R&D and the Timing of Product Switching

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R&D and the Timing of Product Switching Naohiko Wakutsu∗ Institute of Innovation Research Hitotsubashi University Kunitachi, Tokyo, Japan 186-8603 May 25, 2012

Abstract The incentives for R&D of a multi-product firm and the timing of product switching are examined in a two-product Cournot model of duopoly with process innovation. The two firms are an innovator and an imitator and both supply a market with an existing homogeneous product. The innovator is also a prototype producer of a new product, which is an imperfect substitute to the existing old product. Time is divided into two stages. Stage 2 is the production stage. In stage 1, the imitator copies the innovator’s initial technology while the innovator carries out further cost-reducing process R&D. A new R&D model is proposed, which adopts a search-theoretic framework to formulate the stochastic nature of the R&D process and can be used to extend many leading deterministic models in the literature. By dealing with multi-product R&D and product selection together, this paper presents new insights into the timing of product switching. Keywords: R&D, innovation, multi-product firm, product switching JEL classification: O31, L13, D21

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Introduction

Data assert that overlapping product life-cycles are common in many industries. Bernard, Redding and Schott (2010) recently found that more than one-half of U.S. manufacturing ﬁrms changed their product mix every ﬁve years between 1972 and 1997, that many of the changes were within industries, and that virtually all product additions occurred using a ﬁrm’s existing facilities and not as a result of mergers and acquisitions.1 ∗

E-mail address: [email protected] They found that 97% of product additions occurred using a firm’s existing facilities and fewer than 1% were accounted for by M&A. Similarly, Broda and Weinstein (2007) found that 92% of product creations occurred within existing firms. 1

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A stylized overlapping product life-cycle may be described as follows. A new product is ready to launch by the time of the maturity of an existing old product. Given the initial introduction of a new product by some innovating ﬁrm, more ﬁrms add it to their product mix, as it becomes a dominating product. An old product eventually dies out because of the cessation of production by a large majority of ﬁrms. How can it be, from a ﬁrm’s point of view, in some sense optimal that it starts R&D in the hope of introducing a new product while still operating in the market of an old one? New product development no doubt brings a ﬁrm certain complications in its problem. For instance, no decision should be made unless the possible cannibalization that the new product could cause is fully considered. Nonetheless, the commonality of overlapping product cycles suggests that the optimality of such decisions should hold in some relatively wide range of parametric space. The incentives for R&D of a multi-product ﬁrm and the timing of product switching are examined in a two-product Cournot model of duopoly with process innovation. The two ﬁrms are an innovator and an imitator. Both ﬁrms supply a market with an existing homogeneous product. The innovator is also a prototype producer of a new product, which is an imperfect substitute to the old product. Time is divided into two stages. Stage 2 is the production stage. In stage 1, the imitator copies the innovator’s initial technology while the innovator carries out further cost-reducing process R&D. The issues are important for two reasons. First, both product switching and R&D are common for a large majority of ﬁrms in all sectors. Second, both play prominent roles in economic progress. Bernard, Redding and Schott (2010) found that roughly one-third of the increase in real U.S. manufacturing shipments between 1972 and 1997 was due to net added and dropped products by continuing ﬁrms. Also, there is broad agreement among economists that R&D is a major source of economic growth (Romer (1990)). A main contribution of this paper is that these issues are dealt with together in a single model, yielding a new insight into the timing of product switching, which will be explained below. To the best of our knowledge, the literature usually treats process innovation and product selection separately, a noteworthy exception being Lin (2007). Another contribution is that a new model of R&D is proposed, which is stochastic and can be used to extend many leading deterministic models such as D´Aspremont and Jacquemin (1988), Spence (1984) Vives (2008) and others. Three primary questions are addressed. (i) How is an innovator’s decision on new product introduction related to his production cost for the new product and his and the opponent’s production costs for the existing product? (ii) How is the innovator’s decision on stopping the production of the existing product related to his production cost of the new 2

product and his and the opponent’s production costs of the existing product? (iii) How does a shock to the innovator’s initial production cost for the existing product change his R&D expenditure for the new and existing products? Does this change, and thereby a shock to his current production cost of the old product, foster or deter his product switching? The following conclusions emerge. (i) An innovative ﬁrm’s incentives for product additions and deletions are both negatively correlated to its production cost for the new product, and are positively correlated to its production cost for the existing old product, given the opponent’s production cost for the old product. (ii) The timing of the innovator’s introduction of the new product is not aﬀected by the level of the opponent’s production cost, but its decision on dropping the old product while introducing the new one could be. (iii) A positive shock to the innovator’s initial production cost for the old product has a diﬀerent eﬀect on its R&D expenditures and thereby its timing of product switching, according to the levels of its initial production cost for the old product and its current R&D expenditures. This insight is new, since if there was no R&D stage, a positive correlation between the innovator’s product switching and its production cost for the old product should imply that an increase in its initial production cost for the old product causes more product switching. However, process innovation could deter the innovative ﬁrm’s product switching. Also, this implies that a common productivity shock can cause various responses in product switching among ﬁrms. Bernard, Redding and Schott (2010) found that product additions and deletions are positively correlated across ﬁrms (that is, while some ﬁrms add, others drop). This model’s prediction provides an explanation for why such a positive correlation is common. The theoretical literature on process R&D is large and includes two classes of models. The ﬁrst is the exponential discovery models used by Loury (1979), Dasgupta and Stiglitz (1980), Reinganum (1982), Grossman and Shapiro (1986), Doraszelski (2003) and others.2 In a dynamic setting, this assumes that the outcome of an R&D project at each instant in time is either success or failure, so that the chance of a ﬁrm’s “discovering” a successful outcome by a given date is described by the exponential distribution. With an additional assumption that a successful outcome rewards a ﬁrm with a reduced production cost, this model examines a ﬁrm’s R&D incentives through this assumed stochastic relationship between a variable expenditure and the time at which a successful innovation arrives. The second is comprised of the models of D´Aspremont and Jacquemin (1988), Spence (1984), Vives (2008) and others. This does not assume that an R&D project has a binary outcome. A common speciﬁcation is that the outcome is a reduced production cost. Usually, this outcome is certain and a ﬁrm’s R&D incentives are examined through this 2

For a survey of exponential discovery models, Reinganum (1989).

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assumed deterministic relationship between a variable R&D expenditure and a known reduced production cost. There are only a few models in the latter class that incorporate uncertainty. Notable among such exceptions is the model due to Judd (2003). He considers a multi-stage R&D project. In order that the R&D project be completed, it must be “moved” from some initial location by a given distance d ∈ (0, ∞). Ex ante how large a “jump” occurs from a given R&D expenditure is not known. However, the likelihood of a large jump is higher, the larger is a ﬁrm’s R&D expenditure. He formalizes this idea by assuming that a higher R&D expenditure causes a stochastic dominant shift to the cumulative probability distribution that governs the relationship between a ﬁrm’s R&D investment and its outcome. Our R&D model assumes that the outcome of an R&D project is a candidate for a reduced production cost. The candidate can be much higher than or much lower than a ﬁrm’s initial production cost or somewhere in-between. Therefore, the model is in the second class and includes uncertainty. The way in which we incorporate uncertainty is similar to that of Judd (2003). A distinctive feature of the literature is that it almost exclusively focuses on a singleproduct ﬁrm. Among a few exceptions are Lambertini (2003) and Lin (2004). Assuming that the outcome of an R&D project for each product is a known reduced production cost for that product, they evaluate the incentives for R&D of a multi-product monopolist against the social optimum, with and without scope economies in R&D. The theoretical literature on multi-product ﬁrms is large.3 The issues on product selection have been discussed in various models. For instance, in a model with horizontally diﬀerentiated products, Brander and Eaton (1984) examine the product-line choices of two ﬁrms that are able to commit to product lines before competing in prices. In a model of vertical product diﬀerentiation, Mussa and Rosen (1978) study the product-line decisions of a price-discriminating monopolist. In a model of duopoly, Champsaur and Rochet (1989) characterize an equilibrium where one ﬁrm sells a high-quality product and the other a low-quality product. To the best of our knowledge, the literature usually treats process innovation and product selection separately, a noteworthy exception being Lin (2007). Adopting a similar model to Lin (2004), Lin (2007) studies how a change in the product mix to be oﬀered inﬂuences a multi-product monopolist’s incentives for process innovation. The rest of this paper is organized as follows. Section 2 explains the model. Section 3 characterizes a post-R&D equilibrium. In Section 4, an innovating ﬁrm’s optimal R&D 3

For an overview of the theory of multi-product firms in oligopolistic or perfectly competitive environments, see MacDonald and Slivinski (1987), Okuguchi and Szidarovsky (1990) and Fraja (1994).

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spending is characterized. Section 5 concludes.

2

The Model

Two proﬁt-seeking ﬁrms are Cournot competitors, labeled 1 and 2. Firm 1 is an innovator and ﬁrm 2 is an imitator. Both supply a market with an existing homogeneous product, labeled A. Firm 1 also produces a prototype of a new product, labeled B. Time is divided into two stages. Stage 2 is a production stage. In stage 1, the imitator copies ﬁrm 1’s initial technology while ﬁrm 1 carries out cost-reducing process R&D. The market demand system for these products is ] [ ] [ q1A + q2A max {0, aA − pA + bpB } = , max {0, aB + bpA − pB } q1B where pj > 0, 0 < aj < ∞, j = A, B and 0 < b < 1. Here q1A denotes ﬁrm 1’s output level of product A and pA is its price level. q1B , q2A and pB are analogous. The restriction 0 < b < 1 implies that the two products are imperfect substitutes, so a unit change in the price of product B has less eﬀect on the quantity demanded of product A than does a unit change in the price of product A. They become closer substitutes as b approaches to unity. The inverse demand system is then ] ] [ [ ] [ max {0, (aA + aB b − q1A − q2A − bq1B )/(1 − b2 )} pA pA (q1A + q2A , q1B ) , (1) = = max {0, (aA b + aB − bq1A − bq2A − q1B )/(1 − b2 )} pB pB (q1A + q2A , q1B ) where (q1A , q1B ) ≥ 0 and q2A ≥ 0. The model is organized in two stages. Stage 2 is the production stage. Both ﬁrms have constant marginal and average costs. Let c1j , j = A, B, be ﬁrm 1’s ﬁnalized marginal cost of producing product j. Given q2A and c1 = (c1A , c1B ), ﬁrm 1’s proﬁt from producing products A and B is Π1 (q1A , q1B ; q2A , c1 ) =

∑

[pj (q1A + q2A , q1B ) − c1j ]q1j .

(2)

j=A,B

Denote ﬁrm 2’s marginal cost of producing product A by c2A . Then ﬁrm 2’s production proﬁt is Π2 (q2A ; q1A , q1B , c2A ) = [pA (q1A + q2A , q1B ) − c2A ]q2A .

(3)

Before production, ﬁrm 1 engages in process R&D to reduce its initial marginal costs of producing products A and B. Let kj be the initial marginal cost of producing product j. To ensure that product A is an existing product while product B is just newly invented and not yet marketed, suppose 0 < kA ≤ αA and αB < kB < ∞, where αA = (aA + aB b)/(1 − b2 ),

αB = (aA b + aB )/(1 − b2 ). 5

At the beginning of stage 1, ﬁrm 1 chooses an R&D expenditure rj spent for product j. Given rj , the ﬁrm receives an outcome zj , a candidate for its ﬁnalized marginal cost c1j of product j, at the end of stage 1 and selects the smaller of kj and zj . Thus c1j = min{zj , kj }. R&D is a gamble that is modeled as a stochastic process described by a cumulative distribution function (cdf) G(zj |rj ) deﬁned over a non-degenerate ﬁnite interval [0, k]. Any non-negative expenditure rj on R&D for product j gives ﬁrm 1 a single purely random draw from [0, k]. So, the likelihood of a lower candidate zj depends on the level of funding invested in R&D for product j. The advantage of a higher R&D expenditure is that it gives a more favorable cdf G(zj |rj ) from which to sample, in the sense that if rj′ < rj′′ then G(zj |rj′ ) ≤ G(zj |rj′′ ) for every zj ∈ (0, k). Speciﬁcally, G(zj |rj ) is described by the following conditional probability density function (pdf) { 1/c(rj ), 0 ≤ zj ≤ c(rj ), (4) g(zj |rj ) = c(rj ) < zj ≤ k. 0, Here c is a ﬁnite C 2 function of rj ≥ 0 satisfying c(0) > 0, c′ (rj ) < 0 and c′′ (rj ) > 0 for all rj . In other words, the conditional pdf g(zj |rj ) has the properties of ∂g(zj |rj ) > 0, ∂rj

∂ 2 g(zj |rj ) <0 ∂rj2

for 0 ≤ zj ≤ c(rj ). Thus, G(zj |rj ) is strictly ﬁrst-order stochastic dominant with respect to rj . Moreover, the stochastic shift caused by a unit increase in rj become smaller as the value of rj rises. An interpretation of the supposition c(0) > 0 may be that there exists at least one R&D project that is implementable for free to ﬁrm 1 so that the ﬁrm investing nothing may still reduce the cost of production. A simple example of function c, on which we shall work in a later section, includes c(rj ) =

θ(k + k rj ) rj + θ

(5)

where k and θ are positive constants with θ k < k. Under this speciﬁcation, we observe lim c(rj ) = θk < ∞,

c(0) = k > 0, c′ (rj ) =

rj →∞

θ(θ k − k) < 0, (rj + θ)2

c′′ (rj ) =

2θ(θ k − k) > 0, (rj + θ)3

so all the stated properties hold. In stage 1, ﬁrm 2 copies ﬁrm 1’s endowed technology of producing product A. In this paper, we consider a polar case where ﬁrm 2 is a perfect costless copier so that c2A = kA . 6

Both ﬁrms are fully informed about the market demand for both products. Firm 1’s ﬁnalized marginal costs will be common knowledge before production begins. Firm 1 is fully informed about the stochastic process that governs R&D as well as the marginal cost that ﬁrm 2 will obtain by the end of stage 1. Given r = (rA , rB ) and k = (kA , kB ), ﬁrm 1’s net expected proﬁt from ﬁnancing R&D is ∫

c(rB∫ ) c(rA )

−rA − rB +

π(min{zA , kA }, kB , kA ) dG(zA |rA )dG(zB |rB ).

