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Dynamic product innovation and production decisions under quality authorization
Accepted Manuscript Dynamic product innovation and production decisions under quality authorization Zhi Li, Jian Ni PII: DOI: Reference:
S0360-8352(17)30577-6 https://doi.org/10.1016/j.cie.2017.12.011 CAIE 5015
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
6 July 2016 5 December 2017 9 December 2017
Please cite this article as: Li, Z., Ni, J., Dynamic product innovation and production decisions under quality authorization, Computers & Industrial Engineering (2017), doi: https://doi.org/10.1016/j.cie.2017.12.011
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Dynamic product innovation and production decisions under quality authorization
Zhi Li Tel.: +86 20 39322212 E-mail: [email protected] Guangdong Provencal Key Lab of Computer Integrated Manufacturing, Guangdong University of Technology Guangzhou 510006, China
** Jian Ni Corresponding author Tel.: +86 28 87352835 Fax: +86 28 87092129 E-mail: [email protected] School of Finance, Southwestern University of Finance and Economics Chengdu 611130, China
1
Dynamic product innovation and production decisions under quality authorization
Abstract: We consider the joint product innovation and production decisions of a manufacturing firm under the existence of quality authorization from a 3rd party, which has gained growing popularity in recent years. A dynamic control model is developed to analyse the effects of quality authorization. Combining the techniques of Pontryagin maximum principle and backward induction, the optimal decisions on production and investment in product innovation before and after attaining the quality authorization is analysed. The analytical solution of the optimal investment and production decisions is derived providing that the time for the firm to obtain the quality authorization is known. To fully solve the firm’s optimization problem, an iterative algorithm is then introduced to calculate the best time of attaining the quality authorization. We find that although the firm should have a continuous and incremental improvements in product quality, there can be jumps in the optimal production and investment levels. While the investment in product innovation is higher before obtaining the quality authorization than that after obtaining the quality authorization, the production is lower before obtaining the quality authorization. Moreover, the firm should attain the quality authorization sooner under a less costly the product innovation investment, a lower depreciation rate due to ageing of technology, a smaller production cost, and a lower interest rate. Finally, we compare our results with existing studies and discuss the managerial implications. Keywords: product innovation, production decision, quality authorization, optimal control
1. Introduction Innovation is crucial for the sustainable growth of firms, it can dynamically boost the profitability of firms’ products. For instance, an upgraded computer software that calculates more rapidly or synchronizes data better can help an IT firm to attract more user, gain larger market share, and harvest higher profit. From an economic perspective (Saha 2007), the upgrading of the software can be perceived as product innovation – a type of innovation activities that aim to make the product better. In fact, the economic literature on innovation distinguishes between two basic types of innovation, namely, product innovation and process innovation (Chenavaz 2012). Product innovation is known as an activity that aims to improve the cumulative product quality and make better products (Lambertini and Mantovani 2009, Pan and Li 2016), while process innovation is understood as an effort to reduce the production cost (Lambertini and Orsini 2015, Li and Ni 2016). In this 2
paper, we develop a dynamic control model of product innovation under the existence of quality authorization from a 3rd party. The optimal decisions of the firm’s production quantity and investment in product innovation are analyzed. The objective is to investigate the effects of the existence of the quality authorization on the production and investment choices of the firm. The scheme of quality authorization has gained popularity over the past few years. One typical example is the Q-mark scheme launched in Hong Kong, which audits and endorses products and services for firms with production plants in Hong Kong, the mainland China, as well as Macau. The Q-mark scheme grants quality authorization to those firms with an effective production and quality management system capable of producing consistent quality products. It can be applied by any firms on a voluntary basis. Firms successfully passing the quality audit from the Q-mark council will be entitled to place the Q-Mark labels on their products, which offer consumers the confidence when making purchases. Another recent example is the quality authorization scheme under the China-Korea free trade agreement (CKA), which also provides the guarantee of high quality to consumers buying imported goods. In fact, there are several benefits for firms attaining the quality authorization. First, consumers these days are better educated, affluent and concerned about the product quality. To maintain a healthy lifestyle, they welcome products and services that have guaranteed quality. Second, many insurance firms such as Pacific Insurance Co., Ltd. will provide product quality guarantee insurance only to firms obtaining the quality authorization. Finally, in international trades between China and Korea, the time required for customs clearance in exporting products is much less (2-3 days versus 7-14 days) for firms attaining the quality authorization according to the CKA rule. The quality authorization scheme should affect the firms’ investment behaviour in product innovation that improves product quality (e.g., Lambertini and Orsini 2015, Pan and Li 2016). To address the growing popularity of quality authorization nowadays, we develop a dynamic model of the firm’s product innovation and production decisions under the voluntary quality authorization scheme. The purpose is to analyze the firm’s investment behavior in product innovation given the quality authorization. The optimal production and investment choices of the firm before and after attaining the authorization is examined. Because the value function of the firm obtained after the quality authorization is an input of the dynamic control problem before the quality authorization, the firm’s optimization problem is analyzed by the synergic use of backward induction and Pontryagin maximum principle, as being analogous to the approach taken by Li (2013). Once the time of obtaining the quality authorization is given, both the optimal production and investment decisions can 3
then be expressed in analytical forms. To obtain the optimal decisions of the firm, an iterative algorithm is introduced to identify the best time of attaining the quality authorization. We then analyze the model dynamics through numerical means and report several new findings, for instance, the firm should have a continuous improvements in product quality but may have a discontinuous process in product innovation investment. The rest of the paper is organized as follows. Section 2 provides a review of the related literature. Section 3 introduces the basic framework of the dynamic model. Section 4 then analyses the dynamic model using backward induction. The numerical results are presented in section 5. Section 6 concludes the paper.
2. Literature review Product innovation which aims to make the product better has become a fundamental source of firm growth (Chenavaz 2012), and the related literature is rich. Spence (1975) and Mussa and Rosen (1978) firstly discussed the role of product quality in affecting the optimal monopolistic strategy. Since then, several papers have discussed the impact of monopoly power on product quality under static settings (e.g., Besanko et al. 1987, Lambertini 2006). Recently, Lambertini and Orsini (2015) further extended the analysis to a dynamic control model of product and process innovation. They found that the monopolistic power depresses investment in product innovation. Another line of research then concerns the relationship between product and process innovation, where the process innovation refers to the activity that aims to reduce the production costs. For example, Lin and Saggi (2002) investigated the relationship between process and product innovation in a three stage model. Mantovani (2006) further proposed a dynamic model to study complementarity between market-enhancing product innovation and cost-reducing process innovation in a monopoly setting. Lambertini and Mantovani (2009) considered a dynamic control model to study the dynamic behavior of a multiproduct monopolist investing both in process innovation and in product innovation. Chenavaz (2012) developed a product and process innovation model to determine the optimal product price, the product and process innovation investment strategy, given a time-varying demand conditions. In a recent work, Pan and Li (2016) extended the model of Chenavaz (2012) to a model where the firm’s cost functions of product and process innovation depend on both the innovation investments and the knowledge accumulations of product and process innovation. Further, Zhang et al. (2015) proposed a research model to study the joint effects of social capital on mass customization capability and product innovation capability. Choi et al. (2016) considered the strategic innovation policies and suggested that a new paradigm wherein product and process innovations are pursued concurrently instead of sequentially can guarantee a firm’s sustainable growth. Guimarães et al. (2016) 4
conducted a survey on the performance of spending organizational resources on produce innovation in Brazilian furniture firms. This paper is also related to the broad literature on firms’ dynamic R&D investment activities. Minniti (2010) studied the R&D composition by using a model where the horizontal and vertical innovations simultaneously take place; through the model analysis, the relationship between the product market competition and the leading-edge growth is also investigated. Matsumura et al. (2013) showed that R&D investment will not always be a decreasing function of the intensity of market competition in the absence of spillover. Chrystie et al. (2013) introduced a standard model of strategic R&D with spillovers in R&D inputs, and found that duopoly firms engaged in a standard two-stage game of R&D and Cournot competition will end up in a prisoner’s dilemma situation for their R&D decisions, whenever spillover effects and R&D costs are relatively low. In a recent contribution, EI Ouardighi and Tapiero (2014) analyzed the firms’ R&D with stock-dependent spillover, under the assumption that a firm’s R&D is production-cost-reducing and can benefit from part of the competitor’s R&D stock without payment. Later, Shibata (2014) investigated the R&D investment strategies for firms under R&D spillover in various market structures, while Xing (2014) studied the optimal choices of R&D risk under network externalities by analyzing a Hotelling spatial model. Addessi et al. (2014) then conducted both theoretical and empirical studies to investigate the impacts of R&D activities on the use of external flexibility. This paper analyses the dynamic investment policy of product innovation for firms under quality authorization by analyzing a dynamic control model and the associated dynamic state and costate variables with the well-known Pontryagin maximum principle. This type of dynamic models, which involve both the state and costate variables, has been widely applied in the literature to study the dynamic decision-making of firms. For instance, Cellini and Lambertini (2002) developed a dynamic model to analyze firms’ product differentiation strategies, with the level of product differentiation serving as a state variable. Lambertini and Zaccour (2014) alternatively introduced a dynamic model on the competition of firms with advertising. The authors treated the goodwill resulting from advertising as a state variable and found that the aggregate expenditure on goodwill takes a U-shape. In another contribution, Jørgensen and Zaccour (2014) provided a review on the studies using dynamic game-theoretic models of cooperative advertising. Then, Martín-Herrán et al. (2011) examined the effectiveness of franchise systems with a dynamic game-theoretic model on the strategic interactions between franchisors and franchisees, whereas Martín-Herrán and Taboubi (2015) investigated the price coordination 5
among supply chain members in a dynamic perspective. Recently, Yang et al. (2015) introduced a dynamic control model to study the dynamic trade credit and preservation technology allocation for deteriorating products, with the inventory of the products treated as a state variable. Zhang et al. (2016) further developed a dynamic model that can accommodate common resource constraints for analyzing the pricing, service and preservation technology investments policy of deteriorating items. This paper has several key features that are distinct from the existing literature. First, we develop a new dynamic control model of production decision and product innovation that can incorporate the effects of the existence of quality authorization. To our knowledge, this paper is the first research that addresses the effects of quality authorization on the production and investment decisions of the firm. Since the model is new, we develop an iterative algorithm to optimize the planned time for the firm to attain the quality authorization, with which the optimal production and investment decisions of the firm can be obtained. Second, we show that although the firm should have a continuous and incremental improvements in product quality, there can be jumps in the optimal production and investment levels over time; in particular, there is a need to substantially accelerate the product innovation investment before quality authorization. As compared with existing studies (e.g., Lambertini and Mantovani 2009, Hasnas et al. 2014, Pan and Li 2016) which support the smoothing of investment paths for innovation, our results instead suggest that it is not always optimal to smooth the product innovation path. Third, we find that the firm should attain the quality authorization sooner when the product innovation investment is less costly, the unit production cost is smaller, the depreciation rate due to ageing of technology is lower, the interest rate is lower, and/or product price is more sensitive to product quality. We believe that these results can complement the existing literature on product innovation and help advance our understanding on the dynamics of product innovation investment.
