Technology Licensing in a Stackelberg Duopoly

3.1 Technology Licensing

In part 3.1 and 3.2, we will consider the technology transfer model under Stackelberg market structure without and with managerial delegation.

In part 3.1, we will introduce the basic technology transfer model under Stackelberg duopolistic competition of Kabiraj (2005), which taking a two-stage game model to discuss the technology transfer actions between the two competition firms, and then we will try to combine the managerial delegation system designation with this model to see the results would change or not in part 3.2.

Kabiraj (2005) considered a basic technology transfer model under Stackelberg duopolistic competition, taking a two-stage game model to discuss the technology transfer actions between the two competition firms.

In the first stage, the patent-holding firm sets up the license contract of license fee, and the other firm’s manager decides to accept the contract or not. In the second stage, the two firms compete in quantities under the Stackelberg competition market.

Figure 3 Game Stages of Technology Transfer

Time

Stage I Stage II

Patent-holding firm sets up the license

contract of license fee. Firm’s manager chooses the quantity under Stackelberg competition.

Suppose firm 1 is the leading firm, and firm 2 is the following firm, and consider a homogeneous duopoly Stackelberg model, each firm’s profit function isπi =Ri −C

( )

qi , i =1, 2. Revenue function is

R

_i

= pq

_i, and the inverse demand function is

p = a − ( q

1

+ q

2

)

, cost function isCi

( )

qi =ciqi. Solving the first order condition to derive firm 2’s reaction function.

) 2(

) 1

( _{1 1 2}

2

q a q c

q = − −

(3-1)

Because firm 1 is the leading firm, so it can substitute firm 2’s reaction function (3-1) into its profit function, then under the first order condition, we can have the Stackelberg equilibrium quantities are

) 2

2( 1

2 1

*

1

a c c

q = − +

and ( 2 3 )

4 1

2 1

*

2

a c c

q = + −

(3-2)

and the equilibrium profits are

2 2 1

*

1 ( 2 )

8

1

a − c + c

=

π

and ^*₂ ( 2 ₁ 3 ₂)²

16

1

a + c − c

=

π

(3-3)

Kabiraj (2005) considered two cases as follows,

3.1.1 Leading firm is the patent-holder

3.1.1.1 Fixed fee

Assume firm 1’s marginal costc₁₌c_−ε, firm 2’s marginal costc2 =c, 0<ε<(a−c), which implies that firm 1 owns the producing cost advantage from a non-drastic innovation.

equilibrium quantities and profits (the superscript NL denotes ‘no licensing’), we can have

The maximum license fee firm 1 can charge firm 2 is what will make firm 2 indifferent between licensing and not licensing the new technology. In the case that licensing occurs,

So the equilibrium quantities and profits of each firm under fixed-fee licensing are

)

3.1.1.2 Royalty fee

Following the basic settings and results in the above, if firm 1 license its technology to firm 2 at a fixed royalty rater, and the amount of royalty firm 2 pays will depend on the quantity firm 2 will produce by using the technology patent. In this case, firm 1’s unit production cost is

c − ε

, firm 2’s unit production cost is

c − ε + r

if it licenses from firm 1 and c if it does not license. Note that the maximum royalty rate firm 1 can charge obviously cannot exceed

ε (i.e.,

0

≤ r ≤ ε

).

In the market competition stage, firm 1 and firm 2 will choose

q and

₁

q to maximize

₂

2 1

+ rq

π

and

π . And in the licensing stage, firm 1 will choose r to maximize

₂

π

1

+ rq

2. If the innovation is non-drastic, the optimal royalty rate

r = ε

, and the equilibrium quantities

2 2

1

ε +

=a−c

q^R and

4 2

2

ε

−

=a−c

q^R (3-9) and their equilibrium profits

4 ) 2 (

8 ) 2

( ²

1

ε π R = a−c+ ε + r a−c−

and

16 ) 2

( ²

2

π R = a−c− ε

(3-10)

3.1.1.3 Comparison: Fixed fee versus royalty fee

Evaluate the superiority of a fixed fee licensing versus a royalty licensing in the Stackelberg model, if the leading firm is the patent-holder,

1 0

1^F

− π

^R

<

π

(3-11)

Hence, the patent-holding firm would prefer royalty fee to fixed fee when it chooses the best patent licensing policy.

3.1.2 Following firm is the patent-holder

3.1.2.1 Fixed fee

Assume firm 1’s marginal cost

c

₁

= c

, firm 2’s marginal cost

c

₂

= c − ε

, 0<ε<(a−c), which implies that firm 2 owns the producing cost advantage from a non-drastic innovation.