0

0

Here, π(c) is ﬁrm 1’s post-R&D equilibrium proﬁt when both ﬁrms’ ﬁnalized marginal costs are c = (c1A , c1B , c2A ). For expositional simplicity, we assume kB = k and also αj < k for j = A, B. The assumptions that there are only two products, that the market demand for each product is linear, that there is only one ﬁrm doing R&D and that the probability function that governs an stochastic R&D outcome is uniform are strong simpliﬁcations. However, the analysis of even this relatively simple setting is quite complex. The following sections add notation and an detailed analysis of each stage, beginning with stage 2.

3

Production

At this stage, the two ﬁrms are Cournot competitors. Each knows both ﬁrms’ ﬁnalized marginal costs as well as the market demand functions for each product. Also, there is no further opportunity to reduce the marginal costs. The ﬁrms are in a position to choose production strategies.

3.1

Best Response Correspondence

For a given k, let C1 = [0, k]2 and let Q2 = [0, ∞). Given q2A ∈ Q2 and c1 ∈ C1 , ﬁrm 1’s proﬁt from producing products A and B is Π∗1 (q2A ; c1 ) ≡

max

q1A ≥0,q1B ≥0

Π1 (q1A , q1B ; q2A , c1 )

(6)

where Π1 (q1A , q1B ; q2A , c1 ) is speciﬁed in (2). Denote the best response correspondence of ∗ ∗ ﬁrm 1 by (q1A (q2A ; c1 ), q1B (q2A ; c1 )) whenever it exists. Lemma 1. Π1 is strictly jointly concave in (q1A , q1B ) on [0, ∞)2 .

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Proposition 1. For a given q2A ∈ Q2 and a given ∗ ∗ spondence (q1A (q2A ; c1 ), q1B (q2A ; c1 )) exists and is [ ∗ ] q1A (q2A ; c1 ) = ∗ q1B (q2A ; c1 )     if 0       [ ]    [aA + aB b − q2A − (1 − b2 )c1A ] /2   if   0 [ ]   0   if  2  [a A b + aB − bq2A − (1 − b )c1B ] /2     [ ]    (a − q − c + bc ) /2 A 2A 1A 1B   if  (aB + bc1A − c1B ) /2

c1 ∈ C1 , ﬁrm 1’s best response corre-

{ { { {

q2A > (1 − b2 )(αA − c1A ) q2A > (1 − b2 )(αB − c1B )/b q2A ≤ (1 − b2 )(αA − c1A ) q2A > (1 − b2 )(αB − c1B )/b q2A > (1 − b2 )(αA − c1A ) q2A ≤ (1 − b2 )(αB − c1B )/b q2A ≤ (1 − b2 )(αA − c1A ) q2A ≤ (1 − b2 )(αB − c1B )/b.

∗ ∗ Moreover, q1A (q2A ; c1 ) and q1B (q2A ; c1 ) are function that are continuous on Q2 × C1 and are non-increasing with respect to q2A .

An intuition is simple and familiar. Since ∂Π1 (0)/∂q1A = (1 − b2 )(αA − c1A ) − q2A , the condition q2A > (1 − b2 )(αA − c1A ) speciﬁes a case where ﬁrm 1’s initial marginal proﬁt from producing product A is negative for a given q2A and a given c1 . Analogous interpretations apply to other inequalities. Clearly, ﬁrm 1’s initial marginal proﬁt from producing product j decreases with q2A given c1 . Proposition 1 states that ﬁrm 1 produces nothing of product j if ﬁrm 2’s choice of output level is large enough to make ﬁrm 1’s initial marginal proﬁt from producing product j negative. If initial production is proﬁtable, then it is optimal to produce product j until the marginal proﬁt from that product is zero. Similarly, let Q1 = [0, ∞)2 and let C2 = [0, k]. For a given (q1A , q1B ) ∈ Q1 and a given c2A ∈ C2 , ﬁrm 2’s proﬁt from producing product A is Π∗2 (q1A , q1B ; c2A ) ≡ max Π2 (q2A ; q1A , q1B , c2A ). q2A ≥0

(7)

∗ Denote by q2A (q1A , q1B ; c2A ) the best response correspondence of ﬁrm 2 whenever it exists.

Proposition 2. Given (q1A , q1B ) ∈ Q1 and c2A ∈ C2 , ﬁrm 2’s best response correspondence ∗ q2A (q1A , q1B 1; c2A ) exists and is ∗ (q1A , q1B ; c2A ) = q2A { 0 if ] [ aA + aB b − q1A − bq1B − (1 − b2 )c2A /2 if

q1A + bq1B > (1 − b2 )(αA − c2A ) q1A + bq1B ≤ (1 − b2 )(αA − c2A ).

∗ Moreover, q2A (q1A , q1B 1; c2A ) is a function that is continuous on Q1 × C2 and is non-

increasing with respect to q1j , j = A, B. 8

The intuition is similar to that of Proposition 1. Since ∂Π2 (0)/∂q2A = (1 − b2 )(αA − c2A ) − q1A − bq1B , it says that ﬁrm 2 produces nothing if ﬁrm 2’s initial marginal proﬁt from producing product A is negative given (q1A , q1B ) and c2A . Propositions 1 and 2 show that each ﬁrm responds less aggressively to an increase in the production by the opponent. Production levels are therefore strategic substitutes for both ﬁrms. This very nature of the Cournot competition in production then implies that, given c2 , ﬁrm 2 is more likely to start producing product A if ﬁrm 1 drops product A from the product mix. Hence a correlation of product additions and deletions among ﬁrms is positive, as Bernard, Redding and Schott (2010) recently observed.

3.2

An Equilibrium

For a given k, let C = C1 ×C2 and Q = Q1 ×Q2 . A post-R&D equilibrium proﬁle determined by Cournot competition given c = (c1A , c1B , c2A ) ∈ C will be denoted by q ∗∗ (c) ∈ Q if it exists:  ∗∗    ∗ ∗ ∗∗ ∗∗ q1A (c) q1A (q2A (q1A , q1B ; c2A ); c1A , c1B ) ∗∗ ∗ ∗ ∗∗ ∗∗ . (c) =  q1B (q2A (q1A , q1B ; c2A ); c1A , c1B ) q ∗∗ (c) = q1B (8) ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ q2A (c) q2A (q1A (q2A ; c1A , c1B ), q1B (q2A ; c1A , c1B ); c2A ) As known, operationally it is obtained as a solution of the system of (the best-response) equations in Propositions 1 and 2. According to the value of c, there are eight alternative cases to emerge. These are: (i) q ∗∗ (c) ∈ 0, (ii) q ∗∗ (c)∈{0}×{0}×(0, ∞), (iii) q ∗∗ (c)∈(0, ∞)×{0}×{0}, (iv) q ∗∗ (c)∈{0}×(0, ∞)×{0}, (v) q ∗∗ (c)∈(0, ∞)×(0, ∞)×{0}, (vi) q ∗∗ (c)∈(0, ∞)×{0}×(0, ∞), (vii) q ∗∗ (c)∈{0}×(0, ∞)×(0, ∞), and (viii) q ∗∗ (c)∈(0, ∞)×(0, ∞)×(0, ∞). For the full characterization of an equilibrium proﬁle q ∗∗ (c) given c, let us ﬁrst deﬁne C(u,v,w) for u, v, w ∈ {0, +} by the following subset of C: C(0,0,0) = {c ∈ C : N1 (c) < 0, N5 (c) < 0, N8 (c) < 0} , C(0,0,+) = {c ∈ C : N3 (c) < 0, N7 (c) < 0, N8 (c) > 0} , C(+,0,0) = {c ∈ C : N1 (c) ≥ 0, N6 (c) < 0, N9 (c) < 0} , C(0,+,0) = {c ∈ C : N2 (c) < 0, N5 (c) ≥ 0, N10 (c) < 0} , C(+,+,0) = {c ∈ C : N2 (c) ≥ 0, N6 (c) ≥ 0, N9 (c) < 0} , C(+,0,+) = {c ∈ C : N3 (c) ≥ 0, N6 (c) < 0, N9 (c) ≥ 0} , C(0,+,+) = {c ∈ C : N4 (c) < 0, N7 (c) ≥ 0, N10 (c) ≥ 0} , C(+,+,+) = {c ∈ C : N4 (c) ≥ 0, N6 (c) ≥ 0, N9 (c) ≥ 0} .

9

(9)

Here Ni (c), i = 1. . .10 is a function of c ∈ C speciﬁed as follows: N1 (c) = aA + aB b − (1 − b2 )c1A , N2 (c) = aA − c1A + bc1B , N3 (c) = aA + aB b − 2(1 − b2 )c1A + (1 − b2 )c2A , N4 (c) = 2aA − aB b − (4 − b2 )c1A + 3bc1B + 2(1 − b2 )c2A , N5 (c) = aA b + aB − (1 − b2 )c1B , N6 (c) = aB + bc1A − c1B ,

(10)

N7 (c) = aA b + (2 − b2 )aB − 2(1 − b2 )c1B + b(1 − b2 )c2A , N8 (c) = aA + aB b − (1 − b2 )c2A , N9 (c) = aA + aB b + (1 − b2 )c1A − 2(1 − b2 )c2A , N10 (c) = (2 − b2 )aA + aB b + b(1 − b2 )c1B − 2(1 − b2 )c2A . The right-side panels in Figure 1 illustrate the graphs of Ni (c) = 0, i = 1. . .10 in twodimensional space by holding constant one of the variables c1A , c1B and c2A . The parameter values used are aA = aB = 10 and b = 0.6 (so that αA = αB = 25). The right-side panel in Figure 1a displays the graphs of Ni (c) = 0, i = 1. . .10 in (c1A , c1B )-space for c2A = 20. Note that the graph of N4 (c) = 0 moves to the right as c2A increases. Also, we denote the intersection of the graphs of N2 (c) = 0 and N4 (c) = 0 by s = (sA , sB ) and that of the graphs of N2 (c) = 0 and N6 (c) = 0 by t = (sA , sB ). The right-side panel in Figure 1b illustrates the graphs of Ni (c) = 0, i = 1. . .10 in (c1A , c2A )-space for c1B = 18. Note that the graphs of N2 (c) = 0, N4 (c) = 0 and N6 (c) = 0 move to the right as c2A rises. The right-side panel in Figure 1c depicts the graphs of Ni (c) = 0, i = 1. . .10 in (c1B , c2A ) -space for c1A = 15. Note that the graphs of N2 (c) = 0, N4 (c) = 0 and N6 (c) = 0 move to the right as c1A rises. The left-side panels in Figure 1 illustrate C(u,v,w) in two-dimensional spaces by holding constant one of the variables c1A , c1B and c2A . The parameter values used are aA = aB = 10 and b = 0.6, the same as before. The left-side panel in Figure 1a depicts C(u,v,w) in (c1A , c1B ) -space for c2A = 20. The dotted vertical line is c1A = 20. The left-side panel in Figure 1b illustrates C(u,v,w) in (c1A , c2A )-space for c1B = 18, and the left-side panel in Figure 1c displays C(u,v,w) in (c1B , c2A )-space for c1A = 15. Figure 2 is a further illustration of C(u,v,w) in (c1A , c1B )-space for various values of c2A such as 25, 15, 12 and 2 with the same parameter values of aA = aB = 10 and b = 0.6. The dotted vertical lines are c1A = c2A . As c2A changes, the set of nonempty C(u,v,w) ’s varies, since the graph of N4 (c) = 0 moves to the right as c2A rises while those of N2 (c) = 0 10

and N6 (c) = 0 do not move. We denote the intersection of the graphs of N2 (c) = 0 and N6 (c) = 0 by s = (sA , sB ), and the intersection of those of N4 (c) = 0 and N6 (c) = 0 by t = (tA , tB ). Note sA < c2A < tA for every c2A ∈ (0, αA ). From (9), clearly any two distinct C(u,v,w) ’s are disjoint. Moreover, as demonstrated in Figures 1 and 2, the union of C(u,v,w) ’s is C. Therefore, the sets C(u,v,w) in (9) are a partition of C. Remark 1. Any two distinct C(u,v,w) ’s are disjoint. Also, the union of C(u,v,w) ’s is C. Using this speciﬁcation of C(u,v,w) ’s, the following proposition represents an equilibrium proﬁle q ∗∗ (c) given c. Proposition 3. Given c ∈ C, an equilibrium proﬁle q ∗∗ (c) exists and is q ∗∗ (c) =  0        0       0    2  [aA + aB b − (1 − b )c2A ]/2        [aA + aB b − (1 − b2 )c1A ]/2       0      0        0    [a b + a − (1 − b2 )c ]/2  A B 1B     0     [aA − c1A + bc1B ]/2 [aB + bc1A − c1B ]/2      0       2 2  [a + a b − 2(1 − b )c + (1 − b )c ]/3  A B 1A 2A      0    2 2  [aA + aB b + (1 − b )c1A − 2(1 − b )c2A ]/3        0     [aA b + (2 − b2 )aB − 2(1 − b2 )c1B + b(1 − b2 )c2A ]/(4 − b2 )     [(2 − b2 )aA + aB b + b(1 − b2 )c1B − 2(1 − b2 )c2A ]/(4 − b2 )        [2aA − aB b − (4 − b2 )c1A + 3bc1B + 2(1 − b2 )c2A ]/6       [aB + bc1A − c1B ]/2    2 2 [aA + aB b + (1 − b )c1A − 2(1 − b )c2A ]/3

if

c ∈ C(0,0,0)

if

c ∈ C(0,0,+)

if

c ∈ C(+,0,0)

if

c ∈ C(0,+,0)

if

c ∈ C(+,+,0)

if

c ∈ C(+,0,+)

if

c ∈ C(0,+,+)

if

c ∈ C(+,+,+)

where C(u,v,w) for u, v, w ∈ {0, +} is a subset of C deﬁned by (9). ∗∗ is non-decreasing with c1j ′ q ∗∗ (c) is a function and is continuous on C. Moreover, q1j ∗∗ and non-increasing with c2A , while q2A is non-decreasing with c2A and is non-increasing with c1j , where j = A, B and j ′ = A, B. 11

Given c ∈ C, let π(c) be ﬁrm 1’s post-R&D equilibrium proﬁt so that ∗∗ π(c) = Π∗1 (q2A (c); c1 ),

(11)

∗∗ where Π∗1 is deﬁned in (6) and q2A is deﬁned in (8). The next propositions characterize the ﬁrm’s post-R&D equilibrium proﬁt π(c) given c ∈ C.

