## Abstract

We study a quantity-setting duopoly with homogeneous products in which two firms first make their cost-reducing R&D investments, and then compete in quantities. When making its R&D investment, each firm is uncertain about its R&D outcome. Its new marginal cost is probabilistically determined later, but before the firm chooses its output level. When choosing its output level, each firm has private information regarding its own new marginal cost. We develop the observable-investments and the unobservable-investments models. We compare the outcomes of these two main models, and perform comparative statics of them with respect to each of the parameters, respectively. As variations, we consider the observable-investments and the unobservable-investments model based on a price-setting duopoly with product differentiation.

This is a preview of subscription content, log in to check access.

## Notes

- 1.
Baik and Lee (2007) define a

*simultaneous-move game with sequential moves*between two parties, and come up with a solution technique for it. They define it as a game in which each party has two sequential moves, the first action chosen by each party is hidden from the other party, and the parties choose their second actions simultaneously. - 2.
As will be clear shortly, each firm’s type is determined by its realized marginal cost.

- 3.
Brander and Spencer (1983) assume in the strategic model that the precise effect of each firm’s R&D investment on cost reduction is known to the rival firm when the firms choose their output levels.

- 4.
- 5.
Thomas (1997), Sengupta (2016), and Chatterjee et al. (2019) consider duopoly models in which each firm’s new marginal cost resulting from R&D investment is hidden from the rival firm when the firms compete in the product market. Vives (2002) considers a Cournot market in which firms first decide whether to enter the market or not, and then compete in quantities. In his model, when making its entry decision, each firm is uncertain about its production cost; when choosing its output level, each firm has private information regarding its production cost. Bagnoli and Watts (2010) consider a Cournot duopoly in which each firm has private information regarding its production cost when the firms compete in the product market.

- 6.
Quadratic R&D cost functions are used in, for example, d’Aspremont and Jacquemin (1988), Aghion et al. (2001), Lin and Saggi (2002), Haaland and Kind (2008), Tishler and Milstein (2009), Bourreau and Dogan (2010), Ishida et al. (2011), Lin and Zhou (2013), Milliou and Pavlou (2013) and Chang and Ho (2014).

- 7.
We will make further restrictive assumptions on the parameters later, during the analysis (see footnote 20).

- 8.
We use a capital letter to denote a random variable, and the corresponding small letter to denote a value of the random variable.

- 9.
We assume that \({c}_{{M}}/\delta \) is sufficiently large that (\({c}_{{M}}-\delta {x}_{{i}})>0\) in a relevant range of firm

*i*’s R&D investments. - 10.
The greater firm

*i*’s R&D investment, the better the probability distribution of its new marginal cost. Here “better” could be taken to mean that one distribution first-order stochastically dominates the other. - 11.
Let

*Y*be a random variable with a Pareto distribution with the lower bound parameter \({y}_{{m}}\) and the tail length shape parameter \(\alpha \), where \({y}_{{m}}>0\) and \(\alpha >0\). Then its cumulative distribution function is \(F(y)=[1-({y}_{{m}}/y)^\alpha ]\)\({I}_{{[ y}_{{m}}{, }\infty {)}}(y)\). This paper assumes that 1\(/{C}_{{i}}\) has a Pareto distribution with the lower bound parameter \(1/({c}_{{M}}-\delta {x}_{{i}}\)) and the tail length shape parameter k. - 12.
Throughout the paper, when we use

*i*and*j*at the same time, we mean that \({i}\ne {j}\). - 13.
For the notion and definition of a Bayesian game, see, for example, Osborne (2004).

- 14.
The reaction function of type \({c}_{{i}}\) of firm

*i*shows its best response to every possible “vector” of output levels, one for each type of firm*j*, that all types of firm*j*might choose. - 15.
Note that the second-order condition is satisfied for every maximization problem in this paper; however, for concise exposition, we do not state it explicitly in each case.

- 16.
We assume at this point that

*a*is sufficiently larger than \({c}_{{M}}\), in order to have that \({q}^{{N}}_{{i}}({c}_{{i}})>0\) for all \({c}_{{i}}\in (0,\,{c}_{{M}}-\delta {x}_{{i}}\)] (see footnote 20). - 17.
At the Nash equilibrium of the Bayesian game, the output level of each type of firm

*i*is the best response to the output levels of all the types of firm*j*, one for each type. - 18.
Note that each firm forms its belief about the rival firm’s R&D investment when it chooses its R&D investment.

