3.2 The Decisions of Firms to Merge and Engage in R&D Activity
3.2.2 Only Public Firm Engages in R&D Activity
Under this regime, we denote the equilibrium results of non-merger case by the superscript BN and label the outcomes of merger scheme by the superscript BM .
Case I：Public and private firms do not merge
By using the same methodology (i.e. backward induction) to find the SPNE, public firm respectively maximizes (3.2) while private firm maximizes (3.1) to determine the optimal outputs as well as R&D investment in Stage 2 and Stage 1, we have the following Cournot-Nash equilibriums:
1 1
0 b
Zone I
*
s>sw
s
Figure 3.4 Illustration of lemma 3.3
s* w
s* *p Zone II
sw*> >s s**p
Zone III s<sp**
0.367
S=0.6397
S=0.2606
36
We commonly get that public firm has higher output than private one no matter the parameter values b is. However, only public firm engages in cost-reducing R&D to reduce its production cost enlarges the output gap between two firms14. Also note that public firm has higher profit than private one if the degree of product differentiation is small enough.
(i.e. b>0.1255)
Case II：Public and private firm merge and set up a multi-product firm
Both choose the outputs and R&D investment by maximizing (3.3) in Stage 2 and Stage 1 respectively. We solve the equilibrium results by using backward induction method.
We get：
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We then analyze whether public firm and private firm have incentive to merge together and set up a multi-produc firm which is owned by two sectors, using the same methodology from section 3.2.1 and dividing into two parts of view of public’s as well as private’s to analyze and obtain the Figure 3.5.
On the view of public firm, if the percentage of share is larger than the threshold value, swB , public firm would like to merge with private one, otherwise, if the shares of
public firm is lower than swB , he has no incentive to merge. On the other view of private firm, private firm prefer to merge with public firm provided that private firm has high enough percentage of shares of multi-product firm (s < s Bp ). Comparing Figure 3.5 with
1
Figure 3.5 Illustration of Lemma 3.4
B
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Figure 3.4, where both firms engage in cost-reducing R&D. Only public firm engages in process innovation activity lower its incentive to the merger, at the meantime, private firm has a higher incentive to merge with the public one. In Zone I, public firm wants to merge while the private firm does not want to merge. Neither public firm nor private firm wants to merge in Zone II. However, in Zone III, only private firm wants to merge. We have the lemma 3.4.
Lemma 3.4: Under the condition that only public firm engages in R&D activity but private firm doesn’t, firms still would not merge and set up a multi-product firm.
3.2.3 Only private firms engage in R&D activity
We also use the same method (backward induction) to get the Cournot-Nash equilibrium outcomes under the following two cases of the decision of firm to merge and use the superscript PN and PM to denote the non-merge as well as merger regime respectively.
Case I：Public and private firm do not merge
1 4 2
(2 6 78 225 1008 2880 3888 11124)
4( 24 126)
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Public firm has higher output than private one even only private firm engage in cost-reducing R&D. However, the profitability of private one is larger than public firm, since private firm is profit-maximizer, engaging in R&D activity would reduce the production cost makes it have higher profit than public one.
Case II：Public and private firm merge and set up a multi-product firm
0 2 2 2
Once firms decide to merge and set up a multi-product firm, the optimal equilibriums are obtained by maximizing the objective function (3.3) which is jointly owned by two firms. Hence, public and private’s aggregate output, profit as well as social welfare are exchange of the merger case in section 3.2.2, only public firm engages in R&D activity and set up a multi-product firm.
Redo the same steps according to the regime (i) whether the firms want to merge while only private firm carries out R&D activity and obtain Figure 3.6. For the perspective of public firm, if public firm has higher percentage of shares in the multi-product firm (s>swP),
it would want to merge with private one, while s<swP, public firm would rather compete with private one. On the other hand of private firm’s view, private firm would like to merge and build up a multi-product firm provided that it has high enough percentage of share after merging (i.e. s < sPp ).
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Zone I illustrates that only public firm likes to merge, whereas private has no incentive to merge due to the lower percentage of shares of multi-product firm. Neither public firm nor private firm wants to merge in Zone II. However, in Zone III, only private firm wants to merge but public firm does not want to merge. Figure 2.6 tells that unless private firm has large percentage of shares than 78 percent after merger, or it never wants to merge. However, public firm has higher incentive to merge with private one and set up a multi-product firm compared with the Figure 3.4, both firms engage in cost-reducing R&D.
Lemma 3.5: Provided that only private firm engages in R&D to reduce its production cost, it increases the incentive to merge of the public firm but decreases the merger motion of private one. To sum up, both firms still would not merge and set up a multi-product firm.