Proposition 4. For a given c ∈ C, π is speciﬁed by π(c) =  0       0    [ ]   2 2 2 2 1  (a + a b) /(1 − b ) − 2(a + a b)c + (1 − b )c  A B A B 1A 1A 4   [ ]   1  (aA b + aB )2 /(1 − b2 ) − 2(aA b + aB )c1B + (1 − b2 )c1B 2  4  [ 2   1  (aA + 2aA aB b + aB 2 )/(1 − b2 )  4   − 2aA c1A − 2aB c1B − 2bc1A c1B + c1A 2 + c1B 2 ]  [ ]2  2 2 1   2 (1−b2 ) aA + aB b − 2(1 − b )c1A + (1 − b )c2A  3   [ ]2  2 2 2  1  a b + (2 − b )a − 2(1 − b )c + b(1 − b )c A B 1B 2A 2 2 2  (4−b ) (1−b )   [  1  (4aA 2 + 9aB 2 + 8aA aB b − 5aB 2 b2 )/(1 − b2 )   36     − (16aA − 2aB b)c1A − 18aB c1B + 8(aA + aB b)c2A − 18bc1A c1B      − 16(1 − b2 )c1A c2A + (16 − 7b2 )c1A 2 + 9c1B 2 + 4(1 − b2 )c2A 2 ]

if

c ∈ C(0,0,0)

if

c ∈ C(0,0,+)

if

c ∈ C(+,0,0)

if

c ∈ C(0,+,0)

if

c ∈ C(+,+,0)

if

c ∈ C(+,0,+)

if

c ∈ C(0,+,+)

if

c ∈ C(+,+,+) .

Moreover, π is continuous on C.

Proposition 5. π is diﬀerentiable on C a.e and satisﬁes ∂π(c)/∂c1A = ] 1[ 2 −a − a b + (1 − b )c A B 1A  2    1  [−a + c1A − bc1B ]   2[ A ] 4 −aA − aB b + 2(1 − b2 )c1A − (1 − b2 )c2A 3  [ ]   2 2 1  (−8a + a b) + (16 − 7b )c − 9bc − 8(1 − b )c  A B 1A 1B 2A 18    0 ∂π(c)/∂c1B =

12

if

c ∈ C(+,0,0)

if

c ∈ C(+,+,0)

if

c ∈ C(+,0,+)

if

c ∈ C(+,+,+)

elsewhere

(12)

] 1[ −aA b − aB + (1 − b2 )c1B  2    1  [−aB − bc1A + c1B ]   2 [ ] 2 2 2 4 −a b − (2 − b )a + 2(1 − b )c − b(1 − b )c A B 1B 2A 2 2 (4−b )    1  [−aB − bc1A + c1B ]  2    0

if

c ∈ C(0,+,0)

if

c ∈ C(+,+,0)

if

c ∈ C(0,+,+)

if

c ∈ C(+,+,+)

(13)

elsewhere

and ∂π(c)/∂c2A =  [ ] 2  aA + aB b − 2(1 − b2 )c1A + (1 − b2 )c2A  9   [ ]   2b2 2 aA b + (2 − b2 )aB − 2(1 − b2 )c1B + b(1 − b2 )c2A (4−b ) [ ] 2 2 2  a + a b − 2(1 − b )c + (1 − b )c  A B 1A 2A 9    0

if c ∈ C(+,0,+) if c ∈ C(0,+,+) if c ∈ C(+,+,+)

(14)

elsewhere.

π(c) is non-increasing with c1j , j = A, B and is non-decreasing with c2A . Figure 3a draws ﬁrm 1’s indirect proﬁt function π(c) in (c1A , c1B )-space for c2A = 20 with the same parameter values of aA = aB = 10 and b = 0.6. Figures 3b and 3c depict π(c) in (c1A , c2A )- and (c1B , c2A )-space for c1B = 18 and c1A = 15 respectively. As we have established, π is jointly continuous in C and exhibits the stated monotonicity. Figure 3d is a restriction of π(c) for c1B = 13 and c2A = 20 with aA = aB = 10 and b = 0.6, the same as before. Less obvious from Figure 3a but more easily seen from Figure 3d is that π(c) is smooth and convex in C(u,v,w) but not in C. In this ﬁgure, the restriction of π(c) is kinked at c1A = 15 and 17.3, the boundary between C(+,+,0) and C(+,+,+) and that between C(+,+,+) and C(0,+,+) , respectively. It is locally concave at the boundary between C(+,+,+) and C(+,+,0) but not at the boundary between C(+,+,0) and C(+,+,+) . These are summarized as a remark below. Remark 2. π(c) is convex on C(u,v,w) but not on C. Figure 4 displays a plot of ∂π(c)/∂ciA , i = 1, 2, for given c1A and c2A and for given c1B and c2A , respectively. Speciﬁcally, in Figure 4a, c1A = 17 and c2A = 20 and in Figure 4b, c1B = 17 and c2A = 20. The parameter values are aA = aB = 10 and b = 0.6, the same as before. Note that ∂π(c)/∂c1A is monotonic (non-increasing) in c1B , but ∂π(c)/∂c2A is not in c1A . As we will discuss in a later section, these kinks in π(c) on C play important roles in determining ﬁrm 1’s optimal R&D expenditures and thereby inﬂuencing its optimal timing of product switching.

13

3.3

Product Switching

Proposition 3 characterizes ﬁrm 1’s equilibrium decision on product switching, given c. How is the timing of new product introduction by ﬁrm 1 related to its production cost for product B and its and ﬁrm 2’s production costs for product A? First of all, recalling c1A ≤ c2A , it should be noted that in Figures 1a and 2, the timing of ﬁrm 1’s new product introduction is described by the graph of N6 (c) = 0. If c1 is located below the line, then the ﬁrm introduces product B; and otherwise does not. As can be seen, this is located above the 45-degree line. For a given c2A , this implies that it is not necessary that ﬁrm 1’s production cost for product B is at least as low as that for product A. Second, the graph of N6 (c) = 0 is upward-sloping for any c2A . For a given c2A , this implies that ﬁrm 1’s incentives for introducing product B are stronger, the lower is its production cost c1B for product B, given c1A . Also, for a given c2A , the incentives are stronger, the higher is its production cost c1A for product A, given c1B . Therefore, ﬁrm 1’s product addition is more likely when c1A is high and c1B is low, given c2A . Since the graph of N6 (c) = 0 does not move as c2A changes, these ﬁrm 1’s decisions on new product introduction are not aﬀected by the level of ﬁrm 2’s production cost c2A for product A. That is, if ﬁrm 1 chooses (not) to add product B given c1 when c2 = c′2 for some c′2 , then it also chooses (not) to add product B given c1 when c2 = c′′2 > c′2 . This is the last point to make. How is the timing of ﬁrm 1’s deletion of the existing product related to his production cost for product B and its and ﬁrm 2’s production costs for the existing old product, product A? First of all, given c1A ≤ c2A , in Figures 1a and 2 the timing of ﬁrm 1’s product deletion is described by the upper-envelope of the graphs of N2 (c) = 0 and N4 (c) = 0. If c1 is located below this envelope, then the ﬁrm ceases the production of product A while introducing product B. Otherwise, it does not stop producing product A. As can be seen, this is located below the 45-degree line. For a given c2A , this implies that ﬁrm 1’s does not drop product A until its production cost c1B for product B becomes lower than that for product A. Second, when c2A ≤ aA , ﬁrm 1’s product deletion is not an equilibrium outcome. In Figure 2, this is described by case (d). Therefore, a very low production cost of ﬁrm 2 removes ﬁrm 1’s incentives for product dropping. Next, suppose c2A > aA . Then, according to the value of c1 , it is possible that ﬁrm 1 ceases the production of product A while introducing product B. The graphs of N2 (c) = 0 and N4 (c) = 0 are both upward-sloping for any c2A . For a given c2A , this implies that ﬁrm 1’s incentives for ceasing the production of product A while introducing product B are stronger, the lower is its production cost c1B for product B, given c1A . Also, for a given c2A , the incentives are stronger, the higher is its production cost c1A for product A, given c1B . Therefore, when c2A is not very low, ﬁrm 1’s product deletion while introducing the new product is more likely 14

when c1A is high and c1B is low. As c2A falls, the graph of N4 (c) = 0 shifts to the left. This implies that, in contrast to the case of new product introduction, ﬁrm 1’s decision of product deletion is aﬀected by the level of the opponent’s production cost of product A if c2A > aA . Speciﬁcally, it is possible that a lower c2A increases ﬁrm 1’s incentives for stopping the production of product A while introducing product A, given c1 . This may be illustrated more clearly by using an example. Look at Figure 2b and 2c. If c = (10, 3, 15), then ﬁrm 1 produces both products (since c is above the envelope line). However, if c = (10, 3, 12), then ﬁrm 1 drops product A and concentrates on the production of product B (since c is below the envelope line). In summary, ﬁrm 1’s incentives for product addition and deletion are both negatively correlated to its production cost for the new product, product B, and are positively correlated to its production cost for the existing product, product A, given ﬁrm 2’s production cost c2A for product A. The timing of ﬁrm 1’s introduction of product B is not aﬀected by the level of c2A . However, ﬁrm 1’s decision on stopping the production of product A while introducing product A could be aﬀected by the level of c2A .

4

R&D

Firm 1’s initial marginal costs of producing products A and B are k = (kA , kB ) with kA < αA and αB < kB = k. The question addressed in this section is how much the ﬁrm should invest in R&D in the hope of reducing its marginal costs.

4.1

An Optimal R&D Expenditure

Once the R&D outcome of each product is known, ﬁrm 1’s equilibrium proﬁt level is π(c) as speciﬁed in Section 3. Ex ante and given c2A = kA , the ﬁrm’s expected net proﬁt from an R&D expenditure r = (rA , rB ) is ϕ(k) ≡ max −rA − rB + E(r; k). r>0

(15)

Here, E(r; k) is ﬁrm 1’s expected gross (i.e., exclusive of R&D expenditure) proﬁt from a given R&D expenditure r: c(rB ∫ ) c(rA )

∫

π(min{zA , kA }, zB , kA ) dG(zA |rA )dG(zB |rB ).

E= 0

0

The ﬁrm’s expected reward from R&D investment is thus a reduced marginal cost with all of the attendant enhancements to proﬁtability described in section 3.

15

By taking its derivative with respect to rj , the marginal gross beneﬁt to ﬁrm 1 from an additional dollar spent on R&D for product j, denoted by ej (r; k) ≡ ∂E/∂ rj , is calculated as ∫

c(rA∫) c(rB )

∂g(zA |rA ) eA = [π(min{zA , kA }, zB , kA ) − π(c(rA ), zB , kA )] dG(zB |rB ) dzA ∂rA 0 0 ∫ c(rB∫) c(rA ) ∂g(zB |rB ) eB = [π(min{zA , kA }, zB , kA ) − π(min{zA , kA }, c(rB ), kA )] dG(zA |rA ) dzB . ∂rB 0 0 The following lemma establishes that a higher R&D expenditure for product j brings ﬁrm 1 a higher expected gross proﬁt. Lemma 2. ej (r; k) is strictly positive for all r. While the ﬁrm’s expected gross proﬁt increases with its R&D expenditure for product j, the rate of change in E(r; k) with respect to rj diminishes as rj increases. It is because the stochastic shift caused by a unit increase in rj become smaller as the value of rj rises. This is the content of the following lemma. Lemma 3. ej (r; k) is strictly decreasing with respect to rj a.e. Therefore, at some point the marginal expected gross proﬁt must reduce to below unity. Since the marginal cost of R&D expenditure for product j is constant at unity, the marginal expected net proﬁt −1 + ej (r; k) must eventually be negative. Consequently, the ﬁrm’s expected net proﬁt −rA − rB + E(r; k) is negative for suﬃciently large r. These are illustrated in Figure 5, which depicts the ﬁrm’s expected proﬁt for aA = aB = 10 and b = 0.6, the same parameter values as before, and c(rj ) speciﬁed in (5). As can be seen in Figures 5a and 5b, E(r; k) increases with rj at a diminishing rate, and −rA − rB + E(r; k) < 0 for large r. An optimal R&D expenditure that maximizes ﬁrm 1’s net expected proﬁt in (15) will ∗ ∗ be denoted by r∗ = (rA (k), rB (k)). Let µ∗ = (µ∗A (k), µ∗B (k)) be the Lagrange multiplier implied by the solution of the program. As revealed in the next proposition, an optimal R&D investment should equalize as nearly as possible the marginal expected reward with the marginal cost that is unity. Theorem 1. An optimal expenditure r∗ exists. If ej (0; k) ≤ 1, then rj∗ = 0. Otherwise, rj∗ > 0. Moreover, if rj∗ > 0, then ej (r∗ ; k) = 1. Otherwise, µ∗j = −1 + ej (r∗ ; k). An economic meaning of Theorem 1 is as follows. ej (0; k) could be interpreted as the marginal expected gross proﬁt to ﬁrm 1 from initial R&D investment in product j. Therefore, it says that the ﬁrm invests nothing in R&D for both products, if its marginal 16

expected gross proﬁt from initial investment is at least as large as unity for both products. If its initial marginal expected proﬁt is larger than unity for product j but less then unity for the other product, then it makes a positive R&D investment in only product j. An optimal investment level for product j equalizes the marginal expected proﬁt with its marginal cost. If its marginal expected gross proﬁt is positive for both products, then the ﬁrm ﬁnances positive R&D expenditures for both products until its marginal expected proﬁts from R&D investment for both products are equal to zero. As usual, the Lagrange multiplier µ∗j (k) may be interpreted as the shadow price of ﬁrm 1’s initial technology kA of producing product j. The ﬁrm’s initial R&D incentive for product j is thus positively related to the level of ej (0; k). For instance, if eA (0; k) is larger than unity, then the ﬁrm’s R&D incentive for product A is strong enough to start making a positive R&D investment for product A. Bernard, Redding and Schott (2010) recently ﬁnd that product switching is more likely in productive ﬁrms than less-productive. Does the model accord with this observation? That is, how the model predicts the dependence of the ﬁrm’s R&D incentives on, among other things, the level of its production cost kA initially endowed for producing product A? In the next subsection, such comparative statics analyses are conducted.

