Discussion Papers In Economics
And Business

Environmental Technology Transfer in a Cournot
Duopoly: The Case of Fixed-Fee Licensing
Akira Miyaoka
Discussion Paper 14-08

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEnvironmental Technology Transfer in a Cournot
Duopoly: The Case of Fixed-Fee Licensing
Akira Miyaoka
Discussion Paper 14-08

February 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEnvironmental Technology Transfer in a Cournot
Duopoly: The Case of Fixed-Fee Licensing∗
Akira Miyaoka†

Abstract
This study considers a Cournot duopoly market in which a clean firm can transfer
its less polluting technology to a dirty firm through a fixed-fee licensing contract. We
analyze the impacts of emissions taxes on the incentives of firms to transfer technology as
well as on the total pollution level, and examine the properties of the optimal emissions
tax policy. We first show that higher emissions taxes weaken incentives for technology
transfer and that this can lead to a perverse increase in the level of total pollution. We
then compare the optimal emissions tax when technology licensing is possible with thatwhen licensing is infeasible and show that the relationship between the optimal tax rate
and the degree of the initial technology gap between firms when licensing is possible canbe the opposite of that when licensing is infeasible.

JEL Classification: L13, L24, Q58
Keywords: Technology transfer; Cournot duopoly; Pollution; Emissions tax∗

This paper previously circulated under the title “Emission Taxes and Environmental Technology Transfer.”
I especially thank Shingo Ishiguro, Kenzo Abe, Noriaki Matsushima, Katsuya Takii, Hiroaki Ino, Hiroshi Kitamura, Shuichi Ohori, Takayoshi Shinkuma, Leonard F.S. Wang, Akihiko Yanase, and the conference participantsat the Japanese Association for Applied Economics, the Japanese Economic Association, and the Society for
Environmental Economics and Policy Studies for helpful discussions and comments. I gratefully acknowledge
financial support from the Japan Society for the Promotion of Science and the Global COE program “Human
Behavior and Socioeconomic Dynamics” of Osaka University. The usual disclaimer applies.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043,
Japan. E-mail: jge013ma@mail2.econ.osaka-u.ac.jp.

1 Introduction
To mitigate against environmental degradation, an important concern for policy makers isthe diffusion of environmentally friendly technology. Although there are several channels
through which environmental technology can be distributed, one of the most important is thetransfer of technology among firms through licensing contracts. In fact, private firms often
strategically license their superior technologies to both domestic and foreign rivals. For example, the famous Japanese automobile manufacturer Toyota Motor Corporation entered alicensing agreement with another automobile manufacturer, Mazda Motor Corporation, thustransferring its superior environmental technology. This also occurs in the chemical industry, where some leading firms in the polyethylene market, such as British Petroleum (BP)Chemicals and Dow Chemical, have licensed their less polluting technology to other firms.
In this study, we analyze the effects of emissions taxes and the properties of the optimal
tax rate in the presence of environmental technology transfer between firms through fixed-fee
licensing. We consider a Cournot duopoly model in which one firm has a cleaner technology
and emits less pollution in its production process than another firm. Before the competition
stage, the clean firm can transfer its superior technology to the dirty firm via a licensing
contract. If technology transfer is successful, the dirty firm obtains a clean technology inexchange for a fixed licensing fee.
In this setting, we first analyze the effects of an emissions tax on the incentives of firms totransfer technology as well as on the level of total pollution. We show that a higher emissions
tax makes the transfer of technology less likely and can lead to a perverse increase in the total
pollution level. We then explore the properties of the optimal emissions tax when licensing
is possible by comparing it with the case when licensing is infeasible. We find that thepossibility of licensing can reverse the relationship between the optimal emissions tax rate
and the extent of the initial technology gap between firms: as a dirty technology becomes
more polluting, the optimal emissions tax is (weakly) decreasing when licensing is possible,
while it is increasing when licensing is infeasible.
Although a few recent studies investigate environmental technology transfer via licensing contracts, most of them consider international technology transfers between domestic and1foreign firms (Iida and Takeuchi 2009, 2011; Qiu and Yu 2009; Asano and Matsushima forthcoming).1 In contrast, the present study explores technology transfers between two domestic
firms, which may also be an important policy concern. In our setting, since an emissions tax
set by the government is levied on both the licenser and licensee firms, the effects and properties of the optimal tax policy differ from those of the studies mentioned above, in which a
tax set by the government of one country is only levied on either the licenser or licensee firmthat is located in that country.
The most similar study to ours is Chang et al. (2009), which focuses on the licensing ofless polluting technologies between two domestic firms and compares two licensing methods,
fixed-fee and royalty licensing. Although their analysis is similar to ours, they mainly focus
on the case of royalty licensing. The present study complements their work by providing
a more detailed analysis of fixed-fee licensing. We provide a full characterization of the
optimal emissions tax in the case of fixed-fee licensing by considering not only an interior
solution but also a corner solution, which is not included in their study. We also compare
the optimal tax rates when technology licensing is and is not possible and clarify the novel
properties of the optimal emissions tax in the presence of technology licensing, neither ofwhich is mentioned in the work of Chang et al.
Our study is also related to the literature on optimal emissions taxes under oligopolisticcompetition (see Requate (2006) for a survey). In particular, Simpson (1995) analyzes the
optimal emissions tax in an asymmetric duopoly setting, in which the environmental technologies of both firms are exogenously fixed. In contrast to his study, we assume that the
dirty firm can obtain cleaner technology through a licensing contract and show that the possibility of technology transfer can alter the properties of the optimal emissions tax set by thegovernment.
The remainder of this study is organized as follows. Section 2 sets up the model andderives firm equilibrium outputs and profits. Section 3 considers the conditions under which
technology transfer occurs. Section 4 examines the relationship between the emissions tax
rate and the level of total pollution. Section 5 derives a socially optimal emissions tax rate
1

