of the feedback equilibrium. From Eqs. (3) and (22), we have:

(22)

Proposition 6. If3ρ > γ + ε +

XiS, OLNE >XiS, FNE ,

j

then the equilibrium output based on precommitment is identical to the equilibrium output that is contingent upon the state of experience, qiS, OLNE =qiS, FNE . The balance between the forgetting rate and the spillover rate determines if the firms will scale down their production when observing their state of experience, qiS, FNE < qiS, OLNE or scale it up, qiS, FNE > qiS, OLNE . For example, when the efficiency of a firm’s experience accumulation is low (the organizational forgetting is too high) compared to the experience spillover from the competitor,

If

b (a − ε ) , a−δ−ρ

the static Cournot equilibrium output based on the γ−b<− steady state marginal cost is greater than its steady state counterpart S , FNE b (a − ε ) ; otherwise, if then qiCN C = c S, FNE >q γ − b ≥ − a − δ − ρ, i S , FNE

.■ i Note that given that all conditions of Propositions 2 and 7 are S , FNE S , OLNE and qiCN C = c S, FNE q

qiCN

≤q C = c S, FNE

i

=0

i

=0

not imply qiS, OLNE >qiS, FNE because the steady state experience under open-loop equilibrium is generally different from that under the closed loop equilibrium and therefore the marginal costs at the two equilibria are different. Based on Propositions 1 and 6, we find that a−ε qiS, OLNE 1, which, when a − δ − ρ > 0 boils down to ρ + δ > ε and when a − δ − ρ < 0 , reduces to ρ + δ < ε . Recalling the steady state condition from Proposition 1, we conclude as follows.

|

Proposition 8. Assuming that the steady state conditions ofProposition 6 hold and either a > δ + ρ > ε , or a < δ + ρ < ε , then the steady state conditions of Proposition 1 hold and:

qiS, OLNE
|

XiS, OLNE
ciS, OLNE >ciS, FNE andqiCN

C = c S, OLNE


i

CN

. C = c S, FNE

Fig. 2. Phase-portrait diagram in the state space.

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i.e., δ + ρ > ε , the steady state FNE output declines, qiS, FNE < qiS, OLNE . Note that although both proprietary learning and spillovers encourage greater outputs, the rate of proprietary learning, γ, does not explicitly affect the pivotal condition (27). The proprietary learning impact, however, is observed through the steady state condition of Proposition 1. Indeed, a high value of γ may not leave much space to ensure the ⎛ δ + 2ρ ⎞ steady state condition, so that 3ρ > γ ⎜ δ + ρ ⎟ + ε > 3ε . As a result, only ⎠ ⎝ δ + ρ > ε will hold. That is, when the proprietary learning rate is high, then either qiS, FNE < qiS, OLNE holds or the firms do not reach the steady state. An extreme case of such an effect is found when there is no spillover at all, as stated below.

R = ρ − b −a > 0 . Fig. 3(c)-(d) show that the firms’ instantaneous steady state profits and marginal costs intersect at the same pivotal point of ε = 0.51. An important observation here is that the firms, by scaling down (before the pivotal point) and up (after that point) their production in contingent equilibrium compared to the precommitment one (respectively keeping higher/lower marginal costs) always achieve higher profit. In other words, production adjustment based on current state of experience by either downplaying or aggravating the competition ensures higher profits compared to the myopic precommitment. Fig. 4 presents the equilibrium outputs for a common case of higher proprietary learning (up to γ = 0.47) and lower spillovers ε = 0.3. Accordingly, unlike Fig. 3 that illustrates the effect of ε and consequently the pivotal condition, Fig. 4 illustrates the effect of γ on the equilibrium of which the pivotal condition is independent. Clearly, this is a case where the efficiency of experience accumulation is low (i.e., organizational forgetting is high) compared to the experience spillover from the competitor δ + ρ = 0.51 > ε = 0.3, which entails qiS, FNE < qiS, OLNE . Notably, the greater the proprietary learning rate, the stronger the adjustment of the output in contingent equilibrium and the greater the profit advantage under this equilibrium. The detrimental effect of organizational forgetting on steady state outputs and steady state profits is shown in Fig. 5. The effect of spillover rate values on the profits accumulated over a hundred time units is shown in Fig. 6. The result confirms that an FNE is more profitable than an OLNE. As for the steady state profits in Fig. 3(c), the difference between FNE and OLNE cumulative profits increases with the spillover rate, most notably when the pivotal condition δ + ρ < ε prevails. In order to compare the OLNE and FNE from a social perspective, we assess the social welfare, which is computed as: Social welfare (SW ) = Joint profits (J )+ Consumer surplus (CS ), where the Consumer surplus CS at time t is

