Choosing between Innovation and Imitation
in a Model of International Rivalry
Kenji Fujiwara
Kwansei Gakuin University

Koji Shimomura
Kobe University

Kar-yiu Wong
University of Washington
November 13, 2006

Abstract
This paper constructs a model of international rivalry to examine the competition in terms of technology and output between firms in two trading
countries. We analyze how the less eﬃcient firm chooses endogenously between innovation and imitation, and how its decision may aﬀect the market
equilibrium and profits of the firms. Conditions under which the less eﬃcient
firm is able to catch up or to surpass the technology of the advanced firm
have been derived. The implications of removing the endogeneity of thisdecision of the less eﬃcient firms are also examined.

1

Introduction

Globalization in recent years has greatly enhanced trade and technology diffusion among countries. This phenomenon has brought to government planners and economists many contentious issues.1 Among these issues is theone about international transfer of technology and the appropriate responsesof the governments. With firms producing similar products in various trading countries, the improvement of technologies and possible international
spillover of technologies would aﬀect the competitiveness of these firms andthe welfare of their countries. Governments and economists are thus eager
to find out the right policies to improve the welfare of their own countries.
There is a large literature on the output-technology competition amongfirms in diﬀerent countries. In terms of how firms compete in technology,
papers in this literature can be divided into two categories: those that assume
that all firms are choosing innovation optimally, and those that examine thecompetition between some firms that innovate and some firms (usually inother countries) that imitate.
One of the earlier papers that examines how international rival firms
choose innovation is Spencer and Brander (1983). They also examine theroles of R&D subsidies provided by one or both governments. Bagwell andStaiger (1992, 1994) investigate the robustness of such R&D subsidies in
the presence of uncertainty, and argue that Spencer and Brander’s (1983)
argument for the strategic R&D subsidy is not confined to a deterministic
environment.
The innovation-imitation relations among firms have drawn more attention in the literature. For example, Segerstrom (1991) constructs a model
1

Bhagwati (2004) makes an extensive and stimulating argument on globalization andwelfare in the modern world.

1

of economic growth in which some firms innovate to improve the quality of
their product and some others imitate. Using a dynamic North-South model,
Grossman and Helpman (1991) conclude that the North’s R&D activity isenhanced by the existence of the imitating South. In a similar framework,
Helpman (1993) discusses the eﬀects of enforcement of intellectual property
rights.2 Dinopoulos and Syropoulos (1998), taking account of the more efficient firm’s imitation-blocking and the less eﬃcient firm’s appropriating
activities, identify the conditions under which these ‘redistributive’ expenditures become positive. Dinopoulos et al. (2006) construct a Schumpeterian
growth model of international trade and examine the eﬀect of patent protection on the rate of innovation and imitation. Liao and Wong (2004), also
assuming an innovation-imitation relation between developed countries anddeveloping countries, examine to what extent the governments of developing
countries would be willing to provide intellectual property rights protection
and investigate the roles and the shortcomings of the TRIPS agreement.
These papers all start with the presumption that less eﬃcient firms are
given a particular channel through which they improve their technologies:
either innovation or imitation. This seems to be a serious limitation in manycases because in more general models the less eﬃcient firms should havethe option of choosing the optimal way to improve their technologies. In
other words, the choice between innovation and imitation should be given
to these firms, instead of arbitrarily assigning a technology-improving pathto them. To see how serious this limitation is, we just have to note thatmany firms in many emerging countries started by imitating the technologies
of firms in developed countries, but over time switched to developing theirown technologies. In such a process, they gradually caught up with the2

Helpman’s (1993) model is recently extended and reexamined by Mondal and Gupta(2006).

2

advanced technologies and even surpassed them. The switch from imitation
to innovation is missing in the papers mentioned above.
One very early exception that endogenizes the decision of a firm between
innovation and imitation is Shimomura and Wong (2001). Using a dynamic
model of international rivalry, they examine the conditions for innovationand imitation by a less eﬃcient firm, and investigate the circumstances underwhich the firm will choose to innovate or imitate. They show some interestingcases in which the less eﬃcient firm chooses imitation initially but switches toinnovation in the future. There are also cases presented in the paper in which
the less eﬃcient firm always chooses to imitate and never finds it profitable
to innovate.
The purpose of this chapter is to oﬀer another framework to examine thechoice between innovation and imitation.3 Using a simple model of oligopolistic competition, we endogenize the less eﬃcient firm’s choice between innovation and imitation. We also investigate the conditions under which the lesseﬃcient firm chooses to catch up or surpass the advanced firm in terms oftechnology. This model and the analysis allow us to answer one question: If
the less eﬃcient firm will choose to innovate, what happens if the firm is assumed to imitate instead? This question is important because many papers
just assume the innovation-imitation relations between two firms without
checking whether the less eﬃcient may in fact choose to innovate instead, if
it is allowed to. If one assumes that the less eﬃcient imitates but it prefers
to innovate, what kind of error may one commit?
The rest of this chapter is organized as follows. Section 2 describes themodel with two firms, one in each country. The firms, with diﬀerent ini3