3. The basic framework We consider a dynamic control problem over continuous time t [0, ) , where at any instant a firm needs to make both the production decision and the decision of investment in product innovation. Product innovation can be understood as an improvement of the product quality (Chenavaz 2012). Use x(t ) to denote the product quality of the firm. There is a voluntary quality authorization scheme that the firm can apply. If the firm succeeds in passing the quality audit and attaining the quality authorization, the product’s profitability will be improved, either from the better public perception of the product quality (e.g., the Q-mark case) or from the supports from governments and insurance firms (e.g., the CKA case). However, the firm can attain the quality 6
authorization only when x(t ) reaches a required quality level x . At time t , the firm can invest in product innovation (i.e., quality-enhancing technology) via the instantaneous investment k (t ) to increase x(t ) . With the investment, the differential equation which describes the evolution of x(t ) over time is given by the following form:
x(t ) k (t ) x(t ) where parameter
(1)
0 is the depreciation rate due to ageing of technology. According to Lambertini and
Orsini (2015), the instantaneous cost function of investing in product innovation is C (k (t ))
1 2 ak (t ) , 2
where a is a positive parameter. The convex cost function implies increasing marginal costs and decreasing returns to innovative activity. The parameter a is the inverse measure of product innovation investment efficiency. The firm produces a single item, and consumers vary in their willingness to pay and their concern for product quality. Let q(t ) be the production quantity while c be the unit production cost. The firm’s production cost function is cq(t ) . Following Lambertini and Orsini (2015), we assume that there is no stock, all the demand is satisfied, and all the production is sold. According to Chenavaz (2012) and Pan and Li (2016), we have the following inverse demand function:
p(t ) A q(t ) x(t )
(2)
Now consider the firm’s profit, which can be shifted by the quality authorization. Then, the firm’s instantaneous profit before attaining the quality authorization ( x(t ) x ) is given by
1 2
(t ) [ A c q(t ) x(t )]q(t ) ak 2 (t )
(3)
After attaining the quality authorization ( x(t ) x ), the profitability of the product is boosted ( AH A ), leading to the following instantaneous profit of the firm:
1 2
(t ) [ Ah c q(t ) x(t )]q(t ) ak 2 (t )
(4)
Then, the firm’s optimization problem is the following:
7
T 1 max J (T , q(t ), k (t )) e rt {[ A c q(t ) x(t )]q(t ) ak 2 (t )}dt e rTV1 ( x ) 0 {T , q ( t ), k ( t )} 2 x(t ) k (t ) x(t ) S .t. x(0) x0 , x(T ) x
(5)
where V1 ( x1 ) is the discounted stream of the firm’s profits after obtaining the quality authorization, which is obtained by solving the following optimization problem:
1 e rt {[ Ah c q(t ) x(t )]q(t ) ak 2 (t )}dt q ( t ), k ( t ) 0 2 x(t ) k (t ) x(t ) S .t. x(0) x1
V1 ( x1 ) max
(6)
Next section, we will investigate the optimal investment strategy before and after attaining the quality authorization.
4. Model and analysis The dynamic model is analysed using backward induction. The first step is to analyse the firm’s optimization problem after attaining the quality authorization. The second step is to analyse the optimization problem before attaining the quality authorization. Then, we investigate the effect of the quality authorization by comparing the firm’s investment behaviour before or after attaining the quality authorization. 4.1. The optimization problem after attaining the quality authorization After attaining the quality authorization ( x1 x ), the firm’s objective is to determine the optimal investment level k (t ) and optimal production quantity q(t ) such that the discounted stream of net revenues is maximized, which can be formulated as follows:
1 e rt {[ Ah c q(t ) x(t )]q(t ) ak 2 (t )}dt 2 x(t ) k (t ) x(t ) S .t. x(0) x1
V1 ( x1 ) max
q ( t ), k ( t ) 0
(7)
The corresponding current value Hamiltonian function writes
1 H1 [ Ah c q(t ) x(t )]q(t ) ak 2 (t ) 1 (t )[k (t ) x(t )] 2
(8)
where 1 (t ) is the dynamic costate variable associated with the state x(t ) after attaining the quality authorization. 8
From the Hamiltonian, the first-order conditions are
H1 ak (t ) 1 (t ) 0 1 (t ) ak (t ) k (t )
(9)
H1 1 A c 2 q(t ) x(t ) 0 q(t ) [ Ah c x(t )] q(t ) 2
(10)
The associated costate equation for 1 (t ) is the following
1 (t ) r1 (t )
H1 1 (t ) (r )1 (t ) q(t ) x(t )
with transversality condition lim 1 (t )e t
rt
(11)
0.
Substituting expressions (9) and (10) into equation (11) gives the following differential equation, which describes the dynamic investment level k (t ) :
2 x(t ) ( Ah c) 2a 2a
(12)
From equation (12), one can immediately obtain that
k (t ) 2 k (t ) 0 and (r ) 0 , which x(t ) 2a k (t )
k (t ) (r )k (t )
imply that after attaining the quality authorization, the rate at which the investment in product innovation increases is higher for a relatively lower quality and/or a higher investment. Moreover, one can combine the differential equations (12) and (1) to solve for k (t ) and x(t ) after attaining the quality authorization. Use a superscript “*” to denote “optimal solution” while a subscript “a” to denote “after attaining the quality authorization”. We have the following proposition.
Proposition 1. When
(r ) 2 / (2a ) , the dynamic system admits an unique stable state ( x* , k * )
after
the
k*
attaining
quality
authorization,
where
x*
( Ah c) 2a (r ) 2
,
and
1 [ ( Ah c) (2a1 2 ) x* ] . Moreover, the optimal investment strategy ka* (t ) 2a (r 1 ) *
in production innovation as well as the associated quality state xa (t ) over time are as follows:
9
xa* (t ) x* ( x1 x* )e1t * 2 ( x1 x* ) 1t * ka (t ) k 2a (r ) e 1
(13)
r r 2 4[ (r ) 2 / (2a )] where 1 0. 2
Proof: First, from differential equations (1) and (12), the steady state must solve the following system of equations:
k (t ) x(t ) 0 2 ( r ) k ( t ) x(t ) ( Ah c) 0 2a 2a *
*
*
*
which admits a unique solution ( x , k ) . Next, we examine the stability of ( x , k ) . The Jacobian matrix for differential equations (1) and (12) is:
x x J k x
x k 2 k 2a k
1 r
Because the trace is Tr ( J ) r 0 , a necessary and sufficient condition for ( x , k ) to be stable (in a *
saddle
point
sense)
is
that
the
determinant
is
negative:
*
| J | (r )
2 0 , i.e., 2a
(r ) 2 / (2a ) . Next, we move on to solve for the optimal investment strategy. One can verify that the two eigenvalues of the Jacobian matrix J are:
1
r r 2 4[ (r ) 2 / (2a )] r r 2 4[ (r ) 2 / (2a )] 0 and 2 0 2 2 t
2t
Hence, one can write down the general solution for k (t ) as follows: k (t ) C0 C1e 1 C2e
C1 ,
C2
are
three
constants
to
1 (t ) ak (t ) aC0 aC1e t aC2e t . 1
2
be
determined.
Since
2 10
Then
from
equation
(9),
, where C0 , we
have
r r 2 4[ (r ) 2 / (2a )] r 2
and
1 0 , from the transversality condition lim 1 (t )e rt lim aC2e(2 r )t 0 . Thus, we must have C2 0 . t
t
Now we write down the general solution of x(t ) as follows: x(t ) B0 B1e1t B2e2t , where B0 , B1 ,
B2 are three constants to be determined.. From equation (1), it is easy to see that B2 0 given that C2 0 . Substituting the expressions of the general solutions k (t ) C0 C1e1t and x(t ) B0 B1e1t into (1) and (12), invoking the initial condition x(0) x1 , and simplifying, we get the following system of linear equations for C0 , C1 , B0 , and B1 :
C0 B0 0 C ( ) B 0 1 1 1 2 B0 ( Ah c) (r )C0 2a 2a B0 B1 x1
2 ( x1 x* ) from which we can solve that B0 x , B1 x1 x , C0 k , and C1 . Thus, the 2a (r 1 ) *
*
*
optimal investment strategy after attaining the quality authorization follows from expression (13). End of proof. From Proposition 1, the optimal production decision over time can be calculated from expression (10):
qa* (t )
1 [( Ah c x* ) ( x1 x* )e1t ] . Thus, the steady state production quantity after attaining the 2
* quality authorization is q
relationship
holds
between
Ah c x* . Moreover, one can easily verify that the following linear 2 xa* (t )
and
ka* (t )
,
that
is,
ka* (t ) xa* (t ) ,
where
( Ah c) 2a1 x* 2 and . We can exploit this linear relationship to derive the 2a (r 1 ) 2a (r 1 )
firm’s value function V1 ( x) after attaining the quality authorization. To this end, we need to formulate the Hamilton–Jacobi–Bellman (HJB) equation for the control problem (7):
V ( x) 1 rV1 ( x) ( Ah c q x)q ak 2 (k x) 1 2 x Substituting (10) and (13) into (14), we get 11
(14)
rV1 ( x)
V ( x) 1 1 ( Ah c x)2 a( x)2 [ ( ) x] 1 4 2 x
(15)
According to Dawid et al. (2015), one can guess that V1 ( x) f 0 f1 x f 2 x 2 . Substituting this expression into equation (15) then yields
r ( f 0 f1 x f 2 x 2 )
1 1 ( Ah c x)2 a( x)2 [ ( ) x]( f1 2 f 2 x) 4 2
(16)
from which we can obtain an expression of V1 ( x) , which is stated in Proposition 2.