4.2

Comparative Statics

Given k, a particularly important problem is studying how the ﬁrm’s R&D portfolio r∗ = ∗ ∗ (rA , rB ) changes in response to a shock to the current state of technology kA of producing product A. Answering this comparative statics question helps us in turn deduce the optimal timing from the ﬁrm’s point of view for starting R&D in the hope of introducing a new product, even though it may cannibalize the proﬁt accrued to the existing product. As established below, the answer to this question depends on the manner in which the marginal expected gross proﬁt to ﬁrm 1 will be altered by a unit change in the parameter value of kA , or the sign of ∂ej /∂kA , j = A, B. Theorem 2 below is the comparative statics result where the ﬁrm currently ﬁnances a positive R&D expenditure only for one product. Theorem 2. Let j = A, B and j ′ ̸= j. If rj∗ > 0 and rj∗ ′ = 0, then sgn( ∂rj∗ /∂kA ) = sgn( ∂ej (r∗ ; k)/∂kA ) and sgn( ∂µ∗j ′ /∂kA ) = sgn( ∂ej ′ (r∗ ; k)/∂kA ). Assuming that the ﬁrm’s current R&D expenditure for product j is non-negative and that for the other product, zero, Theorem 2 says that the rate of change in rj∗ with respect to kA is positively related to the sign of ∂ej (r∗ ; k)/∂kA if the ﬁrm’s investment level in R&D for product j is positive. Otherwise, the rate of change in the shadow price µ∗j in response to an increase in kA is positively related to the sign of ∂ej (r∗ ; k)/∂kA . 17

The next theorem is the comparative statics result where the ﬁrm’s current R&D expenditure is strictly positive for both products. Theorem 3. Suppose r∗ ≫ 0. Then,

∑ j=A,B

∂ej (r∗ ; k)/∂kA · ∂rj∗ /∂kA ≥ 0.

The content of Theorem 3 is summarized by Table 1. Here is listed the sign of ∂r∗ /∂kA = ∗ ∗ (∂rA /∂kA , ∂rB /∂kA ) that is not optimal when ∂e(r∗ ; k)/∂kA = (∂eA (r∗ ; k)/∂kA , ∂eB (r∗ ; k)/∂kA ) has the stated sign. For instance, if ∂eA (r∗ ; k)/∂kA > 0 and ∂eB (r∗ ; k)/∂kA = 0, then it is not optimal for the ﬁrm to reduce its R&D investment for product A. Any response in its ∗ R&D investment for product B can be optimal. Thus, rB may or may not increase. ∂eA ∂eB \ ∂kA ∂kA

+ 0 −

+ (−,−) ( ? ,−) (+,−)

0 (−, ? ) (?,?) (+, ? )

− (−,+) ( ? ,+) (+,+)

Table 1: The suboptimal sign of ∂r∗ /∂kA when r∗ ≫ 0

What is the eﬀect of a unit increase in kA on ej (r; k), or the sign of ∂ej (r; k)/∂kA ? As established in Propositions 6–11, the sign of ∂ej (r; k)/∂kA interestingly varies according to the levels of rA , rB and kA . Proposition 6. Suppose that kA is very small so that sA < 0 and sB < 0 hold. Then, ∂eA (r; k)/∂kA is positive if rA is large so that c(rA ) is close to or smaller than kA . Moreover, it tends to decline as rA falls if rA is relatively small so that c(rA ) > kA holds. Proposition 6 speciﬁes how a unit increase in kA changes the ﬁrm’s marginal expected R&D proﬁt eA (r; k) for product A when kA is very small so that sA < 0 and sB hold, i.e., when kA < αA /2 holds. It says that when kA is very small, eA (r; k) tends to increase with kA if the ﬁrm’s current R&D expenditure rA for product A is relatively large. Moreover, its increasing rate tends to decline as rA falls. So, eA (r; k) may or may not increase with kA if rA becomes suﬃciently small. Therefore, an increase in kA could a qualitatively diﬀerent impact on eA (r; k) according to the level of rA . An intuition is stated as follows. If kA is very small, then both ﬁrms will produce non-zero output levels of product A in the production stage. See, for instance, Figure 2d where kA = 2 and αA /2 = 12.5. When rA is small, there may be a substantial chance that the two ﬁrms has the same production cost equal to kA . Then, an increase in kA has two opposing marginal eﬀects on ﬁrm 1’s maximized production proﬁt π(c). On one hand, it 18

lowers π(c) by directly raising ﬁrm 1’s production cost for product A. On the other hand, it increases π(c) by indirectly reducing ﬁrm 2’s choice of output level. As long as the direct eﬀect outweighs the indirect eﬀect, the net eﬀect is negative. And to the extent that this is the case, this decreases eA (r; k) as well. Alternatively, if rA is relatively large so that c(rA ) is close to kA , then there is a substantial chance that the ﬁrm’s ﬁnalized production cost for product A is strictly lower than kA . In this case, an increase in kA only has an indirect eﬀect on π(c), which raises π(c) and so does eA (r; k). Proposition 7. Suppose that kA is relatively small so that sA > 0 and sB < 0 hold. Then, ∂eA (r; k)/∂kA is zero if rA is large so that c(rA ) < sA holds. Moreover, it is negative if rA is relatively large so that c(rA ) is larger than but close to sA , and tends to increase and then falls as rA declines. Proposition 7 speciﬁes the eﬀect of a unit increase in kA on the ﬁrm’s marginal expected R&D proﬁt eA (r; k) for product A when kA is relatively small so that sA > 0 and sB < 0 hold, i.e., when αA /2 < kA < (αB + aA )/2 holds. It says that eA (r; k) could increase with kA if the ﬁrm’s current R&D expenditure rA is relatively small (so that c(rA ) is not apart from kA ), decreases with kA if rA is relatively large (so that c(rA ) is larger than but close to sA ), and is not inﬂuenced by an increase in kA if rA is very large (so that c(rA ) < sA holds). Therefore, the eﬀect of a unit increase in kA on eA (r; k) changes with the level of rA . Note that the eﬀect is somewhat reversed from that in Proposition 6. The reason is stated as follows. When kA is relatively small, it is possible for ﬁrm 1 to force ﬁrm 2 to exit out of the market of product A and become a monopolist. If ﬁrm 1’s R&D expenditure rA for product A is suﬃciently large (so that c(rA ) < sA holds), then this arises with certainty. Clearly the monopoly proﬁt does not depend on ﬁrm 2’s production cost. Hence, an increase in kA should have no eﬀect on eA (r; k). If rA is relatively large, the ﬁrm may have a good chance to be a monopolist by successfully lowering its production cost from kA by some signiﬁcant amount. However, as long as c(rA ) < sA , there still exists a chance of its failing to do so. An increase in kA raises this latter prospect. Since the proﬁt is larger in monopoly than in duopoly, this reduces ﬁrm 1’s expected production proﬁts and so is eA (r; k). If rA is not this large, there may be a substantial chance that ﬁrm 2 keeps operating in the market of product A. In addition, if c(rA ) is relatively close to kA , then it is highly possible that ﬁrm 1’s ﬁnalized production cost for product A is strictly lower than kA . To the extent that this is the case, an increase in kA raises π(c) through its indirect eﬀect, which in turn increases eA (r; k) as well. It is inferred that this last scenario becomes more probable when kA is large. The reason is that if kA is large, even a very small rA could be enough for ﬁrm 1 to expect to have 19

a lower production cost for product A. Once it successfully lowers its production cost, an increase in kA only has the indirect eﬀect on π(c) and raises eA (r; k). As can be seen, this is reﬂected in the next proposition. Proposition 8. Suppose that kA is relatively large so that sA > 0 and sB > 0 hold. Then, ∂eA (r; k)/∂kA is zero if rA or rB is large so that c(rA ) < sA or c(rB ) < sB holds. Otherwise, it is negative if rA is relatively large so that c(rA ) is larger than but close to sA , and tends to increase and then falls as rA declines. Proposition 8 speciﬁes the eﬀect of a unit increase in kA on eA (r; k)/kA when kA is relatively large so that sA > 0 and sB > 0 hold, i.e., when kA > (αB + aA )/2 holds. It says among others that an increase in kA has no eﬀect on eA (r; k) if rB is large so that c(rB ) < sB holds. The reason is that if kA is relatively large, then a large R&D expenditure for product B makes ﬁrm 1 to be a single-product monopolist of product B or a multi-product monopolist. In either case, the ﬁrm’s production proﬁt π(c) does not depend on ﬁrm 2’s production cost. Hence, an increase in kA should have no eﬀect on eA (r; k).

What is the eﬀect of a unit change in kA on eB (r; k)? As before, we consider several sub-cases according to the parameter value of kA . To ease the notation, let m denote the intercept to the horizontal axis of the graph of N4 (c) = 0 in the (c1A , c1B )-space, as depicted in Figures 2b–d and 7a–b. That is, m ≡ (2aA − aB b)/(2 + b2 ). Note that this is the value 1A of kA solving kA = n1A 4 (0, kA ), where n4 is a function deﬁned in (19). Proposition 9. Suppose that kA is very small so that kA < m Then, ∂eB (r; k)/∂kA is zero if rA is large so that c(rA ) < kA holds, and is positive otherwise. Proposition 9 speciﬁes how a unit increase in kA aﬀects ﬁrm 1’s marginal expected R&D proﬁt eB (r; k) for product B when kA is very small so that kA < m holds. It says that eB (r; k) does not change with kA if rA is large, and tends to increase with kA otherwise. The reason is stated as follows. If kA is small, ﬁrm 2 will produce product A. If rA is large, so will ﬁrm 1. Moreover, it will obtain c1A < kA with certainty. Consequently, an increase in kA only has an indirect eﬀect on π(c), which increases π(c) by raising ﬁrm 2’s production cost. According to the outcome zB , ﬁrm 1 may or may not be a multi-product ﬁrm. However, as (14) shows, this does not change the ﬁrm’s marginal production proﬁt with respect to c2A , ∂π(c)/∂c2A . Hence, an increase in kA has no eﬀect on eB (r; k). On the other hand, if rA is relatively small, then there may be a substantial chance that ﬁrm 1 fails to obtain a lower production cost than kA for producing product A. In this case, an increase in kA has two opposing eﬀects π(c). On one hand, it raises π(c) by 20

raising ﬁrm 2’s production cost. On the other hand, it decreases π(c) by directly raising its production cost for product A. Although the latter eﬀects outweighs the former, an increase in kA reduces the chance that the ﬁrm fails to reduce its production cost from kA . Hence, a unit increase in kA raises eB (r; k). Proposition 10. Suppose that kA is relatively small so that kA > m and sB < 0 hold. Then, ∂eB (r; k)/∂kA is zero if rA is large so that c(rA ) < m holds, is negative if rA is relatively large so that m < c(rA ) < kA , and tends to be positive if rA is relatively small so that c(rA ) > kA holds. Propositions 10 speciﬁes the eﬀect of a unit increase in kA on eB (r; k) when kA is relatively small so that kA > m and sB < 0 holds, i.e., when m < kA < (αB + aA )/2 holds. It says among other things that eB (r; k) is not aﬀected by a unit increase in kA if rA is large. The reason is that when kA is relatively small, ﬁrm 2 will not stop producing product A unless ﬁrm 1’s production cost for product A is dropped drastically to sA or smaller. The latter arises with certainty if rA is large so that c(rA ) < sA holds. Since the monopolist’s proﬁt is independent of exiting ﬁrm 2’s production cost, eB (r; k) does not change with kA . The rest of the statement and an intuition are similar to those of Proposition 9. Proposition 11. Suppose that kA is relatively large so that sA > 0 and sB > 0 hold. Then, ∂eB (r; k)/∂kA is zero if rA or rB is large so that c(rA ) < sA or c(rB ) < sB holds. Otherwise, it tends to be negative if rB is relatively large so that c(rB ) is larger than but close to sB . Proposition 11 speciﬁes how a unit increase in kA aﬀects eB (r; k) when kA is relatively large. It says that eB (r; k) may or may not increase with kA if both rA and rB are small, decrease with kA if rB is relatively large and rA , relatively small, and is not inﬂuenced by an increase in kA if rA or rB is signiﬁcantly large. An intuition is explained as follows. When kA is relatively large, it is possible that ﬁrm 1 forces ﬁrm 2 to exit out of the market of product A and becomes a single-product monopolist or a multi-product monopolist. If rA or rB is signiﬁcantly large, then this arises with certainty. In that case, ﬁrm 1’s production proﬁt π(c) is independent of the production cost of ﬁrm 2. Hence, an increase in kA has no eﬀect on eB (r; k). Suppose that rA is not signiﬁcantly large. Then, even if rB is only relatively large, ﬁrm 1 may have a good chance to be a single-product monopolist of product B or a multi-product monopolist by forcing ﬁrm 2 to exit out of the market. However, as long as c(rB ) > sB holds, there still exists a chance for failure. An increase in kA raises this latter possibility. Since the proﬁt is larger in monopoly than in duopoly, this reduces π(c) 21