The present framework is based on that in Shimomura and Wong (2001), except that
it is a static one. This allows us to investigate the rivalry between and strategies of thefirms in more detail.

3

tial technology levels, attempt to improve their technology and use the newtechnology to produce a homogeneous product to be exported to the rest of
the world. Section 3 examines the case in which both firms innovate before
producing the product. Thus they compete directly in terms of technology
and output. Section 4 turns to the case in which the more eﬃcient firm
innovates while the other one imitates. Section 5 examines the conditions
for catching up and surpassing in the innovation-innovation case and in theinnovation-imitation case. Section 6 endogenizes the innovation/imitation
decision of the initially less eﬃcient firm. Section 7 concludes.

2

The Model

Consider a model consisting of three countries labelled Home, Foreign, andROW (the Rest of the World). Both Home and Foreign has a firm producing
a homogeneous product, which is exported to ROW. There is no demand forthe product in Home or Foreign, and the demand in ROW can be represented
by a linear function
p = a − q,(1)

where p is the market price and q the demand. The value of a is positive andsuﬃciently large so that the market can support both the Home firm and theForeign firm.
Initially the Home firm is able to produce the product using a marginal
cost, c0 , which is independent of the output level. The corresponding initial
marginal cost of the Foreign firm is c∗0 .4 We assume that initially the Home
firm has a more advanced technology; i.e.,
Assumption 1. c0 < c∗0 .
4Unless stated otherwise, the variables of Foreign are distinguished by asterisks. Without loss of generality, we assume that the fixed costs of the firms are negligible.

4If we denote the output of the Home (Foreign) firm by x (x∗ ), which isthe same as the export of the product by the country, in equilibrium we haveq = x + x∗ .

(2)

The two firms compete in terms of technology and output. In termsof output, they compete in a Cournot-Nash fashion. In terms of technology,
both of them are able to improve their technology, or to reduce their marginal
costs, through one of the two channels: innovation and imitation.
Innovation requires the intentional expenditure on resources, which results in the improvement of technology. More specifically, we assume thatafter spending an amount of eg2 /2, e > 0 (e∗ g ∗2 /2, e∗ > 0), the Home (Foreign) firm is able to reduce its marginal cost by an amount of g (g∗ ). Variablee (e∗ ) can be interpreted as the cost of innovation.5
Define two variables:
z =
z

∗

a − c∗0a − 2c∗0 + c0

a − c∗0=.
a − 2c∗0 + c0

(3a)
(3b)

Since c∗0 > c0 , equations (3) imply that z > 1 and 1 > z ∗ > 0. To guaranteethat if a firm chooses to innovate, the level of innovation is positive, we makethe following assumption:
Assumption 2. e > z, e∗ > z ∗ .
If both firms choose to innovate, their new marginal costs become, re5

Variable e is interpreted as the cost of innovation because eg 2 /2 is the cost the homefirm has to pay in order to reduce its marginal cost by g. As will be shown later, an increasein e will lower the firm’s output, level of innovation, and profit.

5

spectively,
c1 = c0 − gc∗1 = c∗0 − g ∗ .
Imitation exists when a firm chooses to learn from the other firm. For
example, if the Foreign firm imitates, its new marginal cost becomesc∗1 = c∗0 − θ(c∗0 − c0 ),(4)where θ ∈ (0, 1).6 We assume that imitation requires no costs.

Since initially the Home firm has a lower marginal cost, it has nothing tolearn from the Foreign firm. This means that for the Home firm, imitation is
the same as inaction, spending zero resources on innovation and thus getting
zero innovation. We can thus reduce the present analysis to the following
two cases: (A) Both firms innovate; and (B) The Home firm innovates while
the Foreign firm imitates. These two cases are analyzed below.