The licensing of cost-reducing innovations has been extensively analyzed in the industrial organization
literature (Gallini and Winter 1985; Katz and Shapiro 1985; Kamien and Tauman 1986; Marjit 1990; Wang
1998, 2002).

2and discusses its properties. Section 6 provides concluding remarks. The proofs of someresults are provided in the Appendix.

2 The model
We consider a market consisting of two heterogeneous firms, one clean (firm 1) and one dirty(firm 2), that produce a homogeneous product. Firm 1 emits e1 units of pollution per unit ofoutput, whereas firm 2 emits a higher level of pollution, i.e., e2 > e1 , per unit of output. For
simplicity, the production costs of both firms are assumed to be zero. The inverse demand
function is given by P = 1 − (x1 + x2 ), where P denotes the market price, xi denotes the output
level of firm i ∈ {1, 2}.
We employ a three-stage game. In Stage 1, the government sets a (uniform) emissions
tax to maximize social welfare. In Stage 2, firm 1, which has a clean technology, decides
whether to license its superior technology to firm 2. If technology transfer occurs, both firms
1 and 2 have a clean technology. Otherwise, firm 2’s technology remains dirty. In Stage 3,
both firms compete à la Cournot given the technology inherited from Stage 2.
We solve the game backwards. First, we derive the Cournot equilibrium in Stage 3 when
the technology transfer has occurred in Stage 2. In this case, since both firms have a clean
technology, firm i’s profit (gross of licensing fee) in Stage 3 is given by:
Πi = (1 − x1 − x2 )xi − te1 xi ,i = 1, 2,

where t denotes an emissions tax imposed per unit of emissions. Assuming a symmetricequilibrium, the equilibrium output and profit are, respectively,xiT =1 − e1 t
,
3

πTi =

(1 − e1 t)2,9

i = 1, 2.

(1)

Next, we derive the Cournot equilibrium when firm 1’s clean technology has not beentransferred to firm 2 in Stage 2. In this case, the profit of firm i in Stage 3 is given by:
Πi = (1 − x1 − x2 )xi − tei xi ,i = 1, 2.

Note that since firm 2’s technology remains dirty, firm 2 emits e2 units of pollution per unit3

of output. Then, the equilibrium output and profit are, respectively,
1 − (2e1 − e2 )t
1 − (2e2 − e1 )t
,
x2N =
;
3
3
(1 − (2e2 − e1 )t)2
(1 − (2e1 − e2 )t)2
, π2N =
.
π1N =
9
9
x1N =

(2)
(3)

When the emissions tax rate is too high (t ≥ t¯(e1 , e2 ) ≡ 1/(2e2 − e1 )), firm 2 exits the market
and firm 1 becomes a monopoly. In this case, the firms’ equilibrium output and profit are,respectively,x1M =1 − e1 t,
2x2M = 0;π1M =(1 − e1 t)2,4

π2M = 0.

(4)

3 Licensing and technology transfer
In this section, we analyze the licensing stage. This study focuses on a fixed-fee licensing
contract and assumes that firm 1 (licenser) has all the bargaining power. At this stage, firm
1 first offers firm 2 a fixed licensing fee F, which is independent of firm 2’s output, in a
take-it-or-leave-it manner. If firm 2 accepts this offer, then firm 1 charges F and licenses its
clean technology to firm 2. If firm 2 rejects the offer, licensing does not occur and firm 2’s
technology remains dirty.
First, if 0 ≤ t < t¯(e1 , e2 ), since firm 2’s equilibrium profit without licensing is π2N , themaximum licensing fee firm 1 can charge is F = πT2 − π2N . Then, if licensing occurs, firm 1’s
total profit isπT1 + F = πT1 + (πT2 − π2N ) =(1 − e1 t)2 4(1 − e2 t)(e2 − e1 )t
+
.
9
9

(5)