Corollary 2. Let ρ > a + b and the steady state conditions of Propositions 1 and 6 hold. If there are no spillovers between the competitors, ε = 0, then:

qiCN

C = c S, FNE

< qiCN

C = c S, OLNE


S , OLNE

i

>qiS, FNE .■

Clearly, the greater the spillover, the stronger is the likelihood of switching in (27), from scaling down the production to scaling up in the contingent equilibrium relative to the open-loop one. To verify the results of Theorem 1, derived for unknown values of coefficients a and b of the affine feedback rule (22), we construct the Hamiltonian-JacobiBellman equations:

δVi = Max qi (α − Q − c + γXi + εXj ) qi +

∂Vi ∂V (qi − ρXi ) + i (qj − ρXj ) ∂Xi ∂Xj (28)

where Vi is firm i ’s value function, i , j = 1, 2, i ≠ j . Then, the ∂Vj ∂V equilibrium outputs for symmetric conditions, ∂X = ∂Xi , are: j

i

1⎛ ∂V ⎞ qi = ⎜α − c + γXi + εXj + i ⎟ , 3⎝ ∂Xi ⎠

(29a)

⎤ 1⎡ ⎛ ∂V ⎞ Q = ⎢2 ⎜α − c + i ⎟ + (γ + ε )(Xi + Xj ) ⎥ , 3⎣ ⎝ ∂Xi ⎠ ⎦ i , j = 1, 2,

α

CS (t ) = 2

1⎛ ∂V ⎞ ⎡ ∂V ⎤ ⎜α − c + γXi + εXj + i ⎟ ⎢α − c + (2γ − ε ) Xi − (γ − 2ε ) Xj + i ⎥ 9⎝ ∂Xi ⎠ ⎣ ∂Xi ⎦ 1 ∂Vi ⎡ ∂Vi ⎤ ∂Vi + = 0. ⎥ − ρXi ⎢α − c + εXi − (3ρ − γ ) Xj + 3 ∂Xj ⎣ ∂Xi ⎦ ∂Xi (30) Next, in line with (22), we substitute the value function:

1 (2e + γ ), 3

b=

1 [α − (α − 2q (t ))]2q (t ) = 2(q (t ))2 2

(31) 7. Conclusions

into (30), so that:

a=

q (u ) du =

In Fig. 7, the steady state consumer surplus (a) and social welfare (b) are depicted for different values of the spillover rate. Clearly, the contingent strategy is more socially desirable when the spillover rate is greater than the organizational forgetting. That is, an FNE strategy is more beneficial to both consumers and firms if organizational forgetting is low and experience spillover is high. In the converse case of large organizational forgetting and relatively small experience spillover, firms, in order to maintain higher profits, are still better off adopting a contingent strategy to restrain experience accumulation even though it is detrimental to customers.

(29b)

i ≠ j . Substituting (29) into (28) leads to:

Vi = k + fXi + hXj + eXi2 + dXj2 + gXi Xj

∫α−2q (t )

1 (g + ε ), 3

r=

1 (α − c + f ) 3

Learning and forgetting in a Cournot-type competition creates a dynamics that affects the marginal production costs of firms, their outputs, the market price of their products and the dynamics of accumulating experience. These effects are further influenced by experience spillovers and by the level of information available to the firms. In this paper, we characterize the steady state, stable behavior of competing firms and find that ignoring the dynamics of learning and forgetting may induce much lower competitiveness between firms even if their myopic outputs are determined under steady state marginal costs. We show that both greater proprietary learning and spillovers are pro-competitive, while organizational forgetting limits competitiveness. In addition, under relatively high proprietary learning, a contingent strategy is always less competitive than a precommitmentbased strategy. However, when the proprietary learning is relatively low with respect to the spillover and forgetting rates, different results emerge as a consequence of a simple pivotal steady state condition derived for