Substituting

c

₁

= c

and

c

₂

= c − ε

into Eqs. (3-2) and (3-3) to give the firms’ Stackelberg equilibrium quantities and profits (the superscript NL denotes ‘no licensing’), we can have

2 licensing occurs, both firms will produce at constant unit costc_−ε. So the maximum license fee firm 2 can charge firm 1 is

So the equilibrium quantities and profits of each firm under fixed-fee licensing are

)

2

1 ( )

8

1 ε

π^F = a−c− and

2 ) ) (

2 16(

1 2

2

ε ε

π a c

c

F = a− − + − (3-16)

Combining (3-33) and (3-36) can obtain that π₁^F ≥π₁^NL if and only if 0

< ε ≤

(

a − c

).

3.1.2.2 Royalty fee

Following the basic settings and results in the above, if firm 2 license its technology to firm 1 at a fixed royalty rater, and the amount of royalty firm 1 pays will depend on the quantity firm 1 will produce by using the technology patent. In this case, firm 2’s unit production cost is

c − ε

, firm 1’s unit production cost is

c − ε + r

if it licenses from firm 2 and c if it does not license. Note that the maximum royalty rate firm 2 can charge obviously cannot exceed

ε (i.e.,

0

≤ r ≤ ε

).

In the market competition stage, firm 1 and firm 2 will choose

q and

₁

q to maximize

₂ π1 andπ₂ +rq₁. And in the licensing stage, firm 2 will choose r to maximizeπ₂ +rq₁.

If the innovation is non-drastic, the optimal royalty rate

r = ε

, and the equilibrium quantities

1 2

ε

−

=a−c

q^R and

4 3

2

ε +

=a−c

q^R (3-17)

and their equilibrium profits

8 )

( ²

1

π R = a−c−ε

and

2 ) (

16 ) 3

( ²

2

ε π R = a−c+ ε +r a−c−

(3-18)

3.1.2.3 Comparison: Fixed fee versus royalty fee

Evaluate the superiority of a fixed fee licensing versus a royalty licensing in the Stackelberg model, if the following firm is the patent-holder,

2 0

2^F −π^R <

π (3-19)

Hence, whether the patent-holding firm is Stackelberg leader or follower, it would prefer royalty fee to fixed fee when it chooses the best patent licensing policy.

3.2 Technology Licensing with Delegation

In this part, we will extend the basic Stackelberg technology transfer model of Kabiraj (2005) which we introduce in part 3.1, taking a three-stage game model to discuss the technology transfer actions between the two competing firms with managerial delegation designation.

In the first stage, the patent-holding firm chooses the licensing fee to sell to the patent-buyer. In the second stage, each firm’s owner chooses the incentive parameter to encourage their managers to compete harder in the market. In the third stage, the two firms compete in quantities under the Stackelberg competition market.

We follow the settings of managerial delegation of Sklivas (1987) and Fershtman and Judd (1987). To attain the profits and market power, owner will measure his manager’s performance according to a function

g , which combines firm’s profits (

_i

π ) and revenues

_i (

R ). The higher is

_i

g , the higher is manager i ’s bonus or the lower is the likelihood that he

_i will be fired, so

g could represent manager i ’s incentives. For simplicity, we make

_i

g as

_i a linear combination of profits and revenues, and the marginal costs are constant.

Stage I Stage III

Figure 4 Delegating Game Stages of Technology Transfer.

Time Stage II

Firm’s owner chooses the

incentive parameter. Firm’s manager chooses the quantity in the Stackelberg market Patent-holder chooses

the license fee to sell,

Max g_i=

λ

_i

π

'

( q

1,

q

2

) ( +

1

− λ

_i

) ( R

_i

q

1,

q

2

) ( )

i i

( )

i

i q q C q

R −λ

= 1, 2 , i =1, 2

And we also consider two cases: Leading firm is the patent-holder or following firm is the patent-holder.

3.2.1 Leading firm is the patent holder

3.2.1.1 Fixed fee

Using backward-induction to solve this licensing game. In the third stage, the two firms’

managers compete in market according to the performance indicator

g .Suppose firm 1 is the

_i leading firm, and firm 2 is the follower, and consider a homogeneous duopoly Stackelberg model. Solving the first order condition under profit-maximization condition, we can derive firm 2’s reaction function.

) 2(

) 1

( _{1 1 2 2}

2

q a q c

q = − − λ

(3-20)

Because firm 1 is the leading firm, so it can substitute firm 2’s reaction function (3-20) into its profit function, then under the first order condition, we can have the Stackelberg equilibrium quantities are

) 2

2( ) 1 ,

( _{1 2 1 1 2 2}

1

a c c

q λ λ = − λ + λ

and ( 2 3 )

4 ) 1 ,

( _{1 2 1 1 2 2}

2

a c c

q λ λ = + λ − λ

(3-21)

And substituting Eq. (3-21) to the profit function of delegation stage and taking the first order condition, we can obtain each firm’s optimal incentive parameter is

1

=

1

the equilibrium quantities are

)

and the equilibrium profits are

2

which implies that firm 1 owns the producing cost advantage from a non-drastic innovation.

