This article models the intertemporal behaviour of a firm that sets product prices and simultaneously invests in R&D. The model shows that the dynamic pricing rule follows the evolution of the production cost and is independent of the evolution of the product quality. Thus, process innovation, which reduces production cost, is the main determinant of a firm's pricing policy over time. Moreover, the firm invests more in process innovation over time at the expense of product innovation. Hence, the model explains the decrease in the cost of production and in the price of technological products throughout their life cycle.Dynamic pricing, R&D, optimal control

Régis Chenavaz

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 3 
  
Dynamic pricing rule and R&D 
  
 
 
Régis Chenavaz  
ESG Management School 
Abstract 
This article models the intertemporal behaviour of a firm that sets product prices and simultaneously invests in R&D. 
The model shows that the dynamic pricing rule follows the evolution of the production cost and is independent of the 
evolution of the product quality. Thus, process innovation, which reduces production cost, is the main determinant of 
a firm's pricing policy over time. Moreover, the firm invests more in process innovation over time at the expense of 
product innovation. Hence, the model explains the decrease in the cost of production and in the price of technological 
products throughout their life cycle.
I thank Ekaterina Besson, Octavio Escobar, and Christophe Schalck for extremely helpful comments. 
Citation: Régis Chenavaz, (2011) ''Dynamic pricing rule and R&D'', Economics Bulletin, Vol. 31 no.3 pp. 2229-2236. 
Submitted: Jun 06 2011.   Published: August 04, 2011. 
 
     Economics Bulletin, 2011, Vol. 31 no.3 pp. 2229-2236
1. Introduction
Most rms continually increase the quality of product (product innovation) and decrease the
production cost (process innovation). An enhanced handwritting recognition is an example
of product innovation, whereas a cheaper production technique is an example of process
innovation. Simultaneously to its innovation policy, a rm sets its intertemporal pricing
policy. The literature identies two general stylized facts. First, the prices of technological
products decrease over the life cycle (Adner and Levinthal 2001). Second, the rms prioritise
product innovation at the beginning of the life cycle and process innovation at the end
(Utterback and Abernathy 1975, Klepper 1996).
This papers seeks the determinants of dynamic pricing policy for a rm that invests
in both product and process innovation. Process innovation impact is clear: When the
production cost decreases, the rm can lower the product price. Product innovation impact
is ambiguous: When the product quality increases, the rm can increase the markup by
increasing the product price; markup eect is positive. Alternatively, the rm can increase
the sales by decreasing the product price; sales eect is negative.
Dynamic pricing literature studies the analytical properties of demand functions (Chenavaz
et al. 2011). Therefore, the results are robust among dierent subclasses of demand func-
tions (Kalish 1983, Chatterjee 2009). As the products are supposed innovant, this litera-
ture has not explicitly dealt with innovation (Kalish 1983, Chatterjee 2009, Chenavaz and
Leloup 2011). Innovation literature shows that consumers' preferences leads to the reparti-
tion between product and process innovation eorts (Adner and Levinthal 2001) and that
the interaction between technological development and diverse consumer demands leads to
technological change (Saha 2007). However, innovation literature ignored the question of
pricing (Athey and Schmutzer 1995, Mantovani 2006).
This work models the optimal behaviour of a monopolist that prices a product dynam-
ically and achieves an arbitrage in the allocation of R&D between product innovation and
process innovation. This research incorporates both the demand functions from the pricing
literature and the evolution of quality and cost from the innovation literature.
The main result is that the dynamic of product price follows the dynamic of production
cost and is independent of the dynamic of product quality. Thus, process innovation is the
sole determinant of the dynamic pricing policy. Compared to the literature examining pric-
ing and innovation simultaneously (Bayus 1995, Teng and Thompson 1996, V or os 2006), we
give a managerial implication that is very simple to apply: The rm adopts a pricing policy
that simply imitates the dynamic of the cost of production. Moreover, our analytical results
are stronger than those based on numerical simulations (Bayus 1995, Adner and Levinthal
2001, Saha 2007). Another contribution from this work is that the model jointly explains
two stylized facts: During the life cycle, process innovation relatively increases while product
innovation relatively decreases and product price decreases.
1
2230Economics Bulletin, 2011, Vol. 31 no.3 pp. 2229-2236
2. The model
2.1 Demand, production and R&D
This paper studies a monopoly and develops an optimal control model. The time t 2 [0;T]
is continuous and the temporal horizon T is nite. At each instance of t, the monopolist
simultaneously chooses the price and the distribution of R&D allocated to product and
process innovation.
The current demand (or current production)
:
x(t) is of class C2 and depends jointly on
the price p(t) 2 R+ and the quality q(t) 2 R+:
:
x(t) = f (p(t);q (t)); x(0) = x0; (1)
where
:
x(t) 
dx(t)
dt .
We omit now the variables from the functions when they are obvious. The current demand
decrease with price and increase with quality: (
@f
@p  0;
@f
@q  0). The cost of production is
connected to the unit cost c(t) 2 R+ as well as the quantities produced
:
x(t). Cost 
ow is
thus given by c(t)
:
x(t).
The current prot  (t), with values in R, is as follows:
 (t) = (p   c)
:
x:
The rm has a budget R 2 R+ devoted to R&D. It chooses the allocation of R between
product innovation and process innovation. This situation characterises the case where
the total R&D budget is determined in advance by the administrative board and where,
downstream, the director determines its allocation between the dierent types of innovation.
The amounts allocated at a given time t to product innovation and process innovation are
rq (t) and rc (t). Thus,
rq + rc = R: (2)
The product quality increases with product innovation rq and evolves autonomously
according to the rate q 2 [ 1;1]. When q > 0, q captures the technological obsolescence
over time (Lambertini and Mantovani, 2009). Moreover, when q < 0, the process improves
over time (Bayus 1995, V or os 2006), and any improvement is cumulative (Saha 2007).
With the coecient of eectiveness for product innovation 
q 2 R+, the dynamic of
quality is
:
q = 
q ln(rq)   qq; q (0) = q0: (3)
In a similar manner, c 2 [ 1;1] and 
c 2 R+ are the rate of autonomous degradation
and the coecient of eectiveness for process innovation. According to (2), the dynamic of
cost is
:
c = cc   
c ln(R   rq); c(0) = c0: (4)
Without loss of generality, R is such that ln(rq)  0 and ln(R   rq)  0, so that the
impact of the innovation is positive.
2
2231Economics Bulletin, 2011, Vol. 31 no.3 pp. 2229-2236
2.2 Model analysis
Let p(t) and rq (t) represent pricing and product innovation strategies, at time t;8t 2 [0;T]
and let r  0 be the interest rate. Thus, the monopoly's problem is to determine the pricing
p and innovation r
q strategies that maximize prot over time. We have
 
