Cooperative R&D with durable goods

This paper analyzes the effect of the durability of the good produced by a duopolistic industry on research and development investment in the presence of spillovers. We show that the critical spillover level from which cooperation in R&D increases the level of investment is higher when firms produce durable goods and sell at least some units of their output than when firms produce non-durable goods. Moreover, with R&D cooperation investment is highest with renting firms and lowest with renting–selling firms. These findings indicate that R&D cooperation is more difficult to justify when firms produce durable goods in the presence of intertemporal inconsistency problems.

1. Agreements between two or more firms which restrict competition are prohibited by Article 101(1) of the Treaty, subject to some limited exceptions.

2. The prohibition enacted by Article 101(1) of the EC Treaty does not apply to any agreement that meets Article 101(3) conditions. The Art. 101(3) exemption is normally available where the agreement or practice in question generates economic benefits in terms of lower prices, better quality products or services, faster innovation, etc. that are passed on to consumers, do not impose non essential restrictions and do not eliminate competition completely.

3. According to Katz (1986), “spillovers” means research done by one firm which can be used by another even if the latter is not given permission (i.e. does not purchase a license) to use the inventive output.

4. See De Bondt (1996) for an earlier review of the relevant literature.

5. Saggi and Vettas (2000) analyze sales and leasing strategies in a three-period quantity-setting durable good asymmetric duopoly and find that an increase in the unit cost of a given firm implies a higher ratio of leased units to sales.

6. The model can be considered as an extension to an oligopolistic case of the monopolistic case considered by Bulow (1982), adding the decisions of firms on their R&D levels.

7. Amir et al. (2008) introduce the criterion that investing in one research laboratory should be at least as efficient for a firm as investing in several independent laboratories that benefit mutually from spillovers.

8. Given that the second period is also the last, renting out is identical to selling in that period. We assume that the units rented out in the first period are rented out again in the second.

9. We assume throughout the paper that the parameters are such that each optimization problem has interior solutions.

10. Appendix A.1 demonstrates that the level of investment that maximizes social welfare (defined as the sum of the consumer surplus and the profits of the firms), \(x^{**}=\frac{\left( 1+\beta \right) \left( 2a-A\right) -4b\delta }{4b\gamma -\left( 1+\beta \right) ^{2}}\), is higher than the level under all the commercial practices considered here.

11. \(C_{i}^{t}\) represents the total (first period plus second period) production costs of firm i.

12. As \(\frac{\partial \pi _{j}^{s}}{\partial q_{2j}}\left( \frac{\partial q_{2j}}{\partial x_{i}}-\frac{\partial q_{2j}}{\partial q_{1j} }\frac{\partial q_{1j}}{\partial x_{i}}\right) =-bq_{1j}\left( \frac{135\beta -73}{192b}\right) \), it emerges that the intertemporal inconsistency effect 2 is negative if \(\beta >\frac{73}{135}\).

13. Sagasta and Saracho (2008) analyze the effects of durability on the incentives to merge and find that it can be optimal from the viewpoint of social welfare and consumer surplus not to allow firms to sell their output because of the incentives that they have to merge.

14. The level of investment that would be obtained in a two-period repeated Cournot game that arises with non-durable goods when firms do not cooperate in R&D is \(x_{i}^{*nd}=\frac{4\left( a-A\right) \left( 2-\beta \right) -9b\delta }{9b\gamma -4\left( 1+\beta \right) \left( 2-\beta \right) }\) and when firms cooperate in R&D is \({\hat{x}}_{i}^{nd}=\frac{4\left( a-A\right) \left( 1+\beta \right) -9b\delta }{9b\gamma -4\left( 1+\beta \right) ^{2}}\). Hence, as \({\hat{x}} _{i}^{nd}-x_{i}^{*nd}=\frac{36\left( \gamma \left( a-A\right) -\delta \left( 1+\beta \right) \right) \left( 2\beta -1\right) b}{\left( 9b\gamma -4\left( \beta +1\right) ^{2}\right) \left( 9b\gamma -4\left( 2-\beta \right) \left( 1+\beta \right) \right) }\), \({\hat{x}}_{i}^{nd} >x_{i}^{*nd}\) if and only if \(\beta >\frac{1}{2}.\)

15. When firms have identical technology and their marginal costs of production are constant and positive, the quantity rented out in each period by a durable goods firm coincides with the quantity produced by a non-durable good firm that faces, instead, the inverse demand function \(p=\alpha -\theta Q\), with \(\alpha =2a\) and \(\theta =2b\). Nevertheless, in our model production costs depend on the R&D level at firms, so it can be shown that the R&D level and the quantity produced by a non-durable good firm are higher than the quantity rented out in each period by a durable good firm.