In the presence of cost-reducing R&D activity with product differentiation, firm engages in R&D lower its production cost makes it lower the incentive to merge no matter the rival engages in R&D activity or not, however, firm doesn’t engage in R&D prefers the
1 1
0
Zone I
P
s>sw
s
Figure 3.6. Illustration of Lemma 3.5
P
s w
Zone II swP > >s sPp
Zone III s<spP
S=0.2298 S=0.5653
S=0.2777
P
sp
b
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merger case no matter the rival engages in R&D activity or not. In a duopoly market, regardless of only one firm (public firm or private firm) engages in R&D or both firms engage in R&D, since engages in R&D activity exist a benefit of cost-saving makes both firms do not like the merger case. We get the following proposition.
Proposition 3.1: Once firm engages in process innovation activity, both firms would not merge and set up a multi-product firm.
3.3 Comparison and discussion of the results
In order to evaluate whether engage in R&D would boost merger incentive or not, we note the benchmark case in the absence of R&D which is obtained from Bárcena-Ruiz and Garzón (2003) and denoted their results of non-merger and merger by the superscript ∗N and M∗ respectively.
Case I：Public and private firm do not merge
*
Case II：Public and private firm merger and set up a multi-product firm
* * *
Comparing three regimes listed above with the benchmark case neither firm engages in
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R&D activity of premerger case, we obtain the following corollary 3.1.
Corollary 3.1： The ranking of the optimal quantities, profitability and social welfare among different regimes under the premerger case are q0BN >q0N >q0*N >q0PN ;
*
1 1 1 1
PN N N BN
q >q >q >q ; π0BN >π0N >π*0N >π0PN ; π1PN >π1N >π1*N >π1BN; WN >WBN>WPN>W*N.
Carrying out R&D activity exists a benefit of cost-saving that lower its production cost, even if engages in R&D activity is costly, the cost it saves is larger than R&D spending.
Therefore, firm engages in R&D would increase its own output and then profit. That is, regardless of on the view of public firm or private firm, only itself engages in R&D activity but rival firm does not engage in it has the highest output as well as profit among these four regimes, the second ranking is both firms engage in R&D and neither firm engages in R&D is in third order, however, the situation itself does not engage in R&D, while rival firm engages in it results in the lowest quantity and profitability of itself. It is not doubtful to get these results. In addition, for the view of social welfare, that is the sum of consumer surplus and producer surplus, both firms engage in R&D activity has the highest total outputs and profits which make the social welfare the highest, the middle two ranking are only one firm engage in R&D and only public firm engages in R&D has higher social welfare than the condition only private one engages in it. Furthermore, the lowest is both firms do not carry out R&D.
Once one of firm engages in cost-reducing R&D, firm would not merge and set up a multi-product firm. Hence, we examine the effect of product differentiation (b ), differentiating all equilibrium results with respect to the parameter b , we get the comparative static results of non-merger regime, increasing the degree of product differentiation, firms’ output, profitability as well as social welfare are all increasing. We get that product differentiation setting not only beneficial to the overall social welfare but also
43
This paper postulated that there are one public firm and one private firm in the industry with producing heterogeneous products. We examined three regimes, both firms engage in R&D, only one part of firm (public firm or private firm) engages in R&D activity, and decided whether a private firm and a public firm would like to merge and set up a multi-product firm which is owned by two parts of firms. Although engages in R&D activity is costly, whereas it could reduce the production cost more than the cost it generate.
Hence, in the presence of cost-reducing R&D activity with product differentiation, firm engages in R&D would have lower production cost prefers non-merger case no matter the rival engages in R&D or not, however, firm does not engage in R&D prefers merger case no matter the rival engages in R&D activity or not. To sum up, once one part of firm engages in process innovation activity would increases its own outputs as well as profits, moreover,
Table 3.1 Comparative Static Results of Non-merger Scheme
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this act also boost the social welfare makes public firm and private firm have no incentive to merge and set up a multi-product firm.
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CHAPTER FOUR: COST-REDUCING R&D INVESTMENT AND SUBSIDIZATION IN A DIFFERENTIATED MIXED DUOPOLY
In recent years, many firms adopt R&D activities to promote their competitiveness.
However, a proactively technological innovation takes a tremendous of time to develop;
moreover, it is also accompanied with high risk and uncertainty. Thus, fewer firms are willing to take such a risky task unless they were be supported or granted patents by the government. Thus, government plays an important role to encourage firms to do R&D to boost the overall social welfare. We divide the government subsidization into four regimes, which are unsubsidized policy, R&D subsidized policy, Output subsidized policy and mix-subsidized policy composed of R&D and output subsidization. We aim to examine whether the government subsidization is essential to the firms. If the subsidized policy is needed, which types of subsidization is beneficial to the firms and overall social welfare.