Proposition 2. The firm’s value function after attaining the quality authorization is V1 ( x) f 0 f1 x f 2 x 2 , where the coefficients f 0 , f1 , and f 2 are
f0
a 2 ( Ah c)2 ( Ah c) 2a 2 2 2 ( 2 2a 2 ) 2r 4r 2r (r ) 4r (r )(r 2 2 )
f1
( Ah c) 2a 2 ( 2 2a 2 ) 2 (r ) 4 (r )(r 2 2 )
f2
2 2a 2 4 (r 2 2 )
Proof: From (16), the following equations for f 0 , f1 , and f 2 hold:
( Ah c) 2 2a 2 rf f 0 1 0 4 ( Ah c) a 0 (r ) f1 2 f 2 2 2 2a 2 ( r 2 2 ) f 0 2 4 from which we can solve for the expressions of f 0 , f1 , and f 2 , as stated in Proposition 2. End of proof. 4.2. The optimization problem before attaining the quality authorization Next, we turn to investigate the firm’s optimization problem before attaining the quality authorization. To proceed, we firstly suppose that the firm obtains the quality authorization at time T , then for t [0, T ) , the 12
dynamic control problem now writes
1 e rt {[ A c q(t ) x(t )]q(t ) ak 2 (t )}dt e rTV1 ( x ) q ( t ), k ( t ) 0 2 x(t ) k (t ) x(t ) S .t. x(0) x0 , x(T ) x
V0 (T ) max
T
(17)
Note that the dynamic control problem (17) is similar to (7) except for two key differences: (i) the profitability of the product is lower before attaining the quality authorization ( A Ah ); (ii) there is a terminal value
e rTV1 ( x ) which represents the discounted streams of profits after attaining the quality authorization at time T ; (iii) there is a terminal state condition x(T ) x , meaning that the firm indeed obtains the quality authorization at time T . The importance of dynamic control problem (17) is that it is a simplified version of the optimization problem (5), that is, V0 (T ) max J (T , q(t ), k (t )) . q ( t ), k ( t )
The corresponding current value Hamiltonian is
1 H 0 [ A c q(t ) x(t )]q(t ) ak 2 (t ) 0 (t )[k (t ) x(t )] 2
(18)
where 0 (t ) is the dynamic costate variables associated with the state x(t ) before attaining the quality authorization. From the Hamiltonian function, the first-order conditions and costate equation are
H1 ak (t ) 1 (t ) 0 1 (t ) ak (t ) k (t )
(19)
H1 1 A c 2 q(t ) x(t ) 0 q(t ) [ A c x(t )] q(t ) 2
(20)
0 (t ) r0 (t )
H 0 0 (t ) (r )0 (t ) q(t ) x(t )
(21)
Because of the terminal value e rTV1 ( x ) as well as the terminal state condition x(T ) x , the transversality
condition is
0 (T )
V1 ( x) f1 2 f 2 x , where is a constant. x x x
Substituting expressions (19) and (20) into equation (21), we can obtain a differential equation of k (t ) :
k (t ) (r )k (t )
2 x(t ) ( A c) 2a 2a
(22) 13
Note that equation (22) is similar to equation (12) except that Ah is replaced by A . Since
k (t ) 0 , the A
fact that A Ah implies that the incentive for the firm to invest in product innovation is higher before attaining the quality authorization than after attaining the quality authorization. Next, use a subscript “b” to denote “before attaining the quality authorization”. Proposition 3 below states the optimal investment strategy before obtaining the quality authorization.
Proposition 3. Before attaining the quality authorization, the optimal investment strategy kb* (t ) in production innovation as well as the associated quality state xb* (t ) over time are as follows:
xb* (t ) xˆ g1e1t g 2e2t * 1t 2t kb (t ) xˆ g1 ( 1 )e g 2 ( 2 )e where
g1
2
(23)
r r 2 4[ (r ) 2 / (2a )] 2
xˆ
,
( A c) 2a (r ) 2
,
(e2T 1)( x0 xˆ ) ( x x0 ) ( x x0 ) (1 e1T )( x0 xˆ ) g , and . 2 (e2T e1T ) (e2T e1T )
Proof: From the proof of Proposition 1, we know that 1 and 2 are the two eigenvalues of the Jacobian matrix J . Considering the initial condition x(0) x0 , one can write x(t ) as follows:
x(t ) ( x0 g1 g2 ) g1e1t g2e 2t where
g1 and
g2
are two constants to be determined. From equation (1), we can write
k (t ) x(t ) x(t ) . Substituting this expression into equation (22), and simplifying, we get:
x(t ) rx(t ) [ (r ) from which we
xˆ
(1 ) 2 (1 ) ]x(t ) ( A c) 2a 2a
can obtain a
linear
equation of
g1 and
g2 :
g1 g2 x0 xˆ , where
( A c) . Then, from the terminal state condition, we have another linear equation: [2a (r ) 2 ]
x(T ) x0 g1 (e1T 1) g2 (e2T 1) x . We can then solve g1 and g 2 from these two linear 14
equations: g1
(e2T 1)( x0 xˆ ) ( x x0 ) ( x x0 ) (1 e1T )( x0 xˆ ) g and . It then follows that 2 (e2T e1T ) (e2T e1T )
xb* (t ) xˆ g1e1t g2e2t ,
and
from
equation
(1)
we
have
an
expression
for
kb* (t ) :
kb* (t ) xb* (t ) xb* (t ) xˆ g1 ( 1 )e1t g2 (2 )e2t . End of proof. From Proposition 3, one can obtain the optimal production decision over time from expression (20):
qb* (t )
1 [( A c xˆ ) g1e1t g 2e2t ] . In addition, from the transversality condition we can then 2
determine the value of : a xˆ 2 f 2 x f1 ag1 ( 1 )e1T ag2 (2 )e2T . 4.3. The overall optimal solution Having obtained the analytical expressions of xb* (t ) , kb* (t ) , and qb* (t ) , the next step is to solve the problem
max J (T , q(t ), k (t )) . Unfortunately, J (T , q(t ), k (t )) is a functional on q(t ) and k (t ) , so this is a
{T , q ( t ), k ( t )}
maximization problem on function space. As there is no guarantee that J (T , q(t ), k (t )) is concave, the optimal (T , q(t ), k (t )) cannot be identified directly from the first-order condition. Therefore, inspired by the algorithms developed by Zhang et al. (2014) and Zhang (2016), we propose an iterative algorithm to find the optimal solution. (Note that the parameters g1 and g 2 relies on the choice of T , thus one can write them as functions of T as follows: g1 g1 (T ) and g 2 g 2 (T ) ):
Algorithm A1. Step 1: Initialized the algorithm with i 1 , qb,i (t ) qa (t ) , and kb,i (t ) ka (t ) . Set an accuracy parameter *
*
*
*
0 that is sufficiently small. Step 2: Calculate Ti by optimizing the following problem: T
Ti arg max J (T , qb*,i (t ), kb*,i (t )) e rt { 0
T
1 [ A c xˆ g1,i (T )e1t g 2,i (T )e2t ]2 4
a [ xˆ ( 1 ) g1,i (T )e1t ( 2 ) g 2,i (T )e2t ]2 }dt e rTV1 ( x ) 2 and let J i J (Ti , qb,i (t ), kb,i (t )) . *
*
15
Step 3: Solve the dynamic control problem (17) with T Ti to obtain qb ,i 1 (t ) and kb ,i 1 (t ) . *
*
Step 4: Check if Ti Ti 1 , then stop with T * Ti ; otherwise, set i i 1 and return to Step 2.
The convergence of the algorithm A1 can be justified as follows. First, although the concavity of
J (T , q(t ), k (t )) with respect to T given q(t ), k (t ) cannot be proved by an analytical approach, a large number of numerical examples have been done, showing that the corresponding concavity indeed holds. Thus
Ti in Step 2 uniquely exists. On the other hand, the unique existence of qb*,i 1 (t ) and kb*,i 1 (t ) in Step 3 is guaranteed by the concavity of the Hamiltonian (18). Next, from steps 2 and 3 one can easily obtain that
J i J (Ti , qb*,i (t ), kb*,i (t )) max J (Ti , q(t ), k (t )) ( q ( t ), k ( t ))
J (Ti , q
* b ,i 1
* b ,i 1
(t ), k
(24)
(t )) J (Ti 1 , qb*,i 1 (t ), kb*,i 1 (t )) J i 1
Then, from expressions (5) and (20), the following inequality holds: Ti
J i J (Ti , qb*,i (t ), kb*,i (t )) e rt { 0
1 1 [ A c x ]2 a[kb*,i (t )]2 }dt V1 ( x ) 2 2
(25)
1 [ A c x ]2 V1 ( x ) 2r
From inequalities (24) and (25), {J i }iN is a monotonically non-decreasing sequence with upper bound; so the limit lim J i must exist. Therefore, the convergence of the Algorithm A1 is guaranteed. The numerical i
*
solution computed from Algorithm A1 keeps at least the property of local optimality. With the optimal T , the overall investment strategy by piecing together the investment strategy before and after attaining the quality authorization. From propositions 1 and 3, the overall investment strategy goes as follows (use a subscript “o” to indicate “overall solution”):
xˆ g1 ( 1 )e1t g 2 ( 2 )e2t , 0 t T * * (i) Investment in product innovation: ko (t ) * 2 ( x x* ) 1 (t T ) k e , t T* 2 a ( r ) 1
(26)
t t 0 t T* xˆ g1e 1 g 2e 2 , (ii) Product quality state: x (t ) 1 ( t T ) * * , t T* x ( x x )e
(27)
* o
16
1 [( A c xˆ ) g1e1t g 2e2t ], 0 t T * 2 * (iii) Production quantity: qo (t ) 1 [( A c x* ) ( x x* )e1 (t T ) ], t T * h 2
(28)
4.4. Discontinuity in production and investment strategy From the expressions of ko* (t ) , xo* (t ) , and qo* (t ) , one may guess that they might be discontinuous in the neighbourhood of T
*
- the time at which the firm obtains the quality authorization. From Proposition 4 below, *
we see that ko* (t ) and qo* (t ) are indeed discontinuous at time T . Such discontinuity at T
*
highlights the
effect of the quality authorization on the firm’s strategy. Indeed, the existence of quality authorization can stimulate
the
investment
in product
( lim* ko (t ) lim* ko (t ) ) *
innovation
*
t T
t T
but
curb
production
( lim* qo (t ) lim* qo (t ) ) before attaining the quality authorization at time T . *
*
*
t T
t T
*
Proposition 4. In the neighbourhood of time point T , while the quality process xo* (t ) is continuous in time ( lim* xo (t ) lim* xo (t ) ), *
*
t T
t T
there
are
jumps
in
the
investment
and
production
decisions,
i.e.,
lim* ko* (t ) lim* ko* (t ) and lim* qo* (t ) lim* qo* (t ) . t T
t T
t T
t T
lim* xo* (t ) x and lim* xo* (t ) xˆ g1e1T g 2e2T xˆ x xˆ x . Thus, *
Proof: It is clear that
t T
*
t T
lim* xo* (t ) lim* xo* (t ) . t T
t T
Next, we examine the behaviour of xo* (t ) around the time T . Since x xˆ *
*
( Ah A) 0, 2a (r ) 2
by definition the following inequality holds:
lim* xo* (t ) lim* xo* (t ) 1 g1e1T 2 g 2e2T 1 ( x x* ) *
t T
*
t T
1 2T *
(e 0
{( 2 e2T 1e1T )( x* xˆ ) ( 2 1 )e2T [(1 e1T ) x* x e1T x0 ]} *
1T *
e
)
*
*
*
*
It then follows that lim* xo (t ) lim* xo (t ) . Then, from equation (1) we have k (t ) x(t ) x(t ) , which *
t T
*
t T
17
implies that lim* ko (t ) lim* xo (t ) lim* xo (t ) lim* xo (t ) lim* xo (t ) lim* ko (t ) . *
*
t T
t T
*
t T
*
t T
*
t T
*
t T
Finally, from expressions (10) and (20), we have
lim* qo* (t ) t T
1 1 [ A c lim* xo* (t )] [ A c lim* xo* (t )] t T t T 2 2
1 [ Ah c lim* xo* (t )] lim* qo* (t ) t T t T 2
End of Proof.