16. In the full cooperation case, we find that the level of investment with renting firms and with renting–selling firms is \({\tilde{x}} ^{r}={\tilde{x}}^{r-s}=\frac{\left( \beta +1\right) \left( 2a-A\right) -8b\delta }{8b\gamma -\left( \beta +1\right) ^{2}}\), whereas the level of investment with selling firms is \({\tilde{x}}^{s}=\frac{\left( \beta +1\right) \left( a-A\right) -4b\delta }{4b\gamma -\left( \beta +1\right) ^{2}}\). A comparison of these levels of investment gives \({\tilde{x}}^{r}-{\tilde{x}} ^{s}=\frac{\left( \beta +1\right) \left( 4\delta b\left( \beta +1\right) -a\left( \beta +1\right) ^{2}+4Ab\gamma \right) }{\left( 4b\gamma -\left( \beta +1\right) ^{2}\right) \left( 8b\gamma -\left( \beta +1\right) ^{2}\right) }\), which is always positive if we take into account restriction \(A\ge \frac{\left( 1+\beta \right) \left( a\left( 1+\beta \right) -4b\delta \right) }{4b\gamma }\) which ensures positive marginal production costs with renting and renting–selling firms.

17. The quantity used in the second period is the sum of the quantities sold in each period. The quantity used in the first period is the sum of the quantity sold and the quantity rented out in that period.

18. The Routh-Hurwitz stability condition requires that \(\frac{\partial ^{2}\pi _{i}\left( x_{i},x_{j}\right) }{\partial x_{i}^{2}}\frac{\partial ^{2}\pi _{j}\left( x_{i},x_{j}\right) }{\partial x_{j}^{2}}-\frac{\partial ^{2}\pi _{i}\left( x_{i},x_{j}\right) }{\partial x_{i}\partial x_{j}}\frac{\partial ^{2}\pi _{j}\left( x_{i},x_{j}\right) }{\partial x_{j}\partial x_{i}}>0\) (see Hinloopen (2015) for a clarification of the stability condition).

The author would like to thank the editor, Giacomo Corneo, two anonymous reviewers, Ana I. Saracho, Iñaki Aguirre, and Miguel González Maestre for helpful comments and suggestions. The usual disclaimer applies. Financial support from Spain’s Ministerio de Economía y Competitividad (ECO2012-35820 and ECO2015-64467-R (MINECO/FEDER, UE)) and from the Departamento de Educación, Universidades e Investigación del Gobierno Vasco (IT-783-13) is gratefully acknowledged.

Authors and Affiliations

1. Department of Economic Analysis II and Bridge, Faculty of Economics and Business, University of the Basque Country UPV/EHU, Avda. Lehendakari Agirre 83, 48015, Bilbao, Spain

   Amagoia Sagasta

Corresponding author

Correspondence to Amagoia Sagasta.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

APPENDIX A

1.1 A.1 Socially efficient level of R&D

Social welfare is defined as the sum in present value of the consumer surplus and the producer‘s profits. The goal is to find the optimal quantities and the optimal investment level that maximize social welfare. We denote by \(Q_{t}\) the output used in the market in each period t, where \(t=1,2\).^{Footnote 17} Consumer surplus and the profits of firms can be written, respectively, as

So assuming a symmetric interior solution, the efficient level of R&D and the socially efficient amount of output that are obtained from the maximization of the social welfare can be written respectively as

We now demonstrate that the level of investment that maximizes social welfare is higher than the level of investment under all the commercial practices considered, with both R&D cooperation and R&D competition.

With R&D cooperation the following emerges:

The stability conditions^{Footnote 18} and the second order conditions of the maximization problems imply that \(4b\gamma -\left( 1+\beta \right) ^{2}>0\) and \(9b\gamma -\left( \beta +1\right) ^{2}>0\), so it must hold that \(\left( 1+\beta \right) \left( 2a-A\right) >4b\delta \) and \(4b\gamma >\left( 1+\beta \right) ^{2}\). Multiplying these inequalities gives \(\left( 2a-A\right) \gamma >\delta \left( 1+\beta \right) \) and as a result \(x^{**}>{\hat{x}}_{i}^{r}\) . The proof of proposition 2 shows that \({\hat{x}}_{i}^{r}>{\hat{x}}_{i}^{s}>{\hat{x}}_{i}^{r-s}\), so this gives \(x^{**}>{\hat{x}}_{i}^{r}>{\hat{x}}_{i}^{s}>{\hat{x}}_{i}^{r-s}\).