(32)

and obtain a system of six nonlinear algebraic equations in six unknowns, k , f , h , e, d andg. The system, though not tractable analytically, is computationally solvable. We illustrate our numerical analysis with Figs. 3–4 for steady state conditions with ρ = 0.5, α = 50 , c = 40 and δ = 0.01. Considering the case of γ = 0.1 and ε running from 0.35 up to 0.65, from Fig. 3, we observe the effect of the pivotal condition (27), δ + ρ = ε , which holds for our data at ε = 0.51. At this point of proprietary learning rate, two steady state output curves intersect, i.e., qiS, FNE = qiS, OLNE (see Fig. 3(a)). Since the outputs under both types of equilibria are equal, the corresponding static Cournot-Nash equilibria (CF – static game equilibrium under feedback marginal cost and COL static game equilibrium under open-loop marginal cost) intersect as well and they are below the dynamic equilibria. That is, static considerations reduce the firms’ competitiveness. Fig. 3(b) verifies the stability condition ρ > a + b of Theorem 1 in the form of 58

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K. Kogan et al.

Fig. 3. Experience spillover and steady state equilibrium in OLNE and FNE. a) Steady state outputs b) Stability condition c) Steady state profit d) Marginal cost.

is zero. Consequently, organizational forgetting is pro-competitive when spillover opportunities are low (and therefore free riding is limited) and the players’ experience levels are not mutually observable. If the organizational forgetting is small and the spillover rate is large, then experience accumulation and therefore marginal cost reduction are very efficient which is verified if the state of experience is observable. Accordingly, in such cases, a contingent strategy is more competitive than myopic precommitment. These results are summarized in Table 2. Note that more competitive strategies (i.e., those that

pro-competitive strategies. Given a discounting rate value, when forgetting is fully offset by experience spillover from the competitor, the equilibrium output based on precommitment is identical to that contingent on the state of experience. The contingent behavior, however, is naturally more sensitive to any change in operating conditions. As a result, if the forgetting rate augmented by the discounting rate is sufficiently large (larger than the spillover rate value), then precommitment is more competitive than contingent strategy. Notably, this condition is always fulfilled if the spillover rate

Fig. 4. Proprietary learning and steady state output and profits in OLNE and FNE. a) Steady state outputs b) Steady state profits.

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K. Kogan et al.

Fig. 5. Organizational forgetting and steady state output and profits in OLNE and FNE. a) Steady state outputs b) Steady state profits. Table 2 Pro-competitive strategies. Patient players (low discounting rate)

Impatient players (high discounting rate)

No spillover opportunities

Open-loop strategy

Open-loop strategy

High spillover opportunities

Closed-loop strategy if δ + ρ < ε , otherwise Openloop strategy Closed-loop strategy

Open-loop strategy

High forgetting rate Low forgetting rate

Fig. 6. Experience spillover and cumulative profits.

Open-loop strategy

depending on the magnitude of organizational forgetting, spillover opportunities and players’ level of patience as shown in Table 2) may not be detrimental from a profit perspective. When there are no spillover opportunities, pro-competitive strategies result both in lower profits and consumer surpluses. In general, firms are better off in the long run by sharing with each other their state of experience especially if profit is their only strategic goal. This intuitive outcome, however,

involve a greater production rate) do not necessarily lead to higher profits that depend on the information available and are therefore ensured by FNE. We find that when the players are patient, long run competitiveness (which implies hiding or sharing information throughout the game,

Fig. 7. Consumer surplus and social welfare for different spillover rates. a) Consumer surplus b) Social welfare.

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can affect a firm’s competitiveness since greater profits can commonly be gained with a lower production level (thereby a higher product price) under a Cournot-Nash competition. To improve competitiveness, management should promote learning and be more careful about every result obtained with production experience. That is, management needs to work on reducing the detrimental role of firm forgetfulness and cultivating the firm’s proprietary learning rates to regain its competiveness. On the other hand, if either sharing the states of experience between firms is impossible or the firms are impatient, management needs to focus on more efficient use of the competitor’s

experience via spillovers to offset information incompleteness and to improve both profits and competiveness. Finally, we considered a number of limiting assumptions including constant deterministic rates for forgetting and learning. Relaxing these assumptions suggests an important direction for future research. Furthermore, we restricted our study to symmetric firms and equilibria. Therefore the current research can be viewed as a first step in studying a complex problem. Assuming non-symmetric parameters of the competing firms is, thus, an important and yet challenging direction for further research.