Substituting c_{1 =}c_−εand c2 =cinto Eqs. (3-23) to give the firms’ Stackelberg equilibrium quantities and profits in the delegation stage (the superscript NL denotes ‘no licensing’), we can have

The technology transfer may occur when

π

₁^NL

≥ π

₂^NL, so we can have the condition that )

2 ( ) 13 34(

5

a − c ≤ ε ≤ a − c

. If the technology transfer occurs, the maximum license fee firm 1

can charge firm 2 is what will make firm 2 indifferent between licensing and not licensing the new technology. In the case that licensing occurs, both firms will produce at constant unit cost c−ε, and we can derive the maximum fixed license fee firm 1 can charge firm 2 in the licensing stage,

ε ε ε

π ε ε

π

[2( ) ]

4 ) 1 , ( )

,

( ₂

2

− − − − = − −

= c c c c a c

F

^{L NL} (3-27)

So the equilibrium quantities and profits of each firm under fixed-fee licensing are

) 3(

1

1 = a−c+ε

q ^F and ( )

2 1

2 = a−c+ε

q ^F (3-28)

ε ε ε

π [2( ) ]

4 ) 1 18(

1 2

1^F = a−c− + a−c − and ₂ ( 2 )²

12

1 ε

π^F = a−c− (3-29)

Combining (3-26) and (3-29) can obtain that π₁^F ≥π₁^NL if and only if 0

< ε ≤

(

a − c

).

3.2.1.2 Royalty fee

We use backward-induction to solve this licensing game. Suppose firm 2 has bought the patent from firm 1, so their marginal costs is the same, c₁ =c₂ =c−ε , and the royalty licensing fee firm 1 charge firm 2 is rper unit.

In the third stage, the two firms’ managers compete in market according to the performance indicator

g .Suppose firm 1 is the leading firm, and firm 2 is the follower, and

_i

consider a homogeneous duopoly Stackelberg model. The objective function for each firms’

Solving the first order condition under profit-maximization condition, we can derive firm 2’s reaction function.

)]

Because firm 1 is the leading firm, so it can substitute firm 2’s reaction function (3-32) into its profit function, then under the first order condition, we can have the Stackelberg equilibrium quantities in the market competition stage are

]

Substituting Eq. (3-23) and (3-34) to the profit function of delegation stage and taking the first order condition, we can obtain each firm’s optimal incentive parameter is

1

=

1 the equilibrium quantities are

) 3(

) 1

1(

r = a − c + r + ε

q

and ( 2 )

2 ) 1

2(

r = a − c − r + ε

q

(3-36)

Substituting Eqs. (3-35) and (3-36) into firm 1’s profit function to solve the optimal royalty rate in the licensing stage, and we can derive the optimal royalty rate r=ε. Then we can have the equilibrium quantities and profits of each firm,

) 2 3(

1

1 = a−c+ ε

q ^R and ( )

2 1

2 = a−c−ε

q ^R (3-37)

18

) ( 12 )

( ²

1

ε

πR = a−c−ε + a−c and ₂ ( )² 12

1 ε

π^R = a−c− (3-38)

3.2.1.3 Comparison: Fixed fee versus royalty fee

Evaluate the superiority of a fixed fee licensing versus a royalty licensing,

1 0

1^F

− π

^R

<

π

(3-39)

Hence, if the leading firm owns the patent and takes the designation of managerial delegation system, it would still prefer royalty fee to fixed fee when it chooses the best patent licensing policy.

3.2.2 Following firm is the patent-holder

Considering a homogeneous duopoly Stackelberg model, supposing firm 1 is the leading firm, and firm 2 is the follower, and firm 2 owns the patent.

3.2.2.1 Fixed fee

The same with the cases of 3.2.1(Leading firm is the patent holder), we use the backward-induction to solve this licensing game. In the third stage, the two firms’ managers compete in market according to the performance indicator

g . Solving the first order

_i condition under profit-maximization condition, we can derive firm 2’s reaction function.