With R&D competition, we first demonstrate that \(x^{**}>x_{i}^{*r}.\)

The conditions for the equilibrium to be stable ensure that the denominator is positive and it is proved above that \(\left( 2a-A\right) \gamma >\delta \left( 1+\beta \right) \), so it emerges that \(x^{**}>x_{i}^{*r}\).

Second, we demonstrate that \(x^{**}>x_{i}^{*r-s}\).

Taking into account the constraint \(Ab\gamma >\frac{4a\left( 1+\beta \right) \left( 3-2\beta \right) -25b\delta \left( 1+\beta \right) }{25}\) which ensures non-negative production costs (\(x_{i}^{*}+\beta x_{j}^{*}\le A\)) for renting–selling firms with R&D competition and the constraint \(A\gamma <a\gamma -\delta \left( 1+\beta \right) \) which ensures \(x_{i}^{*r-s}>0\), it emerges that \(23b\delta -16a\beta -12a+23Ab\gamma +2ab\gamma -34b\beta \delta -57Ab\beta \gamma +82ab\beta \gamma +4a\beta ^{2}+8a\beta ^{3}-57b\beta ^{2}\delta>\)\(23b\delta -16a\beta -12a+23\left( \frac{4a\left( 1+\beta \right) \left( 3-2\beta \right) -25b\delta \left( 1+\beta \right) }{25}\right) +2ab\gamma -34b\beta \delta -57b\beta \left( a\gamma -\delta \left( 1+\beta \right) \right) +82ab\beta \gamma +4a\beta ^{2}+8a\beta ^{3}-57b\beta ^{2} \delta =\)\(\frac{1}{25}a\left( 25\beta +2\right) \left( 25b\gamma -4\left( \beta +1\right) \left( 3-2\beta \right) \right) >0\) and this expression is positive because the stability conditions hold. \(x^{**}>x_{i}^{*r}\) and \(x^{**}>x_{i}^{*r-s}\) and we show in the proof of proposition 2 that \(x_{i}^{*r}>x_{i}^{*s}\), so it can be concluded that the level of investment that maximizes social welfare is higher than the level of investment under all the commercial practices considered.

1.2 A.2 Proof of Proposition 1.

i) First we demonstrate that \({\hat{x}}_{i}^{r}>{\hat{x}}_{i}^{s}\).

The stability conditions and the second order conditions ensure that the denominator is positive. Thus, \({\hat{x}}_{i}^{r}>{\hat{x}}_{i}^{s}\) if the numerator is positive.

Taking into account the constraint that ensures non-negative production costs for renting firms with R&D cooperation, \(A>\frac{\left( 1+\beta \right) \left( 2a\left( 1+\beta \right) -9b\delta \right) }{9b\gamma }\), it emerges that

\(\left( -32a\left( \beta +1\right) ^{2}+\gamma b\left( 115A+58a\right) +115\delta b\left( \beta +1\right) \right)>\)

\( \left( -32a\left( \beta +1\right) ^{2}+\gamma b\left( 115\left( \frac{\left( 1+\beta \right) \left( 2a\left( 1+\beta \right) -9b\delta \right) }{9b\gamma }\right) +58a\right) +115\delta b\left( \beta +1\right) \right) \)

\(=\frac{58}{9}a\left( 9b\gamma -\left( \beta +1\right) ^{2}\right) >0\) and this expression is positive because the stability conditions hold.

Second, we demonstrate that \({\hat{x}}_{i}^{s}>{\hat{x}}_{i}^{r-s}\)

As the stability conditions and second-order conditions ensure that the denominator is positive, consider the numerator

Taking into account constraint \(Ab\gamma \le \frac{5\gamma ba+4a\left( \beta +1\right) ^{2}-30b\delta \left( \beta +1\right) }{30}\), which ensures \(q_{2i}>q_{1i}^{r}\) for renting–selling firms with R&D cooperation, it emerges that

\(=\frac{13}{30}a(25b\gamma -4\left( \beta +1\right) ^{2})\). And this expression is positive because the stability condition \(b\gamma >\frac{4\left( 1+\beta \right) ^{2}}{25}\) for renting–selling firms with R&D cooperation holds.

ii) First, we demonstrate that \(x_{i}^{*r}>x_{i}^{*s}\).

The stability conditions and the second order conditions ensure that the denominator is positive. Thus, \(x_{i}^{*r}>x_{i}^{*s}\) if the numerator is positive. Taking into account the constraint \(A>\frac{\left( 1+\beta \right) \left( 2a\left( 2-\beta \right) -9b\delta \right) }{9b\gamma }\) which ensures non-negative production costs (\(x_{i}^{*}+\beta x_{j}^{*}\le A\)) for renting firms with R&D competition, it emerges that:

Taking into account the constraint \(b\left( \delta +A\gamma \right) \ge \frac{4a\left( 1+\beta \right) \left( 3-2\beta \right) -25b\delta \beta }{25}\) which ensures non-negative production costs for renting–selling firms with R&D competition and the constraint \(Ab\gamma <\frac{4a\left( 1+\beta \right) \left( 3-2\beta \right) +5ab\gamma -30b\delta \left( 1+\beta \right) }{30}\) which ensures \(q_{2i}>q_{1i}^{r}\) for renting–selling firms with R&D competition, it emerges that

Finally, taking into account the stability condition for renting firms with R&D cooperation, \(b\gamma >\frac{\left( 1+\beta \right) ^{2}}{9}\) it emerges that

\(\frac{1}{450}a\left( 15\,300b\gamma -12\,532\beta +12\,225b\beta \gamma -3716\beta ^{2}+10\,680\beta ^{3}+1864\right) \)

\(>\frac{1}{450}a\left( 15\,300\frac{\left( 1+\beta \right) ^{2}}{9}-12\,532\beta \,{+}\,12\,225\frac{\left( 1+\beta \right) ^{2}}{9}\beta \,{-}\,3716\beta ^{2}\,{+}\,10\,680\beta ^{3}\,{+}\,1864\right) \)

\(=\frac{1}{1350}\left( 36\,115\beta ^{2}-34\,013\beta +10\,692\right) \left( \beta +1\right) a\). And this expression is positive if \(0<\beta <1\).

Second, it needs to be shown that \(x_{i}^{*r}\gtreqless x_{i}^{*r-s}.\)

Consider the following example: \(a=130\), \(A=18\), \(\beta =0.4\), \(\delta =25\), \(\gamma =2\), and \(b=1\). In this case, \(x_{i}^{*r}=10.29\) and \(x_{i}^{*r-s}=9.57\). Therefore, \(x_{i}^{*r}>x_{i}^{*r-s}\) .

However, if \(a=200\), \(A=10\), \(\beta =0.3\), \(\delta =35\), \(\gamma =8\), and \(b=1\), then \(x_{i}^{*r}=4.986\) and \(x_{i}^{*r-s}=5.060\). Therefore, \(x_{i}^{*r}<x_{i}^{*r-s}.\)\(\square \)

1.3 A.3 Proof of Proposition 2.

If firms rentout their output, then

The stability conditions and the second order conditions of the maximization problems imply that \(9b\gamma -\left( 2-\beta \right) \left( 1+\beta \right) >0\) and \(9b\gamma -\left( \beta +1\right) ^{2}>0\), so it must hold that \(\left( 2-\beta \right) \left( 2a-A\right) >9b\delta \) and \(9b\gamma >\left( 2-\beta \right) \left( 1+\beta \right) \). Multiplying these inequalities gives \(\left( 2a-A\right) \gamma >\left( 1+\beta \right) \delta \) and as a result \({\hat{x}}_{i}^{r}-x_{i}^{*r}>0\) if and only if \(\beta >\frac{1}{2} \).

If firms sell their output, then

with

The stability conditions and the second order conditions ensure that the denominator is positive.

First, it is demonstrated that when \(\beta >\frac{260}{476}\), (i) \(x_{i}^{*s}<{\hat{x}}_{i}^{s}\) if \(\beta >\frac{25}{39}\) and (ii) \(x_{i}^{*s}>{\hat{x}}_{i}^{s}\) if \(\beta <\frac{3}{5}\)

1. (i)

   It holds that \(x_{i}^{*s}<{\hat{x}}_{i}^{s}\) if \(H<0\). Taking into account the constraint which ensures positive second period output with R&D cooperation, \(Ab\gamma <\frac{32b\gamma a+7a\left( \beta +1\right) ^{2}-80b\delta \left( \beta +1\right) }{80}\), then

   $$\begin{aligned}&H<264ab\gamma +216b\beta \delta +\left( \frac{32b\gamma a+7a\left( \beta +1\right) ^{2}-80b\delta \left( \beta +1\right) }{80}\right) \\&\qquad \times \left( 476\beta -260\right) -440ab\beta \gamma -260b\delta -11a\left( \beta +1\right) +476b\beta ^{2}\delta \\&\qquad +11a\beta ^{2}\left( \beta +1\right) \\&\quad =\left( -\frac{1}{20}\right) a\left( 39\beta -25\right) \left( 128b\gamma -27\left( \beta +1\right) ^{2}\right) \end{aligned}$$

   As \(\left( 128b\gamma -27\left( \beta +1\right) ^{2}\right) >0\) for the stability condition for selling firms with R&D cooperation, it holds that \({\hat{x}}_{i}^{s}>x_{i}^{*s}\) if \(\beta >\frac{25}{39}\).