and so does eA (r; k). If rB is small, there may be a substantial chance that ﬁrm 2 keeps operating in the market of product A. In that case, an increase in kA raises π(c) through the indirect eﬀect and therefore increases eB (r; k). The chance, however, is not nil that the ﬁrm becomes a monopolist. The higher is this chance, the higher is eB (r; k). An increase in kA reduces this chance, yielding a reduced eB (r; k). Hence, when rB is small, an increase in kA may have a mixed eﬀect on eB (r; k), and which eﬀect dominates cannot be determined in general. Figures 6 and 7 are graphical summaries of Propositions 6–8 and 9–11, respectively. Here, the signs of ∂eA (r; k)/∂kA and ∂eB (r; k)/∂kA are included in the (c(rA ), c(rB ))-space, respectively. As can be seen, the sign varies according to the levels of rA , rB and kA . This various dependence of e(r; k) on the levels of rA , rB and kA has the following implications on the timing of product switching. First, since ﬁrm 1’s product addition and deletion are both negatively correlated to c1B and positively correlated to c1A , a diﬀerent sign of (∂eA (r; k)/∂kA , ∂eB (r; k)/∂kA ) results in a diﬀerent eﬀect of an increase in kA on the timing of product switching. In particular, Theorems 2 and 3 imply that an increase in kA tends to foster ﬁrm 1’s product switching if an increase in kA reduces eA (r; k) and increases eB (r; k), and tends to deter if an increase in kA increases eA (r; k) and decreases eB (r; k). As can be seen, both signs of (−, +) and (+, −) appear in Figures 6 and 7. Hence, an increase in kA can foster and deter the ﬁrm’s product switching, according to the levels of rA , rB and kA . It deserves emphasis, since if there were no R&D stage incorporated in our model, an increase in kA may be predicted to foster ﬁrm 1’s product switching, since the ﬁrm’s incentive is positively correlated to c1A . So, the result can be reversed with process innovation. Secondly, this in turn implies that a common productivity shock to kA causes various responses in product switching among ﬁrms. Bernard, Redding and Schott (2010) ﬁnd that ﬁrms’ product additions and deletions are positively correlated across ﬁrms (that is, while some ﬁrms add, others drop). This provides an explanation for why such a positive correlation is common.

5

Conclusion

This paper examines the R&D incentives of a multi-product ﬁrm and the timing of product switching in a two-product Cournot model of duopoly with process innovation. The two ﬁrms are an innovator and an imitator. Both supply a market with an existing identical product. In addition, the innovator is a prototype producer of a new product. Time is divided into two stages. Stage 2 is the production stage. In stage 1, the imitator copies the 22

innovator’s initial technology while the innovator carries out further cost-reducing process R&D. The model is motivated by the commonality of overlapping product life-cycles in many industries and also by the empirical evidence of the recent studies that product switching is a common practice for a large majority of ﬁrms in all sectors and does not arise as a result of M&A. The issues addressed are: (i) How is an innovator’s decision on new product introduction related to his production cost for the new product and its and the opponent’s production costs for the existing product? (ii) How is the innovator’s decision on stopping the production of the existing product related to its production cost for the new product and its and the opponent’s production costs for the existing product? (iii) How does a shock to the innovator’s initial production cost for the existing product change its R&D expenditure for the new and existing products? Does this shock thereby foster or deter its product switching? The following conclusions emerge. (i) An innovative ﬁrm’s incentives for product addition and deletion are both negatively correlated to its production cost for the new product, and are positively correlated to its production cost for the existing old product, given the opponent’s production cost for the old product. (ii) The timing of the innovator’s introduction of the new product is not aﬀected by the level of the opponent’s production cost, but its dropping decision on the old product while introducing the new one could be. (iii) A positive shock to the innovator’s initial production cost for the old product has a diﬀerent eﬀect on its R&D expenditures and thereby its timing of product switching, according to the levels of its initial production cost for the old product and its current R&D expenditures. This insight is new, since if there was no R&D stage then an increase in its initial production cost for the old product should cause more product switching. This is because the innovator’s product switching is positively correlated to its production cost for the old product. So, an optimal process innovation can deter the timing of product switching. Also, this implies that a common productivity shock can cause various responses in product switching among ﬁrms. Bernard, Redding and Schott (2010) ﬁnd that product additions and deletions are positively correlated across ﬁrms (that is, while some ﬁrms add, others drop). This model’s prediction provides an explanation for why such a positive correlation is common.

23

Appendix A: Proofs of the Results in Section 3 Proof of Lemma 1. Given q2A and c1 , diﬀerentiate Π1 twice with respect to q1j , j = A, B: 2 ∂ 2 Π1 /∂q1j = 2/(1 − b2 ),

∂ 2 Π1 /∂q1A ∂q1B = −2b/(1 − b2 ),

j = A, B.

The claim follows. Proof of Proposition 1. Lemma implies that (6) is a concave programming. Therefore a solution exists and is unique. From the latter, the best response correspondence is a function. Additionally, since it is a concave programming the ﬁrst-order condition is also suﬃcient. Letting (λ1A , λ1B ) be a vector of Lagrangian multipliers, form the Lagrangian: L1 (q1A , q1B , λ1A , λ1B ) ≡ Π1 (q1A , q1B ; q2A , c1 ) + λ1A q1A + λ1B q1B The ﬁrst-order conditions include ∗ ∗ ∂Π1 (q1A , q1B )/∂q1j + λ∗1j ≡ 0

j = A, B

∗ plus the complementary slackness conditions λ∗1j q1j = 0, j = A, B where

∂Π1 (q1A , q1B )/∂q1A = [aA + aB b − 2q1A − q2A − 2bq1B ] /(1 − b2 ) − c1A ∂Π1 (q1A , q1B )/∂q1B = [aA b + aB − 2bq1A − bq2A − 2q1B ] (1 − b2 ) − c1B . ∗ ∗ Solve for (q1A , q1B ) and (λ∗1A , λ∗1B ). The speciﬁcation in the proposition is just this rearranged. ∗ Obviously, q1j is continuous on Q2 × C1 and diﬀerentiable almost everywhere (a.e.) for ∗ j = A, B. By taking the derivative with respect to q2A we observe ∂q1j /∂q2A ≤ 0. Recalling ∗ the continuity on Q2 × C1 , this implies that q1j does not increase with q2A .

Proof of Proposition 2. Omitted. Proof of Proposition 3. First, q ∗∗ is a function on C, since it is a function on C(u,v,w) , u, v, w ∈ {0, +} and since C(u,v,w) are disjoint sets. ∗∗ ∗∗ , j = A, B is continuous on C1 and so is q2A By the Berge Theorem of Maximum, q1j ∗∗ ∗∗ on C2 . Therefore, if q2A is shown to be continuous on C1 as well, so that q2A is continuous on C, then the continuity of q ∗∗ on C follows, since a function of a continuous function is ∗∗ is continuous on C for both j = A, B. We will establish this with continuous, so that q1j an aid of Figure 1. Let ϵ > 0. Given c2A , suppose that (c′1A , c′1B ) is a point at the boundary between C(+,+,0) and C(+,+,+) . We will show ∗∗ ′ ∗∗ ′ (c1A , c′1B , c2A ), lim q2A (c1A − ϵ, c′1B − ϵ, c2A ) = q2A

ϵ→0+

24

∗∗ is continuous at the neighborhood of this boundary for a given c2A since (c′1A , c′1B ) so that q2A is chosen arbitrarily. On the right-hand side,

[ ] ∗∗ ′ q2A (c1A , c′1B , c2A ) = aA + aB b + (1 − b2 )c1A − 2(1 − b2 )c2A /3 = 0 since if (c′1A , c′1B ) is at this boundary then c′1A = [−aA − aB b + 2(1 − b2 )c2A ] /(1 − b2 ). Trivially the left-hand side is equal to zero, so the claim follows. Next suppose that (c′1A , c′1B ) is a point at the boundary between C(0,+,+) and C(+,+,+) . We will show ∗∗ ′ ∗∗ ′ lim q2A (c1A + ϵ, c′1B − ϵ, c2A ) = q2A (c1A , c′1B , c2A ),

ϵ→0+

∗∗ is continuous about the boundary for a given c2A since (c′1A , c′1B ) is arbitrary. so that q2A Note if (c′1A , c′1B ) is at this boundary, then

[ ] c′1B = −2aA + aB b + (4 − b2 )c′1A − 2(1 − b2 )c2A /3b for c′1A ∈ [ (−aA − aB b + 2(1 − b2 )c2A ) /(1 − b2 ), kA ]. On the left-hand side, ∗∗ ′ lim q2A (c1A + ϵ, c′1B − ϵ, c2A ) [ ] = lim+ (2 − b2 )aA + aB b + b(1 − b2 )(c′1B − ϵ) − 2(1 − b2 )c2A /(4 − b2 ) ϵ→0 [ ] = aA + aB b + (1 − b2 )c′1A − 2(1 − b2 )c2A /3

ϵ→0+

∗∗ ′ = q2A (c1A , c′1B , c2A ). ∗∗ Therefore, q2A is continuous at this neighborhood.

Suppose now that (c′1A , c′1B ) is a point at the boundary between C(0,+,0) and C(0,+,+) . We will show ∗∗ ′ ∗∗ ′ lim+ q2A (c1A + ϵ, c′1B − ϵ, c2A ) = q2A (c1A , c′1B , c2A ),

ϵ→0

∗∗ so that q2A is continuous about this boundary since (c′1A , c′1B ) is again arbitrary. Note

[ ] c′1B = −(2 − b2 )aA /b − aB /(1 − b2 ) + 2c2A /b at this boundary, so the right-hand side is ] [ ∗∗ ′ q2A (c1A , c′1B , c2A ) = (2 − b2 )aA + aB b + b(1 − b2 )c′1B − 2(1 − b2 )c2A /(4 − b2 ) = 0. Since the other side is trivially zero, the claim follows. Elsewhere the continuity is obvious. ∗∗ Therefore q2A is continuous on C1 . And therefore q ∗∗ is continuous on C.

25

∗∗ , j = A, B is right- and left- diﬀerentiable with respect to each of the arguments. q1j ∗∗ ∗∗ Where it is diﬀerentiable, ∂q1j /∂c1j ′ ≤ 0 and ∂q1j /∂c2A ≥ 0 for j, j ′ = A, B. At every c ∈ C

the same signs hold for both the right- and the left-derivatives with respect to c1j ′ , j ′ = A, B ∗∗ and to c2A , which are non-positive and non-negative respectively. Therefore q1j , j = A, B is non-increasing with c1j ′ , j ′ = A, B and non-decreasing with c2A . An analogous argument ∗∗ will establish the stated monotonicity of q2A .

Proof of Proposition 4. Direct calculation yields π as in the form given in the proposition. ∗∗ as in (11). From Proposition 1, it is immediate π is a composite function of Π∗1 and q2A ∗∗ that Π∗1 is continuous on Q2 × C1 . As proven above, q2A is continuous on C. Since a composite function of continuous functions are continuous, π is continuous on C.

Proof of Proposition 5. Direct calculation, if appropriate, yields the derivatives of π as in the proposition. For any c ∈ C(u,v,w) , it is easy to observe ∂π(c)/∂c1A ≤ 0, ∂π(c)/∂c1B ≤ 0 and ∂π(c)/∂c2A ≥ 0, except ∂π(c)/∂c1A ≤ 0 for c ∈ C(+,+,+) . To see the latter, we employ the First Envelope Theorem: ∗∗ ∗∗ ∗∗ ∂π ∂Π∗1 ∂q2A ∂Π∗1 q ∗∗ + bq1B ∂q2A 1 ∗∗ ∗∗ ∗∗ = · + = − 1A · − q1A = − (4q1A + bq1B ) ≤ 0. 2 ∂c1A ∂q2A ∂c1A ∂c1A 1−b ∂c1j 3

Together with the continuity on C, these imply that π is non-increasing with c1j , j = A, B and non-decreasing with c2A .