p
;r

q

= argmax
p;rq
Z T
0
e
 rtdt;
under constraints
:
x = f (p;q); x(0) = x0;
:
q = 
q ln(rq)   qq; q (0) = q0;
:
c = cc   
c ln(R   rq); c(0) = c0:
We form the Hamiltonian with the hidden prices e rtq (t); e rtc (t):
H (p;rq;x;q;c;q;c)
= e
 rt ((p   c)f + q (
q ln(rq)   qq) + c (cc   
c ln(R   rq))):
The Hamiltonian maximisation implies the rst-order conditions
@H
@p
= 0 ) p = c  
f
@f
@p
; (5a)
@H
@rq
= 0 ) rq =
R
1  

cc

qq
: (5b)
The maximum principle implies
:
q =  
@H
@q
+ rq; q (T) = 0;
:
c =  
@H
@c
+ rc; c (T) = 0:
Hence,
:
q =  
@f
@q
(p   c)f + (r + q)q; q(T) = 0; (6a)
:
c = f + (r   c)c; c(T) = 0: (6b)
Substituting (5a) in (6a), the solutions are as follows
q (t) =
Z T
t
e
 (r+q)(s t)@f
@q
(p   c)fds;
c (t) =  
Z T
t
e
 (r c)(s t)fds
and imply
q (t)  0; c (t)  0; 8t 2 [0;T]: (7)
3
2232Economics Bulletin, 2011, Vol. 31 no.3 pp. 2229-2236
2.3 Investment in product and process innovation
The evolution of rq over time is given by dierentiating (5b) with respect to t
:
rq =

c

q
 :
cq c
:
q
(q)2


1  

cc

qq
2 :
The rst factor of the numerator and the denominator are positive. Thus, the sign of
:
rq
depends on the sign of the second factor of the numerator. By substituting (6a) and (6b) in
the preceding result, we get
sgn(
:
rq) = sgn