3

Case (A): Foreign Firm Innovating

In this case, both firms innovate. Denote the profits of the Home and Foreign
firms by Πv and Π∗v , respectively, where superscript “v” denotes the variablesin this case. We haveeΠv = [a − (c0 − g) − x − x∗ ]x − g 22e∗Π∗v = [a − (c∗0 − g ∗ ) − x − x∗ ]x∗ − g∗2 .
2
6

(5a)
(5b)

We exclude the two extreme cases in which θ = 0 and θ = 1. The case of θ = 0 is
not interesting because it means no technology spillover. The case of θ = 1 is excluded
because in general a firm can never be as good as the other firm from which it learns.

6Both firms choose the new technology level (marginal cost) first and thencompete in terms of output. Since the present game never involves the problem of time inconsistency, we need not solve the game in a backward way;i.e., the solution obtained by solving the game backward is equivalent tothe solution derived in the first stage. Thus we assume that the Home firm
maximizes (5a) with respect to x and g, whereas the Foreign firm (5b) withrespect to x∗ and g ∗ , both in a Cournot-Nash fashion. The corresponding
first-order conditions for profit maximization are7
x : a − c0 + g − 2x − x∗ = 0(6a)

g : x − eg = 0(6b)x∗ : a − c∗0 + g∗ − x − 2x∗ = 0(6c)g∗ : x∗ − e∗ g ∗ = 0.

(6d)This system is solved as follows. From (6b) and (6d), x = eg and x∗ = eg∗
follow. Then, substituting these into (6a) and (6c), we geta − c0 − (2e − 1)g − e∗ g ∗ = 0
a − c∗0 − eg − (2e∗ − 1)g ∗ = 0.
Solving this system for g and g∗ yields the optimal innovation levels:gv(e∗ − 1)(a − c0 ) + e∗ (c∗0 − c0 )= g (e, e ) =(2e − 1)(2e∗ − 1) − ee∗v∗g∗v = g ∗v (e, e∗ ) =7(e − 1)(a − c∗0 ) − e(c∗0 − c0 ).
(2e − 1)(2e∗ − 1) − ee∗

Throughout the paper, we assume interior equilibria.

7

(7a)
(7b)

Given assumptions 1 and 2, gv > 0 and g ∗v > 0. Combining (7), (6b), and(6d), we get the optimal outputs as follows:¸∙ ∗(e − 1)(a − c0 ) + e∗ (c∗0 − c0 )vv∗
x = x (e, e ) = e(2e − 1)(2e∗ − 1) − ee∗
∙
¸
∗
∗
∗v
∗v
∗
∗ (e − 1)(a − c0 ) − e(c0 − c0 ).
= x (e, e ) = ex(2e − 1)(2e∗ − 1) − ee∗

(8a)
(8b)

Making use of (7) and (8), the profits of the firms as given by (5) reduce to:¸2∙e(2e − 1) (e∗ − 1)(a − c0 ) + e∗ (c∗0 − c0 )v∗
(9a)
Π (e, e ) =2(2e − 1)(2e∗ − 1) − ee∗¸2∙e∗ (2e∗ − 1) (e − 1)(a − c∗0 ) − e(c∗0 − c0 )∗v∗
Π (e, e ) =.
(9b)2(2e − 1)(2e∗ − 1) − ee∗
The above analysis shows that the outputs, innovation levels, and profits arefunctions of the costs of innovation, e and e∗ .
Lemma 1. In the present case with both firms innovating, we have
∂gv
< 0,
∂e
∂xv
< 0,
∂e

∂g∗v
∂g v
∂g∗v
>
0,
<0
>
0,
∂e∗
∂e
∂e∗
∂xv
∂x∗v
∂x∗v
>
0,
>
0,
< 0.
∂e∗
∂e
∂e∗

The proofs of all lemmas, propositions, and corollaries are given in the appendix.
∗v
8
Lemma 2. Πve < 0, Πve∗ > 0, Π∗v
e > 0 and Πe∗ < 0.

Lemmas 1 and 2 make it clear why e and e∗ are interpreted as the costof innovation: An increase in its value a firm faces discourages the firm’s
innovation, output, and profit but encourages those of the other firm.
8

Subindices are used to denote partial derivatives.

84Case (B): Foreign Firm Imitating

Denote the profits of the Home and Foreign firms by Πm and Π∗m , respectively. Since imitation involves zero cost, the profits of the firms areeΠm = [a − (c0 − g) − x − x∗ ]x − g2
2
∗m
∗m
∗
∗
Π
= [a − c1 − x − x ] x .