) 2(

) 1

( _{1 1 2 2}

2

q a q c

q = − − λ

(3-40)

Because firm 1 is the leading firm, so it can substitute firm 2’s reaction function (3-40) into its profit function, then under the first order condition, we can have the Stackelberg equilibrium quantities are

) 2

2( ) 1 ,

( _{1 2 1 1 2 2}

1

a c c

q λ λ = − λ + λ

and ( 2 3 )

4 ) 1 ,

( _{1 2 1 1 2 2}

2

a c c

q λ λ = + λ − λ

(3-41)

And substituting Eq. (3-41) to the profit function of delegation stage and taking the first order condition, we can obtain each firm’s optimal incentive parameter is

1

=

1

λ

and

2 1

2 3

2 2 c

c a+

−

=

λ (3-42)

the equilibrium quantities are

and the equilibrium profits are

2

which implies that firm 2 owns the producing cost advantage from a non-drastic innovation.

Substituting

c

₁

= c

and

c

₂

= c − ε

into Eqs. (3-43) and (3-44) to give the firms’ Stackelberg equilibrium profits (the superscript NL denotes ‘no licensing’), we can have

) indifferent between licensing and not licensing the new technology. In the case that licensing occurs, both firms will produce at constant unit cost c−ε. And we can derive the maximum license fee firm 2 can charge firm 1 in the licensing stage,

ε

So the equilibrium quantities and profits of each firm under fixed-fee licensing are

3.2.2.2 Royalty fee

In the royalty fee licensing case, we also use the backward-induction to solve this licensing game. Suppose firm 1 has bought the patent from firm 2, so their marginal costs is the same, c₁ =c₂ =c−ε , and the royalty licensing fee firm 2 charge firm 1 is rper unit.

In the third stage, the two firms’ managers compete in market according to the performance indicator

g .Suppose firm 1 is the leading firm, and firm 2 is the follower, and

_i consider a homogeneous duopoly Stackelberg model. The objective function for each firms’

manager is

Solving the first order condition under profit-maximization condition, we can derive firm 1’s reaction function.

)]

Because firm 1 is the leading firm, so it can substitute firm 2’s reaction function (3-52) into its profit function, then under the first order condition, we can have the Stackelberg equilibrium quantities in the market competition stage are

]

Substituting Eq. (3-53) and (3-54) to the profit function of delegation stage and taking the first order condition, we can obtain each firm’s optimal incentive parameter is

1

=

1

the equilibrium quantities are

) royalty rate in the licensing stage, and we can derive the optimal royalty rate r_=ε. Then we can have the equilibrium quantities and profits of each firm,

)

18 )

( ²

1

π

_R

= a − c − ε

and

12

) ( 8 )

( ²

2

ε

π

_R

= a − c + ε + a − c

(3-58)

Combining (3-44) and (3-54) can obtain that

π

₂^R

≥ π

₂^NL if and only if

3 ) 0

< ε ≤

(

a − c

.

3.2.2.3 Comparison: fixed fee versus royalty fee

Evaluate the superiority of a fixed fee licensing versus a royalty licensing,

2 0

2^F

− π

^R

<

π

(3-59)

Hence, if the following firm owns the patent and takes the designation of managerial delegation system, it would still prefer royalty fee to fixed fee when it chooses the best patent licensing policy.

Integrate the results of 3.2.1 and 3.2.2, we have proposition 1

Proposition 1. Under Cournot Stackelberg market competition, either the leading firm owns the patent or not, it won’t take the managerial delegation decision anymore.

Because the managerial delegation designation is a system that help firm to win more quantities in market, but for a Stackelberg leading firm, taking the quantity-leader actions and make the emphasis of market share is originally included in leading firm’s market competition decisions, so if the quantity-leader actions can help it win get more quantities in the market, then it may not prefer take the managerial delegation decisions.

Comparing the results of 3.1 and 3.2, we can have the follow propositions,

Proposition 2. When inside technology transfer occurs in Stackelberg market, then either the competing firms take the managerial delegation designation or not, or the patent-holding firm is the leading firm or following firm in the markets, they would prefer royalty fee licensing to fixed fee licensing.

Proposition 3. In Stackelberg market, either the patent-holding firm is the leading firm or is the following firm, technology transfer will be less likely to occur under strategic delegation than under no delegation.

This proposition is the same like the case in Cournot market that under strategic delegation, firms (managers) will behave more aggressively than under standard quantity competition, so it will reduce the incentive for the patent-holding firm to license its innovation to the other firm.

Besides, comparing the results of Stackelberg and Cournot model, we can have the proposition.

Proposition 4. Either the patent-holder is the leading firm or the following firm, and either

the patent-holder takes the managerial delegation designation or not, the licensing fee

(either fixed or royalty fee) that it can gain from transferring the technology to the

patent-buyer under Stackelberg competition will be less than it can get under Cournot

competition.

在文檔中 雙占競爭下管理授權與技術移轉之研究 (頁 22-41)