q +
@f
@q
(p   c)c

f   (q + c)qc

: (8)
Proposition 1 If q + c = 0, then product innovation increases with the hidden price of
quality q and decreases with the hidden price of cost c.
Proof Evident with (8), recalling (7).
Proposition 1 also holds when the rates q and c are suciently small or when the tem-
poral horizon T is suciently short to be approximated by zero. Following Utterback and
Abernathy (1975) and Adner and Levinthal (2001), at the beginning of a product's life cycle
consumer interest in product quality is signicant. In contrast, at the end of the cycle the rm
is concerned with the cost of production. This implication of the model is connected to the
relationship between consumer preferences and the rm's technological abilities. It is coher-
ent with the stylized facts from the empirical innovation literature (Utterback and Abernathy
1975, Klepper 1996). In eect, at the beginning of a cycle, innovation develops the product
so that satises consumers' preferences. Specically, innovation develops the product func-
tionalities. Once the product has met the market expectations, innovation helps the rm to
focus on its internal organization. Process innovation thus increases the markup over time by
reducing the costs of production while maintaining consumers' satisfaction with the product.
2.4 Dynamic pricing rule
The second-order condition on p implies1:
@2H
@p2  0 ) 2  
@2f
@p2f

@f
@p
2  0: (9)
Dierentiating (5a) with respect to t, substituting (5a) in the result and rearranging gives
:
p
0
B
@2  
@2f
@p2f

@f
@p
2
1
C
A =
:
c +
:
q
0
B
@
@2f
@p@qf  
@f
@p
@f
@q

@f
@p
2
1
C
A: (10)
1This proof follows Kalish (1983). The other second-order conditions are veried in the Appendix.
4
2233Economics Bulletin, 2011, Vol. 31 no.3 pp. 2229-2236
For general demand function, (10) veries that the impact of product quality on product
price is ambiguous. From the right-hand side of the numerator, the rst term which measures
the sales eect is negative, while the second term wich measures the markup eect is positive.
The impact of product quality on product price depends therefore on the magnitude of sales
and markup eects.
The specication of the demand function (1) gives stronger results. A relatively gen-
eral and unconstrained specication of a demand function is multiplicatively separable into
functions of price and quality such as
:
x = h(p)l(q); x(0) = x0: (11)
Substituting (11) in (10), price dynamics is
:
p
 
2  
d2h
dp2h
(dh
dp)2
!
=
:
c: (12)
Proposition 2 For a multiplicatively separable demand function, the dynamic of price is
determined by cost dynamic and is independent of quality dynamic.
Proof Immediate with (12).
Variations in cost are re
ected in the price. Furthermore, the markup eect (positive)
and the sales eect (negative) linked to the quality of the good cancel exactly each other
for multiplicatively separable functions. Thus, variations in the quality have no impact on
variations in the price. The curve of the pricing policy follows the curve of the cost dynamic.
When more of the R&D allocation is dedicated to process innovation over time, the cost
decreases continuously (
:
c is negative), implying that the price decreases over time, as in Ad-
ner and Levinthal (2001). The decrease in prices of technological goods is well documented
in the literature (Saha 2007), where this is tied to the decrease in willingness to pay for
increased product quality. Typically, at the beginning of a cycle, consumers are particularly
interested in the characteristics of the good and in the improvement in the quality of the
good. Initially, their willingness to pay is higher for improvements in product quality. Over
time, consumers' willingness to pay decreases; these consumers are satised with the basic
functionalities of the product and are more sensitive to price decreases than to quality in-
creases.
3. Conclusion
This paper develops an optimal control model to study the dynamics of a monopolist con-
fronted simultaneously with the problems of product pricing and dividing the R&D allocation
between product innovation and process innovation.
The rst determinant of the pricing policy is process innovation. Indeed, the price dy-
namic is independent of the evolution of quality and follows the evolution of cost. Thus, the
curve of the pricing policy mimics the curve of the production costs. This result is indepen-
dent of the forms of innovation. In agreement with the literature on technological change,
5
2234Economics Bulletin, 2011, Vol. 31 no.3 pp. 2229-2236
product innovation relative to process innovation is more important at the beginning than at
the end of the product life cycle. Thus, process innovation increases over the life cycle of the
product. Furthermore, the cost of production decreases, followed by a subsequent decrease
in the price.
The managerial implication of the model is that a rm should base the intertemporal
pricing policy solely on the evolution of the production cost. A rm does not consider the
evolution of product quality when setting its pricing policy. Thus, when process innovation
increases over time and lowers the cost of production, the prices decrease, regardless of the
change in product quality. The managerial implications of the model jointly explain the
stylized facts from the literature: the relative increase in process innovation and decrease in
product innovation (Utterback and Abernathy 1975) as well as the decrease in the price of
technological goods (Adner and Levinthal 2001).
Appendix
The second-order condiditon on rq gives :
@2H
@r2
q
 0 )  

