Chapter 3 The Basic Model
3.3 Comparative Statics Analysis
1 )(
2 )(
1 ( 2
3
) 1 )(
2 )(
1 ( 2
3
*
*
ρ ρ β
γ
ρ ρ β
γ
−
−
= −
−
−
= −
j j
i i
k k
(23)
) 1 )(
2 ( 2 1 3
) 1 )(
2 ( 2 1 3
*
*
ρ ρ β γ
ρ ρ β γ
−
− −
=
−
− −
=
j j
i i
k
k (24)
3.3 Comparative Statics Analysis
First, we discuss the influence of spillover effect (ρ ) on quantity of production and marketing, market price, R&D outcome, sales profits, and total profits. To avoided the complicated calculation, according to Eqs. (23) and (24), the two firms have symmetric context of incentive contract, let k=ki =kj and β =βi =βj. The results of comparative statics analysis are as follows:12
12 Notice that while k=ki=kj and β =βi =βj, Q=6γ(a− f)/9γ−2k(1−β)(2−ρ)(1+ρ)>0
2
From Eq. (25), we find that the degree of spillover effect is the key factor while determining its impact on sales, market price, R&D outcome, sales profits and total profits. The results are compiled in Proposition 1.
Proposition 1: If the HQs of the two firms set up incentive contract respectively, when spillover effect is significant (
2
>1
ρ ), the greater the spillover effect, the smaller the
firms’ sales, R&D outcome, sales profits, and total profits will be, but market price will be higher; when spillover effect is of little significant (ρ <
2
1), the greater the
spillover effect is, the higher the firms’ sales, profits and total profits will be greater, but R&D outcome and market price will be lower. When spillover effect is equal to
2 1,
the firms’ sales, market price, R&D outcome, profits and total profits will be constant.
Proposition 1 reveals that under the situation when spillover effect is significant (ρ >
2
1 ), because the two firms know that competitors can reduce their own
production cost through technology spillover, the firms are less willingness to engage in R&D. Hence, R&D outcome decrease and the firms’ production cost will increase,
resulting in fewer market sales, increased market price, fewer sales profits, and then fewer total profits. On the contrary, when the spillover effect is not significant (ρ <
2
1), although the two firms are less inclined to share their R&D outcome with
competitors, they are also less likely to learn competitors’ R&D outcome through technology spillover, which also increases their own R&D cost. Thus, the two firms are also less willingness to engage in R&D activities, and R&D outcome will decrease consequently. Then, all they can do is to increase their output in the competition market, resulting in lower market price as well as higher sales profits and total profits.
Regarding the influence of the parameter of disutility effect of effort (γ ) on sales, market price, R&D outcome, sales profits and total profits, the results of the comparative statics analysis are as follows:
)] 0
Eq. (26) shows that the influence of the parameter of disutility effect for R&D effort on the two firms’ sales, market price, sales profits, R&D outcome, and total profits. The results are compiled in Proposition 2.
Proposition 2: When the head quarters of the two firms set up incentive contract
respectively, the higher the disutility effect for R&D effort is, the lower the two firms’
sales, R&D outcome, profits, and total profits will be, but market price will be higher.
Proposition 2 shows that when the HQs of the two firms provide incentive contract, if the disutility effect for R&D effort is higher, R&D personnel will have to pay higher cost to engage in R&D activities. Hence, R&D outcome will decrease and the firms’ cost will increase, resulting in higher market price, fewer sales in the market, and then decreased sales profits and total profits.
Next, we will go to the discussion of whether the spillover effect should be taken into account when the HQ sets up the context of incentive contract. Aimed at whether spillover effect will affect a vendor’s ratio of R&D budget and ratio of incentive contract. The results of the comparative statics analysis are as follows:
)] 0 1 )(
2 )[(
1 ( 2
) 2 3 ( 3
2
*
− >
−
−
= −
∂
∂
ρ ρ β
ρ γ
ρi i
k (27)
)] 0 1 )(
2 [(
2
) 2 3 ( 3
2
*
− <
−
− −
∂ =
∂
ρ ρ
ρ γ
ρ β
i i
k (28)
According to Eqs. (27) and (28), we conclude as Proposition 3.
Proposition 3: When the a head quarter sets up the context of incentive contract, the
greater the spillover effect is, the higher the share of R&D expenditure paid by head quarter (i.e., β ) and the percentage of selling profit (i.e., k) will be.
Proposition 3 indicates that when the HQ sets up the context of incentive
contract for R&D department, the degree of spillover effect should be taken into account. If the spillover effect increases, when the R&D department engages in R&D activities, it will allow competitor to obtain part of the R&D outcome and reduce production cost due to spillover effect. In order to gain more cost advantages, the firm has to engage in more R&D efforts. Therefore, the firm must provide R&D personnel with many more R&D rewards and R&D budgets.
Finally, we discuss the influence of disutility of R&D effort (γ ) when the HQ sets up the context of incentive contract. The results of the comparative statics analysis are as follows:
) 0 1 )(
2 )(
1 ( 2
* 3
− >
−
= −
∂
∂
ρ ρ β
γi i
k (29)
) 0 1 )(
2 ( 2
* 3
− <
− −
∂ =
∂
ρ ρ γ
β
i i
k (30)
According to Eqs. (29) and (30), the results are compiled in Proposition 4.
Proposition 4: When a head quarter sets up the context of incentive contract for R&D
personnel, if the disutility of R&D effort increases, the firm has to increase the share of R&D expenditure paid by head quarter or the percentage of selling profit in the incentive contract.
Proposition 4 indicates that if the disutility of R&D personnel for every unit of R&D outcome increases, the incentive rewards or R&D budgets have to increase as well. The reason is when the HQ sets up the context of incentive contract, if the R&D personnel must pay more efforts to reach one unit of R&D outcome that means the
disutility of R&D personnel is higher. Therefore, the firm has to provide many more incentive rewards or R&D budgets, so as to compensate R&D personnel for the disutility caused by R&D efforts.