5. Numerical analysis We now employ numerical means to analyse the optimal investment behaviour of the firm under the quality authorization. To get the appropriate parameter values, references were taken from Hasnas et al. (2014) and Dawid et al. (2015). The base-case parameter values are technology),
0.3 (depreciation rate due to ageing of
0.6 (sensitivity of price to quality), 0.1 (sensitivity of price to demand), A 10
(profit profitability before quality authorization), Ah 10.1 (boosted profit profitability after quality authorization), a 1 (investment cost rate), c 2 (production cost), r 0.06 (interest rate), x0 1 (initial quality level), and x 2 (required quality level for quality authorization). Such a parameter setting implies a roughly 1% benefit for the firm from attaining the quality authorization. We firstly plot the profit function J (T , q(t ), k (t )) as a function of the quality authorization time T given the optimal quality and investment paths: q(t ) qb* (t ) and k (t ) kb* (t ) , as shown in Figure 1. It illustrates that the profit function
J (T , q(t ), k (t )) is concave with respect to the quality authorization time T .
Figure 1: The concavity of J (T , q(t ), k (t )) with respect to T 18
From Figure 1, we also observe that the profit function J (T , q(t ), k (t )) achieves its maximal value in the proximity of time T 1 . In fact, using Algorithm A1, we find that the optimal time to attain the quality authorization is T * 1.196 . Then, the corresponding quality state xo* (t ) , the investment level ko* (t ) in product innovation, and the production decision qo* (t ) can be obtained from expressions (26), (27), and (28); the results are plotted in figures 2-4.
Figure 2: The optimal path of the product quality xo* (t )
Figure 3: The optimal path of the product innovation investment ko* (t )
19
Figure 4: The optimal path of the production rate qo* (t ) Figure 2 shows that there is a kink in the optimal path of product quality xo* (t ) at time T * 1.196 . As a result, the speed at which the product quality improves is relatively faster for t T * than it would be for
t T * , implying that the existence of quality authorization can effectively induce the firm to improve the *
product quality. Then from Figure 3, the investment level ko* (t ) in product innovation has a jump at time T , indicating that the firm will optimally scale down the investment level after attaining the quality authorization. Moreover, it is shown that the investment level ko* (t ) increases rather slower for t T * , as compared with the case of t T * . This result in turn suggests that the incentive for the firm to make investment in production innovation is lower after attaining the quality authorization. Figure 4 shows that the production decision qo* (t ) exhibits an opposite pattern as compared with the investment ko* (t ) : the production quantity is substantially lower for t T * as compared with the case of t T * . The result implies that the production decision is curbed before attaining the quality authorization. In all, these results coincide with the analytical finding of Proposition 4. *
From the above analysis, we see that the firm’s optimal investment behaviour is characterized by T , the optimal time to attaining the quality authorization. Next, we conduct a sensitivity analysis to investigate how the changes parameter values should affect the optimal time T analysis results are summarized in Table 1.
20
*
of quality authorization. The sensitivity
Table 1: Sensitivity analysis of changing parameters on T Variation in parameters
a c r x
-20 0.958 1.381 0.988 0.953 1.009 1.164 0.848
-15 1.018 1.334 1.042 1.014 1.050 1.172 0.933
% change value in parameters -10 -5 0 5 10 1.078 1.137 1.196 1.254 1.312 1.288 1.242 1.196 1.152 1.109 1.094 1.146 1.196 1.245 1.291 1.074 1.131 1.196 1.256 1.314 1.095 1.143 1.196 1.254 1.317 1.180 1.188 1.196 1.204 1.212 1.019 1.107 1.196 1.287 1.379
It can be seen from Table 1 that the quality authorization time T
*
15 1.369 1.067 1.336 1.371 1.386 1.219 1.472
*
20 1.425 1.027 1.377 1.427 1.463 1.227 1.568
% changes * in T 39.0% -29.6% 32.5% 39.6% 37.9% 5.3% 60.2%
will increase when depreciation rate
due to ageing of technology, investment cost rate a , and/or production cost c increases. The result means that the firm will attain the quality authorization later when the ageing of technology is faster, the investment is more costly, and/or the unit production cost is higher. Moreover, while the quality authorization time T
*
is increasing with the sensitivity
of price to quality, it is decreasing in the sensitivity of
price to demand, implying that the firm will obtain the quality authorization sooner if the product price is more sensitive to product quality but less sensitive to product supply. Besides, Table 1 also shows that the quality authorization time T
*
will increase slightly as the interest rate r increases. This in turn suggest that a lower
interest rate should encourage investment in product innovation, thus driving the firm to produce high-quality products. This result coincides with the empirical result that a credit crunch, like the one during 2008-2009, can cause firms to scale down or even abandon their investment projects on product innovation (Paunov 2012). Finally, it can be seen that the quality authorization time T
*
is most sensitive to changes in the required
quality level x of quality authorization, and an intuitive explanation is that as the quality level required to attain the quality authorization increases, the firm will spend more time to meet the required quality level.
6. Conclusions In this paper, we have developed a dynamic model of production decision and product innovation that can incorporate the effects of quality authorization. The optimal production and investment decisions before and after attaining the quality authorization is analysed with the technique of backward induction. We show that while the product quality of the firm should admit a continuous and incremental path, there can be jumps in the optimal production and investment levels. The jumps can occur at the time of obtaining the quality authorization, which can then be identified by the iterative algorithm we propose. Once the time of attaining the 21
quality authorization is identified, the optimal investment and production decisions of the firm are then fully determined. Further, through numerical means, we find that the firm should attain the quality authorization sooner under a less costly the product innovation investment, a lower depreciation rate due to ageing of technology, a smaller production cost, a lower interest rate, and in a product market of higher sensitivity to product quality. This study can contribute to the literature from several angles. First of all, we have developed and analyzed a new dynamic model of production decision and product innovation with quality authorization. Although the related literature on product innovation is rich (e.g., Mantovani 2006, Lambertini and Orsini 2015), this paper is, to our knowledge, the first attempt to investigate the effects of the existence of the quality authorization on the production and investment decisions of the firm. Due to the complexity of the model, an iterative algorithm is also developed to solve for the optimal decisions of the firm. Next, we show that the firm’s optimal product innovation investment path is not necessarily smoothing. This result is different from many existing studies (e.g., Lambertini and Mantovani 2009, Hasnas et al. 2014) which suggests that firms’ investments in innovation should be smoothing. In fact, there exists a common belief in corporate finance that firms should smooth their R&D expenditure over time (e.g., Brown and Petersen 2011, Sasidharan et al. 2015, He and Wintoki 2016). In contrast, our analysis finds that under the quality authorization there can be significant jumps along the firms’ optimal product innovation investment paths, which suggests that smoothing the R&D expenditure (i.e., investment in innovation) is not the firm’s optimal choice. Further, our analysis reveals that although the dynamic product quality is continuous over time, the optimal production rate is not continuous (so is the product price). The finding is distinct from the results of many existing studies such as Chenavaz (2012), Pan and Li (2016), and Li and Ni (2016). This finding thus challenges the common intuition that a firm’s optimal production and/or pricing policy depends mainly on its current product quality and production cost, as well as the current market conditions (e.g., the price elasticity of demand); in fact, the investment behavior of product innovation is also an important determinant of the firm’s production and pricing policies. Our results have several managerial implications. First, it is important for firms’ managers to pay attention to the quality authorization schemes in deciding on the firms’ investment policy in product innovation. In fact, firms under quality authorization will be better off by substantially accelerating the product innovation investment at the beginning, and ever more so for manufacturing firms characterized by low production cost 22
and low ageing rate of technology. As a result, when deciding a firm’s product innovation investment policy, the manager should carefully consider the planned time for quality authorization and synergically take into account not only the characteristics of the innovation process itself, but also the costs involved in manufacturing process and the financial environment, even when the firm is not financially constrained. Further, under quality authorization managers should also restrain the desire to smooth the R&D expenditure for innovation, though this is a common practice among manufacturing firms (Brown and Petersen 2011). Finally, when deciding on a firm’s production and pricing policies, managers are suggested to take into account not only the current quality level of the product, but also the current investment level for product innovation which affects the quality level in the future. This paper has some limitations that may motivate future research. A possible extension to investigate the effects of quality authorization on product innovation along the supply chain. Due to complexity of this topic, it is not covered in this paper. It should be interesting to examine how the existence of quality authorization will shape the organization of the product innovation investments of the supply chain players (e.g., manufacturer and the supplier), and whether the double marginalization problem along the supply chain will be mitigated or exacerbated under the quality authorization. Another possible extension is to consider the effects of learning by doing in product innovation. Learning by doing, although not considered in our model, is an important factor that can accelerate the product innovation by accumulating knowledge and experience along the innovation process. In fact, how to adequately describe the role of learning by doing in the model is a challenging task, which deserves future research. Acknowledgements: We would like to thank the Editor and three anonymous referees for their insightful comments and suggestions for the revision of the paper. This work was supported by the National Natural Science Foundation of China (71601159, 51405089) and the Science and Technology Planning Project of Guangdong Province (2015B010131008).