The rest of the paper is organized as follows. In Section 4.1 is the basic model setting and then derive four sub-sections of government policies. In Section 4.2, we explore the effects of product differentiation among distinct schemes and compare the equilibrium solutions in Section 4.3. Finally, concluding remarks is presented in Section 4.4.
4.1 The Basic Model
An industry is composed of one welfare maximizing public firm and one profit maximizing private firm (denoted by 0 and 1 respectively) which producing differentiated products. The market inverse demand function is given by：Pi = − −a qi bqj,i j, =0,1. i≠ , j where b is the degree of product differentiation. If b= products are homogenous 1,
46
products and they are completely substitute, whereas b= product is independent goods. 0, In order to avoid the extreme cases, we assume that 0< <b 1, that we only focus on the substitute case and postulate that both firms engage in cost-reducing R&D activity but in the absence of spillover effect. Following the production cost is specified as
( ) 2, 0,1
i i i i
C = −c x q +q i= which is followed by Gil-Moltó et al. (2006)15 which exists a diminishing return to scale by introducing a quadratic term in the production cost in order to avoid the case of natural monopolies. However, engaging in cost-reducing R&D activity is costly and R&D cost function is defined as D x( )i =xi2, 0,1.i= We consider four subsidization schemes: (i) Unsubsidized policy (we denote the case by the superscript NS), (ii) R&D subsidized regime and express this case by the superscript RS, (iii) Output subsidized scheme that we represent the case by the superscript QS and (iv) mix-subsidized policy which are compose of R&D as well as output subsidies and denote the case by the superscript MS.
Pq C x without government subsidize
Pq C x s x with R D subsidy where sx and sq denote the per-unit subsidy to R&D level and output respectively, and the social welfare function is given by
0 1
W C S without government subsidize W C S subsidy with government subsidize
π π
15 Gil-Moltó et al. (2006) investigated the use of subsidies to R&D in a mixed and a private oligopoly markets.
47
The consumer surplus for differentiated good is given by 1 02 12 0 1
( )
CS= 2 q +q +bq q . In the unsubsidized policy (Section 4.1.1) and subsidized policies (Section 4.1.2, 4.1.3 and 4.1.4), private firm is a profit-maximizer who chooses the output and R&D investment by maximizing (4.1.1), whereas the public firm is the welfare-maximizer who chooses the output as well as R&D by maximizing the welfare function (4.2.1 or 4.2.2). All the results are presented in each sub-section below.
There is a three-stage game in subsidize policy: In Stage 1 the government chooses the modes of subsidy and determines the optimal subsidized volume by maximizing the welfare function; in Stage 2, each firm chooses its own level of R&D by maximizing its objective function and then simultaneously determine the output in Stage 3. If the market is absence of government intervention, it becomes a two-stage game: Each firm decide the level of R&D and output by maximizing its objective function in Stage 1 and Stage 2 respectively.
We get the subgame-perfect Nash equilibrium (SPNE) by using backward induction method.
The timing structure of the game is illustrated in Figure 4.1.
Stage 1 Stage 2 Stage 3
Time
Each firm decides the level of R&D,x and 0
x , respectively. 1
Each firm makes the output decision, q 0 and q , respectively. 1
Figure 4.1 Game Structure of Government Subsidization and R&D Activity The government chooses
the subsidization policy and decides the optimal subsidy s∗.
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4.1.1 Unsubsidized regime
As a benchmark case and obtain SPNE by backward induction. In output stage, public firm and private firm decides the output level by maximizing (4.2.1) and (4.1.1) respectively.
We obtain 0 ( 4)( 2 ) 4 0 1
own R&D investment increase its output but decrease rival’s quantity16. At the proceeding R&D stage, firms maximizes (4.2.1) and (4.1.1) respectively and get the optimal R&D,
3 2
Substituting subgame outcomes into each objective function, we have the Cournot-Nash equilibriums as follows.
3 2
(8 22 300 789 3660 9543 12960 33840)( ) 2(2 45 210) .
NS b b b b b b b a c
W b b
− − + + − − + −
= − +
Under the unsubsidized policy with R&D activity, public firm’s R&D investment as well as output are larger than private one17 gives the same result with the unsubsidized mixed oligopoly without R&D activity. (e.g. De Fraja and Delbono, 1989) Since public firm is welfare maximizing firm makes it have higher R&D than private one, furthermore, carrying out more R&D have more benefit of cost-reducing which result in higher output.