From (3) and (5), we have that πT1 + F > π1N if and only if 0 ≤ t ≤ tˆ(e1 , e2 ) ≡ 2/(5e2 − 3e1 ).
Therefore, firm 1 licenses its technology if 0 ≤ t ≤ tˆ(e1 , e2 ) but does not if tˆ(e1 , e2 ) < t <t¯(e1 , e2 ).
Second, if t ≥ t¯(e1 , e2 ), when technology licensing does not occur, firm 2 exits the market
and its equilibrium profit is π2M = 0. Then, the maximum licensing fee firm 1 can charge is
F = πT2 − π2M , and firm 1’s total profit under licensing becomes
πT1 + F = πT1 + (πT2 − π2M ) =42(1 − e1 t)2
.
9

(6)

Comparing (4) and (6), we always obtain πT1 +F < π1M . Therefore, for t ≥ t¯(e1 , e2 ), firm 1 does
not have an incentive to license its clean technology and consequently becomes a monopoly.
To summarize these results, we have the following proposition.
Proposition 1. Firm 1 licenses its clean technology if and only if 0 ≤ t ≤ tˆ(e1 , e2 ).
[Figure 1 about here.]
The reason that licensing does not occur under a high emissions tax is as follows. Notethat technology licensing occurs if and only if the joint profit of the two firms with licensing,
πT1 + πT2 , is higher than that without licensing, π1N + π2N (or π1M + π2M ). When the tax rate ishigh and there is no licensing, firm 2’s market share is very small (or zero) and firm 1 hasan almost (or complete) monopoly. In this case, the joint profit of the two firms is close (orequal) to the monopoly profit of firm 1. In contrast, when licensing occurs, the joint profit
becomes smaller than that without licensing because firm 2 obtains a clean technology andthe market becomes more competitive. Therefore, under a high emissions tax, firm 1’s clean
technology is not licensed.2 In Figure 1, the shaded area indicates the pair (t, e2 ) such thattechnology transfer takes place.

4 The level of pollution
In this section, we focus on the level of aggregate pollution, denoted by E, and examine therelationship between the pollution level and emissions tax rate. First, if 0 ≤ t ≤ tˆ(e1 , e2 ),licensing occurs and both firms have a clean technology. Therefore, by using (1), the level ofpollution in this case, E T (t), becomes
E T (t) = e1 (x1T + x2T ) =22e1 (1 − e1 t)
.
3

(7)

As shown in Chang et al. (2009), if we assume a per-unit output royalty instead of a fixed licensing fee,since firm 1 (licenser) can manipulate the output of firm 2 (licensee) through a per-unit royalty rate, the former
can extract all the incremental profit of the latter while maintaining the latter’s output at the same level as it
would be without licensing. Therefore, for any emissions tax rate, technology licensing via royalty licensing
always occurs in equilibrium, and firm 1 always prefers royalty licensing to fixed-fee licensing. However, there
are situations in which it is difficult for firm 1 to use royalty licensing. For example, once firm 2 has the licensed
technology, it may be able to imitate the technology and produce output with the imitation, thereby avoiding
per-unit charges. In this case, firm 1 is restricted to fixed-fee licensing, which is independent of firm 2’s output(Katz and Shapiro 1985).

5

Next, if tˆ(e1 , e2 ) < t < t¯(e1 , e2 ), the technology is not transferred but firm 2 is active in themarket. Since firm 2’s technology remains dirty and the outputs are given by (2), the level ofpollution in this case, E N (t), is given by
E N (t) = e1 x1N + e2 x2N =(e1 + e2 ) − 2(e21 − e1 e2 + e22 )t
.
3

(8)

Finally, if t ≥ t¯(e1 , e2 ), firm 2 exits the market and firm 1 becomes a monopoly. By using (4),the level of pollution in this case, E M (t), is given by
E M (t) = e1 x1M =e1 (1 − e1 t).
2

From these equations, it is easy to see that a higher emissions tax t decreases the level ofaggregate pollution in each case. However, since an emissions tax above tˆ(e1 , e2 ) undermines
the incentive for technology to be transferred between firms, as shown in Proposition 1, ahigher emissions tax can have a perverse effect on aggregate pollution. More precisely, we
obtain the following result.
Proposition 2. When e2 > 3e1 , the level of pollution jumps to t = tˆ(e1 , e2 ) and the relationshipbetween the emissions tax and the level of pollution is non-monotonic.
[Figure 2 about here.]
Figure 2 illustrates the relationship between the emissions tax and the pollution level when
(e1 , e2 ) = (1, 5). The failure of technology licensing caused by a higher emissions tax has two
opposing effects on total emissions. Although it reduces the total output of both firms, it
increases firm 2’s pollution level per unit of output. When the initial technology gap between
firms is sufficiently large (e2 > 3e1 ), even a relatively low emissions tax prevents technology
licensing. At such a low tax rate, the former effect is smaller and dominated by the latter.
Therefore, an increase in the emissions tax above tˆ(e1 , e2 ) results in a perverse increase intotal emissions.
A similar result is also obtained by Roy Chowdhury (2008), but his result is driven by
the endogeneity of the market structure. In his study, under a high emissions tax, firms opt
for Cournot competition, whereas under a low tax rate, firms prefer to form joint ventures.
6

Therefore, an increase in the emissions tax can trigger a regime switch from joint ventures to
Cournot competition, which makes the market more competitive and can cause the pollution
level to increase.