23
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26
Highlights:
1. We model production decision and product innovation under quality authorization 2. The model is analyzed using Pontryagin maximum principle and backward induction 3. An iterative algorithm is developed to support the decision making of the firm 4. There can be jumps in the innovation investment and production rates over time
27
S0360-8352(17)30577-6 https://doi.org/10.1016/j.cie.2017.12.011 CAIE 5015
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
6 July 2016 5 December 2017 9 December 2017
Please cite this article as: Li, Z., Ni, J., Dynamic product innovation and production decisions under quality authorization, Computers & Industrial Engineering (2017), doi: https://doi.org/10.1016/j.cie.2017.12.011
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Dynamic product innovation and production decisions under quality authorization
Zhi Li Tel.: +86 20 39322212 E-mail: [email protected] Guangdong Provencal Key Lab of Computer Integrated Manufacturing, Guangdong University of Technology Guangzhou 510006, China
** Jian Ni Corresponding author Tel.: +86 28 87352835 Fax: +86 28 87092129 E-mail: [email protected] School of Finance, Southwestern University of Finance and Economics Chengdu 611130, China
1
Dynamic product innovation and production decisions under quality authorization
Abstract: We consider the joint product innovation and production decisions of a manufacturing firm under the existence of quality authorization from a 3rd party, which has gained growing popularity in recent years. A dynamic control model is developed to analyse the effects of quality authorization. Combining the techniques of Pontryagin maximum principle and backward induction, the optimal decisions on production and investment in product innovation before and after attaining the quality authorization is analysed. The analytical solution of the optimal investment and production decisions is derived providing that the time for the firm to obtain the quality authorization is known. To fully solve the firm’s optimization problem, an iterative algorithm is then introduced to calculate the best time of attaining the quality authorization. We find that although the firm should have a continuous and incremental improvements in product quality, there can be jumps in the optimal production and investment levels. While the investment in product innovation is higher before obtaining the quality authorization than that after obtaining the quality authorization, the production is lower before obtaining the quality authorization. Moreover, the firm should attain the quality authorization sooner under a less costly the product innovation investment, a lower depreciation rate due to ageing of technology, a smaller production cost, and a lower interest rate. Finally, we compare our results with existing studies and discuss the managerial implications. Keywords: product innovation, production decision, quality authorization, optimal control
1. Introduction Innovation is crucial for the sustainable growth of firms, it can dynamically boost the profitability of firms’ products. For instance, an upgraded computer software that calculates more rapidly or synchronizes data better can help an IT firm to attract more user, gain larger market share, and harvest higher profit. From an economic perspective (Saha 2007), the upgrading of the software can be perceived as product innovation – a type of innovation activities that aim to make the product better. In fact, the economic literature on innovation distinguishes between two basic types of innovation, namely, product innovation and process innovation (Chenavaz 2012). Product innovation is known as an activity that aims to improve the cumulative product quality and make better products (Lambertini and Mantovani 2009, Pan and Li 2016), while process innovation is understood as an effort to reduce the production cost (Lambertini and Orsini 2015, Li and Ni 2016). In this 2
paper, we develop a dynamic control model of product innovation under the existence of quality authorization from a 3rd party. The optimal decisions of the firm’s production quantity and investment in product innovation are analyzed. The objective is to investigate the effects of the existence of the quality authorization on the production and investment choices of the firm. The scheme of quality authorization has gained popularity over the past few years. One typical example is the Q-mark scheme launched in Hong Kong, which audits and endorses products and services for firms with production plants in Hong Kong, the mainland China, as well as Macau. The Q-mark scheme grants quality authorization to those firms with an effective production and quality management system capable of producing consistent quality products. It can be applied by any firms on a voluntary basis. Firms successfully passing the quality audit from the Q-mark council will be entitled to place the Q-Mark labels on their products, which offer consumers the confidence when making purchases. Another recent example is the quality authorization scheme under the China-Korea free trade agreement (CKA), which also provides the guarantee of high quality to consumers buying imported goods. In fact, there are several benefits for firms attaining the quality authorization. First, consumers these days are better educated, affluent and concerned about the product quality. To maintain a healthy lifestyle, they welcome products and services that have guaranteed quality. Second, many insurance firms such as Pacific Insurance Co., Ltd. will provide product quality guarantee insurance only to firms obtaining the quality authorization. Finally, in international trades between China and Korea, the time required for customs clearance in exporting products is much less (2-3 days versus 7-14 days) for firms attaining the quality authorization according to the CKA rule. The quality authorization scheme should affect the firms’ investment behaviour in product innovation that improves product quality (e.g., Lambertini and Orsini 2015, Pan and Li 2016). To address the growing popularity of quality authorization nowadays, we develop a dynamic model of the firm’s product innovation and production decisions under the voluntary quality authorization scheme. The purpose is to analyze the firm’s investment behavior in product innovation given the quality authorization. The optimal production and investment choices of the firm before and after attaining the authorization is examined. Because the value function of the firm obtained after the quality authorization is an input of the dynamic control problem before the quality authorization, the firm’s optimization problem is analyzed by the synergic use of backward induction and Pontryagin maximum principle, as being analogous to the approach taken by Li (2013). Once the time of obtaining the quality authorization is given, both the optimal production and investment decisions can 3
then be expressed in analytical forms. To obtain the optimal decisions of the firm, an iterative algorithm is introduced to identify the best time of attaining the quality authorization. We then analyze the model dynamics through numerical means and report several new findings, for instance, the firm should have a continuous improvements in product quality but may have a discontinuous process in product innovation investment. The rest of the paper is organized as follows. Section 2 provides a review of the related literature. Section 3 introduces the basic framework of the dynamic model. Section 4 then analyses the dynamic model using backward induction. The numerical results are presented in section 5. Section 6 concludes the paper.
2. Literature review Product innovation which aims to make the product better has become a fundamental source of firm growth (Chenavaz 2012), and the related literature is rich. Spence (1975) and Mussa and Rosen (1978) firstly discussed the role of product quality in affecting the optimal monopolistic strategy. Since then, several papers have discussed the impact of monopoly power on product quality under static settings (e.g., Besanko et al. 1987, Lambertini 2006). Recently, Lambertini and Orsini (2015) further extended the analysis to a dynamic control model of product and process innovation. They found that the monopolistic power depresses investment in product innovation. Another line of research then concerns the relationship between product and process innovation, where the process innovation refers to the activity that aims to reduce the production costs. For example, Lin and Saggi (2002) investigated the relationship between process and product innovation in a three stage model. Mantovani (2006) further proposed a dynamic model to study complementarity between market-enhancing product innovation and cost-reducing process innovation in a monopoly setting. Lambertini and Mantovani (2009) considered a dynamic control model to study the dynamic behavior of a multiproduct monopolist investing both in process innovation and in product innovation. Chenavaz (2012) developed a product and process innovation model to determine the optimal product price, the product and process innovation investment strategy, given a time-varying demand conditions. In a recent work, Pan and Li (2016) extended the model of Chenavaz (2012) to a model where the firm’s cost functions of product and process innovation depend on both the innovation investments and the knowledge accumulations of product and process innovation. Further, Zhang et al. (2015) proposed a research model to study the joint effects of social capital on mass customization capability and product innovation capability. Choi et al. (2016) considered the strategic innovation policies and suggested that a new paradigm wherein product and process innovations are pursued concurrently instead of sequentially can guarantee a firm’s sustainable growth. Guimarães et al. (2016) 4
conducted a survey on the performance of spending organizational resources on produce innovation in Brazilian furniture firms. This paper is also related to the broad literature on firms’ dynamic R&D investment activities. Minniti (2010) studied the R&D composition by using a model where the horizontal and vertical innovations simultaneously take place; through the model analysis, the relationship between the product market competition and the leading-edge growth is also investigated. Matsumura et al. (2013) showed that R&D investment will not always be a decreasing function of the intensity of market competition in the absence of spillover. Chrystie et al. (2013) introduced a standard model of strategic R&D with spillovers in R&D inputs, and found that duopoly firms engaged in a standard two-stage game of R&D and Cournot competition will end up in a prisoner’s dilemma situation for their R&D decisions, whenever spillover effects and R&D costs are relatively low. In a recent contribution, EI Ouardighi and Tapiero (2014) analyzed the firms’ R&D with stock-dependent spillover, under the assumption that a firm’s R&D is production-cost-reducing and can benefit from part of the competitor’s R&D stock without payment. Later, Shibata (2014) investigated the R&D investment strategies for firms under R&D spillover in various market structures, while Xing (2014) studied the optimal choices of R&D risk under network externalities by analyzing a Hotelling spatial model. Addessi et al. (2014) then conducted both theoretical and empirical studies to investigate the impacts of R&D activities on the use of external flexibility. This paper analyses the dynamic investment policy of product innovation for firms under quality authorization by analyzing a dynamic control model and the associated dynamic state and costate variables with the well-known Pontryagin maximum principle. This type of dynamic models, which involve both the state and costate variables, has been widely applied in the literature to study the dynamic decision-making of firms. For instance, Cellini and Lambertini (2002) developed a dynamic model to analyze firms’ product differentiation strategies, with the level of product differentiation serving as a state variable. Lambertini and Zaccour (2014) alternatively introduced a dynamic model on the competition of firms with advertising. The authors treated the goodwill resulting from advertising as a state variable and found that the aggregate expenditure on goodwill takes a U-shape. In another contribution, Jørgensen and Zaccour (2014) provided a review on the studies using dynamic game-theoretic models of cooperative advertising. Then, Martín-Herrán et al. (2011) examined the effectiveness of franchise systems with a dynamic game-theoretic model on the strategic interactions between franchisors and franchisees, whereas Martín-Herrán and Taboubi (2015) investigated the price coordination 5
among supply chain members in a dynamic perspective. Recently, Yang et al. (2015) introduced a dynamic control model to study the dynamic trade credit and preservation technology allocation for deteriorating products, with the inventory of the products treated as a state variable. Zhang et al. (2016) further developed a dynamic model that can accommodate common resource constraints for analyzing the pricing, service and preservation technology investments policy of deteriorating items. This paper has several key features that are distinct from the existing literature. First, we develop a new dynamic control model of production decision and product innovation that can incorporate the effects of the existence of quality authorization. To our knowledge, this paper is the first research that addresses the effects of quality authorization on the production and investment decisions of the firm. Since the model is new, we develop an iterative algorithm to optimize the planned time for the firm to attain the quality authorization, with which the optimal production and investment decisions of the firm can be obtained. Second, we show that although the firm should have a continuous and incremental improvements in product quality, there can be jumps in the optimal production and investment levels over time; in particular, there is a need to substantially accelerate the product innovation investment before quality authorization. As compared with existing studies (e.g., Lambertini and Mantovani 2009, Hasnas et al. 2014, Pan and Li 2016) which support the smoothing of investment paths for innovation, our results instead suggest that it is not always optimal to smooth the product innovation path. Third, we find that the firm should attain the quality authorization sooner when the product innovation investment is less costly, the unit production cost is smaller, the depreciation rate due to ageing of technology is lower, the interest rate is lower, and/or product price is more sensitive to product quality. We believe that these results can complement the existing literature on product innovation and help advance our understanding on the dynamics of product innovation investment.