qq
r2
q
+

cc
(R   rq)2  0: (13)
Because of (7), this condition is satised. The third condition of the second-order for the
maximisation of H is:
@2H
@p2
@2H
@r2
q
  (
@2H
@p@rq
)
2  0
) (2  
@2f
(@p2)2
@f
@p
f)( 

qq
r2
q
+

cc
(R   rq)2)  0;
that is satised thanks to (9) and (13).
References
[1] Athey, S. and A. Schmutzler (1995) "Product and Process Flexibility in an Innovation
Environment" The RAND Journal of Economics 26(4), 557-574.
[2] Adner, R. and D. Levinthal (2011) "Demand Heterogeneity and Technology Evolution:
Implications for Product and Process Innovation"Management Science 47(5), 611-628.
[3] Bayus, B. (1995) "Optimal Dynamic Policies for Product and Process Innovation"
Journal of Operations Management 12(3), 173-185.
[4] Chatterjee, R. (2009) "Strategic Pricing of New Product and Services"In Handbook of
Pricing Research in Marketing Ed. V. Rao, EE.
6
2235Economics Bulletin, 2011, Vol. 31 no.3 pp. 2229-2236
[5] Chenavaz, R., L.-P. Carrier, L. Etienne and C. Paraschiv (2011) "Dynamic
Pricing in Management Science" Journal of Economic Studies and Research
DOI:10.5171/2011.283.
[6] Chenavaz, R. and B. Leloup (2011) "Les d eterminants de la politique de tarication
dynamique sur un march e biface : Le cas des logiciels de type client-serveur" Revue
d'Economie Industrielle 135, 1-26.
[7] Kalish, S. (1983) "Monopolist Pricing with Dynamic Demand and Production Cost"
Marketing Science 2(2), 135-159.
[8] Klepper, S. (1996) "Entry, Exit and Innovation Over the Product Life Cycle"American
Economic Review 86(3), 562-583.
[9] Lambertini, L. and A. Mantovani (2009) "Process and Product Innovation by a Mul-
tiproduct Monopolist: A Dynamic Approach" International Journal of Industrial Or-
ganization 27(4), 508-518.
[10] Mantovani, A. (2006) "Complementarity Between Product and Process Innovation in
a Monopoly Setting" Economics of Innovation and New Technology 15(3), 219-234.
[11] Saha, S. (2007) "Consumer Preferences and Product and Process R&D" The RAND
Journal of Economics 38(1), 250-268.
[12] Sahal, D. (1985) "Technological Guideposts and Innovation Avenues" Research Policy
14(2), 61-82.
[13] Teng, J. and G. Thompson (1996) "Optimal Strategies for General Price-Quality De-
cision Models of New Products with Learning Production Costs" European Journal of
Operational Research 93(3), 476-489.
[14] Utterback, J. and W. Abernathy (1975) "A Dynamic Model of Process and Product
Innovation" Omega 3 3(6), 639-656.
[15] V or os, J. (2006) "The Dynamics of Price, Quality and Productivity Improvement De-
cisions" European Journal of Operational Research 170(3), 809-823.
7
2236

A Dynamic Model of Process and Product Innovation&quot;

Complementarity Between Product and Process Innovation in a Monopoly Setting&quot;

Consumer Preferences and Product and Process R&D&quot;

Demand Heterogeneity and Technology Evolution:

Dynamic Pricing in Management Science&quot;

Entry, Exit and Innovation Over the Product Life Cycle&quot;American Economic Review

Les d eterminants de la politique de tari dynamique sur un march e biface : Le cas des logiciels de type client-serveur&quot;

Monopolist Pricing with Dynamic Demand and Production Cost&quot;

Optimal Dynamic Policies for Product and Process Innovation&quot;

Optimal Strategies for General Price-Quality Decision Models of New Products with Learning Production Costs&quot;

Process and Product Innovation by a Multiproduct Monopolist: A Dynamic Approach&quot;

Product and Process Flexibility in an Innovation Environment&quot; The RAND

Strategic Pricing of New Product and Services&quot;In Handbook of Pricing Research

Technological Guideposts and Innovation Avenues&quot;

The Dynamics of Price, Quality and Productivity Improvement Decisions&quot;

Dynamic pricing rule and R&D

http://www.accessecon.com/Pubs/EB/2011/Volume31/EB-11-V31-I3-P202.pdf

Dynamic pricing rule and R&D

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics