Firm’s Objective Function and Product and Process
R&D
Souresh Saha∗
Abstract
Firms undertake different kinds of R&D activities. They do product R&D (R&D
aimed at improving the quality of existing products, and creating new products). They
also do process R&D (R&D aimed at lowering the cost of making existing and new
products). Moreover, firms often do both product and process R&D simultaneously. As
far as the objective of firms is concerned, this need not be limited to profit-maximization
only. Rather, firms may have a broader objective, where they care about profits as well
as consumer surplus. This paper studies effects of a firm having a general objective
function (that takes into consideration both profits and consumer surplus) on its
product and process R&D choices, and corresponding implications.
I consider product and process R&D choices of firms in an infinite horizon set-up
with discrete time. Firms in my framework can simultaneously do both product and
process R&D in every period, face a discrete-choice model of consumer demand with
vertical product differentiation, and maximize a discounted, weighted sum of their
profits and consumer surplus over the infinite time horizon.
I show how process and product R&D differ from each other in my framework,
and the role of a firm’s objective function in this regard. I characterize how in my
framework, the choice of process R&D in total R&D - R&D composition - by a firm
varies over time, and also how process and product R&D choices, process and product
R&D productivity, and the choice of R&D composition vary across firms that differ in
size but are otherwise similar. I also compare process and product R&D choices across
firms that differ in their objective function, and illustrate effects of providing general
fiscal incentives for R&D (incentives given for any R&D, regardless of whether it is
product or process R&D) to firms.
JEL Classification Number(s): O33, O32, L20
Keywords: General objective function; Process R&D; Product R&D; R&D incentives
∗
Brunel Business School, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK;
E-mail: [email protected]; Phone: +44 (0)1895 267696
1
1
Introduction
R&D choices by firms are of vital importance, not just to individual firms themselves but
to economies and nations as a whole (Vives, 2008; Greenhalgh and Rogers, 2010). This,
together with the often-held notion that R&D choices by private firms differ significantly from
corresponding socially optimal choices, has in fact over the years led governments all over the
world, in both developed as well as developing nations, to engage in various efforts to promote
R&D activities (Mansfield, 1986; Hall and Van Reenen, 2000; Klette et al., 2000; Bloom et
al., 2002; Whalley and Zhou, 2007; Hall and Maffioli, 2008; Özcelik and Taymaz, 2008).
Studies regarding R&D have typically looked at choices of overall R&D levels by profitmaximizing firms. However, as discussed in greater detail below, it is increasingly being
recognized that firms (a) undertake different kinds of R&D activities, and (b) may have a
broader objective than profit-maximization alone.
Within R&D, firms undertake different kinds of R&D activities. Firms do product R&D,
defined as R&D aimed at trying to improve the quality of existing products, and creating
new products. Firms also do process R&D, defined as R&D aimed at improving existing
processes, and creating new processes so as to lower the cost of making existing and new
products. Further, firms typically do both product and process R&D simultaneously (Capon
et al., 1992; Landau and Rosenberg, 1992), and do so in an incremental fashion and on a
continuing basis rather than in a one-off manner (Kline and Rosenberg, 1986; Bayus, 1995).
The need to understand not just overall R&D efforts but also its composition in terms
of the division between product and process R&D is reflected for example in the growing
theoretical literature on issues related to simultaneous process and product R&D choices by
firms (Athey and Schmutzler, 1995; Cohen and Klepper, 1996a; Eswaran and Gallini, 1996;
Klepper, 1996; Lambertini and Orsini, 2000; Lin and Saggi, 2002; Lambertini, 2003, 2004;
Rosenkranz, 2003; Bandyopadhyay and Acharyya, 2004; Lin, 2004; Mantovani, 2006; Saha,
2007; Lambertini and Mantovani, 2009; Plehn-Dujowich, 2009; Chenavaz, 2011). Some of
these studies have considered one-off product and process R&D choices by firms, while others
have considered product and process R&D choices by firms over time.
It is also increasingly being recognized that firms need not have only profit-maximization,
or alternatively welfare-maximization as their objective. Rather, firms may have an objective
2
broader than that of only profit-maximization, where they care, at least to some extent, about
consumer surplus as well as profits, without necessarily being welfare-maximizing.
This broader objective may be due to one or more, fairly commonly observed factors like
(a) the implementation by firms, investor-owned or otherwise, of the concept of “stakeholder
society” (Tirole, 2001; Ghosh and Mitra, 2012; Willner, 2012), (b) partial government
ownership of firms (Matsumura, 1998), and (c) firms having organizational forms other than
that of investor-owned firms, such as various kinds of non-profits and co-operatives, and
the corresponding allocation of residual control rights in an environment with incomplete
contracts (Hansmann, 1996; Hart and Moore, 1996; Hart et al., 1997; Glaeser and Shleifer,
2001; Levin and Tadelis, 2005; Herbst and Prüfer, 2011).
Moreover, the broader objective need not be the same across all firms. Instead, firms may
differ in the extent to which they care about profits and consumer surplus owing to differences
in the extent to which they care about various stakeholders and what the stakeholders care
about, the extent of government control, and/or their organizational form.
It would perhaps be natural at this point to ask how important it might be to consider
firms that have a broader objective than profit-maximization alone. There are increasing
trends of (i) the concept of corporate social responsibility (CSR) becoming both popular
among consumers as well as a mainstream business activity (Berger et al., 2007; Besley
and Ghatak, 2007; Kitzmueller and Shimshack, 2012; Kopel and Brand, 2012), and (ii)
privatization, usually partial, of state-owned enterprises around the world (Megginson, 2005;
Chen et al., 2009; Wang and Chen, 2011). These two trends suggest that the number of
firms that care not just about profits but also, though perhaps not necessarily to the same
extent, about consumer surplus is only likely to increase, and that too perhaps rapidly, in
coming years.1
To summarize the discussion so far, it is important within R&D to consider not just overall
R&D levels but to understand simultaneous process and product R&D choices by firms over
time. It is also likely that firms with an objective broader than that of profit-maximization
1
Note that the number and contribution of firms, both now and previously, with an objective function
broader than that of profit-maximization alone is not insignificant to begin with, and moreover this is true in
R&D-intensive sectors as well (Bartlett et al., 1992; Hansmann, 1996; Anderson et al., 1997; Nilsson, 2001;
Oehmke, 2002; Godø et al., 2003; Case, 2005; Heisey et al., 2005; Goering, 2008b; Bühler and Wey, 2010;
Gil-Moltó et al., 2011; Marini and Zevi, 2011; Ohnishi, 2011; Kopel and Marini, 2012).
3
alone will become increasingly more common. Given the overall importance of R&D activities
to individual firms as well as to society as a whole, it thus seems imperative that we consider
these two aspects of firm behavior together. More specifically, we must try to understand
effects of a firm having a general objective function - i.e., an objective function that takes into
consideration, though not necessarily to the same extent, both profits and consumer surplus on its process and product R&D choices over time, and corresponding policy implications.
However, to the best of my knowledge, R&D choices by firms with an objective broader
than that of profit-maximization alone, in scenarios where they can simultaneously do both
product and process R&D over time has not been studied till date. There are two separate
strands of theoretical literature that come closest in this regard. One is the literature
discussed earlier on issues related to simultaneous process and product R&D choices by
firms. This literature however has only considered the case of profit-maximizing firms, and in
some cases (Eswaran and Gallini, 1996; Lambertini and Orsini, 2000; Lin and Saggi, 2002;
Lambertini, 2003; Rosenkranz, 2003) corresponding welfare implications. There is another
body of theoretical work that has considered issues related to R&D choices by firms with
objective functions different from that of only profit-maximization (Delbono and Denicolo,
1993; Lambertini, 1998; Poyago-Theotoky, 1998; Nishimori and Ogawa, 2002; Ishibashi and
Matsumura, 2006; Goel and Haruna, 2007; Cato, 2008a, 2008b; Heywood and Ye, 2009;
Bühler and Wey, 2010; Gil-Moltó et al., 2011; Kesavayuth and Zikos, 2013; Luo, 2013). This
second literature however has considered one-off choices regarding in most cases only process
innovations, or in some cases only product innovations.
In this paper, I study product and process R&D choices by firms, and corresponding
implications, for the case where firms have a general objective function and can do both
product and process R&D simultaneously over time.
The rest of the paper is as follows. Section 2 lays down the basic framework. Section 3
contains the findings of this paper. Section 4 concludes. All proofs are in the Appendix.
2
2.1
The Model and Preliminaries
The Model
I consider the case of a single firm that sells a non-durable product in an infinite horizon
set-up. This firm is alone in the market for the product, and is unaffected by choices or
4
actions of any other firms.2
In my framework, time is discrete, and there are infinitely many time periods denoted by
t, where t = 0, 1, 2, . . . , .
The firm is characterized by two parameters, namely q, and c (these, as well as the
objective function of the firm are described in detail later).
There is a numeraire good, and price of and utility from the product, costs, revenues,
and expenditures are all measured in terms of this numeraire good.
The demand for the firm’s product is as follows.
I assume that in each period, the firm faces an exogenously given mass of potential buyers.
This mass of potential buyers is the same in all periods, and is normalized to be 1.
I model consumer preferences using a discrete-choice model of consumer demand with
vertical product differentiation (Mussa and Rosen, 1978; Maskin and Riley, 1984). In my
framework, the parameter q stands for the quality of the product sold by the firm. All
potential buyers, and the firm agree on the value of q. Each potential buyer has a finite,
non-negative preference parameter θ, where θ ∈ [θ1 ,θ2 ]. Also, in each period, every potential
buyer buys either a single unit or none of the product. The net utility to a buyer with
preference parameter θ from buying one unit of the product of quality q from the firm at
price P is θq − P .
A buyer’s purchasing decision regarding the product in any period does not affect his/her
utility or choice set in any way in subsequent periods. Also, each potential buyer has a
reservation utility of 0 in every period. Hence, in any period, a potential buyer buys one
unit of the product from the firm if the resultant net utility from doing so is non-negative.
The value of θ differs across buyers. I use F (.) to denote the distribution of θ across
all potential buyers. F (.) is exogenously given, and time-invariant, and admits a pdf f (.),
where f (.) > 0, ∀ θ ∈ [θ1 ,θ2 ].
The firm faces no potential threat of entry, and cannot discriminate between various
potential buyers.
On the production side, the firm has a production function with a constant marginal
cost, and zero fixed costs. The parameter c denotes the marginal cost of the firm, and its
2
Such a set-up would for example be consistent with an industry having identical and independent
submarkets where there is at most one firm in each submarket (Bresnahan et al., 1997; Sutton, 1998).
5
value is independent of the quality, namely q, of the firm’s product.
The firm can undertake both process R&D (i.e., try to lower the value of c), and product
R&D (i.e., try to increase the value of q)3 in every period. It does not have any financial
constraints with respect to its R&D expenditures.
Both cost reductions, and quality improvements add up over time.
Outcomes of R&D efforts undertaken by the firm are deterministic, and as follows.
I call (q,c), the characteristic vector of the firm, and denote by (qt , ct ) the characteristic
vector that the firm starts off with in any period t.
I use Etc to denote the process R&D expenditure, and Etq to denote the product R&D
expenditure of the firm in any period t. Then, at the end of period t, the firm has a per-unit
cost given by ct+1 = ct − Act g(Etc ), and product quality given by qt+1 = qt + Aqt g(Etq ).4 Here,
g(.) is an exogenously given R&D production function that is common to both process, and
product R&D. Act and Aqt are the R&D productivity, and also reflect the extent of technological opportunity for process and product R&D respectively in period t. (Aqt , Act ) ∈ <2++ ,
∀t ≥ 0, and are exogenously given.5 I make further assumptions about g(.), and (Aqt , Act )t=∞
t=0
in subsection 2.2.
As far as initial values of quality and per-unit cost are concerned, the firm starts off
in period 0 with an exogenously given characteristic vector (q0 , c0 ), where (q0 , c0 ) ∈ <2++ .
Subsequently, the values of q and c for the firm evolve over time, as described above,
depending on the exogenously given R&D opportunities, and the firm’s R&D efforts.
As for output, in any period t, the firm undertakes production after the realization of the
3
Product quality often has multiple dimensions, and consumers may differ in their needs along these
different dimensions. Also, firms often try to, at times simultaneously, improve product quality in several
different directions. One way of interpreting my assumption of one-dimensional quality, and product R&D as
trying to improve this one-dimensional variable is to think of q as a composite index for quality, and product
R&D as improving this composite index. Alternatively, it can be thought of as the firm trying to identify, and
improve the commonalities in the product features that most consumers value (Kim and Mauborgne, 1997).
4
This functional form of returns to R&D has been used elsewhere in the literature (Klepper, 1996). It is
also analogous to the form of returns assumed in Sutton (1998).
5
Note that the values of Act , and Aqt can be different from each other in any period, and can also vary
over time. Thus, although I use the same function g(.) in the modeling of both process, and product R&D
for analytical tractability, returns to process and product R&D in any period need not be the same, and can
also vary (both in absolute and relative terms) over time.
6
outcome of its R&D efforts in that period. Thus, in any period t, the firm sells a product of
quality qt+1 that it produces at a per-unit cost of ct+1 (where qt+1 and ct+1 are as described
above). The firm does not have any capacity constraints regarding production.
As far as the objective of the firm is concerned, the firm has a discount parameter
β ∈ [0,1), and a non-negative weight α that it assigns to consumer surplus, and maximizes
the discounted sum of [profits + α(consumer surplus)] over the entire time horizon.
Here, α = 0 corresponds to the case of the standard profit-maximizing firm.
However, as discussed in the Introduction, the firm may care not just about profits but
about consumer surplus as well. Moreover, if the firm cares about both profits and consumer
surplus, then it may care equally about the two (α = 1; i.e., maximize welfare), or it may
assign a lower (0 < α < 1) or higher (α > 1) weight to consumer surplus than to profits.6
In fact, the formulation of the objective function of the firm being used here can accommodate a variety of objective functions corresponding to different scenarios that have been
used in the literature (Matsumura, 1998; Lien, 2002; Goering, 2007, 2008a, 2008b; Kopel
and Brand, 2012; Lin and Matsumura, 2012; Willner, 2012).
Throughout this paper, I consider the fraction of its potential buyers that the firm chooses
to sell its product to in any period t (the degree of market penetration). I denote this by
dt . Clearly, dt ∈ [0, 1]. Note that this is equivalent to looking at the choice of output by the
firm. Also, given the assumptions made regarding demand for the firm’s product, it follows
that if in any period the firm sells a product of quality q and decides to sell its product to a
fraction d of its potential buyers, then it receives a per-unit price of [F −1 (1 − d)]q. Further,
the corresponding consumer surplus equals
R θ2
F −1 (1−d) [θ
− F −1 (1 − d)]qf (θ)dθ in such a case.
I use Γ to denote {(q0 , c0 ), [θ1 , θ2 ], F (.), β, α, g(.), [Act , Aqt ]t≥0 }. Γ is the set of all building
blocks of my model.
Given Γ, suppose the firm chooses the R&D expenditure sequence [Etc , Etq ]t≥0 . Then
at the end of any period t, it has a per-unit cost ct+1 = {c0 −
6
Pk=t
k=0
Ack g(Ekc )}, and quality
Welfare maximization is often assumed to be the goal of a public or a regulated firm. More generally, it
is typically assumed that a government or a regulator (and hence a public or a regulated firm) cares at least
equally if not more about consumer surplus than profits (Armstrong and Sappington, 2007). Also, note that
even if a government or a regulator cares more about consumer surplus than about profits, a firm with mixed
ownership that is jointly owned by the private and public sectors may depending on the extent of government
control end up assigning a higher or a lower weight to consumer surplus than to profits (Matsumura, 1998).
7
qt+1 = {q0 +
Pk=t
k=0
Aqk g(Ekq )}. Given this characteristic vector (qt+1 , ct+1 ), it chooses dt ∈ [0, 1]
to maximize the value of [product market profit + α(consumer surplus)] in period t; i.e., it
chooses dt ∈ [0, 1] such that
dt ≡ argmaxd∈[0,1] [(qt+1 F −1 (1 − d) − ct+1 )d + αqt+1 {
R θ2
F −1 (1−d) [θ
− F −1 (1 − d)]f (θ)dθ}].7
Note that
R θ2
[(qt+1 F −1 (1 − d) − ct+1 )d + αqt+1 {
F −1 (1−d) [θ
− F −1 (1 − d)]f (θ)dθ}]
=
[(1 − α)dF −1 (1 − d) + α
R θ2
F −1 (1−d) θf (θ)dθ]qt+1
− dct+1 .
Hence, the dynamic optimization problem faced by a firm in my framework is given by
max
X
[Etc ,Etq ]t≥0 t≥0
t
β [(1−α)dt F
−1
k=t
X
k=t
X
q
q
q
c
c
c
(1−dt )+αS(dt )]{q0 + Ak g(Ek )}−dt {c0 − Ak g(Ek )}−Et −Et
k=0
k=0
(1)
where S(d) ≡
Z θ2
F −1 (1−d)
θf (θ)dθ, ∀d ∈ [0, 1], and
dt ≡ argmaxd∈[0,1] [(1−α)dF −1 (1−d)+αS(d)]{q0 +
k=t
X
Aqk g(Ekq )}−d{c0 −
k=0
k=t
X
Ack g(Ekc )}, ∀t ≥ 0.
k=0
I next make some assumptions so as to ensure that in my framework, optimal product
and process R&D expenditure choices of the firm are positive but finite in all periods, and
its dynamic optimization problem is well-defined.
2.2
Assumptions regarding Γ, and some preliminary dynamics
First, I make some assumptions regarding F (.) and α.
I assume (i) θ is uniformly distributed over [θ1 , θ2 ], and (ii) α < 2. These two assumptions
together imply strict concavity of [(1 − α)dF −1 (1 − d) + αS(d)] in d, ∀d ∈ [0, 1]. This means
that in my framework, the firm always faces a downward-sloping “marginal revenue” curve
(since, [(1 − α)dF −1 (1 − d) + αS(d)]qt+1 is the “revenue function” faced by the firm in any
period t), and has a unique choice as far as optimal output in any period is concerned.
7
Thus, ∀t ≥ 0, dt denotes the optimal choice of output (or equivalently, the degree of market penetration)
in period t for the firm given its choice of process and product R&D expenditures ([Etc , Etq ]t≥0 ) over time.
8
Second, I make some assumptions regarding g(.), the R&D production function common
to both product and process R&D.
I assume: (i) g(0) ≥ 0, (ii) g(.) is monotonically increasing, strictly concave, and differentiable in E, ∀E ∈ <+ , (iii) limE→0 g 0 (E) = ∞, and (iv) limE→∞ g 0 (E) = 0. This in turn
implies that returns to both product and process R&D in each period have these properties
as well. These are standard assumptions regarding returns to R&D, and are usually made
in the R&D literature.
Third, I make some assumptions in order to ensure that product and process R&D
expenditure choices, improvements in product quality, and reductions in per-unit cost are
well-behaved in my framework.
Here, productivity of process R&D (namely, Act in period t), and that of product R&D
(namely, Aqt in period t), can vary over time. I assume that both Act and Aqt are finite, ∀t ≥ 0.
This, together with the assumption that limE→∞ g 0 (E) = 0, ensures that the firm chooses
to spend a finite amount on process or product R&D in any period.
I define h(.) ≡ [g 0 ]−1 (.). Given the assumptions regarding g(.) made above, h(.) is
a well-defined function on (0, ∞). I define
maxd∈[0,1] [(1−α)dF −1 (1−d)+αS(d)], and
P
P
t≥0
t≥0
1−β
Aqt g(h( γ(α)A
q )) ≡ ∆qu , where γ(α) ≡
t
)) ≡ ∆cu . ∆qu is an upper bound
Act g(h( 1−β
Ac
t
for the improvement in product quality, while ∆cu is an upper bound for the reduction in
per-unit cost over the entire time horizon. I assume: (i) both ∆qu , and ∆cu are finite, and
(ii) ∆cu ≤ c0 . This ensures that in my framework (a) total cost reduction, and total quality
improvement over the entire time horizon are both finite (i.e., per-unit cost is always greater
than a finite lower bound ≡ cmin = c0 − ∆cu , while product quality is always less than a
finite upper bound ≡ qmax = q0 + ∆qu ), and (b) per-unit cost is always non-negative.
Given these assumptions, the dynamic optimization problem of the firm as described by
equation (1) above is well-defined.
I make some additional assumptions to facilitate my analysis.
I assume: θ2 q0 > c0 . This implies that in my framework, the firm definitely chooses
to sell a positive amount of output in period 0 (i.e., d0 > 0). Also, this and assumptions
made regarding returns to product and process R&D together imply that in my framework,
(i) the firm spends a positive amount on both product and process R&D in every period,
(ii) there is an increase in product quality, and a reduction in per-unit cost across any
9
two consecutive time periods (i.e., qt+1 > qt , while ct+1 < ct , ∀t ≥ 0), and (iii) the firm
sells a positive amount of output in each period (i.e., dt+k > 0, ∀k > 0). I also assume:
[(2 − α)θ1 − (1 − α)θ2 ]qmax < cmin . This implies dt < 1, ∀t ≥ 0, i.e., the market is never
completely saturated. It also implies that in my framework, the firm chooses an output
sequence that is increasing over time, i.e., 0 < dt < dt+1 < 1, ∀t ≥ 0. The two assumptions
let me present my findings simply without having to state qualifiers as to what happens if
(i) the firm sells no output initially, or (ii) the firm sells to all its potential buyers from some
point of time onwards.8
I also make an assumption regarding ratios of marginal returns from R&D and R&D
expenditures. I assume: g(.) is such that
g 0 (E2 )
g 0 (E1 )
>
g 0 (E4 )
g 0 (E3 )
⇒
E2
E1
<
E4
,
E3
∀ (E1 , E2 , E3 , E4 ) ∈ <4++ .
I need this assumption for analytical tractability only. It allows me to analytically derive
results regarding choices by firms of the ratio of process to product R&D expenditure. The
economic intuition corresponding to such results is, in fact, independent of this assumption.
The significance of the assumption made in the paragraph above is as follows. It says
that if E1 , E2 , E3 , and E4 are values for R&D expenditures, then g(.) has the feature that
a higher(lower) ratio of marginal returns from R&D implies a lower(higher) ratio of R&D
expenditures. If (E1 , E2 , E3 , E4 ) is such that E3 ≤ E1 and E4 ≥ E2 , or E3 ≥ E1 and
E4 ≤ E2 , with at least one strict inequality in either case, then this property, in fact, follows
simply from the strict concavity of g(.). The assumption ensures this property holds when
(E1 , E2 , E3 , E4 ) is such that E4 > E2 and E3 > E1 , or E4 < E2 and E3 < E1 , as well.
It would be natural to ask if there are R&D functions that satisfy all the assumptions
that have been made. If g(.) is of the form: g(E) = E λ + k, where λ ∈ (0,1) and k ≥ 0 (this,
in fact, is an often-used functional form for returns to R&D), then the basic assumptions
regarding g(.), and the assumption regarding ratios of marginal returns from R&D and R&D
expenditures hold. Further, the assumptions that both ∆qu and ∆cu are finite, and ∆cu ≤ c0
hold in such a case, if c0 is high enough and both Act and Aqt → 0 sufficiently quickly as t → ∞.
8
From a real-life point of view, it seems highly unlikely that a firm will ever find it optimal to sell its
product to all of its potential buyers. Further, findings of the paper remain qualitatively the same even if
one or both of these scenarios occur.
10
3
Findings
I begin by considering the difference between process and product R&D in my framework.
3.1
Process and Product R&D: Economic Difference
As set out in subsection 2.1, equation (1) describes the dynamic optimization problem faced
by the firm in my framework. This, and the envelope theorem together imply that as far as
R&D expenditures are concerned, the following two FONCs must hold in every period FONC for Etc [process R&D]: [
X
β k dt+k ]Act g 0 (Etc ) = 1,
(2)
k≥0
FONC for Etq [product R&D]: [
X
β k dt+k {(1−α)F −1 (1−dt+k )+α(S(dt+k )/dt+k )}]Aqt g 0 (Etq ) = 1.
k≥0
(3)
The reason as to why the FONC for optimal choices of
Etc
and
Etq
in any period t are as
given by equations (2) and (3) above respectively is as follows.
Let us begin by considering the firm’s choice of process R&D expenditure. Suppose that
in some period t, the firm has already spent the amount E c on efforts to lower its per-unit
cost. If the firm then decides to spend one more (less) unit on process R&D in period t,
then this will ceteris paribus lead to a decrease (increase) in its per-unit cost by an amount
of [Act g 0 (E c )] in period t, and in all subsequent periods. The benefit (loss) to the firm from
this cost reduction (increase), evaluated in period t while keeping its output in period t and
all subsequent periods unchanged, is the discounted sum of the product of its (a) output,
and (b) benefit (loss) per-unit output from the change in per-unit cost (namely, Act g 0 (E c ) the amount of reduction (increase) in per-unit cost), from period t onwards. At the optimal
choice of process R&D expenditure in any period, a unit increase (decrease) in process R&D
expenditure must result in a corresponding benefit (loss) of exactly 1. Moreover, by the
envelope theorem, any change in Etc has no effect on the optimal choice of d in any period
from period t onwards. Hence, the optimal process R&D expenditure choice of the firm in
any period must satisfy equation (2).
As for product R&D, ceteris paribus, the benefit (loss) per-unit output to the firm, in
any period t + k where k ≥ 0, from a unit increase (decrease) in product R&D expenditure
in period t if it has already spent E q in that period in efforts to improve the quality of its
11
product is given by [{(1−α)x(the marginal buyer’s WTP) + α(the average WTP of buyers)}
for each unit of quality improvement in period t + k]x[Aqt g 0 (E q )]. The same logic as outlined
in the preceding paragraph for process R&D then leads to equation (3).
The following key economic difference thus emerges in my framework between process
and product R&D. We find that the per-unit output return to process R&D is independent
of consumer preferences, and hence of who the firm sells its product to. The per-unit output
return to product R&D however depends on both the marginal buyer’s WTP, and the average
WTP of buyers for quality improvements (with the extent to which it depends on each of
these two factors being determined by the objective function of the firm),9 and thus on who
the firm sells its product to. Formally,
Proposition 1. (Economic difference between process and product R&D) For a given sequence
of output sold by the firm over time, in any period,
(i) returns from process R&D depend only on the sequence of outputs from that period on,
and the process R&D productivity in that period,
whereas
(ii) returns from product R&D depend on the sequence of outputs from that period on,
product R&D productivity in that period, and a weighted sum (where the weights are
determined as per the objective function of the firm) of the marginal buyer’s WTP and
the average WTP of buyers for quality improvements in that and subsequent periods.
I next consider how the choice of R&D composition (the ratio of process R&D expenditure
to total R&D expenditure) by the firm varies over time in my framework.
3.2
R&D composition over time
I define ect ≡ Etc /(Etc + Etq ), ∀t ≥ 0. Thus, ect represents the share of process R&D in the
firm’s total R&D expenditure in period t.
Changes over time in the opportunity for process relative to that for product R&D (in my
framework, this corresponds to changes over time in the exogenously given values of Act /Aqt )
will obviously have an effect on how the choice of R&D composition by the firm varies with
time. I abstract away from such technological changes in order to focus exclusively on how
9
I develop implications of the difference in objective function across firms in subsection 3.4.
12
consumer preferences, through its different effects on returns to process and product R&D
(as discussed in the previous subsection), affect the choice of R&D composition by the firm
over time. I find that
Proposition 2. (R&D composition: Evolution over time) Consider the firm as described
in Section 2, and facing the dynamic optimization problem as given by equation (1), and
00
0
00
0
suppose ∃ periods t , t , such that Act00 /Aqt00 = Act0 /Aqt0 , where t > t ≥ 0.
Then, ect00 > ect0 (i.e., if the firm faces the same relative technological opportunity for process
and product R&D, then it does relatively more of process R&D over time).
It has been observed in several industries that firms do relatively more of process R&D
over time (Klepper, 1996; Filson, 2001). Proposition 2 shows that this can be the case even
if firms have a more general objective function than profit maximization alone.
The intuition underlying Proposition 2 is as follows.
In my framework, the firm while deciding about process and product R&D expenditures
in any period considers the corresponding benefits that it receives in that and subsequent
periods from such activities. Note that the benefit to the firm per-unit output in any period
from a unit cost reduction relative to that from a unit quality improvement is given by
1/[(1 − α) times the marginal buyer’s WTP + α times the average WTP of buyers for each
unit of quality improvement]. Also, as discussed in Section 2, in my framework, the firm
sells to a greater fraction of its potential buyers (i.e., achieves greater market penetration)
over time. It turns out that [(1 − α) times the marginal buyer’s WTP + α times the
average WTP of buyers for each unit of quality improvement] decreases as the extent of
market penetration increases.10 This in turn implies that over time, given the same relative
technological opportunity for process and product R&D, the firm relatively speaking benefits
more from process than from product R&D and hence does relatively more of process R&D.
I next consider how differences in firm size affect their R&D choices in my framework.
10
Note that both the marginal buyer’s WTP, and the average WTP of buyers for each unit of quality
improvement decrease as the degree of market penetration increases. Hence, (1 − α) times the marginal
buyer’s WTP + α times the average WTP of buyers for each unit of quality improvement is decreasing
in the degree of market penetration for α ∈ [0, 1]. For α ∈ (1, 2), the firm puts a negative weight on the
marginal buyer’s WTP for improvements in product quality. However, even in such a case, (1 − α) times the
marginal buyer’s WTP + α times the average WTP of buyers for each unit of quality improvement turns
out to be decreasing in the degree of market penetration.
13
3.3
Effects of difference in firm size
Throughout this subsection, I consider two firms - Firm L and Firm S - that differ in size
(measured in terms of output), and compare their choices of amount of process and product
R&D expenditures and R&D composition, as well as their R&D productivity (defined as the
amount of innovation per unit of R&D expenditure).
I assume that both Firm L and Firm S are in a set-up as described in Section 2, face the
dynamic optimization problem as given by equation (1), and make their process and product
R&D expenditure and output choices over time accordingly.
Also, throughout this subsection, all variables related specifically to Firm i in any period t
will have the subscript (i, t) (i if the variable is time-invariant), where i ∈ {L, S}, and t ≥ 0.
I also assume that (a) except for the initial characteristic vector, both Firm L and Firm S
have the same value for all other components of Γ (namely, [θ1 , θ2 ], F (.), β, α, g(.), [Act , Aqt ]t≥0 ),
and (b) Firm L is always larger than Firm S from some period onwards. Thus, a comparison
across Firm L and Firm S corresponds to comparing two firms of different size (measured
in terms of output), from different industries or from different submarkets within the same
industry, that have the same objective function, and face similar demand conditions and
R&D opportunities.
Regarding process and product R&D expenditure and R&D productivity, I find that
Proposition 3. (Firm size, amount of process and product R&D expenditure, and R&D
0
productivity) Consider Firm L and Firm S as described above, and suppose ∃ some t such
that dL,t0 +k > dS,t0 +k , ∀k ≥ 0 (i.e., Firm L sells more output than Firm S in every period
0
from some period t onwards). Then, ∀k ≥ 0,
r
r
r
r
r
r
(a) EL,t
> ES,t
; while (b) Art0 +k g(EL,t
)/EL,t
< Art0 +k g(ES,t
)/ES,t
; r ∈ {c, q},
0
0
0
0
0
0
+k
+k
+k
+k
+k
+k
0
(i.e., in every period from period t onwards, (the larger firm) Firm L, does more of, but has a
lower R&D productivity for, both process and product R&D, than (the smaller firm) Firm S).
The intuition underlying Proposition 3 is as follows.
From equation (2), it is immediately obvious that in my framework there are scale effects
in returns to process R&D. Also, [(1 − α) times the marginal buyer’s WTP + α times the
average WTP of buyers for each unit of quality improvement]x[degree of market penetration]
increases as the extent of market penetration increases over the range of values of market
14
penetration chosen by a firm given its optimization problem. Hence, from equation (3) it
follows that there are scale effects in returns to product R&D as well. Consequently, when
both firms have the same opportunities for process and product R&D, (the larger firm)
Firm L gets greater benefit from conducting both process and product R&D, and therefore
ends up doing more of both than (the smaller firm) Firm S.
Regarding R&D productivity, in my framework diminishing returns to R&D implies that
the amount of innovation per unit of R&D expenditure and the amount of R&D expenditure
are inversely related. Since (the larger firm) Firm L does more of both process and product
R&D, it has a lower R&D productivity for both than (the smaller firm) Firm S.11
Regarding a comparison of the choice of R&D composition across Firm L and Firm S, I
begin by considering the special case where both firms are completely myopic (i.e., β = 0).
I find that
Proposition 4. (Firm size and R&D composition: Special case - β = 0) Consider Firm L and
0
Firm S as described at the beginning of this subsection, and suppose ∃ some t such that
dL,t0 +k > dS,t0 +k , ∀k ≥ 0 (i.e., Firm L sells more output than Firm S in every period from
0
some period t onwards), and that β = 0 (i.e., both firms are completely myopic).
Then, ∀k ≥ 0, ecL,t0 +k > ecS,t0 +k ,
0
(i.e., in every period from period t onwards, (the larger firm) Firm L does relatively more
of process R&D than (the smaller firm) Firm S).
The intuition underlying Proposition 4 is as follows.
Here, since β = 0, both firms while deciding about process and product R&D expenditure
in any period consider the corresponding benefit that they receive from such activities in that
period only. As discussed earlier (following Proposition 2), in my framework, the benefit to
a firm per-unit output in any period from a unit cost reduction relative to that from a unit
quality improvement is given by the value of 1/[(1 − α) times the marginal buyer’s WTP +
11
Here, I am considering the case where the firm has a general objective function, and can possibly care
about both profits as well as consumer surplus. The standard profit-maximizing firm is a special case, namely
α = 0, in my framework. Thus, Propositions 1-3 here generalize corresponding findings in Saha (2007), which
considers the case of a profit-maximizing firm only, regarding the economic distinction between process and
product R&D, the evolution of R&D composition over time, and the effect of firm size on process and product
R&D expenditure choices and corresponding R&D productivity.
15
α times the average WTP of buyers for each unit of quality improvement] in that period.
Firm L being larger than Firm S implies that it has greater market penetration. In my
framework, [(1 − α) times the marginal buyer’s WTP + α times the average WTP of buyers
for each unit of quality improvement] is decreasing in the extent of market penetration.
Consequently, (the larger firm) Firm L relatively speaking gets more benefit from process
compared to that from product R&D, and hence finds it optimal to do relatively more of
process R&D than (the smaller firm) Firm S.
I next compare the choice of R&D composition across Firm L and Firm S for the general
case when they are non-myopic (i.e. β 6= 0). Before proceeding further,
I need to define a
P
j=k
j
(β d
)
t,k
Pj=0 j i,t+j .
variable, st,k
i (t ≥ 0, k ≥ 0, and i ∈ {L, S}), as follows: ∀k ≥ 0, si ≡
(β di,t+j )
j≥0
st,k
i
Note that
is simply the share of (discounted) output of Firm i (where i ∈ {L, S})
over periods t to t + k in its total (discounted) output from period t onwards. I find that
Proposition 5. (Firm size and R&D composition: General case - β 6= 0) Consider Firm L
0
and Firm S as described at the beginning of this subsection, and suppose ∃ some t such that
dL,t0 +k > dS,t0 +k , ∀k ≥ 0 (i.e., Firm L sells more output than Firm S in every period from
0
some period t onwards), and that β 6= 0 (i.e., the firms are non-myopic).
Further,
(a) if ∀k ≥ 0,
d
0
L,t +k+1
d
0
L,t +k
≥
d
0
S,t +k+1
dS,t+k
,12 then ecL,t0 +k > ecS,t0 +k , ∀k ≥ 0;
or
0
0
(b) if for any k, where k ≥ 0, stS +k,l ≥ stL +k,l , ∀l ≥ 0,13 then ecL,t0 +k > ecS,t0 +k .
Proposition 5 shows that even when the two firms are non-myopic, under certain sufficient
conditions regarding how the output sequences of the two firms evolve over time, (the larger
firm) Firm L does relatively more of process R&D than (the smaller firm) Firm S.
The intuition behind Proposition 5 is as follows.
As far as a comparison between (the larger firm) Firm L and (the smaller firm) Firm S
0
of incentives regarding the choice of R&D composition in any period t + k, where k ≥ 0, is
12
This condition implies that (the larger firm) Firm L grows at the same or at a faster rate than (the
0
smaller firm) Firm S in every period from period t + 1 onwards.
13
This condition can be interpreted as (the larger firm) Firm L growing at the same or at a faster rate
than (the smaller firm) Firm S over “later periods”.
16
concerned, what matters is how the following two factors vary across the two firms: (i) the
0
benefit per-unit output in any period t +k+l, where l ≥ 0, from a unit cost reduction relative
0
to that from a unit quality improvement in period t + k, and (ii) the relative contribution of
0
different periods from period t + k onwards in the overall stream of returns to process and
0
product R&D expenditures undertaken in period t + k.
As for the first factor, since (the larger firm) Firm L is always larger than (the smaller
0
0
firm) Firm S from period t onwards, the benefit per-unit output in any period t + k + l
0
from a unit cost reduction relative to that from a unit quality improvement in period t + k
is always larger for (the larger firm) Firm L. This gives (the larger firm) Firm L an incentive
to do relatively more of process R&D than (the smaller firm) Firm S.
As for the second factor, suppose later periods have a higher contribution for (the smaller
firm) Firm S compared to that for (the larger firm) Firm L in the overall stream of returns
0
from process and product R&D expenditures in period t + k. Then, since (as discussed in
subsection 3.2) over time a firm relatively speaking benefits more from process than from
product R&D, this provides an incentive to (the smaller firm) Firm S to do relatively more
of process R&D than (the larger firm) Firm L.
To sum up, of the two factors mentioned above, the first always gives (the larger firm)
Firm L an incentive to do relatively more of process R&D than (the smaller firm) Firm S,
while the second can potentially give (the smaller firm) Firm S an incentive to do relatively
more of process R&D than (the larger firm) Firm L.14 Hence, with β 6= 0, an additional (as
compared to Proposition 4) sufficient condition (as in Proposition 5(a) or (b)) regarding how
output sequences of the two firms evolve over time is needed to analytically derive the result.
However, it is worth noting that even if neither of the additional sufficient conditions in
Proposition 5 is satisfied, it does not mean that the result will necessarily not hold.
In fact, what is needed in order to have (the smaller firm) Firm S do relatively more of
process R&D than (the larger firm) Firm L, i.e., for the second factor to be not only working
in the opposite direction but also being larger in magnitude than the first factor, is that
14
In contrast, in the scenario of Proposition 4, since β = 0, the first factor plays a limited role (the firms
while deciding about process and product R&D expenditures in any period, take into account corresponding
benefits arising from such activities in that period only and not in subsequent periods), while the second
factor is absent.
17
Firm S be initially much smaller, but later on suddenly get close enough in size to Firm L.15
This is highly unlikely given that (a) both firms here face the same demand conditions, and
the same opportunities for process and product R&D, and (b) the positive interrelationship
in my framework between output and process as well as product R&D.16
Thus, Proposition 5 will likely hold even if the sufficient conditions in it are not satisfied.
It has been empirically observed that compared to a smaller firm, a larger firm typically
(a) conducts more R&D (Grilliches, 1984; Cohen et al., 1987; Holmes et al., 1991; Cohen
and Klepper, 1996b), (b) has lower R&D productivity (Acs and Audretsch, 1988, 1991),
and (c) conducts relatively more of process R&D (Mansfield, 1981; Pavitt et al., 1987;
Scherer, 1991; Cohen and Klepper, 1996a). Propositions 3-5 show these features can hold
even if firms have a more general objective function than profit maximization alone.
I next consider how differences in the objective function across firms can affect their R&D
choices.
3.4
Effects of difference in objective function
Throughout this subsection, I consider two firms - Firm P and Firm G - that differ in their
objective function (i.e., in their value of α, the weight assigned to consumer surplus), and
compare their process and product R&D expenditure choices.
Here, all variables related specifically to Firm i in any period t will have the subscript (i, t)
(i if the variable is time-invariant), where i ∈ {P, G}, and t ≥ 0.
I make the following assumptions regarding the two firms.
I assume both Firm P and Firm G are in a set-up as described in Section 2.
Also, except for the value of α (note that αP 6= αG ) and perhaps the initial characteristic
vector, both Firm P and Firm G have the same value for all other components of Γ (namely,
[θ1 , θ2 ], F (.), β, g(.), [Act , Aqt ]t≥0 ).
Firm P faces the dynamic optimization problem as given by equation (1), and chooses
its process and product R&D expenditures and output over time accordingly.
15
16
I provide an illustrative example in this regard in the Appendix.
The positive effect of output on process and product R&D expenditures has been discussed earlier
(Proposition 3, and the discussion that follows). The positive effect of reductions in per-unit cost, and/or
improvement in product quality on output follows from how dt has been defined, and assumptions regarding
F (.) and α.
18
Firm G on the other hand produces the same output sequence over time as chosen
by Firm P (i.e., Firm G produces the output sequence [dP,t ]t≥0 ), and given this chooses
q
c
|[dP,t ]t≥0 and EG,t
|[dP,t ]t≥0
its process and product R&D expenditures (which I denote as EG,t
respectively to indicate that these are choices made by Firm G, given that it produces the
output sequence [dP,t ]t≥0 ) so as to maximize the value of its corresponding objective function.
In other words, Firm P solves the dynamic optimization problem as given by equation (1)
q
c
(where in terms of notation, we have EP,t
, EP,t
, dP,t , and αP instead of Etc , Etq , dt , and α
respectively).
On the other hand, Firm G solves:
max
q
X
c |
[EG,t
[dP,t ]t≥0 ,EG,t |[dP,t ]t≥0 ]t≥0 t≥0
β [(1 − αG )dP,t F −1 (1 − dP,t ) + αG S(dP,t )]{qG,0 +
t
k=t
X
q
Aqk g(EG,k
|[dP,t ]t≥0 )}
k=0
−dP,t {cG,0 −
k=t
X
c
Ack g(EG,k
|[dP,t ]t≥0 )}
−
q
EG,t
|[dP,t ]t≥0
−
c
EG,t
|[dP,t ]t≥0
k=0
where as before S(d) ≡
Z θ2
F −1 (1−d)
θf (θ)dθ, ∀d ∈ [0, 1].
One way to think about the two firms is that Firm P stands for a private firm (that may
be partially but is not fully owned by the government),17 while Firm G stands for a similar
(in terms of demand conditions and R&D opportunities) firm that does exactly what the
government (or equivalently a regulator, or a social planner, or a policymaker) wants.18,19
17
As mentioned in the Introduction, partial government ownership of firms is quite commonly observed,
and moreover seems to be on the rise (Matsumura, 1998; Megginson, 2005; Chen et al., 2009; Wang and
Chen, 2011).
18
The discussion in this subsection is applicable to a comparison across any pair of firms that differ in
their objective functions. Here, Firm P stands for an existing firm in an industry whose objective function
cannot be changed, while Firm G is representative of actual (for example of a different kind say in terms
of the ownership structure, or from another country) or hypothetical firms deemed to be pursuing a more
desirable aim than the existing firm.
19
Much of the literature assumes that a government wants to maximize welfare (this corresponds to αG = 1
in my framework). However, this need not always be the case. For example, the objective of a government
may differ from that of welfare maximization due to political reasons (Baron, 1988). Hence, I do not restrict
my analysis to that of a welfare-maximizing Firm G. Also, the usual comparison between choices made by
a profit-maximizing to that by a welfare-maximizing firm is a special case, namely αP = 0 and αG = 1, in
my framework.
19
(4)
The reason behind assuming that Firm G produces exactly the same output sequence
over time as Firm P is as follows.
Even if both Firm P and Firm G have the same characteristic vector in some period, if
both firms are allowed to choose their output as per their objective function, then they will
choose different outputs due to the difference in their objective functions.20
As discussed before (equations (1)-(3)), the output sequence of a firm plays a role in both
its process and product R&D expenditure choices; further, output and process and product
R&D expenditure choices of a firm are interrelated. Thus, both the difference in objective
functions and output choices will play a role in any comparison of the process and product
R&D expenditure choices by a private firm with what the government would want in such a
scenario.
However, unless we are in the highly unlikely scenario where a government can force a
private firm to make the output choices that the government wants, a relevant comparison
between R&D choices by a private firm and R&D choices that the government would want
will likely entail making such a comparison for the case where both firms choose the same
output sequence over time. The assumption made regarding the output choices of Firm P
and Firm G allow for precisely such a “second-best comparison”.21
Comparing process and product R&D expenditure choices across Firm P and Firm G, I
find that
Proposition 6. (Difference in objective: Second-best comparison of R&D choices) Consider
Firm P and Firm G as described at the beginning of this subsection. Then ∀t ≥ 0,
q
q
c
c
|[dP,t ]t≥0 , if αP < (>)αG ,
< (>)EG,t
(a.i) EP,t
= EG,t
|[dP,t ]t≥0 , while (a.ii) EP,t
(i.e., in every period, both Firm P and Firm G choose the same process R&D expenditure,
while Firm P does less (more) product R&D than Firm G provided Firm P assigns a lower
(higher) weight to consumer surplus than Firm G); and
(b)
20
21
q
q EP,t+1
EG,t+1 |[dP,t ]
t≥0
− 1
>
q
q EP,t
EG,t |[dP,t ]
t≥0
− 1,
In fact, the firm with a higher α will choose a higher output in such a case.
For further discussion as to why such a second-best comparison might be more meaningful than a “first-
best” comparison that also takes the difference in output choices into account, see for example Bhattacharya
and Mookherjee (1986), Mankiw and Whinston (1986), Suzumura and Kiyono (1987), Tirole (1988),
Suzumura (1992), Leahy and Neary (1997), Amir et al. (2011).
20
(i.e., the “inoptimality” in the product R&D choice by Firm P in comparison to the choice
by Firm G increases over time).
The intuition behind Proposition 6 is as follows.
From equations (2) and (3), it follows that here, for a given opportunity for process and
product R&D and consumers’ WTP for quality improvements, the incentive for a firm for
process R&D in any period depends only on the output sequence from that period onwards
and is independent of how much the firm cares about consumer surplus, while that for
product R&D depends on the output sequence from that period onwards and how much the
firm cares about consumer surplus. Moreover, for a given output sequence, the incentive
for a firm for product R&D depends on the weights (which add up to 1) that it assigns to
the marginal buyer’s WTP and the average WTP of buyers for quality improvements. In
my framework, the latter always exceeds the former, and also the more a firm cares about
consumer surplus the greater the weight it assigns to the latter and lesser to the former.
Hence, given that Firm P and Firm G are producing the same output sequence over time,
both choose to undertake the same amount of process R&D in any period, while the firm
that cares more about consumer surplus does more product R&D (Proposition 6(a)).
As far as the evolution of the ratio of the product R&D choice by Firm P to that by
Firm G over time is concerned, it turns out that this ratio becomes smaller (larger) over time
if αP < (>) αG , provided S(d)/dF −1 (1 − d) increases over time. In my framework, it turns
out that S(d)/dF −1 (1 − d) is increasing in d, ∀d ∈ [0, 1]. Also, as discussed in subsection 2.2,
Firm P sells to a larger fraction of its potential buyers over time. Hence, if αP < (>) αG ,
then it turns out that not only does Firm P conduct less (more) product R&D than Firm G
q
q
|[dP,t ]t≥0
/EG,t
in every period, but that it does so to a greater extent over time (in that EP,t
decreases (increases) over time). Proposition 6(b) thus shows that in this sense, regardless
of whether Firm P cares more or less about consumer surplus than Firm G, there is an
increasing inoptimality over time in the choice of product R&D by Firm P compared to the
choice of product R&D by Firm G.
Given the difference demonstrated above in the choice of product R&D expenditure across
a private firm (Firm P) and what the government wants (Firm G), it would be natural to
consider possible effects of R&D-related policies in my framework.
As discussed in the Introduction, governments around the world have been trying to
21
boost R&D efforts in general, and by firms in particular. Fiscal incentives for R&D (either
as R&D subsidies, or as R&D tax incentives/credits) are commonly used in this regard. Such
incentives for R&D are typically general in nature (in that they are applicable to any R&D
activity, and not just to say process or product R&D only).
I consider the case where Firm P receives general fiscal incentives for R&D (positive or
negative), per-unit R&D expenditure regardless of whether it is product or process R&D, over
time. I denote such fiscal incentives for R&D by [δt ]t≥0 , where δt < 1, ∀t. Thus, when Firm P
receives general fiscal incentives for R&D of [δt ]t≥0 , its actual expenditure for either process
or product R&D in any period t is (1-δt ), for every unit spent on process or product R&D.22
I denote the output, process, and product R&D expenditure choices of Firm P when it
receives general fiscal incentives for R&D, per-unit R&D expenditure over time of [δt ]t≥0 by
q
c
|[δt ]t≥0 , and EP,t
|[δt ]t≥0 respectively.
dP,t |[δt ]t≥0 , EP,t
Fiscal incentives for R&D would also lead to a change in the output sequence produced
by Firm P. A second-best analysis of the effects of general fiscal incentives for R&D would
r
r
entail a comparison of EP,t
|[δt ]t≥0 with EG,t
|dP,t |[δt ]t≥0 , where r ∈ {c, q}. In order to facilitate
such a comparison, I find it useful to consider process and product R&D expenditure choices
q
c
|dP,t |[δt ]t≥0 and EP,t
|dP,t |[δt ]t≥0 respectively) if it
that Firm P would make (which I denote by EP,t
produced the output sequence dP,t |[δt ]t≥0 , but did not receive any general fiscal incentives for
R&D. I can then carry out a second-best analysis of the effects of general fiscal incentives for
r
r
r
r
|dP,t |[δt ]t≥0 ,
|dP,t |[δt ]t≥0 with EG,t
|[δt ]t≥0 with EP,t
|dP,t |[δt ]t≥0 , and (b) EP,t
R&D by comparing (a) EP,t
where r ∈ {c, q}.
r
r
As far as the second comparison, namely that of EP,t
|dP,t |[δt ]t≥0 with EG,t
|dP,t |[δt ]t≥0 , where
r ∈ {c, q}, is concerned, from the same idea as that behind Proposition 6(a), I find that
c
c
c
c
|dP,t |[δt ]t≥0 , if αP < (>)αG . This
EP,t
|dP,t |[δt ]t≥0 = EG,t
|dP,t |[δt ]t≥0 , while EP,t
|dP,t |[δt ]t≥0 < (>) EG,t
leads to the following findings regarding a second-best analysis of the effects of general fiscal
incentives for R&D.
Proposition 7. (Effects of general fiscal incentives for R&D: Second-best analysis) Suppose,
Firm P receives general fiscal incentives for R&D, per-unit R&D expenditure over time of
[δt ]t≥0 as described above. Then, in any period t, where t ≥ 0
22
Here, I am allowing fiscal incentives for R&D to vary with time; also, the firm could be receiving positive
fiscal incentives (such as subsidies) in some, and negative fiscal incentives (such as taxes) in other periods.
22
r
r
(a) if δt > (<) 0, then EP,t
|[δt ]t≥0 > (<) EP,t
|dP,t |[δt ]t≥0 ; r ∈ {c, q},
while
q
q
c
c
|dP,t |[δt ]t≥0 /EP,t
|dP,t |[δt ]t≥0 .
|[δt ]t≥0 /EP,t
|[δt ]t≥0 = EP,t
(b) EP,t
Proposition 7(a) shows that while a general fiscal incentive (disincentive) for R&D will
lead to Firm P doing more (less) of product R&D, it will also result in Firm P doing more
(less) of process R&D as well. Proposition 7(b) shows that in proportional terms, the amount
of increase or decrease in both product and process R&D will be the same!
The underlying intuition is as follows.
A general fiscal incentive (disincentive) for R&D lowers (raises) the actual cost of both
process and product R&D. Thus, in this case, the FONC for process and product R&D
expenditure for Firm P in any period will be as in equations (2) and (3) respectively but
with (1-δt ) instead of 1 on the right-hand side. In other words, a general fiscal incentive
(disincentive) for R&D raises (lowers) the marginal return from spending one more unit
on both process and product R&D by the same proportion, and hence leads to the same
proportional increase (decrease) in both (compared to if Firm P produced the output sequence
dP,t |[δt ]t≥0 , but did not receive any general fiscal incentives or disincentives for R&D).
The significance of Propositions 6 and 7 is as follows.
Proposition 6(a) shows that a private firm [Firm P] in a second-best sense (i.e., corresponding to the output sequence that it chooses to produce) does the same amount of
process, but less or more (depending on whether αP is greater than or less than αG ) of
product R&D compared to what the government would want it to do.
Proposition 7 shows that given the output sequence being produced, general fiscal incentives for R&D have no effect on the choice of R&D composition by a firm (which is determined
by the output sequence and the objective of the firm)! This is of particular significance given
concerns regarding perhaps socially inoptimal choices of R&D composition by firms (Cohen
and Klepper, 1996a). While general fiscal incentives for R&D can be used to alter product
R&D choices by a private firm [Firm P], it will also lead to an equal (in proportional terms,
and in the same direction) change in process R&D away from its optimal (in second-best
terms; i.e., corresponding to the output sequence that it chooses to produce) level. In other
words, in terms of a second-best analysis (i.e., corresponding to the output sequence that a
private firm chooses to produce), the more a general fiscal incentive or disincentive for R&D
23
rectifies the product R&D choice of a private firm [Firm P], the more inoptimality it will
cause with regard to its process R&D choice.
Thus, Propositions 6 and 7 together highlight the need for taking into account the fact
that firms do different kinds of R&D, and of considering policies accordingly rather than
thinking of R&D in general. They suggest the need for considering measures, if possible,
focused specifically towards affecting only the product R&D choice of firms. Moreover,
Proposition 6(b) shows that any R&D-related measures must also, if possible, consider the
increasing inoptimality in product R&D choices by a private firm over time.