- 19.
Using a technique similar to the one for solving for \({K}^{{N}}_1\) and \({K}^{{N}}_2\) in “Appendix A”, we obtain \({M}_{{i}}({x}_1\), \({x}_2)={a}/3-{k}({c}_{{M}}-2\delta {x}_{{i}}+\delta {x}_{ {j}})/3({k}+1)\) for

*i*, \({j}=1\), 2 with \({i}\ne {j}\). - 20.
We have assumed in Sect. 3 that \(\underline{{S}}<{c}_{{M}}/{a}<\mathrm{min}\{\bar{{S}}, 1\}\), and in Sect. 4 that \(\underline{{U}}<{c}_{{M}}/{a}<\bar{{U}}\). Because \(\underline{{U}}<\underline{{S}}<\bar{{U}}<\bar{{S}}\), we assume in this section that \(\underline{{S}}<{c}_{{M}}/{a}<\bar{{U}}\).

- 21.
Note that, in a standard quantity-setting duopoly with homogeneous products, a firm with a higher marginal cost chooses a smaller output level in equilibrium than the rival firm.

- 22.
We cannot determine the sign of the comparative statics of \({CS}^{{*}}\) or \({SW}^{{*}}\) with respect to each of the parameters, without assuming specific values for the other parameters.

- 23.
- 24.
Recall from Lemma 1 that \(\pi ^{{*}}_{{i}}({c}_{{i}}\)) increases as \({c}_{{i}}\) decreases.

- 25.
In the model with observable R&D investments, each firm knows the rival firm’s new marginal cost when choosing its price. However, in the model with unobservable R&D investments, each firm does not know the rival firm’s new marginal cost when choosing its price.

## References

Aghion P, Harris C, Howitt P, Vickers J (2001) Competition, imitation and growth with step-by-step innovation. Rev Econ Stud 68(3):467–492

Bagnoli M, Watts SG (2010) Oligopoly, disclosure, and earnings management. Account Rev 85(4):1191–1214

Bagwell K, Staiger RW (1994) The sensitivity of strategic and corrective R&D policy in oligopolistic industries. J Int Econ 36(1–2):133–150

Baik KH (2016) Endogenous group formation in contests: unobservable sharing rules. J Econ Manag Strategy 25(2):400–419

Baik KH, Kim J (2014) Contests with bilateral delegation: unobservable contracts. J Inst Theor Econ 170(3):387–405

Baik KH, Lee S (2007) Collective rent seeking when sharing rules are private information. Eur J Polit Econ 23(3):768–776

Baik KH, Lee D (2012) Do rent-seeking groups announce their sharing rules? Econ Inq 50(2):348–363

Baik KH, Lee D (2019) Decisions of duopoly firms on sharing information on their delegation contracts. Rev Ind Organ (Forthcoming)

Bessen J, Maskin E (2009) Sequential innovation, patents, and imitation. RAND J Econ 40(4):611–635

Bourreau M, Dogan P (2010) Cooperation in product development and process R&D between competitors. Int J Ind Organ 28(2):176–190

Brander JA, Spencer BJ (1983) Strategic commitment with R&D: the symmetric case. Bell J Econ 14(1):225–235

Bustos P (2011) Trade liberalization, exports, and technology upgrading: evidence on the impact of MERCOSUR on Argentinian firms. Am Econ Rev 101(1):304–340

Chaney T (2008) Distorted gravity: the intensive and extensive margins of international trade. Am Econ Rev 98(4):1707–1721

Chang MC, Ho Y-C (2014) Comparing Cournot and Bertrand equilibria in an asymmetric duopoly with product R&D. J Econ 113(2):133–174

Chatterjee R, Chattopadhyay S, Kabiraj T (2019) R&D in a duopoly under incomplete information. Int J Econ Theory 15(4):341–359

d’Aspremont C, Jacquemin A (1988) Cooperative and noncooperative R&D in duopoly with spillovers. Am Econ Rev 78(5):1133–1137

Delbono F, Denicolo V (1991) Incentives to innovate in a Cournot oligopoly. Q J Econ 106(3):951–961