3. The basic framework We consider a dynamic control problem over continuous time t [0, ) , where at any instant a firm needs to make both the production decision and the decision of investment in product innovation. Product innovation can be understood as an improvement of the product quality (Chenavaz 2012). Use x(t ) to denote the product quality of the firm. There is a voluntary quality authorization scheme that the firm can apply. If the firm succeeds in passing the quality audit and attaining the quality authorization, the product’s profitability will be improved, either from the better public perception of the product quality (e.g., the Q-mark case) or from the supports from governments and insurance firms (e.g., the CKA case). However, the firm can attain the quality 6
authorization only when x(t ) reaches a required quality level x . At time t , the firm can invest in product innovation (i.e., quality-enhancing technology) via the instantaneous investment k (t ) to increase x(t ) . With the investment, the differential equation which describes the evolution of x(t ) over time is given by the following form:
x(t ) k (t ) x(t ) where parameter
(1)
0 is the depreciation rate due to ageing of technology. According to Lambertini and
Orsini (2015), the instantaneous cost function of investing in product innovation is C (k (t ))
1 2 ak (t ) , 2
where a is a positive parameter. The convex cost function implies increasing marginal costs and decreasing returns to innovative activity. The parameter a is the inverse measure of product innovation investment efficiency. The firm produces a single item, and consumers vary in their willingness to pay and their concern for product quality. Let q(t ) be the production quantity while c be the unit production cost. The firm’s production cost function is cq(t ) . Following Lambertini and Orsini (2015), we assume that there is no stock, all the demand is satisfied, and all the production is sold. According to Chenavaz (2012) and Pan and Li (2016), we have the following inverse demand function:
p(t ) A q(t ) x(t )
(2)
Now consider the firm’s profit, which can be shifted by the quality authorization. Then, the firm’s instantaneous profit before attaining the quality authorization ( x(t ) x ) is given by
1 2
(t ) [ A c q(t ) x(t )]q(t ) ak 2 (t )
(3)
After attaining the quality authorization ( x(t ) x ), the profitability of the product is boosted ( AH A ), leading to the following instantaneous profit of the firm:
1 2
(t ) [ Ah c q(t ) x(t )]q(t ) ak 2 (t )
(4)
Then, the firm’s optimization problem is the following:
7
T 1 max J (T , q(t ), k (t )) e rt {[ A c q(t ) x(t )]q(t ) ak 2 (t )}dt e rTV1 ( x ) 0 {T , q ( t ), k ( t )} 2 x(t ) k (t ) x(t ) S .t. x(0) x0 , x(T ) x
(5)
where V1 ( x1 ) is the discounted stream of the firm’s profits after obtaining the quality authorization, which is obtained by solving the following optimization problem:
1 e rt {[ Ah c q(t ) x(t )]q(t ) ak 2 (t )}dt q ( t ), k ( t ) 0 2 x(t ) k (t ) x(t ) S .t. x(0) x1
V1 ( x1 ) max
(6)
Next section, we will investigate the optimal investment strategy before and after attaining the quality authorization.
4. Model and analysis The dynamic model is analysed using backward induction. The first step is to analyse the firm’s optimization problem after attaining the quality authorization. The second step is to analyse the optimization problem before attaining the quality authorization. Then, we investigate the effect of the quality authorization by comparing the firm’s investment behaviour before or after attaining the quality authorization. 4.1. The optimization problem after attaining the quality authorization After attaining the quality authorization ( x1 x ), the firm’s objective is to determine the optimal investment level k (t ) and optimal production quantity q(t ) such that the discounted stream of net revenues is maximized, which can be formulated as follows:
1 e rt {[ Ah c q(t ) x(t )]q(t ) ak 2 (t )}dt 2 x(t ) k (t ) x(t ) S .t. x(0) x1
V1 ( x1 ) max
q ( t ), k ( t ) 0
(7)
The corresponding current value Hamiltonian function writes
1 H1 [ Ah c q(t ) x(t )]q(t ) ak 2 (t ) 1 (t )[k (t ) x(t )] 2
(8)
where 1 (t ) is the dynamic costate variable associated with the state x(t ) after attaining the quality authorization. 8
From the Hamiltonian, the first-order conditions are
H1 ak (t ) 1 (t ) 0 1 (t ) ak (t ) k (t )
(9)
H1 1 A c 2 q(t ) x(t ) 0 q(t ) [ Ah c x(t )] q(t ) 2
(10)
The associated costate equation for 1 (t ) is the following
1 (t ) r1 (t )
H1 1 (t ) (r )1 (t ) q(t ) x(t )
with transversality condition lim 1 (t )e t
rt
(11)
0.
Substituting expressions (9) and (10) into equation (11) gives the following differential equation, which describes the dynamic investment level k (t ) :
2 x(t ) ( Ah c) 2a 2a
(12)
From equation (12), one can immediately obtain that
k (t ) 2 k (t ) 0 and (r ) 0 , which x(t ) 2a k (t )
k (t ) (r )k (t )
imply that after attaining the quality authorization, the rate at which the investment in product innovation increases is higher for a relatively lower quality and/or a higher investment. Moreover, one can combine the differential equations (12) and (1) to solve for k (t ) and x(t ) after attaining the quality authorization. Use a superscript “*” to denote “optimal solution” while a subscript “a” to denote “after attaining the quality authorization”. We have the following proposition.
Proposition 1. When
(r ) 2 / (2a ) , the dynamic system admits an unique stable state ( x* , k * )
after
the
k*
attaining
quality
authorization,
where
x*
( Ah c) 2a (r ) 2
,
and
1 [ ( Ah c) (2a1 2 ) x* ] . Moreover, the optimal investment strategy ka* (t ) 2a (r 1 ) *
in production innovation as well as the associated quality state xa (t ) over time are as follows:
9
xa* (t ) x* ( x1 x* )e1t * 2 ( x1 x* ) 1t * ka (t ) k 2a (r ) e 1
(13)
r r 2 4[ (r ) 2 / (2a )] where 1 0. 2
Proof: First, from differential equations (1) and (12), the steady state must solve the following system of equations:
k (t ) x(t ) 0 2 ( r ) k ( t ) x(t ) ( Ah c) 0 2a 2a *
*
*
*
which admits a unique solution ( x , k ) . Next, we examine the stability of ( x , k ) . The Jacobian matrix for differential equations (1) and (12) is:
x x J k x
x k 2 k 2a k
1 r
Because the trace is Tr ( J ) r 0 , a necessary and sufficient condition for ( x , k ) to be stable (in a *
saddle
point
sense)
is
that
the
determinant
is
negative:
*
| J | (r )
2 0 , i.e., 2a
(r ) 2 / (2a ) . Next, we move on to solve for the optimal investment strategy. One can verify that the two eigenvalues of the Jacobian matrix J are:
1
r r 2 4[ (r ) 2 / (2a )] r r 2 4[ (r ) 2 / (2a )] 0 and 2 0 2 2 t
2t
Hence, one can write down the general solution for k (t ) as follows: k (t ) C0 C1e 1 C2e
C1 ,
C2
are
three
constants
to
1 (t ) ak (t ) aC0 aC1e t aC2e t . 1
2
be
determined.
Since
2 10
Then
from
equation
(9),
, where C0 , we
have
r r 2 4[ (r ) 2 / (2a )] r 2
and
1 0 , from the transversality condition lim 1 (t )e rt lim aC2e(2 r )t 0 . Thus, we must have C2 0 . t
t
Now we write down the general solution of x(t ) as follows: x(t ) B0 B1e1t B2e2t , where B0 , B1 ,
B2 are three constants to be determined.. From equation (1), it is easy to see that B2 0 given that C2 0 . Substituting the expressions of the general solutions k (t ) C0 C1e1t and x(t ) B0 B1e1t into (1) and (12), invoking the initial condition x(0) x1 , and simplifying, we get the following system of linear equations for C0 , C1 , B0 , and B1 :
C0 B0 0 C ( ) B 0 1 1 1 2 B0 ( Ah c) (r )C0 2a 2a B0 B1 x1
2 ( x1 x* ) from which we can solve that B0 x , B1 x1 x , C0 k , and C1 . Thus, the 2a (r 1 ) *
*
*
optimal investment strategy after attaining the quality authorization follows from expression (13). End of proof. From Proposition 1, the optimal production decision over time can be calculated from expression (10):
qa* (t )
1 [( Ah c x* ) ( x1 x* )e1t ] . Thus, the steady state production quantity after attaining the 2
* quality authorization is q
relationship
holds
between
Ah c x* . Moreover, one can easily verify that the following linear 2 xa* (t )
and
ka* (t )
,
that
is,
ka* (t ) xa* (t ) ,
where
( Ah c) 2a1 x* 2 and . We can exploit this linear relationship to derive the 2a (r 1 ) 2a (r 1 )
firm’s value function V1 ( x) after attaining the quality authorization. To this end, we need to formulate the Hamilton–Jacobi–Bellman (HJB) equation for the control problem (7):
V ( x) 1 rV1 ( x) ( Ah c q x)q ak 2 (k x) 1 2 x Substituting (10) and (13) into (14), we get 11
(14)
rV1 ( x)
V ( x) 1 1 ( Ah c x)2 a( x)2 [ ( ) x] 1 4 2 x
(15)
According to Dawid et al. (2015), one can guess that V1 ( x) f 0 f1 x f 2 x 2 . Substituting this expression into equation (15) then yields
r ( f 0 f1 x f 2 x 2 )
1 1 ( Ah c x)2 a( x)2 [ ( ) x]( f1 2 f 2 x) 4 2
(16)
from which we can obtain an expression of V1 ( x) , which is stated in Proposition 2.
Proposition 2. The firm’s value function after attaining the quality authorization is V1 ( x) f 0 f1 x f 2 x 2 , where the coefficients f 0 , f1 , and f 2 are
f0
a 2 ( Ah c)2 ( Ah c) 2a 2 2 2 ( 2 2a 2 ) 2r 4r 2r (r ) 4r (r )(r 2 2 )
f1
( Ah c) 2a 2 ( 2 2a 2 ) 2 (r ) 4 (r )(r 2 2 )
f2
2 2a 2 4 (r 2 2 )
Proof: From (16), the following equations for f 0 , f1 , and f 2 hold:
( Ah c) 2 2a 2 rf f 0 1 0 4 ( Ah c) a 0 (r ) f1 2 f 2 2 2 2a 2 ( r 2 2 ) f 0 2 4 from which we can solve for the expressions of f 0 , f1 , and f 2 , as stated in Proposition 2. End of proof. 4.2. The optimization problem before attaining the quality authorization Next, we turn to investigate the firm’s optimization problem before attaining the quality authorization. To proceed, we firstly suppose that the firm obtains the quality authorization at time T , then for t [0, T ) , the 12
dynamic control problem now writes
1 e rt {[ A c q(t ) x(t )]q(t ) ak 2 (t )}dt e rTV1 ( x ) q ( t ), k ( t ) 0 2 x(t ) k (t ) x(t ) S .t. x(0) x0 , x(T ) x
V0 (T ) max
T
(17)
Note that the dynamic control problem (17) is similar to (7) except for two key differences: (i) the profitability of the product is lower before attaining the quality authorization ( A Ah ); (ii) there is a terminal value
e rTV1 ( x ) which represents the discounted streams of profits after attaining the quality authorization at time T ; (iii) there is a terminal state condition x(T ) x , meaning that the firm indeed obtains the quality authorization at time T . The importance of dynamic control problem (17) is that it is a simplified version of the optimization problem (5), that is, V0 (T ) max J (T , q(t ), k (t )) . q ( t ), k ( t )
The corresponding current value Hamiltonian is
1 H 0 [ A c q(t ) x(t )]q(t ) ak 2 (t ) 0 (t )[k (t ) x(t )] 2
(18)
where 0 (t ) is the dynamic costate variables associated with the state x(t ) before attaining the quality authorization. From the Hamiltonian function, the first-order conditions and costate equation are
H1 ak (t ) 1 (t ) 0 1 (t ) ak (t ) k (t )
(19)
H1 1 A c 2 q(t ) x(t ) 0 q(t ) [ A c x(t )] q(t ) 2
(20)
0 (t ) r0 (t )
H 0 0 (t ) (r )0 (t ) q(t ) x(t )
(21)
Because of the terminal value e rTV1 ( x ) as well as the terminal state condition x(T ) x , the transversality
condition is
0 (T )
V1 ( x) f1 2 f 2 x , where is a constant. x x x
Substituting expressions (19) and (20) into equation (21), we can obtain a differential equation of k (t ) :
k (t ) (r )k (t )
2 x(t ) ( A c) 2a 2a
(22) 13
Note that equation (22) is similar to equation (12) except that Ah is replaced by A . Since
k (t ) 0 , the A
fact that A Ah implies that the incentive for the firm to invest in product innovation is higher before attaining the quality authorization than after attaining the quality authorization. Next, use a subscript “b” to denote “before attaining the quality authorization”. Proposition 3 below states the optimal investment strategy before obtaining the quality authorization.
Proposition 3. Before attaining the quality authorization, the optimal investment strategy kb* (t ) in production innovation as well as the associated quality state xb* (t ) over time are as follows:
xb* (t ) xˆ g1e1t g 2e2t * 1t 2t kb (t ) xˆ g1 ( 1 )e g 2 ( 2 )e where
g1
2
(23)
r r 2 4[ (r ) 2 / (2a )] 2
xˆ
,
( A c) 2a (r ) 2
,
(e2T 1)( x0 xˆ ) ( x x0 ) ( x x0 ) (1 e1T )( x0 xˆ ) g , and . 2 (e2T e1T ) (e2T e1T )
Proof: From the proof of Proposition 1, we know that 1 and 2 are the two eigenvalues of the Jacobian matrix J . Considering the initial condition x(0) x0 , one can write x(t ) as follows:
x(t ) ( x0 g1 g2 ) g1e1t g2e 2t where
g1 and
g2
are two constants to be determined. From equation (1), we can write
k (t ) x(t ) x(t ) . Substituting this expression into equation (22), and simplifying, we get:
x(t ) rx(t ) [ (r ) from which we
xˆ
(1 ) 2 (1 ) ]x(t ) ( A c) 2a 2a
can obtain a
linear
equation of
g1 and
g2 :
g1 g2 x0 xˆ , where
( A c) . Then, from the terminal state condition, we have another linear equation: [2a (r ) 2 ]
x(T ) x0 g1 (e1T 1) g2 (e2T 1) x . We can then solve g1 and g 2 from these two linear 14
equations: g1
(e2T 1)( x0 xˆ ) ( x x0 ) ( x x0 ) (1 e1T )( x0 xˆ ) g and . It then follows that 2 (e2T e1T ) (e2T e1T )
xb* (t ) xˆ g1e1t g2e2t ,
and
from
equation
(1)
we
have
an
expression
for
kb* (t ) :
kb* (t ) xb* (t ) xb* (t ) xˆ g1 ( 1 )e1t g2 (2 )e2t . End of proof. From Proposition 3, one can obtain the optimal production decision over time from expression (20):
qb* (t )
1 [( A c xˆ ) g1e1t g 2e2t ] . In addition, from the transversality condition we can then 2
determine the value of : a xˆ 2 f 2 x f1 ag1 ( 1 )e1T ag2 (2 )e2T . 4.3. The overall optimal solution Having obtained the analytical expressions of xb* (t ) , kb* (t ) , and qb* (t ) , the next step is to solve the problem
max J (T , q(t ), k (t )) . Unfortunately, J (T , q(t ), k (t )) is a functional on q(t ) and k (t ) , so this is a
{T , q ( t ), k ( t )}
maximization problem on function space. As there is no guarantee that J (T , q(t ), k (t )) is concave, the optimal (T , q(t ), k (t )) cannot be identified directly from the first-order condition. Therefore, inspired by the algorithms developed by Zhang et al. (2014) and Zhang (2016), we propose an iterative algorithm to find the optimal solution. (Note that the parameters g1 and g 2 relies on the choice of T , thus one can write them as functions of T as follows: g1 g1 (T ) and g 2 g 2 (T ) ):
Algorithm A1. Step 1: Initialized the algorithm with i 1 , qb,i (t ) qa (t ) , and kb,i (t ) ka (t ) . Set an accuracy parameter *
*
*
*
0 that is sufficiently small. Step 2: Calculate Ti by optimizing the following problem: T
Ti arg max J (T , qb*,i (t ), kb*,i (t )) e rt { 0
T
1 [ A c xˆ g1,i (T )e1t g 2,i (T )e2t ]2 4
a [ xˆ ( 1 ) g1,i (T )e1t ( 2 ) g 2,i (T )e2t ]2 }dt e rTV1 ( x ) 2 and let J i J (Ti , qb,i (t ), kb,i (t )) . *
*
15
Step 3: Solve the dynamic control problem (17) with T Ti to obtain qb ,i 1 (t ) and kb ,i 1 (t ) . *
*
Step 4: Check if Ti Ti 1 , then stop with T * Ti ; otherwise, set i i 1 and return to Step 2.
The convergence of the algorithm A1 can be justified as follows. First, although the concavity of
J (T , q(t ), k (t )) with respect to T given q(t ), k (t ) cannot be proved by an analytical approach, a large number of numerical examples have been done, showing that the corresponding concavity indeed holds. Thus
Ti in Step 2 uniquely exists. On the other hand, the unique existence of qb*,i 1 (t ) and kb*,i 1 (t ) in Step 3 is guaranteed by the concavity of the Hamiltonian (18). Next, from steps 2 and 3 one can easily obtain that
J i J (Ti , qb*,i (t ), kb*,i (t )) max J (Ti , q(t ), k (t )) ( q ( t ), k ( t ))
J (Ti , q
* b ,i 1
* b ,i 1
(t ), k
(24)
(t )) J (Ti 1 , qb*,i 1 (t ), kb*,i 1 (t )) J i 1
Then, from expressions (5) and (20), the following inequality holds: Ti
J i J (Ti , qb*,i (t ), kb*,i (t )) e rt { 0
1 1 [ A c x ]2 a[kb*,i (t )]2 }dt V1 ( x ) 2 2
(25)
1 [ A c x ]2 V1 ( x ) 2r
From inequalities (24) and (25), {J i }iN is a monotonically non-decreasing sequence with upper bound; so the limit lim J i must exist. Therefore, the convergence of the Algorithm A1 is guaranteed. The numerical i
*
solution computed from Algorithm A1 keeps at least the property of local optimality. With the optimal T , the overall investment strategy by piecing together the investment strategy before and after attaining the quality authorization. From propositions 1 and 3, the overall investment strategy goes as follows (use a subscript “o” to indicate “overall solution”):
xˆ g1 ( 1 )e1t g 2 ( 2 )e2t , 0 t T * * (i) Investment in product innovation: ko (t ) * 2 ( x x* ) 1 (t T ) k e , t T* 2 a ( r ) 1
(26)
t t 0 t T* xˆ g1e 1 g 2e 2 , (ii) Product quality state: x (t ) 1 ( t T ) * * , t T* x ( x x )e
(27)
* o
16
1 [( A c xˆ ) g1e1t g 2e2t ], 0 t T * 2 * (iii) Production quantity: qo (t ) 1 [( A c x* ) ( x x* )e1 (t T ) ], t T * h 2
(28)
4.4. Discontinuity in production and investment strategy From the expressions of ko* (t ) , xo* (t ) , and qo* (t ) , one may guess that they might be discontinuous in the neighbourhood of T
*
- the time at which the firm obtains the quality authorization. From Proposition 4 below, *
we see that ko* (t ) and qo* (t ) are indeed discontinuous at time T . Such discontinuity at T
*
highlights the
effect of the quality authorization on the firm’s strategy. Indeed, the existence of quality authorization can stimulate
the
investment
in product
( lim* ko (t ) lim* ko (t ) ) *
innovation
*
t T
t T
but
curb
production
( lim* qo (t ) lim* qo (t ) ) before attaining the quality authorization at time T . *
*
*
t T
t T
*
Proposition 4. In the neighbourhood of time point T , while the quality process xo* (t ) is continuous in time ( lim* xo (t ) lim* xo (t ) ), *
*
t T
t T
there
are
jumps
in
the
investment
and
production
decisions,
i.e.,
lim* ko* (t ) lim* ko* (t ) and lim* qo* (t ) lim* qo* (t ) . t T
t T
t T
t T
lim* xo* (t ) x and lim* xo* (t ) xˆ g1e1T g 2e2T xˆ x xˆ x . Thus, *
Proof: It is clear that
t T
*
t T
lim* xo* (t ) lim* xo* (t ) . t T
t T
Next, we examine the behaviour of xo* (t ) around the time T . Since x xˆ *
*
( Ah A) 0, 2a (r ) 2
by definition the following inequality holds:
lim* xo* (t ) lim* xo* (t ) 1 g1e1T 2 g 2e2T 1 ( x x* ) *
t T
*
t T
1 2T *
(e 0
{( 2 e2T 1e1T )( x* xˆ ) ( 2 1 )e2T [(1 e1T ) x* x e1T x0 ]} *
1T *
e
)
*
*
*
*
It then follows that lim* xo (t ) lim* xo (t ) . Then, from equation (1) we have k (t ) x(t ) x(t ) , which *
t T
*
t T
17
implies that lim* ko (t ) lim* xo (t ) lim* xo (t ) lim* xo (t ) lim* xo (t ) lim* ko (t ) . *
*
t T
t T
*
t T
*
t T
*
t T
*
t T
Finally, from expressions (10) and (20), we have
lim* qo* (t ) t T
1 1 [ A c lim* xo* (t )] [ A c lim* xo* (t )] t T t T 2 2
1 [ Ah c lim* xo* (t )] lim* qo* (t ) t T t T 2
End of Proof.