Before I end this subsection, a further remark is in order.
Here, I have considered the case where only general fiscal incentives for R&D are used.
As discussed earlier in this subsection (see the discussion following equation (4)), in my
framework Firm P will choose a different, and thus an inoptimal output sequence compared to
what Firm G would choose. Policymakers could potentially try to rectify this inoptimality in
the choice of output sequence by providing appropriate output subsidies. However, from the
intuition underlying Propositions 6 and 7 (these have been discussed following the respective
Propositions), it follows that the inoptimality, in the second-best sense, of R&D choices of
Firm P demonstrated above will remain, even if output incentives are used either on their
own or together with general R&D incentives.
4
Conclusion
In this paper, I have studied how a firm having a general objective function (that considers
both profits and consumer surplus) affects its product and process R&D choices, and corresponding implications. Within this context, I have shown how a firm’s objective function
plays a role in the difference between process and product R&D, characterized how the
choice of process R&D in total R&D by a firm varies over time, and developed cross-sectional
implications regarding firm size and process and product R&D choices by firms. I have also
illustrated some effects of providing general fiscal incentives for R&D (incentives given for
any R&D, regardless of whether it is product or process R&D) to firms.
Appendix
I begin with two observations that I will use in the proof of some of my results.
24
Observation 1 (O1): In my framework, [(1-α)F −1 (1 − d) + α( S(d)
d )] is ↓ in d, ∀d ∈ [0, 1].
Proof of O1: In my framework, since θ is uniformly distributed over [θ1 , θ2 ], it follows that [(1-α)F −1 (1 − d)
+α(S(d)/d)] = [2θ2 − (2 − α)(θ2 − θ1 )d]/2, which is ↓ in d, ∀d ∈ [0, 1], since α < 2, thus proving O1.
Observation 2 (O2): In my framework, S(d)/dF −1 (1 − d) is ↑ in d, ∀d ∈ [0, 1].
Proof of O2: In my framework, since θ is uniformly distributed over [θ1 , θ2 ], it follows that S(d)/dF −1 (1−d)
= 0.5 +
θ2
2[θ2 −d(θ2 −θ1 )] ,
which is ↑ in d, ∀d ∈ [0, 1], thus proving O2.
Proof of Proposition 1: Follows directly from equations (2) and (3).
00
0
Proof of Proposition 2: Let A∗ denote the common value of Act /Aqt in periods t and t .
Then, from equations (2) and (3), it follows that in this case,
0
0
g (E c0 )
g
0
−
P
t
(E q0 )
t
=A
∗
g (E c00 )
t
g 0 (E q00 )
l≥0
t
β l {(1−α)dt0 +l F −1 (1−dt0 +l )+αS(dt0 +l )}
P
l≥0
= [P
l≥0
A∗P
β l dt0 +l ][
P
β l {(1−α)dt00 +l F −1 (1−dt00 +l )+αS(dt00 +l )}
,
(5.1)
h
S(d 0 )
l 0 0
−1
(1 − dt0 ) + α d t0 }
β
d
d
t t +k+l {(1 − α)F
l≥0
t
i
S(dt0 +k+l )
−1
−{(1 − α)F (1 − dt0 +k+l ) + α d 0
} .
(5.2)
β l dt0 +l
l≥0
−
P
l≥0
β l dt00 +l
Pk=t00 −t0 P
l≥0
k=1
β l dt00 +l ]
0
00
0
Since dt is ↑ in t, ∀t ≥ 0, and t > t , O1 and equation (5.2) ⇒
(E1 , E2 , E3 , E4 ) ∈ <4++ ,
g 0 (E2 )
g 0 (E1 )
g 0 (E4 )
g 0 (E3 )
>
g (E c0 )
t
g 0 (E q0 )
>
t
⇒
E2
E1
E4
E3 ,
<
t +k+l
0
g (E c00 )
t
g 0 (E q00 )
t
⇒ ect00 > ect0 , since ∀
thus proving Proposition 2.
Proof of Proposition 3: Here, ∀t ≥ 0, the FONC for process and product R&D for Firm i, i ∈ {L, S} are
r
given by equations (2) and (3) respectively, with di,t+k and Ei,t
where r ∈ {c, q} in place of dt+k and Etr . In
0
any period t + k, k ≥ 0, comparing the FONC for process R&D across Firm L and Firm S and for product
r
r
R&D across Firm L and Firm S, since dL,t0 +k > dL,S 0 +k , ∀k ≥ 0, it follows that g 0 (EL,t
) < g 0 (ES,t
),
0
0
+k
+k
where r ∈ {c, q}. This leads to Proposition 3(a), since g(.) is monotonically increasing and strictly concave
in E, ∀E ∈ <+ . Proposition 3(b) follows from (a) and strict concavity of g(.).
0
0
q
c
Proof of Proposition 4: Here, since β = 0, it follows from equations (2) and (3) that ∀t ≥ 0, g (Ei,t
)/g (Ei,t
)
= (Aqt /Act )[(1 − α)F −1 (1 − di,t ) + α(S(di,t )/di,t )], where i ∈ {L, S}. This and O1 imply that ∀k ≥ 0,
0
0
0
0
q
q
c
c
g (EL,t
)/g (EL,t
) < g (ES,t
)/g (ES,t
), since dL,t0 +k > dS,t0 +k , ∀k ≥ 0, ⇒ ecL,t0 +k > ecS,t0 +k ,
0
0
0
0
+k
+k
+k
+k
∀k ≥ 0, since ∀ (E1 , E2 , E3 , E4 ) ∈ <4++ ,
g 0 (E2 )
g 0 (E1 )
>
g 0 (E4 )
g 0 (E3 )
⇒
E2
E1
<
E4
E3 ,
thus proving Proposition 4.
Proof of Proposition 5: Here, it follows from equations (2) and (3) that ∀t ≥ 0,
0
c
g (EL,t
)
q
g 0 (EL,t
)
Aq
= Atc
0
g (E c )
− g0 (ES,t
q
)
S,t
P
l
l≥0
β {(1−α)dL,t+l F −1 (1−dL,t+l )+αS(dL,t+l )}
P
t
= [P
l≥0
l≥0
Aqt /Act
β l dL,t+l ][
β l dL,t+l
P
P
l≥0
βl d
S,t+l ]
P
−
l≥0
β l {(1−α)dS,t+l F −1 (1−dS,t+l )+αS(dS,t+l )}
P
l≥0
l≥0 (β
2l
β l dS,t+l
P
l≥0
,
(6.1)
n
S(dL,t+l )
}
dL,t+l dS,t+l ) {(1 − α)F −1 (1 − dL,t+l ) + α dL,t+l
−{(1 − α)F −1 (1 − dS,t+l ) + α
+
P
l0 ≥l+1β
l+l
0
o
S(dS,t+l )
dS,t+l }
n
S(d
0)
dL,t+l0 dS,t+l [{(1 − α)F −1 (1 − dL,t+l0 ) + α d L,t+l0 }
L,t+l
−{(1 − α)F −1 (1 − dS,t+l ) + α
25
S(dS,t+l )
dS,t+l }]
−dL,t+l dS,t+l0 [{(1 − α)F −1 (1 − dS,t+l0 ) + α
S(dS,t+l0 )
}
o
S(dL,t+l )
−{(1 − α)F −1 (1 − dL,t+l ) + α dL,t+l
}] . (6.2)
dS,t+l0
0
(a) For any t = t + k, where k ≥ 0:
P
(i) O1 ⇒ each term within l≥0 in the RHS of (6.2) is negative, since dL,t0 +k > dS,t0 +k , ∀k ≥ 0.
P P
(ii) For terms within l≥0 l0 ≥l+1 in the RHS of (6.2), note that
O1 ⇒ [{(1 − α)F −1 (1 − dL,t+l0 ) + α
S(dL,t+l0 )
dL,t+l0
} − {(1 − α)F −1 (1 − dS,t+l ) + α
S(dS,t+l )
dS,t+l }]
is always negative,
since dL,t0 +k > dS,t0 +k , ∀k ≥ 0.
Now, there can be two cases.
0
Case 1: (l, l ) is such that dL,t+l ≥ dS,t+l0
S(d
0)
S(d
)
L,t+l
⇒ [{(1 − α)F −1 (1 − dS,t+l0 ) + α d S,t+l0 } − {(1 − α)F −1 (1 − dL,t+l ) + α dL,t+l
}] ≥ 0 (by O1),
S,t+l
P P
0
and thus the entire term within l≥0 l0 ≥l+1 for such an (l, l ) is negative.