Appendix B: Proofs of the Results in Section 4.1 Proof of Lemma 2. To begin with, recall that π is strictly decreasing in c1j , j = A, B for c1j substantially small (see Figures 1 and 2 as well as (12)–(13)). Therefore, the proﬁt diﬀerence π(min{zA , kA }, zB , kA )−π(c(rA ), zB , kA ) is positive for zA suﬃciently small. Likewise, π(min{zA , kA }, zB , kA ) − π(min{zA , kA }, c(rB ), kA ) > 0 for zB suﬃciently small. Since g(zj |rj ) = c(rj )−1 > 0 and ∂g(zj |rj )/∂rj = −c ′ (rj )c(rj )−2 > 0, the integrand in the right-hand side is positive. The claim follows. Proof of Lemma 3. Assume that ej (r; k) is diﬀerentiable with respect to rj at r ≥ 0. If rA is large so that c(rA ) < kA holds, then ∫ c(rA∫) c(rB ) ∂π(c(rA ), zB , kA ) ′ ∂eA ∂g(zA |rA ) =− c (rA )dG(zB |rB ) dzA ∂rA ∂c1A ∂rA 0 0 ∫ c(rA∫) c(rB ) ∂ 2 g(zA |rA ) [π(zA , zB , kA ) − π(c(rA ), zB , kA )] dG(zB |rB ) dzA . + 2 ∂rA 0 0 26

This is negative, since ∂π(c)/∂c1A ≤ 0 with strict inequality when c1A signiﬁcantly low, c ′ (rA ) < 0, ∂g(zA |rA )/∂rA > 0, π(zA , zB , kA ) − π(c(rA ), zB , kA ) ≥ 0 with strict inequality 2 when c1A signiﬁcantly small and ∂ 2 g(zA |rA )/∂rA < 0 Instead, if rA is such that c(rA ) < kA holds, then ∫ c(rA∫) c(rB ) ∂eA ∂π(c(rA ), zB , kA ) ′ ∂g(zA |rA ) =− c (rA )dG(zB |rB ) dzA ∂rA ∂c1A ∂rA 0 0 ∫ c(rA∫) c(rB ) ∂ 2 g(zA |rA ) + [π(min{zA , kA }, zB , kA ) − π(c(rA ), zB , kA )] dG(zB |rB ) dzA 2 ∂rA 0 0 ∫ c(rB ) ∂g(zA |rA ) ′ [π(min{zA , kA }, zB , kA ) − π(c(rA ), zB , kA )] dG(zB |rB ) c (rA ), + ∂rA 0

which is negative since the additional last term is also negative. eA is the right- and the leftdiﬀerentiable at any r ≥ 0. And the same sign holds for the right- and the left-derivative, which is negative. Therefore, eA decreases with rA . For product B the proof is parallel. Proof of Theorem 1. Let 0 < r < ∞. By the Weierstrass Theorem, −rA − rB + E(r; k) has a maximum on [0, r]2 since it is a continuous function on a compact set [0, r]2 to R. If r is signiﬁcantly large then −rA − rB + E(r; k) < 0 for r ∈ [r, ∞)2 , since the rate of increase in E diminishes with rj (as shown in Lemma 3) while −rj declines at a constant rate of unity. For r so chosen, the maximum on [0, r]2 is also the maximizer on [0, ∞)2 . Therefore r∗ , an optimal r, exists. r∗ necessarily satisﬁes the ﬁrst-order condition (FOC) [ ] −1 + eA (r∗ ; k) + µ∗A ≡0 −1 + eB (r∗ ; k) + µ∗B plus the complementary slackness condition µ∗j rj∗ = 0, j = A, B. Hence, (a) ej (r∗ ; k) = 1 − µ∗j ≤ 0 if r∗ = 0; (b) ej (r∗ ; k) = 1 − µ∗j = 0, if r∗ ≫ 0 and (c) ej (r∗ ; k) = 1 − µ∗j = 0 and ej (r∗ ; k) = 1 − µ∗j ′ ≤ 0, if rj∗ > 0 and rj∗ ′ = 0 where j ̸= j ′ . By Lemma 3, this in turn suggests that (a’) ej (0; k) ≤ 1 if r∗ = 0; (b’) ej (0; k) > 1, if r∗ ≫ 0 and (c’) ej (0; k) > 1 and ej ′ (0; k) ≤ 1, if rj∗ > 0 and rj∗ ′ = 0. Given these, the claim is established by contradiction. First, suppose ej (0; k) ≤ 1 and r∗ ̸= 0. Let rj∗ > 0. Then, necessarily ej (0; k) > 1, a contradiction. Therefore, r∗ = 0. The other cases can be considered in similar fashions. Hence the statement holds.

Appendix C: Proofs of the Results in Section 4.2 Proof of Theorem 2. Lemma 3 implies ∂ej (r; k)/∂rj < 0 for every rj where the diﬀerentiation is applicable. Thus, an increment in kA results in an increase (decrease) in rj∗ , if 27

the increase in kA raises (lowers) ej (r; k) at the neighborhood of r = r∗ . An increase in kA has no eﬀect on rj∗ , if the increment in kA does not change ej (r; k) at the neighborhood of r∗ = r. Together with Theorem 1, the claim follows. Proof of Theorem 3. Applying the comparative statics technique due to Silberberg (1974), consider the “primal-dual” problem of the ﬁrm’s R&D program (15): min

r≥0,k>0

ϕ(kA , kB ) + rA + rB − E(rA , rB ; kA , kB ).

The Lagrangian of this primal-dual problem may be L(r, k, µ) = ϕ(k) + rA + rB − E(r; k) + µA rA + µB rB . Let r∗ ≫ 0. Then, necessarily [ ] DL(r,k) (r∗ , k, µ∗ ) = Lr (r∗ , k, µ∗ ) Lk (r∗ , k, µ∗ ) ≡ 0, and

] Lrr (r∗ , k, µ∗ ) Lrk (r∗ , k, µ∗ ) is p.s.d. = Lkr (r∗ , k, µ∗ ) Lkk (r∗ , k, µ∗ ) [

2 D(r,k) L(r∗ , k, µ∗ )

Here, [ ] [ ] Lr (r∗ , k, µ∗ ) = LrA (r∗ , k, µ∗ ) LrB (r∗ , k, µ∗ ) = eA (r∗ , k) eB (r∗ , k) , ] [ ] [ ∂E(r∗ ;k) ∂ϕ(k) ∂E(r∗ ;k) Lk (r∗ , k, µ) = LkA (r∗ , k, µ) LkB (r∗ , k, µ) = ∂ϕ(k) . − − ∂kA ∂kA ∂kB ∂kB A particularly important implication of the second-order necessary condition is that any leading principal minor of Lkk (r∗ , k, µ∗ ) is nonnegative. Hence, LkA kA (r∗ , k, µ∗ ) = ∂ 2 ϕ(k)/∂kA2 − ∂ 2 E(r∗ ; k)/∂kA2 ≥ 0. Rearrange the ﬁrst-order condition and diﬀerentiate it with respect to kA , yielding ∑ ∂ 2 ϕ(k)/∂kA2 = ∂ 2 E(r∗ ; k)/∂kA ∂rj · ∂ 2 rj∗ /∂kA + ∂ 2 E(r∗ ; k)/∂kA2 . j=A,B

Applying this to the above inequality, the claim follows.

For the proofs of Propositions 6–8 Assuming that the diﬀerentiation is applicable, diﬀerentiate eA (r; k) with respect to kA , yielding ] ∫ c(rA∫) c(rB )[ ∂eA ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) = − dG(zB |rB ) dzA . (16) ∂kA ∂c2A ∂c2A ∂rA 0 0 28

if rA is relatively large so that c(rA ) < kA holds, and ] ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − dG(zB |rB ) dzA ∂c2A ∂c2A ∂rA 0 0 ] ∫ c(rA∫) c(rB )[ ∂π(kA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − dG(zB |rB ) dzA + ∂c2A ∂c2A ∂rA kA 0 ∫ c(rA∫) c(rB ) ∂g(zA |rA ) ∂π(kA , zB , kA ) dG(zB |rB ) dzA . (17) + ∂c1A ∂rA kA 0

∂eA = ∂kA

∫ kA∫

c(rB )[

otherwise. Lemma 4. Let c′ = (c′1A , c′1B , c′2A ) and c′′ = (c′′1A , c′′1B , c′′2A ) with c′1A < c′′1A , c′1B = c′′1B and c′2A = c′′2A . Then, ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A has the sign speciﬁed in Table 2. c′ \c′′ C(0,0,+) C(0,+,+) C(0,+,0) C(+,+,+) C(+,0,+) C(+,+,0) C(+,0,0)

C(0,0,+) 0

C(0,+,+)

C(0,+,0)

C(+,+,+)

C(+,0,+)

C(+,+,0)

C(+,0,0)

0 0

0

0 0 + 0

+ + − −

+ + − −

0

+ −

Table 2: The sign of ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A Proof of Lemma 4. For proof, we will establish each of the following: 2 (i) ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A ≥ 0, if (c′ , c′′ ) ∈ C(u,v,w) for any (u, v, w) with strict inequality when C(u,v,w) = C(+,0,+) or C(+,+,+) ; (ii) ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A > 0, if (c′ , c′′ ) ∈ C(+,0,+) × C(+,+,+) ;

(iii) ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A = 0, if (c′ , c′′ ) ∈ C(+,+,0) × C(0,+,0) , if (c′ , c′′ ) ∈ C(0,+,+) × C(0,+,+) or if (c′ , c′′ ) ∈ C(+,0,0) × C(+,+,0) or C(+,0,0) × C(0,+,0) ; (iv) ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A > 0, if (c′ , c′′ ) ∈ C(+,0,+) × C(0,0,+) or if (c′ , c′′ ) ∈ C(+,+,+) × C(0,0,+) ; (v) ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A > 0, if (c′ , c′′ ) ∈ C(+,0,+) × C(0,+,+) or C(+,+,+) × C(0,+,+) ; (vi) ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A < 0, if (c′ , c′′ ) ∈ C(+,0,0) ∪ C(+,+,0) × C(+,0,+) ∪ C(0,+,+) ∪ C(+,+,+) . For proof of part (i), diﬀerentiate ∂π(c)/∂c2A in (14) with respect to c1A where applicable, yielding ∂ 2 π(c)/∂c1A ∂c2A = −4(1−b2 )/9 < 0 if c ∈ C(+,0,+) ∪C(+,+,+) , and ∂ 2 π(c)/∂c1A ∂c2A = 0 elsewhere. Since ∂π(c)/∂c2A ≥ 0 for any c ∈ C(u,v,w) , the claim holds. 29

Prove part (ii). Suppose c ∈ C(+,0,+) and c′′ ∈ C(+,+,+) . Then, (14) implies ∂π(c)/∂c2A = ∂π(c′ )/∂c2A > 0. Since ∂ 2 π/∂c1A ∂c2A > 0 for any c and c′ , the claim follows. Part (iii) is obvious, since ∂π(c)/∂c2A = 0 for c in the stated areas. From (14), we know ∂π(c)/∂c2A > 0 for c ∈ C(+,+,+) ∪ C(+,0,+) , while ∂π(c)/∂c2A = 0 for c ∈ C(0,0,+) . Therefore part (iv) holds. For proof of part (v), ﬁrst note ∂π(c′ )/∂c2A > ∂π(c′′ )/∂c2A for (c′ , c′′ ) ∈ C(+,0,+) × C(0,+,+) . What remains to be shown is ∂π(c′ )/∂c2A > ∂π(c′′ )/∂c2A ,

(c′ , c′′ ) ∈ C(+,+,+) × C(0,+,+) .

(18)

In showing that, of particular concern may be whether (18) holds about the boundary. That is, does it hold that ∂π(c1A , c1B , c2A )/∂c2A > lim+ ∂π(c1A + ϵ, c1B , c2A )/∂c2A ϵ→0

for c at the boundary between C(+,+,+) and C(0,+,+) ? Suppose that c ∈ C(+,+,+) is at the boundary between C(+,+,+) and C(0,+,+) . Let n1A i , i = 1. . .10 be a function of c−1A = 2 (c1B , c2A ) ∈ (−∞, ∞) implicitly deﬁned as follows: Ni (n1A i (c1B , c2A ), c1B , c2A ) = 0,

i = 1. . .10

(19)

where Ni is deﬁned in (10). Then, [ ] 2 2 c1A = n1A 4 (c1B , c2A ) = 2aA − aB b + 3bc1B + 2(1 − b )c2A /(4 − b ). Applying this and collecting the terms, we have [ ] ∂π(c)/∂c2A = 2b aA b + (2 − b2 )aB − 2(1 − b2 )c1B + b(1 − b2 )c2A /3(4 − b2 ). On the right-hand side, we observe lim ∂π(c1A + ϵ, c1B , c2A )/∂c2A ] [ = 2b aA b + (2 − b2 )aB − 2(1 − b2 )c1B + b(1 − b2 )c2A /(4 − b2 )2 .

ϵ→0+

Therefore the required inequality holds for every c at this boundary. Lastly prove part (vi). From (14), ∂π(c)/∂c2A is positive for c ∈ C(+,0,+) ∪ C(0,+,+) ∪ C(+,+,+) and zero for c ∈ C(+,0,0) ∪ C(+,+,0) . Hence the claim holds. Lemma 5. Let c′ = (c′1A , c′1B , c′2A ) and c′′ = (c′′1A , c′′1B , c′′2A ) with c′1A = c′2A = c′′2A < c′′1A and c′1B = c′′1B . Then, ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A + ∂π(c′ )/∂c1A has the sign speciﬁed in Table 3. 30

c′ \c′′ C(0,+,0) C(0,+,+) C(+,+,+) C(+,0,+)

C(0,+,0) 0

C(0,+,+)

C(+,+,+)

C(+,0,+)

C(0,0,+)

0 − −

− −

−

−

Table 3: The sign of ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A + ∂π(c′ )/∂c1A Proof of Lemma 5. Without loss of generality we can write c′ = (kA , c1B , kA ) and c′′ = (c′′1A , c1B , kA ). To begin with, suppose c′ ∈ C(+,0,+) . Then, ∂π(c′ )/∂c2A + ∂π(c′ )/∂c1A < 0 while ∂π(c′′ )/∂c2A > 0 from (12) and (14). Thus, the sign is negative for (c′ , c′′ ) ∈ C(+,0,+) × C(0,+,+) ∪ C(+,+,+) ∪ C(+,0,+) ∪ C(0,0,+) . Next, let c′ ∈ C(+,+,+) be at the boundary between C(0,+,+) and C(+,+,+) . Then [ ] 2 c1B = n1B 4 (kA , kA ) = − 2aA − aB b − (2 + b )kA /3b

(20)

where n1B is deﬁned as in (19). Set ϵ > 0. Then, limϵ→0 (kA + ϵ, c1B , kA ) ∈ C(0,+,+) . 4 Substitute (20) into (12) and (14) and rearrange the terms, yielding ∂π(c′ )/∂c2A + ∂π(c′ )/∂c1A + lim ∂π(kA + ϵ, c1B , kA )/∂c2A ϵ→0 [ ] = −(2 + b2 ) aA + aB b − (1 − b2 )kA /9(4 − b2 ) < 0.