5. Numerical analysis We now employ numerical means to analyse the optimal investment behaviour of the firm under the quality authorization. To get the appropriate parameter values, references were taken from Hasnas et al. (2014) and Dawid et al. (2015). The base-case parameter values are technology),
0.3 (depreciation rate due to ageing of
0.6 (sensitivity of price to quality), 0.1 (sensitivity of price to demand), A 10
(profit profitability before quality authorization), Ah 10.1 (boosted profit profitability after quality authorization), a 1 (investment cost rate), c 2 (production cost), r 0.06 (interest rate), x0 1 (initial quality level), and x 2 (required quality level for quality authorization). Such a parameter setting implies a roughly 1% benefit for the firm from attaining the quality authorization. We firstly plot the profit function J (T , q(t ), k (t )) as a function of the quality authorization time T given the optimal quality and investment paths: q(t ) qb* (t ) and k (t ) kb* (t ) , as shown in Figure 1. It illustrates that the profit function
J (T , q(t ), k (t )) is concave with respect to the quality authorization time T .
Figure 1: The concavity of J (T , q(t ), k (t )) with respect to T 18
From Figure 1, we also observe that the profit function J (T , q(t ), k (t )) achieves its maximal value in the proximity of time T 1 . In fact, using Algorithm A1, we find that the optimal time to attain the quality authorization is T * 1.196 . Then, the corresponding quality state xo* (t ) , the investment level ko* (t ) in product innovation, and the production decision qo* (t ) can be obtained from expressions (26), (27), and (28); the results are plotted in figures 2-4.
Figure 2: The optimal path of the product quality xo* (t )
Figure 3: The optimal path of the product innovation investment ko* (t )
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Figure 4: The optimal path of the production rate qo* (t ) Figure 2 shows that there is a kink in the optimal path of product quality xo* (t ) at time T * 1.196 . As a result, the speed at which the product quality improves is relatively faster for t T * than it would be for
t T * , implying that the existence of quality authorization can effectively induce the firm to improve the *
product quality. Then from Figure 3, the investment level ko* (t ) in product innovation has a jump at time T , indicating that the firm will optimally scale down the investment level after attaining the quality authorization. Moreover, it is shown that the investment level ko* (t ) increases rather slower for t T * , as compared with the case of t T * . This result in turn suggests that the incentive for the firm to make investment in production innovation is lower after attaining the quality authorization. Figure 4 shows that the production decision qo* (t ) exhibits an opposite pattern as compared with the investment ko* (t ) : the production quantity is substantially lower for t T * as compared with the case of t T * . The result implies that the production decision is curbed before attaining the quality authorization. In all, these results coincide with the analytical finding of Proposition 4. *
From the above analysis, we see that the firm’s optimal investment behaviour is characterized by T , the optimal time to attaining the quality authorization. Next, we conduct a sensitivity analysis to investigate how the changes parameter values should affect the optimal time T analysis results are summarized in Table 1.
20
*
of quality authorization. The sensitivity
Table 1: Sensitivity analysis of changing parameters on T Variation in parameters
a c r x
-20 0.958 1.381 0.988 0.953 1.009 1.164 0.848
-15 1.018 1.334 1.042 1.014 1.050 1.172 0.933
% change value in parameters -10 -5 0 5 10 1.078 1.137 1.196 1.254 1.312 1.288 1.242 1.196 1.152 1.109 1.094 1.146 1.196 1.245 1.291 1.074 1.131 1.196 1.256 1.314 1.095 1.143 1.196 1.254 1.317 1.180 1.188 1.196 1.204 1.212 1.019 1.107 1.196 1.287 1.379
It can be seen from Table 1 that the quality authorization time T
*
15 1.369 1.067 1.336 1.371 1.386 1.219 1.472
*
20 1.425 1.027 1.377 1.427 1.463 1.227 1.568
% changes * in T 39.0% -29.6% 32.5% 39.6% 37.9% 5.3% 60.2%
will increase when depreciation rate
due to ageing of technology, investment cost rate a , and/or production cost c increases. The result means that the firm will attain the quality authorization later when the ageing of technology is faster, the investment is more costly, and/or the unit production cost is higher. Moreover, while the quality authorization time T
*
is increasing with the sensitivity
of price to quality, it is decreasing in the sensitivity of
price to demand, implying that the firm will obtain the quality authorization sooner if the product price is more sensitive to product quality but less sensitive to product supply. Besides, Table 1 also shows that the quality authorization time T
*
will increase slightly as the interest rate r increases. This in turn suggest that a lower
interest rate should encourage investment in product innovation, thus driving the firm to produce high-quality products. This result coincides with the empirical result that a credit crunch, like the one during 2008-2009, can cause firms to scale down or even abandon their investment projects on product innovation (Paunov 2012). Finally, it can be seen that the quality authorization time T
*
is most sensitive to changes in the required
quality level x of quality authorization, and an intuitive explanation is that as the quality level required to attain the quality authorization increases, the firm will spend more time to meet the required quality level.
6. Conclusions In this paper, we have developed a dynamic model of production decision and product innovation that can incorporate the effects of quality authorization. The optimal production and investment decisions before and after attaining the quality authorization is analysed with the technique of backward induction. We show that while the product quality of the firm should admit a continuous and incremental path, there can be jumps in the optimal production and investment levels. The jumps can occur at the time of obtaining the quality authorization, which can then be identified by the iterative algorithm we propose. Once the time of attaining the 21
quality authorization is identified, the optimal investment and production decisions of the firm are then fully determined. Further, through numerical means, we find that the firm should attain the quality authorization sooner under a less costly the product innovation investment, a lower depreciation rate due to ageing of technology, a smaller production cost, a lower interest rate, and in a product market of higher sensitivity to product quality. This study can contribute to the literature from several angles. First of all, we have developed and analyzed a new dynamic model of production decision and product innovation with quality authorization. Although the related literature on product innovation is rich (e.g., Mantovani 2006, Lambertini and Orsini 2015), this paper is, to our knowledge, the first attempt to investigate the effects of the existence of the quality authorization on the production and investment decisions of the firm. Due to the complexity of the model, an iterative algorithm is also developed to solve for the optimal decisions of the firm. Next, we show that the firm’s optimal product innovation investment path is not necessarily smoothing. This result is different from many existing studies (e.g., Lambertini and Mantovani 2009, Hasnas et al. 2014) which suggests that firms’ investments in innovation should be smoothing. In fact, there exists a common belief in corporate finance that firms should smooth their R&D expenditure over time (e.g., Brown and Petersen 2011, Sasidharan et al. 2015, He and Wintoki 2016). In contrast, our analysis finds that under the quality authorization there can be significant jumps along the firms’ optimal product innovation investment paths, which suggests that smoothing the R&D expenditure (i.e., investment in innovation) is not the firm’s optimal choice. Further, our analysis reveals that although the dynamic product quality is continuous over time, the optimal production rate is not continuous (so is the product price). The finding is distinct from the results of many existing studies such as Chenavaz (2012), Pan and Li (2016), and Li and Ni (2016). This finding thus challenges the common intuition that a firm’s optimal production and/or pricing policy depends mainly on its current product quality and production cost, as well as the current market conditions (e.g., the price elasticity of demand); in fact, the investment behavior of product innovation is also an important determinant of the firm’s production and pricing policies. Our results have several managerial implications. First, it is important for firms’ managers to pay attention to the quality authorization schemes in deciding on the firms’ investment policy in product innovation. In fact, firms under quality authorization will be better off by substantially accelerating the product innovation investment at the beginning, and ever more so for manufacturing firms characterized by low production cost 22
and low ageing rate of technology. As a result, when deciding a firm’s product innovation investment policy, the manager should carefully consider the planned time for quality authorization and synergically take into account not only the characteristics of the innovation process itself, but also the costs involved in manufacturing process and the financial environment, even when the firm is not financially constrained. Further, under quality authorization managers should also restrain the desire to smooth the R&D expenditure for innovation, though this is a common practice among manufacturing firms (Brown and Petersen 2011). Finally, when deciding on a firm’s production and pricing policies, managers are suggested to take into account not only the current quality level of the product, but also the current investment level for product innovation which affects the quality level in the future. This paper has some limitations that may motivate future research. A possible extension to investigate the effects of quality authorization on product innovation along the supply chain. Due to complexity of this topic, it is not covered in this paper. It should be interesting to examine how the existence of quality authorization will shape the organization of the product innovation investments of the supply chain players (e.g., manufacturer and the supplier), and whether the double marginalization problem along the supply chain will be mitigated or exacerbated under the quality authorization. Another possible extension is to consider the effects of learning by doing in product innovation. Learning by doing, although not considered in our model, is an important factor that can accelerate the product innovation by accumulating knowledge and experience along the innovation process. In fact, how to adequately describe the role of learning by doing in the model is a challenging task, which deserves future research. Acknowledgements: We would like to thank the Editor and three anonymous referees for their insightful comments and suggestions for the revision of the paper. This work was supported by the National Natural Science Foundation of China (71601159, 51405089) and the Science and Technology Planning Project of Guangdong Province (2015B010131008).
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Highlights:
1. We model production decision and product innovation under quality authorization 2. The model is analyzed using Pontryagin maximum principle and backward induction 3. An iterative algorithm is developed to support the decision making of the firm 4. There can be jumps in the innovation investment and production rates over time
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