(10a)
(10b)

Maximizing (10a) with respect to x and g and (10b) with respect to x∗ gives
the following optimality conditions
x : a − c0 + g − 2x − x∗ = 0(11a)

g : x − eg = 0

(11b)

∗
x∗ : a − c∗m
1 − x − 2x = 0,

(11c)

∗
∗
where c∗m
1 = c0 − θ(c0 − c0 ). Solving this system immediately yieldsmxe[a − c0 + (1 − θ)(c∗0 − c0 )]= x (e, θ) =3e − 2mx∗m = x∗m (e, θ) =gm = g m (e, θ) =(2e − 1)[a − c∗0 + θ(c∗0 − c0 )] − e(a − c0 )3e − 2a − c0 + (1 − θ)(c∗0 − c0 ),3e − 2

(12a)
(12b)
(12c)

where superscript “m” denotes the variables in the present case. The resulting profits of the firms are∙¸2e(2e − 1) a − c0 + (1 − θ)(c∗0 − c0 )Π (e, θ) =2
3e − 2¾2½(2e − 1)[a − c∗0 + θ(c∗0 − c0 )] − e(a − c0 )∗mΠ (e, θ) =.
3e − 2
m

(13a)
(13b)

The dependence of the outputs, innovation level, and the profits of the firms
are derived as follows.
9Lemma 3. In the present case with the Foreign firm imitating, we have
∂xm
∂xm
∂x∗m
∂x∗m
< 0,
< 0,
> 0,
>0
∂e
∂θ
∂e
∂θ
∂g m
∂g m
< 0,
< 0.
∂e
∂θ

m
∗m
∗m
Lemma 4. Πme < 0, Πθ < 0, Πe > 0 and Πθ > 0.

As in the case of innovation by the less eﬃcient firm, an increase in
the cost of innovation faced by the advanced firm is detrimental to the firm’s
innovation, output, and profit but is good for the other firm. Not surprisingly,
an increase in the degree of imitation, θ, has similar eﬀects because it tends
to help the less eﬃcient firm but hurt the advanced firm.

5Surpassing and Catching Up

Before we explain how the foreign firm chooses between innovation and imitation, we want to derive more properties of the two cases. In particular,
we want to examine how in each of the cases surpassing or catching up ispossible.
Definition (Surpassing). Surpassing by the Foreign firm occurs if and only
if c1 > c∗1 .
Definition (Catching Up). Catching up by the Foreign firm occurs if andonly if c∗0 − c0 > c∗1 − c1 > 0.
In other words, the Foreign firm is said to have surpassed the Home firmin terms of technology if its new marginal cost is lower than that of theHome firm. The Foreign firm is said to be catching up if it is closing the10technology gap between the two firms, but the Home firm still has a moreadvanced technology. We want to derive conditions under which surpassingor catching up is possible.

5.1

Case (A)We now examine the conditions for surpassing and catching-up in case (A),in which the Foreign firm innovates (so does the Home firm).
Proposition 1. Given Assumption 2, if the Foreign firm chooses to innovate, it will surpass the Home firm if and only ife>e∗ (a − c∗0 ).
(a − c0 ) − 3e∗ (c∗0 − c0 )(14)Corollary 1. If the Foreign firm has a cost of innovation not less than thatof the Home firm, e∗ ≥ e, then surpassing is not possible with both firms

innovating.

What the proposition and condition (14) mean is that if the Home firm’s
cost of innovation is suﬃciently higher than that of the Foreign firm, theForeign firm will choose a new marginal cost lower than what the Home firmchooses.
[Figure 1 approximately here.]
Surpassing can be illustrated graphically in Figure 1. Curve SP represents
the following equation:e=e∗ (a − c∗0 ).
(a − c0 ) − 3e∗ (c∗0 − c0 )It passes through the origin. The slope of curve SP is equal to¯de∗ ¯¯[a − c0 − 3e∗ (c∗0 − c0 )]2> 0.
=
de ¯SP(a − c0 )(a − c∗0 )
11

(15)

(16)

When e∗ → 0, the slope of the curve is greater than unity, implying that curve

SP is above the 45◦ line in the small region near the origin (except at theorigin). Equation (16) shows that the slope is monotonically decreasing withe∗ . Curve SP cuts the 45◦ line when e = e∗ , or, from (15), when e = e∗ = 1/3.
For e > 1/3, curve SP is entirely below the 45◦ line. For e > z (Assumption2), surpassing occurs in the region on the right hand side of (below) curveSP.
We now turn to catching up.
Proposition 2. In the present case in which both sides choose to innovate,
when given assumption 1 , catching up occurs if and only ife>e∗ (a − c0 ) + (e∗ − 1)(c∗0 − c0 ).
a − 2c∗0 + c0