(21)

Together with the fact that ∂π(c)/∂c2A is independent of c1A for every c ∈ C(0,+,+) while ∂π(c)/∂c2A + ∂π(c)/∂c1A , strictly decreasing with respect to c1B for any c ∈ C(+,+,+) , (21) implies ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A + ∂π(c′ )/∂c1A < 0 if (c′ , c′′ ) ∈ C(+,+,+) × C(0,+,+) . Moreover, since ∂π(c)/∂c1A is strictly decreasing with respect to c1B for c ∈ C(+,+,+) , (21) also implies ∂π(c′ )/∂c2A −∂π(c′′ )/∂c2A +∂π(c′ )/∂c1A < 0 if (c′ , c′′ ) ∈ C(+,+,+) ×C(+,+,+) . The rest of the claim is trivial. Hence the entire statement holds. Proof of Proposition 6. Suppose ﬁrst that c(rA ) < kA holds. Then, [0, c(rA )]×[0, c(rB )]× {kA } ⊂ C(+,0,+) ∪C(+,+,+) ∪C(0,+,+) holds. Applying Lemma 4, the integrand of ∂eA (r; k)/∂kA in (16) is positive. Hence, so is ∂eA (r; k)/∂kA . Next suppose that c(rA ) is larger than but close to kA . Then, Lemmata 4 and 5 imply that the ﬁrst term of ∂eA (r; k)/∂kA in (17) is positive and the sum of the last two terms, negative. So, in general, the sign of the entire expression is ambiguous. However, if we let c(rA ) approach to kA by increasing rA and consider the left-side limit of ∂eA /∂kA , then the last two terms vanish. So, the left-side limit is positive. Since the same sign should hold as long as c(rA ) is substantially close to kA , ∂eA (r; k)/∂kA tends to be positive. Moreover, as can be seen in Proposition 5, generally the direct eﬀect ∂π(c)/∂c1A should be larger in magnitude than the diﬀerence of the indirect eﬀects ∂π(c)/∂c2A − ∂π(c′ )/∂c2A 31

(where c and c′ are such that c1A = c2A = c′2A < c′1A and c1B = c′1B ), even though ∂π(c)/∂c1A becomes less negative as rA falls (since ∂ 2 π/∂c21A > 0). In addition, the interval [kA , c(rA )] becomes larger as rA falls. These together imply that the eﬀect of the last term in (17) should increase as rA falls. Hence ∂eA (r; k)/∂kA tends to decline as rA falls. Lemma 6. Suppose that sA > 0 and sA < c(rA ) < sA + ϵ holds for some ϵ > 0. Suppose also that c(rB ) > sB if sB > 0 holds. Then, ∂eA (r; k)/∂kA is negative if ϵ is suﬃciently small. Proof of Lemma 6. If sA + ϵ < kA holds, then ∂eA (r; k)/∂kA in (16) is rewritten as c(rA )[

] ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − dzA dG(zB |rB ) ∂c2A ∂c2A ∂rA 0 0 ] ∫ c(rB ) ∫ sA[ ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − dzA dG(zB |rB ) − ∂c2A ∂c2A ∂rA ∆′′ 0 ] ∫ c(rB ) ∫ c(rA )[ ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) + − dzA dG(zB |rB ), ∂c2A ∂c2A ∂rA ∆′′ sA

∂eA = ∂kA

∫ ∆∫ ′′

where ∆′′ ≡ max{0, sB }. Applying Lemma 4, the ﬁrst term is zero, the second, negative and the third, positive. Consider the left-side limit of ∂eA (r; k)/∂kA by letting c(rA ) approach to sA by increasing rA . Then, since the last term vanishes while the second does not, the left-side limit is negative. Apparently the same sign should hold if ϵ is substantially small. Hence the claim follows. Lemma 7. Suppose that sA > 0 and c(rA ) > sA holds. Suppose also that c(rB ) > sB if sB > 0 holds. Then, ∂eA (r; k)/∂kA tends to rise initially and then start declining as rA falls. Proof of Lemma 7. If c(rA ) < kA , then ∂eA (r; k)/∂kA in (16) can be rewritten as ] ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − dzA dG(zB |rB ) ∂c2A ∂c2A ∂rA 0 0 ] ∫ c(rB ) ∫ sA[ ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − − dzA dG(zB |rB ) ∂c2A ∂c2A ∂rA ∆′′ 0 ] ∫ c(rB ) ∫ c(rA )[ ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) + − dzA dG(zB |rB ), ∂c2A ∂c2A ∂rA ∆′′ sA

∂eA = ∂kA

∫ ∆∫ ′′

c(rA )[

where ∆′′ ≡ max{0, sB }. Applying Lemma 4, the ﬁrst term is zero, the second, negative and the third, positive. Although the last two terms are opposite in sign, the last term tends to be more positive as rA falls. (See Figures 1 and 1.) Hence it tends to rise initially as rA falls. 32

On the other hand, if c(rA ) > kA , then ∂eA (r; k)/∂kA in (17) can be rewritten as ] ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − dzA dG(zB |rB ) ∂c2A ∂c2A ∂rA 0 0 ] ∫ c(rB ) ∫ sA[ ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − − dzA dG(zB |rB ) ∂c2A ∂c2A ∂rA ∆′′ 0 ] ∫ c(rB ) ∫ kA[ ∂π(zA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂g(zA |rA ) − dzA dG(zB |rB ) + ∂c2A ∂c2A ∂rA ∆′′ sA ] ∫ c(rB ) ∫ c(rA )[ ∂π(kA , zB , kA ) ∂π(c(rA ), zB , kA ) ∂π(kA , zB , kA ) − + + ∂c2A ∂c2A ∂c1A ∆′′ kA ∂g(zA |rA ) × dzA dG(zB |rB ). ∂rA

∂eA = ∂kA

∫ ∆∫ ′′

c(rA )[

Applying Lemmata 4 and 5, the ﬁrst term is zero, the third, positive and the rest, negative. Although the signs of these terms are mixed, the last term tends to be more positive as rA falls, since the interval [kA , c(rA )] becomes larger. Hence it tends to decline as rA falls. Therefore the claim follows. Proof of Proposition 7. Suppose c(rA ) < sA . Then, Lemma 4 implies that the integrand of ∂eA (r; k)/∂kA in (16) is zero. And so is ∂eA (r; k)/∂kA . The rest of the claim is established by Lemmata 6 and 7. Proof of Proposition 8. The proof is analogous to that of Proposition 7.

For the proofs of Propositions 9–11. Assuming that the diﬀerentiation is applicable, diﬀerentiate eB (r; k) with respect to kA , yielding ] ∫ c(rB ∫ ) c(rA ) [ ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂eB ∂g(zB |rB ) = − dG(zA |rA ) dzB (22) ∂kA ∂c2A ∂c2A ∂rB 0 0 if rA is relatively large so that c(rA ) < kA holds, and ∂eB = ∂kA

] ∂g(zB |rB ) ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) − dG(zA |0) dzB ∂c2A ∂c2A ∂rB 0 0 ] ) c(rA ) [ ∑ ∫ c(rB ∫ ∂π(kA , zB , kA ) ∂π(kA , c(rB ), kA ) + − dG(zA |rA ) ∂ciA ∂ciA kA i=1,2 0

∫

c(rB ∫ ) kA

[

× otherwise. 33

∂g(zB |rB ) dzB (23) ∂rB

Lemma 8. Let c′ = (c′1A , c′1B , c′2A ) and c′′ = (c′′1A , c′′1B , c′′2A ) with c′1A = c′′1A , c′1B < c′′1B and c′2A = c′′2A . Then, ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A has the sign speciﬁed in Table 4.4 c′ \c′′ C(0,0,+) C(+,0,+) C(+,+,+) a C(0,+,+) b C(0,+,+) C(0,+,0) C(+,0,0) C(+,+,0)

C(0,0,+) 0

+ + 0

C(+,0,+)

C(+,+,+)

C(0,+,+)

C(0,+,0)

C(+,0,0)

C(+,+,0)

0 0 − + −

0 − + −

+ + −

0

0 0 0

0 0

Table 4: The sign of ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A Proof of Lemma 8. Diﬀerentiate ∂π/∂c2A in (14) with respect to c1B where applicable, yielding ∂ 2 π/∂c2A ∂c1B ≤ 0 with strict inequality when c ∈ C(0,+,+) . Recalling ∂π/∂c2A ≥ 0 2 in (14), this implies ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A ≥ 0 when (c′ , c′′ ) ∈ C(u,v,w) . Next, consider ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A for (c′ , c′′ ) ∈ C(0,+,+) × C(+,0,+) ∪ C(+,+,+) . To begin with, let c1A = c2A . Without loss of generality, we can write c = (kA , c1B , kA ). Then, if c ∈ C(+,0,+) ∪ C(+,+,+) , (14) implies [ ] ∂π(c)/∂c2A = 2 aA + aB b − (1 − b2 )kA /9. Alternatively, if c ∈ C(0,+,+) , then [ ] ∂π(c)/∂c2A = 2 aA + aB b − (1 − b2 )kA /(4 − b2 ), c1B →s1B + [ ] lim ∂π(c)/∂c2A = 2 aA + aB b − (1 − b2 )kA /(4 − b2 )3 lim1

− c1B →n1B 4 (kA ,kA )

where s(kA ) is the intersection of the graphs of N2 (c) = 0 and N6 (c) = 0 given c2A and n1B 4 is deﬁned as in (??). Compare these three values. Clearly, the ﬁrst is greater than the second but less than the third. Additionally, (14) implies that ∂π(c)/∂c2A is continuous and strictly decreasing with respect to c1B when c ∈ C(0,+,+) . The Intermediate Value Theorem then ensures the existence of cˆ1B such that ] [ ∂π(c)/∂c2A ⋛ 2 aA + aB b − (1 − b2 )kA /9,

4

c1B ⋚ cˆ1B ,

a b In the table, C(0,+,+) refers to the upper part of C(0,+,+) , while C(0,+,+) , to its lower part.

34

which is unique. Therefore, provided c′1A = c′′1A = c′2A = c′′2A = kA , { 1 [s1B , cˆ1B (kA , kA )] × (n1B 4 (kA , kA ), αB ], ′ ′′ ′ ′′ ∂π(c )/∂c2A ⋛ ∂π(c )/∂c2A , (c1B , c1B ) ∈ 1B [ˆ c1B , n1B 2 (kA , kA )] × (n4 (kA , kA ), αB ]. Now consider (c′ , c′′ ) ∈ C(0,+,+) × C(+,+,+) ∪ C(+,0,+) with c′1A = c′′1A < c′2A = c′′2A = kA . Then, ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A < ∂π(kA , c′1B , kA )/∂c2A − ∂π(kA , c′′1B , kA )/∂c2A . This holds, since (14) implies that ∂π(c)/∂c2A is continuous and strictly decreasing with c1A if c ∈ C(+,0,+) ∪C(+,+,+) , while ∂π(c)/∂c2A is continuous and constant with c1A if c ∈ C(0,+,+) . Together these imply that the claim follows. The rest of the claim holds, since π(c)/∂c2A is positive if c ∈ C(+,0,+) ∪ C(0,+,+) ∪ C(+,+,+) and is zero elsewhere, as observed in (14). Lemma 9. Let c′ = (c′1A , c′1B , c′2A ) and c′′ = (c′′1A , c′′1B , c′′2A ) with c′1A = c′′1A = c′2A = c′′2A and c′1B < c′′1B . Then, ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A + ∂π(c′ )/∂c1A − ∂π(c′′ )/∂c1A has the sign speciﬁed in Table 5. c′ \c′′ C(+,0,+) C(+,+,+) C(0,+,+) C(0,+,0)

C(+,0,+) 0 + + +

C(+,+,+)

C(0,+,+)

C(0,+,0)

+ + +

0 −

0

Table 5: The sign of ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A + ∂π(c′ )/∂c1A − ∂π(c′′ )/∂c1A Proof of Lemma 9. Without loss of generality, we can write c′1A = c′′1A = c′2A = c′′2A = kA . First, let (c′ , c′′ ) ∈ C(0,+,+) × C(+,0,+) ∪ C(+,+,+) . Then, (12) and (14) imply ∂π(c′ )/∂c2A + ∂π(c′ )/∂c1A = ∂π(c′′ )/∂c2A > 0, [ ] ∂π(c′′ )/∂c2A + ∂π(c′′ )/∂c1A = −10 aA + aB b − (1 − b2 )kA < 0. Taking the diﬀerence, we have ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A + ∂π(c′ )/∂c1A − ∂π(c′′ )/∂c1A > 0 if (c′ , c′′ ) ∈ C(0,+,+) × C(+,0,+) ∪ C(+,+,+) . Next, suppose (c′ , c′′ ) ∈ C(0,+,0) × C(+,0,+) ∪ C(+,+,+) . Then, (12) and (14) imply ∂π(c′ )/∂c2A + ∂π(c′ )/∂c1A = 0,