Appendix Proof of Proposition 1. To find a steady state, we set Xi̇ =0 , and thereby qi=ρXi , i=1,2 , and substitute non-zero qi from (9) into (3):

α − c +(γ + ε ) Xi + ψi = ρXi 3

(A1)

i = 1,2 . Next differentiating (A1) over an interval of time, we find

ψ̇i 3

= 0 , i = 1,2 , i.e.,

ψi̇ = 0andqi̇ = 0

(A2)

From (7), we get ψi̇ =(δ + ρ) ψi − γqi=0 and ψi =

γqi , δ+ρ

which leads us to rewrite (A1) as:

(δ + ρ)[α − c + Xi (γ + ε )] + γρXi = ρXi 3(δ + ρ)

(A3)

i = 1, 2 . Thus, the OLNE steady state experience is XiS, OLNE (3δ −2γ − ε )2 +12δ (γ + ε ) − (3δ −2γ − ε )

ρ > ρ∼ =

6

⎛ δ +2ρ ⎞ (α − c )(δ + ρ) = (δ + ρ)(3ρ − γ − ε) − γρ , i = 1, 2 , which is feasible if 3ρ > γ ⎜ δ + ρ ⎟ +ε , or equivalently, ⎠ ⎝

, where ρ∼ is a threshold for experience depreciation.

Proof of Proposition 4. Differentiating (16) and substituting Xi=−

3qi̇ − [3(δ + ρ) +ε] qi + (δ + ρ)(α − c ) , (δ + 2ρ)(γ + ε )

we have:

⎡ ργ ⎤ 3q ï = (ε + 3δ ) q i̇ + (δ + ρ) ⎢3ρ − γ − ε − ⎥ q −ρ (δ + ρ)(α −c ) ⎣ δ + ρ⎦ i

(A4)

i = 1, 2 . The solution to (A4) with respect to the boundary condition given by the steady state is qiOLNE = N1 e ⎛ ργ ⎞ 1 1 i = 1, 2 , where B = 3 (δ + ρ) ⎜3ρ − γ − ε − δ + ρ ⎟, A = 3 ε + δ and N1 and N2 are the integration constants. ⎝ ⎠ To find the integration constants,

⎞ 1⎛ ⎜ A2 +4B + A⎟ t 2⎝ ⎠

+N2 e

⎛ ⎞ − 1 ⎜ A2 +4B − A⎟ t 2⎝ ⎠

+qiS, OLNE ,

N1 + N2 = qiOLNE (0)−qiS, OLNE we first solve equation (7) which, when accounting for the transversality conditions, implies N1 = 0 , N2 = N , and: 1⎛ ⎜A −

ψi =

2Nγe 2 ⎝

⎞ A2 +4B ⎟ t ⎠

2(δ + ρ) − A +

A2

Then,

from

γqiS, OLNE

+

δ+ρ

+ 4B

ψi (0) =

1⎡ = 3 ⎢α − c + (γ + ε ) Xi (0) + ⎣ N = β (Xi (0)−XiS, OLNE )

(A5)

2Kγ 2(δ + ρ) −A + A2 +4B 2Kγ

2(δ + ρ) −A + A2 +4B

+

+

γqiS, OLNE δ+ρ

and

qiOLNE (0) =

α − c + γXi (0) + εXj (0) + ψi (0) 3

,

we

have:

N = qiOLNE (0)−qiS, OLNE

γqiS, OLNE ⎤ δ+ρ

S , OLNE . That is, ⎥ − qi ⎦

(A6)

where:

A2 + 4B ]

(γ + ε )[2(δ + ρ) − A +

β=

3[2(δ + ρ) − A +

A2

(A7)

+ 4B ]−2γ

Proof of Proposition 6. At the steady state, we have ψ̇i = 0 and Xi̇ = 0 which then leads to:

(δ + ρ) ψi − γqi +

(δ + ρ) ψi j − εqi +

∂qj ∂Xi

qi −

∂qj ∂Xj

qi −

∂qj ∂Xi

ψi j = 0,

∂qj ∂Xj

(A8a)

ψi j = 0.

(A8b)

Since the problem we study is linear-quadratic, we consider the linear strategies (22). Then, from (A8) and Xi̇ = qi −ρXi=0 , we have

ψi =

ρ ⎡ b (a − ε ) ⎤ ⎢γ − b + ⎥ Xi , δ + ρ⎣ a − δ − ρ⎦

(A9) 61

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K. Kogan et al.

which, with respect to (9), results in:

qi =

ρb (δ + ρ − ε ) ⎤ ⎫ 1⎧ 1 ⎡ ⎨α − c + εXj + ⎢γ (δ + 2ρ) + ⎥ Xi ⎬ = ρXi . 3⎩ δ + ρ⎣ a−δ−ρ ⎦ ⎭

(A10)

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