0
Case 2: (l, l ) is such that dL,t+l < dS,t+l0
⇒ dS,t+l < dL,t+l < dS,t+l0 < dL,t+l0
⇒ [{(1 − α)F −1 (1 − dL,t+l0 ) + α
< [{(1 − α)F −1 (1 − dS,t+l0 ) + α
S(dL,t+l0 )
dL,t+l0
S(dS,t+l0 )
dS,t+l0
} − {(1 − α)F −1 (1 − dS,t+l ) + α
} − {(1 − α)F −1 (1 − dL,t+l ) + α
S(dS,t+l )
dS,t+l }]
S(dL,t+l )
dL,t+l }]
< 0,
which since (dL,t+l0 )(dS,t+l ) ≥ (dL,t+l )(dS,t+l0 ) [this follows from the assumption that
P P
0
∀k ≥ 0] ⇒ that the entire term within l≥0 l0 ≥l+1 for such an (l, l ) is negative.
dL,t0 +k+1
dL,t0 +k
∀k ≥ 0, ⇒ ecL,t0 +k > ecS,t0 +k , ∀k ≥ 0, since ∀ (E1 , E2 , E3 , E4 ) ∈ <4++ ,
0
g (E2 )
g 0 (E1 )
>
⇒
,
g (E c
<
Thus, (i), (ii), and equation (6.2) imply that in the scenario of Proposition 5(a),
dS,t0 +k
0
0
g (E c
)
0
L,t +k
g 0 (E q 0
)
L,t +k
g 0 (E4 )
E2
g 0 (E3 )
E1
dS,t0 +k+1
≥
)
0
S,t +k
g 0 (E q
<
)
0
S,t +k
E4
E3 ,
,
thus
proving Proposition 5(a).
0
0
t +k,l
(b) For any k where k ≥ 0, for which stS +k,l ≥ sL
, ∀l ≥ 0,
S(d
)
P
P
0
S,t +k+l
l
−1
l
l≥0
β dS,t0 +k+l {(1−α)F
(1−dS,t0 +k+l )+α
P
l≥0
P
>
l≥0
d
0
S,t +k+l
β l dS,t0 +k+l
l≥0
l≥0
≥
β dL,t0 +k+l {(1−α)F −1 (1−dS,t0 +k+l )+α
P
l≥0
β l dL,t0 +k+l {(1−α)F −1 (1−dL,t0 +k+l )+α
P
}
)
0
L,t +k+l
}
d
0
L,t +k+l
S(d
d
)
0
S,t +k+l
}
0
S,t +k+l
β l dL,t0 +k+l
S(d
β l dL,t0 +k+l
,
(7)
0
0
t +k,l
where the first weak inequality follows from (i) the assumption that stS +k,l ≥ sL
, ∀l ≥ 0, and (ii) since
{(1 − α)F −1 (1 − dS,t ) + α
S(dS,t )
dS,t }
is decreasing in t, ∀t ≥ 0 (this follows from O1 since dS,t is decreasing
in t, ∀t ≥ 0), while the second strict inequality follows from the fact that {(1 − α)F −1 (1 − dS,t0 +k+l ) +
α
S(dS,t0 +k+l )
dS,t0 +k+l
} > {(1 − α)F −1 (1 − dL,t0 +k+l ) + α
S(dL,t0 +k+l )
dL,t0 +k+l
} (this follows from O1 since dL,t0 +k > dS,t0 +k ,
∀k ≥ 0).
0
0
g (E c
Thus, equations (7) and (6.1) imply that in the scenario of Proposition 5(b),
0
0
g
0
)
0
L,t +k
(E q 0
)
L,t +k
g (E c
<
t +k,l
k ≥ 0, for which stS +k,l ≥ sL
, ∀l ≥ 0, ⇒ ecL,t0 +k > ecS,t0 +k , since ∀ (E1 , E2 , E3 , E4 ) ∈ <4++ ,
⇒
E2
E1
<
E4
E3 ,
)
0
S,t +k
g 0 (E q
)
0
S,t +k
0
g (E2 )
g 0 (E1 )
, for any
>
g 0 (E4 )
g 0 (E3 )
thus proving Proposition 5(b).
Example illustrating the need for an additional sufficient condition in Proposition 5 (a) and
26
(b): Consider a 2-period scenario with periods 0 and 1, and Firm L and Firm S. Everything else is as in
subsection 3.3. Then, using equation (6.2), it turns out that in this case,
0
0
c
g (EL,0
)
0
q
g (EL,0
)
g (E c )
− 0 S,0
q
g(ES,0 )
Aq0
Ac0
=
1
[dL,0 +βdL,1 ][dS,0 +βdS,1 ]
{(1 − α)dL,0 F −1 (1 − dL,0 ) + αS(dL,0 )}(dS,0 + βdS,1 )
+β{(1 − α)dL,1 F −1 (1 − dL,1 ) + αS(dL,1 )}(dS,0 + βdS,1 )
−{(1 − α)dS,0 F −1 (1 − dS,0 ) + αS(dS,0 )}(dL,0 + βdL,1 ) −β{(1 − α)dS,1 F −1 (1 − dS,1 ) + αS(dS,1 )}(dL,0 + βdL,1 ) .
0
0
Now, if we assume that dS,0 = 0 < dL,0 < dL,1 = dS,1 , then
c
g (EL,0
)
q
g 0 (EL,0
)
>
c
g (ES,0
) 23
,
q
g 0 (ES,0
)
implying that Firm S will
in such a case do relatively more of process R&D in period 0 than Firm L.
What is happening here is that Firm S (since it produces no output in period 0) gets returns from period
1 (the last period in this case) only, while Firm L gets returns from both period 0 and period 1, from any
cost reduction or improvement in product quality in period 0. Further, since Firm S despite not producing
any output in period 0, manages to catch up with Firm L in period 1, it ends up doing relatively more of
process R&D than Firm L, whose output grows more gradually over periods 0 and 1.
However, given that both Firm S and Firm L have the same objective, and face the same demand
conditions and R&D opportunities, and the positive interrelationship between output and both process
and product R&D in my framework, such a pattern of evolution of output over time for the two firms
(namely, Firm S initially being much smaller than Firm L, but later on suddenly catching up with or getting
close enough to Firm L) is highly unlikely in my framework, and is excluded by the sufficient condition in
Proposition 5 (a) and (b).24
Proof of Proposition 6: (a) Here, the FONC for process and product R&D for Firm P are given by
equations (2) and (3) respectively with dP,t instead of dt for any t ≥ 0, and αP instead of α. The FONC for
process and product R&D for Firm G are also given by equations (2) and (3) respectively with dP,t instead
of dt for any t ≥ 0, and αG instead of α. Proposition 6(a) follows from comparing the FONC for process
and for product R&D across Firm P and Firm G, and strict concavity of g(.).
(b) Here, comparing the FONC for product R&D across Firm P and Firm G, it turns out that ∀t ≥ 0:
q
EP,t+1
q
EG,t+1
|[dP,t ]t≥0
P
= [P
l≥0
l≥0
−
q
EP,t
q
EG,t
|[dP,t ]t≥0
(αG −αP )β l dP,t dP,t+1+l F −1 (1−dP,t )F −1 (1−dP,t+1+l ){
β l {(1−αP )dP,t+1+l F −1 (1−dP,t+1+l )+αP S(dP,t+1+l )}][
Since ∀ (E1 , E2 , E3 , E4 ) ∈ <4++ ,
g 0 (E2 )
g 0 (E1 )
>
g 0 (E4 )
g 0 (E3 )
⇒
E2
E1
S(dP,t+1+l )
S(dP,t )
−
}
dP,t+1+l F −1 (1−dP,t+1+l )
dP,t F −1 (1−dP,t )
P
l≥0
<
β l {(1−αP )dP,t+l F −1 (1−dP,t+l )+αP S(dP,t+l )}]
E4
E3 ,
.
(8)
Proposition 6(b) follows from O2 and equation
(8).
0
0
23
Here, I have assumed dS,0 = 0 < dL,0 < dL,1 = dS,1 . However, note that
c
g (EL,0
)
0
q
g (EL,0
)
>
c
g (ES,0
)
0
q
g (ES,0 )
will continue
to hold, even if dS,0 6= 0 but is sufficiently less than dL,0 , and dS,1 < dL,1 but dS,1 is close enough to dL,1 .
24
Note that (the 2-period equivalent) sufficient condition of neither Proposition 5 (a) or (b) is satisfied by
the pattern of evolution of output of Firm L and Firm S that has been assumed here.
27
Proof of Proposition 7: Here, the FONC for process and product R&D for Firm P when it receives
general fiscal incentives or disincentives for R&D, per-unit R&D expenditure, over time of [δt ]t≥0 , are given by
equations (2) and (3) respectively, with dP,t |[δt ]t≥0 instead of dt , and (1−δt ) instead of 1 on the right hand side
of the equations. The FONC for process and product R&D for Firm P when it does not receive any general
fiscal incentives or disincentives for R&D, per-unit R&D expenditure, but produces the output sequence
dP,t |[δt ]t≥0 , are given by equations (2) and (3) respectively with dP,t |[δt ]t≥0 instead of dt . Proposition 7(a)
follows from comparing the FONC for process and for product R&D for Firm P across the two cases, since g(.)
is monotonically increasing and strictly concave in E, ∀E ∈ <+ . Proposition 7(b) follows from comparing
0
0
c
g (EP,t
|[δt ]t≥0 )
g
0
q
(EP,t
|[δt ]t≥0 )
with
c
g (EP,t
|dP,t |[δ
g
0
t ]t≥0
q
(EP,t
|dP,t |[δ ]
t t≥0
)
)
(since ∀ (E1 , E2 , E3 , E4 ) ∈ <4++ ,
g 0 (E2 )
g 0 (E1 )
>
g 0 (E4 )
g 0 (E3 )
⇒
E2
E1
<
E4
E3 ).
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