Gal-Or E (1985) Information sharing in oligopoly. Econometrica 53(2):329–343

Gill D (2008) Strategic disclosure of intermediate research results. J Econ Manag Strategy 17(3):733–758

Goel RK, Haruna S (2011) Cost-reducing R&D with spillovers and trade. J Inst Theor Econ 167(2):314–326

Haaland JI, Kind HJ (2008) R&D policies, trade and process innovation. J Int Econ 74(1):170–187

Hall BH, Oriani R (2006) Does the market value R&D investment by European firms? Evidence from a panel of manufacturing firms in France, Germany, and Italy. Int J Ind Organ 24(5):971–993

Ishida J, Matsumura T, Matsushima N (2011) Market competition, R&D and firm profits in asymmetric oligopoly. J Ind Econ 59(3):484–505

Ishii Y (2017) International asymmetric R&D rivalry and industrial strategy. J Econ 122(3):267–278

Kawasaki A, Lin MH, Matsushima N (2014) Multi-market competition, R&D, and welfare in oligopoly. South Econ J 80(3):803–815

Koh P-S, Reeb DM (2015) Missing R&D. J Account Econ 60(1):73–94

Kovenock D, Morath F, Münster J (2015) Information sharing in contests. J Econ Manag Strategy 24(3):570–596

Lin P, Saggi K (2002) Product differentiation, process R&D, and the nature of market competition. Eur Econ Rev 46(1):201–211

Lin P, Zhou W (2013) The effects of competition on the R&D portfolios of multiproduct firms. Int J Ind Organ 31(1):83–91

Melitz MJ, Ottaviano GIP (2008) Market size, trade, and productivity. Rev Econ Stud 75(1):295–316

Milliou C, Pavlou A (2013) Upstream mergers, downstream competition, and R&D investments. J Econ Manag Strategy 22(4):787–809

Nitzan S, Ueda K (2011) Prize sharing in collective contests. Eur Econ Rev 55(5):678–687

Osborne MJ (2004) An introduction to game theory. Oxford University Press, New York

Sengupta A (2016) Investment secrecy and competitive R&D. B.E. J Econ Anal Policy 16(3):1573–1583

Slivko O, Theilen B (2014) Innovation or imitation? The effect of spillovers and competitive pressure on firms’ R&D strategy choice. J Econ 112(3):253–282

Steinmetz A (2015) Competition, innovation, and the effect of R&D knowledge. J Econ 115(3):199–230

Tesoriere A (2015) Competing R&D joint ventures in Cournot oligopoly with spillovers. J Econ 115(3):231–256

Thomas CJ (1997) Disincentives for cost-reducing investment. Econ Lett 57(3):359–363

Tishler A, Milstein I (2009) R&D wars and the effects of innovation on the success and survivability of firms in oligopoly markets. Int J Ind Organ 27(4):519–531

Vives X (1984) Duopoly information equilibrium: Cournot and Bertrand. J Econ Theory 34(1):71–94

Vives X (1989) Technological competition, uncertainty, and oligopoly. J Econ Theory 48(2):386–415

Vives X (2002) Private information, strategic behavior, and efficiency in Cournot markets. RAND J Econ 33(3):361–376

## Acknowledgements

We are grateful to Chris Baik, Anne-Christine Barthel, Jerry Carlino, Hsueh-Jen Hsu, Jin-Hyuk Kim, Jong Hwa Lee, Wooyoung Lim, Daehong Min, seminar participants at Korea University and Yeungnam University, two anonymous referees, and the editor-in-chief of the journal, Giacomo Corneo, for their helpful comments and suggestions. We are also grateful to Hyok Jung Kim and Gwihwan Seol for their excellent research assistance. Earlier versions of this paper were presented at the 87th Annual Conference of the Western Economic Association International, San Francisco, CA, July 2012; and the 78th Annual Meeting of the Midwest Economics Association, Evanston, IL, March 2014. Part of this research was conducted while the first author was Visiting Professor at the University of Illinois at Urbana-Champaign.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

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

## Appendices

### Appendix A: Obtaining the Nash equilibrium of the second-stage subgame

From function (4), we have the following system of simultaneous equations:

and

where

and

Using function (1) and Eqs. (A1) and (A4), we obtain

Using function (1) and Eqs. (A2) and (A3), we obtain

Next, solving this pair of simultaneous equations for \({K}_1\) and \({K}_2\), we obtain

and

Finally, substituting these expressions for \({K}^{{N}}_1\) and \({K}^{{N}}_2\) into Eqs. (A1) and (A2), we obtain the Nash equilibrium of the second-stage subgame, reported in (5).