2. (ii)

   It holds that \(x_{i}^{*s}>{\hat{x}}_{i}^{s}\) if \(H>0\). Taking into account the constraint \(Ab\gamma >\frac{\left( 1+\beta \right) \left( 22a\left( 1+\beta \right) -128b\delta \right) }{128}\) that ensures non-negative production costs, \(x_{i}^{*}+\beta x_{j}^{*}\le A\), then

   $$\begin{aligned}&H>264ab\gamma +216b\beta \delta +\frac{\left( 1+\beta \right) \left( 22a\left( 1+\beta \right) -128b\delta \right) }{128}\left( 476\beta -260\right) \\&\qquad -\,440ab\beta \gamma -260b\delta -11a\left( \beta +1\right) +476b\beta ^{2}\delta +11a\beta ^{2}\left( \beta +1\right) \\&\quad =\left( \frac{11}{16}\right) a\left( 3-5\beta \right) \left( 128b\gamma -27\left( \beta +1\right) ^{2}\right) \end{aligned}$$

As \(\left( 128b\gamma -27\left( \beta +1\right) ^{2}\right) >0\) for the stability condition, it holds that \(x_{i}^{*s}>{\hat{x}}_{i}^{s}\) if \(\frac{260}{476}<\beta <\frac{3}{5}\).

Second, we demonstrate that when \(\beta <\frac{260}{476}\), \(x_{i}^{*s} >{\hat{x}}_{i}^{s}\).

Taking into account the constraint which ensures that \({\hat{x}}_{i}^{s}\,{>}\,0\), \(A\,{<}\,\left( \frac{22a\left( \beta +1\right) -128b\delta }{27\left( \beta +1\right) }\right) \), it holds that \(H>\)\(264ab\gamma +216b\beta \delta +\left( \frac{22a\left( \beta +1\right) -128b\delta }{27\left( \beta +1\right) }\right) b\gamma \left( 476\beta -260\right) -440ab\beta \gamma -260b\delta -11a\left( \beta +1\right) +476b\beta ^{2} \delta +11a\beta ^{2}\left( \beta +1\right) =\)

\(\left( \frac{1}{27\left( \beta +1\right) }\right) \left( 128b\gamma -27\left( \beta +1\right) ^{2}\right) \left( 11a-11a\beta ^{2}+b\delta \left( 260-476\beta \right) \right) >0\). Thus it emerges that \(x_{i}^{*s}>{\hat{x}}_{i}^{s}\) if \(\frac{260}{476}>\beta \).

Taking into account that \(x_{i}^{*s}>{\hat{x}}_{i}^{s}\) if \(\beta <\frac{3}{5}\), that \(x_{i}^{*s}<{\hat{x}}_{i}^{s}\) if \(\beta >\frac{25}{39}\), that \({\hat{x}}_{i}^{s}\) is increasing with \(\beta \) (the denominator is decreasing with \(\beta \) and the numerator is increasing with \(\beta \) as \(22a-27A>0\) when \({\hat{x}}_{i}^{s}>0\)) and that \(x_{i}^{*s}\) is decreasing with \(\beta \) if \(\beta >\frac{54}{130}\) (the numerator is decreasing with \(\beta \) and the denominator is increasing with \(\beta \) if \(\beta >\frac{54}{130}\)), there must be a critical value \(\beta ^{*}\) that satisfies \(\frac{3}{5}<\beta ^{*}<\frac{25}{39}\) such that \({\hat{x}}_{i}^{s}>x_{i}^{*s}\) if and only if \(\beta >\beta ^{*}\).

When firms both rent out and sell their output, then

The conditions that assure the stability of the equilibrium imply that \(25b\gamma -4\left( \beta +1\right) ^{2}>0\) and \(25b\gamma -4\left( \beta +1\right) \left( 3-2\beta \right) >0\), so it must hold that \(4\left( a-A\right) \left( \beta +1\right) >25b\delta \) and \(25b\gamma >4\left( \beta +1\right) ^{2}\). Multiplying these inequalities gives \(\left( a-A\right) \gamma >\delta \left( \beta +1\right) \) and as a result \(\hat{x}_{i}^{r-s}-x_{i}^{*r-s}>0\) if and only if \(\beta >\frac{2}{3}\)\(\square \)

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