] [ ∂π(c′′ )/∂c2A + ∂π(c′′ )/∂c1A = −10 aA + aB b − (1 − b2 )kA < 0. 35

Taking the diﬀerence, we have ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A + ∂π(c′ )/∂c1A − ∂π(c′′ )/∂c1A > 0 if (c′ , c′′ ) ∈ C(0,+,0) × C(+,0,+) ∪ C(+,+,+) . Diﬀerentiate ∂π(c)/∂c1A in (12) with respect to c1B where applicable, yielding  − b, c ∈ C(+,+,0) ,    − 9b, c ∈ C(+,+,+) , ∂ 2 π(c)/∂c1A ∂c1B =    0, elsewhere. As ∂π(c)/∂c1A ≤ 0 by (12), this implies ∂π(c′ )/∂c1A − ∂π(c′′ )/∂c1A ≥ 0 when (c′ , c′′ ) ∈ 2 . Using Lemma 8, this in turn implies ∂π(c′ )/∂c2A − ∂π(c′′ )/∂c2A + ∂π(c′ )/∂c1A − C(u,v,w) 2 ∂π(c′′ )/∂c1A ≥ 0 if (c′ , c′′ ) ∈ C(u,v,w) . By (12), ∂π(c)/∂c1A is negative if c ∈ C(+,0,0) ∪ C(+,+,0) ∪ C(+,0,+) ∪ C(+,+,+) and is zero elsewhere. Applying Lemma 8, the rest of the claim instantly follows. Proof of Proposition 9. Suppose kA < m. If c(rA ) < kA , then ∂eB /∂kA is written in (22). By Lemma 8, the integrand is zero. Hence the claim follows. On the other hand, if c(rA ) ≥ kA , then ∂eB /∂kA is written in (23). By Lemma 8, the integrand of the ﬁrst term is zero. By Lemma 9, that of the second term is positive. Hence, the claim follows. Proof of Proposition 10. If c(rA ) < m holds, then Lemma 8 implies that the integrand of ∂eB /∂kA in (22) is zero. Hence the claim follows. Now suppose that c(rA ) > kA holds. By Lemma 9, the second term in (23) is positive. Rewrite the remaining ﬁrst term as ∫ m∫ 0

0

] ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂g(zB |rB ) − dzB dG(zA |rA ) ∂c2A ∂c2A ∂rB ] ∫ kA∫ ∆[ ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂g(zB |rB ) + − dzB dG(zA |rA ) ∂c2A ∂c2A ∂rB m 0 ] ∫ kA∫ c(rB )[ ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂g(zB |rB ) + − dzB dG(zA |rA ). ∂c2A ∂c2A ∂rB ∆ m

c(rB )

[

1B where ∆ ≡ min{c(rB ), n1B 4 (zA , kA )} or zA in [m, kA ] and n4 deﬁned in (19). Applying Lemma 8, the ﬁrst term and the last term are zero, while the second term

tends to be positive (negative) if kA is relatively close to m (sB is close to zero). Thus, the ﬁrst term in (23) tends to be positive unless kA is such that sB is very close to zero. Hence, the claim follows. Lemma 10. Suppose that sA > 0 and sB > 0 hold. Suppose also that c(rA ) < sA or c(rB ) < sB holds. Then, ∂eB (r; k)/∂kA is zero. 36

Proof of Lemma 10. Suppose that c(rA ) < sA holds. Since this implies c(rA ) < kA , by Lemma 8 the integrand of ∂eB /∂kA in (22) is zero. Hence the claim follows. On the other hand, if c(rB ) < sB holds, then apply Lemmata 8 and 9 to evaluate ∂eB /∂kA in (23). Since the integrands are zero for both terms, the claim follows. Lemma 11. Suppose that sA > 0 and sB > 0 hold. Suppose also that c(rA ) > sA and sB < c(rB ) < sB + ϵ hold for some ϵ > 0. Then, ∂eB (r; k)/∂kA is negative if ϵ is suﬃciently small. Proof of Lemma 11. If kA , rA and rB have the values as in the statement, then ∂eB (r; k)/∂kA in (23) can be rewritten as [

] ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂g(zB |rB ) − dzB dG(zA |rA ) ∂c2A ∂c2A ∂rB 0 0 ] ∫ kA∫ ∆′ [ ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂g(zB |rB ) + − dzB dG(zA |rA ) ∂c2A ∂c2A ∂rB sA 0 ] ∫ kA∫ c(rB ) [ ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂g(zB |rB ) + − dzB dG(zA |rA ) ∂c2A ∂c2A ∂rB sA ∆′ ] ) ∆′[ ∑ ∫ c(rA∫ ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂g(zB |rB ) + − dzB dG(zA |rA ) ∂c ∂c ∂r iA iA B k 0 A i=1,2 ] ) c(rB )[ ∑ ∫ c(rA∫ ∂π(zA , zB , kA ) ∂π(zA , c(rB ), kA ) ∂g(zB |rB ) − dzB dG(zA |0), + ∂c ∂c ∂r ′ 2A 2A B k ∆ A i=1,2

∂eB = ∂kA

∫

sA∫ c(rB )

where ∆′ ≡ min{sB , c(rB )}. By Lemma 8, the ﬁrst term is zero, the second term, negative and the third term, ambiguous in sign. By Lemma 9, the forth term is negative and the ﬁfth term, ambiguous in sign. Both the third term and the ﬁfth term become negligible, as c(rB ) approaches to sB from the right. Hence, ∂eB (r; k)/∂kA becomes negative at the limit as c(rB ) approaches to sB . Apparently, the same sign should hold as long as c(rB ) is signiﬁcantly close to sB . Hence, the claim holds.

References Bernard, Andrew B., Stephen J. Redding, and Peter K. Schott (2010) “Multi-Product Firms and Product Switching,” American Economic Review. forthcoming. Brander, James A. and Jonathan Eaton (1984) “Product Line Rivary,” American Economic Review, Vol. 74, No. 3, pp. 323–334. 37

Broda, Christian and David E. Weinstein (2007) “Product Creation and Destruction: Evidence and Price Implications.” Mimeo. Champsaur, Paul and Jean-Charles Rochet (1989) “Multiproduct Duopolists,” Econometrica, Vol. 57, No. 3, pp. 533–557. Dasgupta, Partha and Joseph Stiglitz (1980) “Industrial Structure and the Nature of Innovative Activity,” Economic Journal, Vol. 90, No. 358, pp. 266–293. D´Aspremont, Claude and Alexis Jacquemin (1988) “Cooperative and Noncooperative R&D in Duopoly with Spillovers,” American Economic Review, Vol. 78, No. 5, pp. 1133– 1137. Doraszelski, Ulrich (2003) “An R&D Race with Knowledge Accumulation,” RAND Journal of Economics, Vol. 34, No. 1, pp. 20–42. Fraja, Giovanni De (1994) “A General Characterization of Multiproduct Cournot Competition,” Bulletin of Economic Research, Vol. 46, No. 2, pp. 171–183. Grossman, Gene M. and Carl Shapiro (1986) “Optimal Dnamic R&D Programs,” RAND Journal of Economics, Vol. 17, No. 4, pp. 581–593. Judd, Kenneth L. (2003) “Closed-Loop Equilibrium in a Multi-Stage Innovation Race,” Economic Theory, Vol. 21, No. 2/3, pp. 673–695. Lambertini, Luca (2003) “The Monopolist’s Optimal R&D Portfolio,” Oxford Economic Paper, Vol. 55, No. 4, pp. 561–578. Lin, Ping (2004) “Process and Product R&D by a Multiproduct Monopolist,” Oxford Economic Papers, Vol. 56, No. 4, pp. 735–743. (2007) “Process R&D and Product Line Deletion by a Multiproduct Monopolist,” Journal of Economics, Vol. 91, No. 3, pp. 245–262. Loury, Glenn C. (1979) “Market Structure and Innovation,” Quarterly Journal of Economics, Vol. 93, No. 3, pp. 395–410. MacDonald, Glenn M. and Alan Slivinski (1987) “The Simple Analytics of Competitive Equilibrium with Multiproduct Firms,” American Economic Review, Vol. 77, No. 5, pp. 941–953. Mussa, Michael and Sherwin Rosen (1978) “Monopoly and Product Quality,” Journal of Economic Theory, Vol. 18, No. 2, pp. 301–317. 38

Okuguchi, Koji and Ferenc Szidarovsky (1990) The Theory of Oligopoly with Multi-Product Firms, Vol. 342 of Lecture Notes in Economics and Mathematical Systems, Berlin: Springer-Verlag. Reinganum, Jennifer F. (1982) “A Dynamic Game of R and D: Patent Protection and Competitive Behavior,” Econometrica, Vol. 50, No. 3, pp. 671–688. (1989) “The Timing of Innovation: Research, Development, and Diﬀusion,” in Richard Schmalensee and Robert Willig eds. Handbook of Industrial Organization, Vol. 1, Amsterdam, North-Holland: Elsevier, pp. 849–908. Romer, Paul M. (1990) “Endogenous Technological Change,” Journal of Political Economy, Vol. 98, No. 2, pp. 71–102. Silberberg, Eugene (1974) “A Revision of Comparative Statics Methodology in Economics, or, How to Do Comparative Statics on the Back of an Envelope,” Journal of Economic Theory, Vol. 7, No. 2, pp. 159–172. Spence, Michael (1984) “Cost Reduction, Competition and Industry Performance,” Econometrica, Vol. 52, No. 1, pp. 101–121. Vives, Xavier (2008) “Innovation and Competitive Pressure,” Journal of Industrial Economics, Vol. 56, No. 3, pp. 419–469.

39

N_5 25 N_7

C_(0,0,+)

25

t

C_(+,0,+)

c_1B 20

c_1B

C_(+,0,0)

20

C_(+,+,+) 15 15 C_(0,+,+) N_6 10 N_10

10

s

C_(+,+,0) 5 5 C_(0,+,0) 0 0

5

10

15

20 c_1A

k_A

0 0

5

10

20 c_1A

15

N_2 N_4

25

N_9

25 N_3 N_1

(a) in (c1A , c1B )-space holding c2A = 20

N_6 N_8 25

25

C_(0,+,0) N_10

C_(+,+,0)

20

c_2A

C_(+,0,0)

20

15

C_(+,+,+)

N_4

15 c_2A

N_9 10 10

C_(0,+,+) 5 C_(+,0,+) N_7

5

0 0

5

10

C_(0,0,+) 0 0

5

10

15 c_1A

20

15

N_3

20 c_1A N_2

25 N_1

25

(b) in (c1A , c2A )-space holding c1B = 18

N_8 25

25 C_(0,+,0)

C_(+,+,0)

C_(+,0,0)

N_9 20 c_2A

20

N_10 15 15

C_(+,+,+)

C_(+,0,+)

c_2A 10 10 C_(0,+,+) N_3

5

5 0 0

C_(0,0,+)

5

10 N_2

0 0

5

10

15 c_1B

20

25

(c) in (c1B , c2A )-space holding c1A = 15

Figure 1: C(u,v,w) and Relevant Loci 40

15

20 c_1B N_7 N_6

25 N_4N_5

C_(0,0,0) 25

25

s=t

C_(0,0,+)

C_(+,0,+) 20

20 C_(+,0,0)

C_(+,0,0)

N_6

N_4

15

15 c_1B N_6

10

10

C_(+,+,+)

C_(+,+,0) C_(+,+,0) 5

N_2

C_(0,+,+)

5

C_(0,+,0)

m

k_A

k_A

0

0 0

5

10

15 c_1A

20

0

25

5

10

(a) c2A = 25

25

15 c_1A

20

25

(b) c2A = 15

25

C_(0,0,+)

C_(0,0,+)

20

C_(+,0,+)

20

N_4

15

C_(+,0,+)

15

N_6

N_6

10

C_(0,+,+)

10 N_4

C_(0,+,+)

C_(+,+,+) C_(+,+,+) 5

5

m

k_A

k_A

0

m

0 0

5

10

15 c_1A

20

0

25

(c) c2A = 12

5

10

15 c_1A

(d) c2A = 02

Figure 2: C(u,v,w) in (c1A , c1B )-space

41

20

25

(a) in (c1A , c1B )-space

(b) in (c1A , c2A )-space

27

26

25

24

23

22

21 14

15

16

17

18

19

c_1A

(c) in (c1B , c2A )-space

(d) a restriction for c1B = 13 and c2A = 20

Figure 3: Proﬁt Function

42

0 2

-1

1.5

1

-2

0.5 -3

0 10

12

14

16

18

20

8

22

(a) i = 1, c ∈ {17}×[8, 23]×{20}

10

12

14

(b) i = 2, c ∈ [6, 15]×{17}×{20}

Figure 4: Non-convexity of π(c) on C: ∂π(c)/∂ciA

(a) gross of R&D expenditures

(b) net of R&D expenditures

Figure 5: Expected proﬁt functions E(rA , rB )

43

(a) kA is large so that sA > 0 and sB > 0 hold

(b) kA is relatively small so that sA > 0 and sB < 0 hold

(c) kA is small so that sA < 0 and sB < 0 hold

Figure 6: The sign of ∂eA (r; k)/∂kA in (c(rA ), c(rB ))-space 44

(a) kA is large so that sA > 0 and sB > 0) hold

(b) kA is relatively small so that kA > m and sB < 0 hold

(c) kA is small so that kA < m and sB < 0 hold

Figure 7: The sign of ∂eB (r; k)/∂kA in (c(rA ), c(rB ))-space 45

The Timing of Takeovers in Growing and Declining ...