### Appendix B: Proof of Proposition 1

The proof of parts (*ii*) and (*iii*) is immediate from Lemmas 1 and 2, and therefore omitted. The proof of part (*vii*) is immediate from parts (*iv*) and (*vi*), and therefore omitted.

### B1. Proof of part (*i*)

and

where \(A\equiv \delta k\{8({k}+1\))(\({k}+2){a}-(8{k}^2+16{k}+9){c}_{{M}}\}\), \(B\equiv 18\gamma ({k}+1)^2({k}+2)-\delta ^2k(16{k}^2+\)\(32{k}+9\)), and \(D\equiv 2\gamma ({k}+2)-\delta ^2k\). Using these, it is straightforward to obtain

The numerator of the right-hand side is positive because \(a>{c}_{{M}}\) by assumption, and \(\{12\gamma ({k}+1)^2({k}+2)-6\delta ^2 k\}>12\gamma k({k}+1\))(\({k}+2\)) due to the assumption that \(\gamma \ge 1\), \(k\ge 1\), and \(0<\delta <1\). The denominator of the right-hand side also is positive because \(B>0\) and \(D>0\) due to the assumption that \(\gamma \ge 1\), \(k\ge 1\), and \(0<\delta <1\). Hence, we obtain \({x}^{{*}}>{x}^{{**}}\).

### B2. Proof of part (*iv*)

and

These expressions can be rewritten as

and

Using these, it is tedious but straightforward to obtain

where

The numerator of the right-hand side of expression (B1) is negative because \({x}^{{*}}>{x}^{{**}}>0\), as shown in part (*i*) of Proposition 1, and \(W<0\) due to the assumption that \(0<\delta <1\), \(k\ge 1\), \({c}_{{M}}>0\), and \(\gamma \ge 1\). The denominator of the right-hand side is positive due to the assumption that \(k\ge 1\). Hence, we obtain \({H}_{{i}}({x}^{{*}}_1, {x}^{{*}}_2)<{G}_{{i}}({x}^{{**}}_1, {x}^{{**}}_2\)).

### B3. Proof of part (*v*)

and

Comparing these, we obtain \(E[ {q}_{{i}}^{{*}}({C}_{{i}})]>E[ {q}_{{i}}^{{**}}({C}_{{i}})]\) because \({x}^{{*}}>{x}^{{**}}>0\) as shown in part (*i*) of Proposition 1. Note that both \(E[ {q}_{{i}}^{{*}}({C}_{{i}}\))] and \(E[ {q}_{{i}}^{{**}}({C}_{{i}})]\) are positive because \(a>{c}_{{M}}\) by assumption.

### B4. Proof of part (*vi*)

and

Let \({Z}={K}^2[32{a}^2({k}+1)^2-K\{16a({k}+1\))(\({k}+3)-(5{k}^2+18{k}+21)K\} ]/144({k}+1)^2\), where *K* is a continuous variable with positive values. Then, it is straightforward to obtain \(dZ/dK>0\). This means that the value of *Z* at \({K}={c}_{{M}}-\delta {x}^{{*}}\) is less than the value of *Z* at \({K}={c}_{{M}}-\delta {x}^{{**}}\) because \({c}_{{M}}-\delta {x}^{{*}}<{c}_{{M}}-\delta {x}^{{**}}\). Therefore, we obtain \({CS}^{{*}}<{CS}^{{**}}\).

## Rights and permissions

## About this article

### Cite this article

Baik, K.H., Kim, S. Observable versus unobservable R&D investments in duopolies.
*J Econ* **130, **37–66 (2020). https://doi.org/10.1007/s00712-019-00679-3

Received:

Accepted:

Published:

Issue Date:

### Keywords

- Observable R&D investment
- Unobservable R&D
- Uncertain R&D outcome
- Private information regarding R&D outcomes
- Cost-reducing R&D investment
- Information sharing

### JEL Classification

- D43
- L13
- C72