(17)Corollary 2. If e∗ ≥ e, catching up is not possible.
Catching up can be illustrated graphically. In Figure 1, curve CV represents equatione∗ (a − c0 ) + (e∗ − 1)(c∗0 − c0 ).
a − 2c∗0 + c0
The slope of curve CV is equal to¯de∗ ¯¯a − 2c∗0 + c0=
> 0.
de ¯CV a − 2c0 + c∗0
e=

(18)

(19)

Equation (19) shows that the slope of curve CV is constant and less thanone. From (18), we see that curve CV cuts the horizontal axis at a point at
which e = −(c∗0 − c0 )/(a − 2c∗0 + c0 ), and cuts the 45◦ line when e = e∗ = 1/3.

It is interesting to compare curve SP and curve CV, as both of them arepositively sloped.
12

Lemma 5. Curves SP and CV intersect at two points, A (e = e∗ = 1/3)
and B (e = e∗ = σ, where σ = (a − c0 )/(a − 2c0 + c∗0 ) < 1). For e ∈ [0, 1/3)

and e ∈ (σ, ∞), curve CV is above curve SP. For e ∈ (1/3, σ), curve CV isbelow curve SP.

The above results are shown in Figure 1. Points A and B are the points
of intersection between curve SP and curve CV. Since we focus on the area inwhich e > z, we can conclude that in this relevant region, curve SP is below
and on the right hand side of curve CV.

5.2

Case (B)Let us now turn to the case in which the Foreign firm chooses to imitate
while the Home firm innovates. We first note that imitation will never makethe Foreign firm as good as the Home firm before Home firm’s innovation.
In other words, if the Foreign firm chooses to imitate, surpassing can never
happen. This result is formally stated and proved below:
Proposition 3. In case (B) with the Foreign firm imitating, surpassing is
not possible.
Proposition 3 is quite intuitive. Surpassing equation (4) implies that atbest the Foreign firm can improve its technology to the initial technology
level of the Home firm.9
Proposition 4. In case (B) with the Foreign firm imitating, catching up
occurs if and only ife>9a − c0 + (θ + 1)(c∗0 − c0 ).
3θ(c∗0 − c0 )We do not consider the unrealistic case in which θ > 1.

13

(20)

Proposition 4 means that by imitating, the Foreign firm may make abigger technology improvement than the Home firm does. This occurs if
the Home firm’s cost of innovation is high as compared with the degree ofimitation. Catching up with the Foreign firm imitating can also be illustrated
graphically. Let us consider the following conditione=a − c0 + (θ + 1)(c∗0 − c0 ),3θ(c∗0 − c0 )(21)which gives the threshold values of e and θ with which the technology gap
between the two firms remains constant. The condition, when given a valueof θ can be represented by a vertical line in the e-e∗ space similar to Figure 1(not shown). The space to the right of the line represents values of e (whengiven θ) that lead to catching up.

6

Choosing between Innovation and Imitation

So far, we have focused the analysis on each of the two cases, assuming that
the Foreign firm either innovates or imitates. We now explain how the firm
will choose between these two actions.
Since the Foreign firm tries to maximize its profit, it innovates if and onlyifΠ∗v (e, e∗ ) > Π∗m (e, θ).

(22)

To see the choice for the Foreign firm, let us consider the following condition:Π∗v (e, e∗ ) = Π∗m (e, θ).
[Figure 2 approximately here.]
14

(23)

Based (22), (23) represents the combinations of e, e∗ , and θ that will makethe Foreign firm indiﬀerent between innovation and imitation. These combinations, when given θ, can be represented by the indiﬀerence locus, labeled
IL, in Figure 2. Totally diﬀerentiate both sides of (23) to give
∗v
∗
∗m
∗m
Π∗v
e de + Πe∗ de = Πe de + Πθ dθ.

(24)

∗m
∗v
∗m
From Lemmas 2 and 4, Π∗ve∗ < 0, while Πe , Πe , and Πθ are both positive.

When given the value of θ, (24) is rearranged to give the slope of curve IL¯∗vde∗ ¯¯Π∗me − Πe=.
(25)
de ¯ILΠ∗ve∗

The sign of the slope is ambiguous. Thus curve IL is positively sloped if andonly if Π∗m< Π∗vee , i.e., if and only if a rise in e will help the Foreign firm

more when it imitates than when it innovates. This case is shown in Figure
2. The area on the right (left) hand side of curve IL represents the values of
e and e∗ that will lead to innovation (imitation) by the Foreign firm.
Since Π∗m
> 0, an increase in θ will shift curve IL down and to theθright, say, I0 L0 . This means that the Foreign firm will more likely choose toimitate. For example, point Q is on curve IL, meaning that the Foreign firm
is indiﬀerent between imitation and innovation. If there is a rise in θ, thecurve shifts to I0 L0 . Point Q now becomes a point above the curve, implyingthat the Foreign firm now prefers to imitate.
[Figure 3 approximately here.]
We can now add in curves SP and CV to the diagram. See Figure 3. By
Lemma 5, for e > z, curve SP is below and to the right of curve CV. We
can add in line CM, which gives the values of e (when given θ) that will help
the Foreign firm catch up when it chooses imitation. We can now identify
15the following regions in the diagram (Innovation or imitation refers to thedecision of the Foreign firm, as the Home firm always innovates.):• V1: innovation with no catching up or surpassing;• V2: innovation with catching up but no surpassing;• V3: innovation with surpassing;• M1: imitation with no catching up;• M2: imitation with catching up.
So far we have been focusing on the choice of the Foreign firm and its
profits. Now let us examine how the profit of the Home firm may be diﬀerent
in diﬀerent cases. In particular, we want to raise and answer one interestingquestion: Suppose that we do not endogenize the decision of the Foreign firm,
as many papers in the literature do, how would the profit of the Home firm
be aﬀected? To see this question more clearly, consider the values of (e, e∗ , θ)that are represented by point R in Figure 2. At this point, the Foreign
firm will choose to innovate because Π∗v (e, e∗ ) > Π∗m (e, θ). If it does, theHome firm’s profit is Πv (e, e∗ ). However, if we assume that the Foreign firm
imitates, the Home firm’s profit will be Πm (e, θ). How can these two profits
of the Home firm be compared? In other words, if one follows the practiceof many papers in the literature and assumes that the Foreign firm imitates,
then what kind of error may one commit?
Before we answer the above questions, let us first present the following 2
lemmas:16

Lemma 6. If the Home firm always chooses its output and level of innovation
optimally, its profit is negatively dependent on the Foreign firm’s output but
independent of whether the Foreign firm innovates or imitates.
This lemma is just a simple application of the Envelope Theorem.
∗mLemma 7. If Π∗v (e, e∗ ) > Π∗m (e, θ), then g∗v > θ(c∗0 − c0 ) and c∗v
1 < c1 .

Lemmas 6 and 7 are used to prove the following proposition, which answers the questions raised earlier.
Proposition 5. If Π∗v (e, e∗ ) > Π∗m (e, θ), then Πv (e, e∗ ) < Πm (e, θ).
The proposition implies that if the Foreign firm prefers to innovate but
is allowed to imitate only, its profit will drop while the profit of the Home
firm will rise. In other words, in the innovation-imitation models, if theForeign firm in fact prefers to innovate but is assumed to imitate, the profitof the Foreign firm could be underestimated but that of the Home firm beoverestimated.

7

Concluding Remarks

Using a simple model of international rivalry, we analyzed how two firms
compete in terms of technology and output. We derived conditions for theless eﬃcient firm to choose between innovation and imitation, and also conditions under which the less eﬃcient firm is able to catch up or to surpass
the technology of the other firm.
Endogenizing the innovation/imitation decision of the less eﬃcient firm
allows us to examine the potential risk of the artificial assumption that theless eﬃcient firm always imitates. We argued that if the less eﬃcient firm
17would choose to innovate, if allowed to, then assuming that it imitates willtend to underestimate its profit but overestimate the profit of the other firm.
In the present paper, we restricted our analysis to a static model in orderto get more insight and results. It would be interesting to examine some dynamic issues such as switching between innovation and imitation over timeusing a dynamic model, as Shimomura and Wong (2001) do. Another approach to analyzing the issues of growth and innovation/imitation switchingis to combine the present framework with a diﬀerential game of duopoly
in which both of the two firms have two alternatives: to innovate and toimitate.10 The latter could be a topic of future research.

10

See, for example, Fershtman and Kamien (1987) and Tsutsui and Mino (1990) forrecent work of analysis using diﬀerential game models of duopoly.

18

Appendix
Proofs of Lemmas and Propositions Stated in this Paper
Proof of Lemma 1. Diﬀerentiating (7a) and (8a) with respect to e and e∗ ,we get∂g v−(3e∗ − 2)[(e∗ − 1)(a − c0 ) + e∗ (c∗0 − c0 )]=<0∂e[(2e − 1)(2e∗ − 1) − ee∗ ]2(e − 1)(a − c∗0 ) − e(c∗0 − c0 )
∂g v
=
>0
∂e∗
[(2e − 1)(2e∗ − 1) − ee∗ ]2∂xv−(2e∗ − 1)[(e∗ − 1)(a − c0 ) + e∗ (c∗0 − c0 )]<0=∂e[(2e − 1)(2e∗ − 1) − ee∗ ]2

∂xv
∂gv
=
e
> 0.
∂e∗
∂e∗

The eﬀects of e and e∗ on g ∗v and x∗v can be obtained in a symmetric way.
Proof of Lemma 2. Diﬀerentiating (9a) and (9b) with respect to e and e∗
respectively yields
Πve
Πve∗
Π∗v
e
Π∗v
e∗

∙
¸2
ψ∗
1
= −
<0
2 (2e − 1)(2e∗ − 1) − ee∗ψψ∗ e(2e − 1)=>0[(2e − 1)(2e∗ − 1) − ee∗ ]3ψψ∗ e∗ (2e∗ − 1)=>0[(2e − 1)(2e∗ − 1) − ee∗ ]3
¸2
∙
ψ
1
= −
< 0,
2 (2e − 1)(2e∗ − 1) − ee∗where the assumption that both g v and g ∗v are positive has been used andψ = (e − 1)(a − c∗0 ) − e(c∗0 − c0 ) and ψ∗ = (e∗ − 1)(a − c0 ) + e∗ (c∗0 − c0 ).
19

Proof of Lemma 3. The response of gm on e and θ, and ∂xm /∂θ are triviallyobtained, while the other results are derived as∂xm−2[a − c0 + (1 − θ)(c∗0 − c0 )]<0=∂e(3e − 2)2∂x∗ma − c0 + (1 − θ)(c∗0 − c0 )>0=∂e(3e − 2)2∂x∗m
(2e − 1)(c∗0 − c0 )
=
> 0.
∂θ
3e − 2This gives Lemma 3.

Proof of Lemma 4. Diﬀerentiating (13a) and (13b) with respect to e and θ,
we have
Πm
=
e
Πm
θ

(2 − 5e)ζ 2<02(3e − 2)3−e(2e − 1)ζ(c∗0 − c0 )<0=3e − 2

=
Π∗m
e
=
Π∗m
θ

2ζ[(2e − 1)ζ ∗ − e(a − c0 )]>0(3e − 2)32(2e − 1)(c∗0 − c0 )[ζ ∗ (2e − 1) − e(a − c0 )]> 0,(3e − 2)2where ζ = a − c0 + (1 − θ)(c∗0 − c0 ) and ζ ∗ = a − c∗0 + θ(c∗0 − c0 ). The signsof the partial derivatives depend on the assumption that the outputs andinnovation level are positive.

Proof of Lemma 5. Equations (15) and (18) imply that the horizontal gap
between curves CV and SP is equal toe∗ (a − c0 ) + (e∗ − 1)(c∗0 − c0 )e∗ (a − c∗0 )−a − c∗0 − (c∗0 − c0 )a − c0 − 3e∗ (c∗0 − c0 )=−(3e∗ − 1)(c∗0 − c0 )[(e∗ − 1)(a − c0 ) + e∗ (c∗0 − c0 )][a − c∗0 − (c∗0 − c0 )][a − c0 − 3e∗ (c∗0 − c0 )]
20

(26)

Setting the numerator in (19) to zero gives two roots: e∗ = 1/3 and e∗ = σ <1. Comparison between (15) and (18) implies that curve CV is above curve
SP when e∗ < 1/3 and when e∗ > σ. In between these two points, curve CV
is below curve SP.
Proof of Lemma 6. We first note that equation (5a) is identical to equation(10a), meaning that the Home firm’s profit function is independent of whether
the Foreign firm innovates or imitates. Its decision depends only on what theForeign firm produces. Its profit function can be written as Π = Π̂(x, g, x∗ ).
If the firm always chooses x and g optimally, then
dΠ
∂Π dx
∂Π dg
∂Π
=
+
+ ∗ = −x < 0.
∗
∗
∗
dx
∂x dx
∂g dx
∂x
This proves the lemma.
Proof of Lemma 7. Suppose that Π∗v (e, e∗ ) > Π∗m (e, θ). If g∗v ≤ θ(c∗0 − c0 ),then innovation can never be a good choice because the Foreign firm canchoose imitation instead, after paying nothing and getting a bigger or equal
technology improvement. So we must have g∗v > θ(c∗0 − c0 ), which implies∗mthat c∗v
1 < c1 .

Proof of Proposition 1. In case (A), with both firms innovating, the new gap
between the firms’ marginal costs is
v
∗
∗v
c∗v
− c0 + gv
1 − c1 = c0 − ge(3e∗ − 1)(c∗0 − c0 ) + (e∗ − e)(a − c∗0 )=,(2e − 1)(2e∗ − 1) − ee∗(27)which in general can be both positive and negative. Given Assumption 2, thev
numerator in (27) is positive. Thus, c∗v
1 − c1 < 0 if and only if the numeratorin (27) is negative, or if and only if−e[a − c0 − 3e∗ (c∗0 − c0 )] + e∗ (a − c∗0 ) < 0.
21

(28)

Condition (28) is equivalent to (14).11
Proof of Proposition 2. To determine whether catching up exists, note thatv∗∗∗v
(c∗0 − c0 ) − (c∗v
− c0 + gv )1 − c1 ) ≡ c0 − c0 − (c0 − g= g ∗v − gve(a − 2c∗0 + c0 ) − e∗ (a − c0 ) − (e∗ − 1)(c∗0 − c0 )=.
(2e − 1)(2e∗ − 1) − ee∗
From the definition, we can see that catching up exists if and only if thenumerator is positive. Rearranging the terms will give the proposition.
Proof of Proposition 3. We note thatmc∗m
= c∗0 − θ(c∗0 − c0 ) − c0 + g m
1 − c1(a − c0 ) + (3e − 1)(1 − θ)(c∗0 − c0 )> 0.
=3e − 2This proves the proposition.
Proof of Proposition 4. With the Foreign firm imitating, the change in the
gap between the marginal costs of the firms is given by
m
∗
∗
∗
m
(c∗0 − c0 ) − (c∗m
1 − c1 ) ≡ c0 − c0 − [c0 − θ(c0 − c0 ) − c0 + g ]= θ(c∗0 − c0 ) − g mθ(3e − 2)(c∗0 − c0 ) − (a − c0 ) − (1 − θ)(c∗0 − c0 ).
=3e − 2The change in the gap is positive if and only if the numerator is positive.
Rearranging the terms will give the proposition.
Proof of Proposition 5. Note first that the two reaction functions given by(6a) and (6b), or (11a) and (11b), of the Home firm depend on x∗ but not on
11

In deriving (28), we have assumed that a − c0 − 3e∗ (c∗0 − c0 ) > 0.

22

whether the Foreign firm innovates or imitates. Substituting (6b) into (6a)(or (11b) into (11a)), we get a function similar to the usual reaction functionof a firm in a duopoly model. If Π∗v (e, e∗ ) > Π∗m (e, θ), then by Lemma 7,∗mc∗v
1 < c1 . This means that if the Foreign firm chooses to innovate but if it

is allowed to imitate only, its new marginal cost will be higher, no matter
what the Home firm chooses. It is well known in the regular duopoly modelthat a rise in the marginal cost of one of the firms will cause it to lower its
output. By lemma 6, the profit of the Home firm will be higher.
Proof of Corollary 1. If e∗ ≥ e, then the numerator in (27) is positive,v
implying that c∗v
1 > c1 .

Proof of Corollary 2. If e∗ ≥ e, the right hand side of (17) can be rearranged
to givee∗ (a − c0 ) + (e∗ − 1)(c∗0 − c0 )(3e∗ − 1)(c∗0 − c0 )∗=e+
a − 2c∗0 + c0
a − 2c∗0 + c0∗
> e,violating condition (17), implying that catching up is not possible.

23

e*

CV

.

.
B

SP

A
0

45 o

1/3

s

z

Figure 1:
Curves SP and CV

e

e*
IL

I’L’

imitation

.

Q

.

R

innovatione
Figure 2:
Curve IL

e*
CM

IL
M2

CV
SP

M1
V1 V2
V3

e
Figure 3:
Innovation and Imitation

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24

[9] Liao, P. and K. Wong (2004) ‘R&D Subsidy, Intellectual Property Rights
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25