49
4.1.2 R&D subsidized regime
We then consider the scenario in which government sets a R&D subsidy depends on the amounts of firms engage in R&D. We obtain subgame perfect Nash equilibriums by backward induction. In the last production stage, we maximize (4.1.2) and (4.2.2) to find the output level:
In R&D stage, we simultaneously find the R&D investment by maximizing (4.1.2) and (4.2.2), we obtain:
If the government subsidizes more R&D subsidy to stir firms engage in more R&D activity. It will increase private firm’s incentives to do more R&D activity, but lower public firm’s incentives to do so18.
At proceeding subsidy stage, we derive the optimal R&D subsidization by maximizing the welfare function. The subsidy is included in the welfare expression as both a component of profits and the government expenditure; even there is no direct effect of the subsidy on the welfare, subsidization plays an indirect effect on welfare through both firms’ R&D levels and outputs. We get the optimal subsidy,
2
Lemma 4.1: In the presence of cost-reducing R&D activity with R&D subsidization, the optimal R&D subsidy is always positive.
50
This Lemma means that the government will always grant a positive R&D subsidy and the more degree of product differentiation, the government will grant more to encourage firms to engage in R&D activity. We summarize the above subgame results and have the following Cournot-Nash equilibriums:
(20 132 203 2816 2726 13212 22113)( ) (4 85 405)
see that the level of R&D and quantity of the public firm exceed the private’s no matter what the parameter is.
4.1.3 Output subsidized regime
We follow the same methodology (backward induction method) of section 4.1.2 and obtain the SPNE as follows:
9 8 7 6 5 4 3 2
0
(2 6 93 267 1578 4284 11952 30780 34560 86400)( ) A ;
51
9 8 7 6 5 4 3 2
0
2(2 6 93 255 1608 4164 12252 30780 34560 86400)( ) A ;
(8 22 372 975 6432 163200 49248 123120 138240 345600)( ) A .
Where 4 216 4491 45324 223020 432000 ; (28 160 2472 14880 91875 600648 1832091 13748640 20136744 195103152 97178976 1756270080 272301552
+
9782562240 6118539120 30780518400 30114892800 41803776000 52254720000)
− b+ b + b − b − b+
If the government subsidizes output to encourage firms engage in cost-reducing R&D activity, the optimal output subsidy is as follows and we have Lemma 4.2.
2 6 4 2
Lemma 4.2: In the presence of cost-reducing R&D activity with output subsidization, the government will always grant a positive subsidy.
We compare the R&D investments and output levels between two firms:
9 8 7 6 5 4 3 2
does more R&D activity than public one regardless of the parameter b is and make former firm exists a higher cost-reducing benefit to produce. Besides that, private firm’s output is outweighs the public’s subject to the degree of product differentiation is large enough (i e b. . 0.437< ). This result reverse the output ranking that obtained in the model with homogeneous goods, firms do not engage in R&D with output subsidy, firms engage in
52
R&D with R&D subsidy or the pure mixed duopoly without the government intervention.
(e.g. DeFraja and Delbono (1989); White (1996) and Gil-Moltó et. al. (2006)) The intuition behind this phenomenon is that if the product differentiation between firms is sufficiently large, it makes the product is more competitive in the market. Moreover, the government grants on output stimulates private firm to do more process innovation activity to reduce its production cost and then increase its output outweighs the public’s.
4.1.4 Both R&D and output subsidized regime
We also follow the same methodology of section 4.1.2 and have the subgame solution in each stage. In last stage, each firm simultaneously chooses the output for a given values of R&D investment and subsidies, we get:
0 1
In the proceeding R&D stage, both firms decide the R&D investments for a given subsidies:
Substituting above subgame equilibriums into welfare function and determine the optimal R&D subsidization as well as output subsidy by maximizing (4.2.2), we get optimal subsidies and lemma 4.3.
2
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Lemma 4.3: In the heterogeneous market with cost-reducing R&D activity, provided that the government simultaneously subsidizes on R&D and output, it will always grant a positive output subsidy but a negative R&D subsidy.
Note that a negative R&D subsidy means the government will assess the R&D tax on firms instead of R&D subsidy. Government urges firms engage more R&D activity, but he assesses R&D tax on firms may be a surprised result. Recalling the finding of Leahy and Neary (1997) who showed that R&D should be taxed, when there are no spillovers or even if spillovers are low enough.
We obtain the subgame perfect Nash equilibriums (SPNE):
0 1
Under the mix-subsidized policies, the government will subsidize on firms’ output but
Under the mix-subsidized policies, the government will subsidize on firms’ output but