5 The optimal emissions tax
In this section, we derive the socially optimal emissions tax rate and explore its properties. In
particular, we focus on the relationship between the optimal emissions tax rate and the extent
of the initial technology gap between two firms. In the analysis below, we therefore consider
e1 to be fixed and interpret e2 ∈ (e1 , ∞) as the extent of the initial technology gap between
two firms.3
In Stage 1, the government sets an emissions tax that maximizes social welfare W, which
consists of the consumer surplus CS , the producer surplus PS , tax revenues T , and the environmental damage D caused by total pollution:
W = CS + PS + T − D.
This study assumes that D = d · E, where d is the constant marginal damage from total
emissions. In addition, because of linear demand, we have CS = (x1 + x2 )2 /2. Below, we
first derive the locally optimal tax rate in each of the following three cases: 0 ≤ t ≤ tˆ(e2 ),tˆ(e2 ) < t < t¯(e2 ), and t ≥ t¯(e2 ). Then, by comparing the maximum level of welfare in eachcase, we derive the globally optimal emissions tax rate.
First, for 0 ≤ t ≤ tˆ(e2 ), in which technology transfer takes place, the government chooses
t to solve the following welfare maximization problem:
max W T (t) ≡t≤tˆ(e2 )

2 

1 T
x1 + x2T + πT1 + πT2 + tE T − dE T .
2

Assuming an interior solution, we obtain the optimal tax rate and the corresponding level ofsocial welfare as follows:
3e1 d − 1
t =
,
2e1
T

3

(1 − e1 d)2
W (t ) =.
2
T

T

We therefore exclude e1 from the arguments of the functions in this section.

7(9)

We assume that 0 < tT < 1/e1 , or, equivalently, that 1/(3d) < e1 < 1/d. This implies that at
this interior solution, it is optimal for the government to levy a positive emissions tax (rather
than provide a subsidy) such that the equilibrium outputs of both firms, given by (1) witht = tT , are positive. Since (9) is valid as long as tT ≤ tˆ(e2 ), the optimal tax when tT > tˆ(e2 ) is
given by tˆ(e2 ).
Second, in the case of t ≥ t¯(e2 ), firm 2 exits the market and firm 1 becomes a monopoly.
In this case, the government solves the following problem:1  M 2
max W M (t) ≡x1 + π1M + tE M − dE M .
t≥t¯(e2 )2Assuming an interior solution, the optimal tax and the corresponding welfare level are, respectively,
2e1 d − 1
(1 − e1 d)2, W M (t M ) =
.
(10)
e1
2
We assume that 0 < t M < 1/e1 , or, equivalently, that 1/(2d) < e1 < 1/d, which implies that attM =this interior solution, it is optimal for the government to impose a positive emissions tax suchthat firm 1’s equilibrium output, given by (4) with t = t M , is positive. Since this assumption
also guarantees that 0 < tT < 1/e1 , we assume that 1/(2d) < e1 < 1/d in the following
analysis. Note that (10) is valid as long as t M ≥ t¯(e2 ). Therefore, when t M < t¯(e2 ), the optimal
emissions tax becomes t¯(e2 ).
Finally, for tˆ(e2 ) < t < t¯(e2 ), technology transfer does not occur but firm 2 is still active inthe market. Then, the government’s problem is as follows:

2 
1 N
x1 + x2N + π1N + π2N + tE N − dE N .
max W N (t, e2 ) ≡2tˆ(e2 )<t≤t¯(e2 )(11)Assuming an interior solution, the optimal tax and the corresponding welfare level are, respectively,
6d(e21 + e22 − e1 e2 ) − (e1 + e2 ),(e1 + e2 )2(e1 + e2 )2 − 2d(e1 + e2 )(e21 + e22 ) + 4d2 (e21 − e1 e2 + e22 )2
W N (t N (e2 ), e2 ) =.
2(e1 + e2 )2t N (e2 ) =

(12)
(13)

Since (12) and (13) are valid as long as tˆ(e2 ) < t N (e2 ) < t¯(e2 ), the maximum level of welfare
could be attained at a corner solution, tˆ(e2 ) +  or t¯(e2 ) − , where  > 0 is an infinitesimallysmall number.
8Now, by comparing the maximized welfare levels in the three cases above, we derive the(globally) optimal emissions tax rate for a given e2 . First, we have the following lemma.
Lemma 1. Suppose that 1/(2d) < e1 < 1/d. Then, the following inequality holds for any
e2 > e1 :maxtˆ(e2 )<t<t¯(e2 )W N (t, e2 ) ≤ max{ max W T (t), max W M (t)}.
0≤t≤tˆ(e2 )t≥t¯(e2 )(14)Lemma 1 implies that from the government’s perspective, it can never be socially optimal
to set an emissions tax rate that allows firm 2 to be active without technology licensing. In
other words, social welfare is maximized under either a duopoly that includes technology
transfer or a monopoly by firm 1. Therefore, in order to derive the socially optimal tax rate,
we only have to compare the levels of welfare under these two conditions. Then, we obtain
the optimal emissions tax under the possibility of technology licensing as follows.
Proposition 3. Suppose that 1/(2d) < e1 < 1/d. Then, when technology licensing is possible,
the optimal emissions tax, t∗ (e2 ), and the resultant licensing decision are given by: Ttif e1 < e2 ≤ ê, licensing occurs,





ˆ
 t(e2 ) if ê < e2 ≤ ẽ, licensing occurs,t∗ (e2 ) = t¯(e2 ) if ẽ < e2 ≤ ē, no licensing,



 tM
if ē < e2 ,no licensing,(15)

where ê, ẽ, and ē are defined, respectively, such that tT = tˆ(ê), W T (tˆ(ẽ)) = W M (t¯(ẽ)), andt M = t¯(ē).
[Figure 3 about here.]
In Figure 3, the bold line represents the relationship between the optimal emissions tax
rate and the initial technology gap between firms, which is measured by firm 2’s initial technology level, e2 . When the environmental technology gap is sufficiently small such thate2 ∈ (e1 , ê], since technology licensing occurs even under a relatively high emissions tax,
the government can induce licensing to occur between firms while setting tT , which is theunconstrained optimal tax rate under licensing. However, as the technology gap widens, thegovernment cannot implement this outcome because for a larger e2 , licensing no longer occurs under tT . Then, for e2 ∈ (ê, ẽ], it is optimal for the government to set a lower tax rate,9tˆ(e2 ), to induce technology licensing. If the technology gap becomes even wider (e2 > ẽ), thegovernment prefers to give up the possibility of technology transfer and drive firm 2 out of
the market. For e2 ∈ (ẽ, ē], since firm 2’s technology is not extremely dirty, the government
must set a sufficiently high emissions tax, t¯(e2 ), in order to induce firm 2 to exit the market.
However, when firm 2 is sufficiently dirty such that e2 > ē, the government can drive firm 2
out of the market by setting t M , which is the unconstrained optimal tax rate under a monopoly
by firm 1.
Then, we compare the optimal tax rates between when technology licensing is possible
and when it is infeasible. When technology licensing is not possible, firm 2’s technology
always remains dirty. In this case, the optimal emissions tax rate is given by the next lemma.
Lemma 2. Suppose that 1/(2d) < e1 < 1/d. Then, when technology licensing is infeasible,
the optimal emissions tax, t∗∗ (e2 ), is given by: N
t (e2 ) if e1 < e2 ≤ e0 ,t¯(e2 ) if e0 < e2 ≤ ē,t∗∗ (e2 ) = 
 tM
if ē < e2 ,(16)

where e0 is defined such that t N (e0 ) = t¯(e0 ).
In Figure 3, the dashed line illustrates the optimal emissions tax given by (16) in theabsence of the possibility of technology licensing.4 If firm 2’s initial technology is not very
dirty (e2 ≤ e0 ), the optimal policy is to set t N (e2 ) and allow both firms to operate in the market.
However, if firm 2’s technology is sufficiently dirty (e2 > e0 ), it is socially desirable to drive
firm 2 out of the market.
Now, from Proposition 3 and Lemma 2, we obtain the following proposition.
Proposition 4. Suppose that 1/(2d) < e1 < 1/d. Then, the relationship between the optimal
emissions tax rate and the degree of the initial technology gap between firms when technology
licensing is possible can be the opposite of that when licensing is infeasible. More precisely,
while t∗∗ (e2 ) is increasing in e2 ∈ (e1 , e0 ], t∗ (e2 ) is (weakly) decreasing in e2 ∈ (e1 , ẽ].
Proposition 4 implies that the possibility of technology licensing can alter the propertiesof the optimal emissions tax. When licensing is infeasible, as firm 2’s initial technology becomes dirtier, it is socially optimal for the government to set a higher emissions tax and shift
4

Depending on the values of e1 and d, ê can be larger than e0 . Figure 3 illustrates the case of ê < e0 .

10market share from the dirty firm (firm 2) to the clean one (firm 1). Therefore, as long as both
firms produce positive outputs for e2 ∈ (e1 , e0 ], the optimal emissions tax t∗∗ (e2 ) is increasingin e2 . In contrast, when technology licensing is possible, the government must choose theemissions tax rate while considering its effect on the possibility of licensing between firms.
In particular, for e2 ∈ (ê, ẽ], since licensing no longer occurs under tT , it is socially optimalfor the government to set a lower emissions tax in order to induce the licensing of technology
between firms. Therefore, while the optimal emissions tax t∗ (e2 ) is constant for e2 ∈ (e1 , ê],it is decreasing for e2 ∈ (ê, ẽ].5

6 Concluding remarks
This study examines the impacts of an emissions tax and the properties of the optimal tax
rate in the presence of environmental technology transfers between firms through fixed-fee
licensing contracts. We show that a higher emissions tax makes technology licensing between
firms less likely and can result in a perverse increase in the level of aggregate pollution. We
also show that the relationship between the optimal emissions tax rate and the degree of theinitial technology gap between firms when licensing is possible can be the opposite of thatwhen licensing is infeasible.
These results clarify that when less-polluting technologies can be transferred between
firms via fixed-fee licensing, there can be a trade-off between a strict environmental policy
and the wide diffusion of superior environmental technologies, which significantly affects theproperties of the optimal emissions tax. Our results imply that in order to design appropriate
environmental policies, governments should pay attention to whether superior environmental
technologies can be diffused in the market in question and through which channels. Otherwise, government policies could have adverse impacts on both the environment and social
welfare.
5
If we consider e2 to be fixed and interpret e1 ∈ (1/(2d), min{e2 , 1/d}) as the extent of the initial technology
gap, the relationship between the optimal emissions tax and the degree of the initial technology gap becomes a
bit more complicated. However, we can also obtain a result similar to Proposition 4 in that case. If technology
licensing is possible, since ∂tT /∂e1 > 0 and ∂tˆ/∂e1 > 0, the optimal emissions tax when both firms are active
is decreasing as e1 becomes smaller. In contrast, from (12), we can confirm that ∂t N /∂e1 < 0 holds for a
sufficiently small e1 . Therefore, if licensing is infeasible, the optimal emissions tax when both firms are active
can be increasing as e1 becomes smaller.

11Appendix
Proof of Proposition 2
The aggregate pollution level increases discontinuously at t = tˆ(e1 , e2 ) if and only if
E T (tˆ(e1 , e2 )) <lim E N (t).

(17)

t→tˆ(e1 ,e2 )From (7), (8), and tˆ(e1 , e2 ) = 2/(5e2 − 3e1 ), we have
E T (tˆ(e1 , e2 )) =10e1 (e2 − e1 );3(5e2 − 3e1 )lim E N (t) =t→tˆ(e1 ,e2 )(e2 − e1 )(7e1 + e2 ).
3(5e2 − 3e1 )By substituting (18) into (17) and rearranging it, we obtain e2 > 3e1 .

(18)
Proof of Lemma 1
To begin the proof, we first introduce the following two lemmas:
Lemma 3. Suppose that 1/(2d) < e1 < 1/d. Then, we havemax W T (t) Q max W M (t)0≤t≤tˆ(e2 )t≥t¯(e2 )if and only if

e2 R ẽ.

(19)

Lemma 4. Suppose that 1/(2d) < e1 < 1/d. Then, t N (e2 ) and W N (t N (e2 ), e2 ) have the following properties:(a) t N (e2 ) is increasing in e2 and lime2 →e1 t N (e2 ) = tT .
(b) lime2 →e1 W N (t N (e2 ), e2 ) = W T (tT ).
(c) There exists e ∈ (e1 , ∞) such that W N (t N (e2 ), e2 ) is decreasing in e2 ∈ (e1 , e] and increasing in e2 ∈ (e, ∞).
Proof of Lemma 3
Since tˆ(e2 ) is decreasing in e2 ∈ (e1 , ∞) and 0 < tˆ(e2 ) < 1/e1 , there existsê =e1 (9e1 d + 1),5(3e1 d − 1)
12

(20)

which satisfies tT = tˆ(ê). Similarly, since t¯(e2 ) is decreasing in e2 ∈ (e1 , ∞) and 0 < t¯(e2 ) <1/e1 , there exists
ē =

de21
,
2de1 − 1(21)which satisfies t M = t¯(ē). Note that from (20) and (21), we haveē − ê =e1 (1 − de1 )(1 + 3de1 )> 0.
5(2de1 − 1)(3de1 − 1)

Then, max0≤t≤tˆ(e2 ) W T (t) and maxt≥t¯(e2 ) W M (t) can respectively be written as follows:( T T
W (t )
if e2 ≤ ê,T
max W (t) =T ˆW (t(e2 )) if e2 > ê,0≤t≤tˆ(e2 )( M
W (t¯(e2 )) if e2 ≤ ē,max W M (t) =W M (t M )if e2 > ē.
t≥t¯(e2 )

(22)
(23)

[Figure 4 about here.]
The two bold lines in Figure 4 illustrate (22) and (23). For e2 ∈ (ê, ē], since tˆ(e2 ) is
decreasing in e2 and tˆ(e2 ) < tT , W T (tˆ(e2 )) is decreasing in e2 . In contrast, for e2 ∈ (ê, ē], sincet¯(e2 ) is decreasing in e2 and t¯(e2 ) ≥ t M , W M (t¯(e2 )) is increasing in e2 . In addition, we have
W T (tT ) = W M (t M ) = (1 − e1 d)2 /2. Then, there exists ẽ ∈ (ê, ē) such that W T (tˆ(ẽ)) = W M (t¯(ẽ)).
More precisely,
q

e1
2
2
ẽ =
9d e1 − 6de1 + 61 + 33de1 − 6 .
60de1 − 25
Therefore, as can also be seen from Figure 4, we obtain (19).

(24)
Proof of Lemma 4(a) By differentiating t N (e2 ) by e2 , we have∂t N (e2 ) (e1 + e2 ) + 18de1 (e2 − e1 )
=
> 0.
∂e2
(e1 + e2 )3(25)

In addition, from (12), it is easy to see that lime2 →e1 t N (e2 ) = tT .
(b) From the right-hand side of (11), it is easy to see that lime2 →e1 W N (t, e2 ) = W T (t). Then,
together with lime2 →e1 t N (e2 ) = tT , we have lime2 →e1 W N (t N (e2 ), e2 ) = W T (tT ).
13

(c) Using the envelope theorem, we havedW N (t N (e2 ), e2 ) ∂W N
(e1 + e2 )(t N )2 + (1 − 6d(2e2 − e1 ))t N + 3d
,
=
=−
de2
∂e2
9
d2 W N (t N (e2 ), e2 ) ∂2 W N ∂t N ∂2 W N=+,∂t∂e2 ∂e2
de22
∂e22

(26)
(27)

where
∂2 W N 18e1 d(e2 − e1 ) + (e1 + e2 )
=
> 0,
∂t∂e2
9(e1 + e2 )∂2 W N t N [6d(e21 + 5e1 e2 + e22 ) + (e1 + e2 )]=> 0.
9(e1 + e2 )2
∂e22

(28)

From (26) and our assumption that 1/(2d) < e1 < 1/d, we havedW N (t N (e2 ), e2 ) d= (e1 d − 1) < 0,e2 →e1de2
2
N N
dW (t (e2 ), e2 )lim
= ∞ > 0.
e2 →∞
de2
lim

(29)

In addition, (25), (27), and (28) lead tod2 W N (t N (e2 ), e2 )
> 0.
de22

(30)

Therefore, from (29) and (30), there exists e ∈ (e1 , ∞) such that W N (t N (e2 ), e2 ) is decreasing in e2 ∈ (e1 , e] and increasing in e2 ∈ (e, ∞).



From Lemma 4(a) and the fact that t¯(e2 ) is decreasing in e2 and 0 < t¯(e2 ) < 1/e1 , it is easy
to see that there exists e0 ∈ (e1 , ∞) such that t N (e0 ) = t¯(e0 ). In the following proof, we deal
with the two cases separately, depending on whether e2 is larger or smaller than e0 .
(a) For e1 < e2 ≤ e0
In order to prove (14), since maxtˆ(e2 )<t<t¯(e2 ) W N (t, e2 ) ≤ W N (t N (e2 ), e2 ) holds, it is sufficient to show that we have W N (t N (e2 ), e2 ) < max0≤t≤tˆ(e2 ) W T (t) for e2 ∈ (e1 , e0 ]. First,
we prove the following lemma:6
Lemma 5. Suppose that 1/(2d) < e1 < 1/d. Then, we have e < e0 < ẽ.
6

The Mathematica file for the proof of this lemma is available from the author upon request.

14

Proof of Lemma 5First, we show that e0 < ẽ. By solving t N (e2 ) = t¯(e2 ) with respect to e2 , we obtain!
e1
1 + 24k − 36k2
0e =1 + 6k + A +
,
(31)
12k
A

1/3
√ √
where A = 108k2 + 36k + 1 + 6k 3 432k4 − 864k3 + 648k2 − 8k − 1
and k =
e1 d. Note that our assumption implies 1/2 < k < 1. Then, from (24) and (31), we
can confirm that e0 < ẽ holds for k ∈ (1/2, 1).
We then show that e < e0 . By substituting e2 = e0 into (26), we have dW N (t N (e0 ), e0 )/de2 .
We can confirm that this is positive for e1 ∈ (1/(2d), 1/d). Therefore, from Lemma 4(c),we have e < e0 .

From Lemmas 4 and 5, W N (t N (e2 ), e2 ) for e2 ∈ (e1 , e0 ] can be depicted as the dashedline in Figure 4. Note that since W N (t¯(e2 ), e2 ) = W M (t¯(e2 )) holds by the definition oft¯(e2 ), we have W N (t N (e0 ), e0 ) = W N (t¯(e0 ), e0 ) = W M (t¯(e0 )). Therefore, from Figure 4, it
can be seen that W N (t N (e2 ), e2 ) < max0≤t≤tˆ(e2 ) W T (t) for e2 ∈ (e1 , e0 ].
(b) For e2 > e0
In this case, since t N (e2 ) > t¯(e2 ) holds, we havemaxtˆ(e2 )<t<t¯(e2 )W N (t, e2 ) = W N (t¯(e2 ) − , e2 ) < W M (t¯(e2 )) ≤ max W M (t),t≥t¯(e2 )(32)

where the last inequality follows from (23) and W M (t¯(e2 )) ≤ W M (t M ) for all e2 .
This completes the proof of Lemma 1.

Proof of Proposition 3This result follows from Lemmas 1 and 3.

15

Proof of Lemma 2
When licensing is infeasible, we must compare max0≤t<t¯(e2 ) W N (t, e2 ) and maxt≥t¯(e2 ) W M (t). In
the case of e2 > e0 , (32) implies that the maximum welfare level is attained under a monopoly
by firm 1. Then, from (23), the optimal emissions tax is given by t¯(e2 ) for e0 < e2 ≤ ē and t M
for e2 > ē.
Next, we consider the case of e2 ≤ e0 . In this case, since t N (e2 ) ≤ t¯(e2 ) holds, we have
max0≤t<t¯(e2 ) W N (t, e2 ) = W N (t N (e2 ), e2 ). On the other hand, since e2 ≤ e0 < ē holds, we havemaxt≥t¯(e2 ) W M (t) = W M (t¯(e2 )). Then, we havemax W N (t, e2 ) = W N (t N (e2 ), e2 ) ≥ W N (t¯(e2 ), e2 ) = W M (t¯(e2 )) = max W M (t).

0≤t<t¯(e2 )t≥t¯(e2 )Therefore, for e2 ≤ e0 , the optimal emissions tax is given by t N (e2 ).

References
Asano, T. and N. Matsushima (forthcoming) “Environmental Regulation and Technology
Transfers” Canadian Journal of Economics.
Chang, M.C., J.L. Hu and G.H. Tzeng (2009) “Decision Making on Strategic Environmental Technology Licensing: Fixed-Fee versus Royalty Licensing Methods” International
Journal of Information Technology & Decision Making 8, 609–624.
Gallini, N.T. and R.A. Winter (1985) “Licensing in the Theory of Innovation” RAND Journal of Economics 16, 237–252.
Iida, T. and K. Takeuchi (2009) “Environmental Technology Transfer via Free Trade” Economics Bulletin 30, 948–960.
Iida, T. and K. Takeuchi (2011) “Does Free Trade Promote Environmental Technology
Transfer?” Journal of Economics 104, 159–190.
Kamien, M.I. and Y. Tauman (1986) “Fees versus Royalties and the Private Value of a
Patent” Quarterly Journal of Economics 101, 471–491.

16

Katz, M. and C. Shapiro (1985) “On the Licensing of Innovations” RAND Journal of Economics 16, 504–520.
Marjit, S. (1990) “On a Non-Cooperative Theory of Technology Transfer” Economics Letters 33, 293–298.
Qiu, L.D. and Z. Yu (2009) “Technology Transfer and the South’s Participation in an International Environmental Agreement” Review of International Economics 17, 409–427.
Requate, T. (2006) “Environmental Policy under Imperfect Competition” in The International Yearbook of Environmental and Resource Economics 2006/2007 by T. Tietenberg
and H. Folmer, Eds., Edward Elgar.
Roy Chowdhury, I. (2008) “Joint Ventures, Pollution and Environmental Policy” Bulletin of
Economic Research 60, 97–121.
Simpson, R.D. (1995) “Optimal Pollution Taxation in a Cournot Duopoly” Environmental
and Resource Economics 6, 359–369.
Wang, X.H. (1998) “Fee versus Royalty Licensing in a Cournot Duopoly Model” Economics
Letters 60, 55–62.
Wang, X.H. (2002) “Fee versus Royalty Licensing in a Differentiated Cournot Duopoly”
Journal of Economics and Business 54, 253–266.

17

1⁄
̅
̂

Technology
is nottransferredTechnology
istransferred

Monopoly

Duopoly0
Figure 1: Area of technology transfer

18

0.73
0.61
0.44

0

0.09 = ̂Figure 2: The level of total pollution when (e1 , e2 ) = (1, 5)

19

∗
∗∗

1⁄
̅

̂
0

̂

̅
̃

Figure 3: The optimal emissions tax

20

max
,

max
̅

̂

̃

̅

Figure 4: The maximized welfare levels

21