Observable versus unobservable R&D investments in duopolies

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.

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Fig. 1
Fig. 2

Notes

  1. 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. 2.

    As will be clear shortly, each firm’s type is determined by its realized marginal cost.

  3. 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. 4.

    One may classify the unobservable-investments model of this paper as a simultaneous-move game with sequential moves, following Baik and Lee (2007), and the nonstrategic model in Brander and Spencer (1983) as a standard simultaneous-move game.

  5. 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. 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. 7.

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

  8. 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. 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. 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. 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. 12.

    Throughout the paper, when we use i and j at the same time, we mean that \({i}\ne {j}\).

  13. 13.

    For the notion and definition of a Bayesian game, see, for example, Osborne (2004).

  14. 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. 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. 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. 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. 18.

    Note that each firm forms its belief about the rival firm’s R&D investment when it chooses its R&D investment.

  19. 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. 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. 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. 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. 23.

    Because the proofs of the comparative statics results presented in Propositions 2 and 3 involve very long mathematical derivations, we do not provide them for concise exposition. But they are available from the authors upon request.

  24. 24.

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

  25. 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.

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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.

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Appendices

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

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

$$\begin{aligned}&{q}_1({c}_1) = 0.5({a}-{c}_1-{K}_2) \end{aligned}$$
(A1)

and

$$\begin{aligned} {q}_2({c}_2) = 0.5({a}-{c}_2-{K}_1), \end{aligned}$$
(A2)

where

$$\begin{aligned} {K}_2\equiv \int _0^{{c}_{{M}}-\delta {x}_2}~{q}_2({c}_2)~{f}_2({c}_2; {x}_2)~{dc}_2 \end{aligned}$$
(A3)

and

$$\begin{aligned} {K}_1~\equiv \int _0^{{c}_{{M}}-\delta {x}_1}~{q}_1({c}_1)~{f}_1({c}_1; {x}_1)~{dc}_1. \end{aligned}$$
(A4)

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

$$\begin{aligned} 2{K}_1+{K}_2 = {a} - {k}({c}_{{M}}-\delta {x}_1)/({k}+1). \end{aligned}$$

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

$$\begin{aligned} 2{K}_2 + {K}_1 = {a} - {k}({c}_{{M}}-\delta {x}_2)/({k}+1). \end{aligned}$$

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

$$\begin{aligned} {K}^{{N}}_1 = {a}/3 - {k}({c}_{{M}}-2\delta {x}_1+\delta {x}_2)/3({k}+1) \end{aligned}$$

and

$$\begin{aligned} {K}^{{N}}_2 = {a}/3 - {k}({c}_{{M}}-2\delta {x}_2+\delta {x}_1)/3({k}+1). \end{aligned}$$

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)

From Lemmas 1 and 2, we have

$$\begin{aligned} {x}^{{*}} = {A}/\{{B}+8\delta ^2{k}^2({k}+2)\}, \end{aligned}$$

and

$$\begin{aligned} {x}^{{**}} = \delta {k}\{2({k}+1)({k}+2){a}-(2{k}^2+4{k}+3){c}_{{M}}\}/ \{3({k}+1)^2{D}+\delta ^2{k}^2({k}+2)\}, \end{aligned}$$

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

$$\begin{aligned} {x}^{{*}} - {x}^{{**}}= & {} \delta {k}({k}+1)({k}+2) \left[ \left\{ 12\gamma ({k}+1)^2({k}+2)-6\delta ^2{k}\right\} {a}\right. \\&\left. -\,12 \gamma {k}({k}+1)({k}+2){c}_{{M}}\right] \\&\div \,\left\{ {B}+8\delta ^2{k}^2({k}+2)\right\} \left\{ 3({k}+1)^2{D}+\delta ^2{k}^2({k}+2)\right\} . \end{aligned}$$

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)

From Lemmas 1 and 2, we have

$$\begin{aligned} {H}_{{i}}({x}^{{*}}_1,{x}^{{*}}_2)= & {} \int _0^{{c}_{{M}}-\delta {x}^{{*}}}\left\{ (2{a}-3{c}_{ {i}})/6+{k}({c}_{{M}}-\delta {x}^{{*}})/6({k}+1)\right\} ^2\\&\times \,\left\{ {kc}^{{k}-1}_{{i}}/({c}_{{M}}- \delta {x}^{{*}})^{{k}}\right\} ~{dc}_{{i}} - \gamma ({x}^{{*}})^2/2, \end{aligned}$$

and

$$\begin{aligned} {G}_{{i}}({x}^{{**}}_1, {x}^{{**}}_2)= & {} \int _0^{{c}_{{M}}-\delta {x}^{{**}}}\left\{ (2{a}-3{c}_{ {i}})/6+{k}({c}_{{M}}-\delta {x}^{{**}})/6({k}+1)\right\} ^2\\&\times \,\left\{ {kc}^{{k}-1}_{{i}}/({c}_{{M}}-\delta {x}^{ {**}})^{{k}}\right\} {dc}_{{i}} - \gamma ({x}^{{**}})^2/2. \end{aligned}$$

These expressions can be rewritten as

$$\begin{aligned} {H}_{{i}}({x}^{{*}}_1, {x}^{{*}}_2)= & {} \left\{ {a}/3+{k}({c}_{{M}}-\delta {x}^{{*}})/6({k}+1)\right\} ^2 \\&-\, \left\{ {a}/3+{k}({c}_{{M}}-\delta {x}^{{*}})/6({k}+1)\right\} \\&\times \,{k}({c}_{{M}}-\delta {x}^{{*}})/({k}+1) + {k}({c}_{{M}}-\delta {x}^{{*}})^2/4({k}+2) - \gamma ({x}^{{*}})^2/2, \end{aligned}$$

and

$$\begin{aligned} {G}_{{i}}({x}^{{**}}_1, {x}^{{**}}_2)= & {} \left\{ {a}/3+{k}({c}_{{M}}-\delta {x}^{{**}})/6({k}+1)\right\} ^2 \\&-\, \left\{ {a}/3+{k}({c}_{{M}}-\delta {x}^{{**}})/6({k}+1)\right\} \\&\times \,{k}({c}_{{M}}-\delta {x}^{{**}})/({k}+1) + {k}({c}_{{M}}-\delta {x}^{{**}})^2/4({k}+2) - \gamma ({x}^{{**}})^2/2. \end{aligned}$$

Using these, it is tedious but straightforward to obtain

$$\begin{aligned} {H}_{{i}}({x}^{{*}}_1, {x}^{{*}}_2) - {G}_{{i}}({x}^{{**}}_1, {x}^{{**}}_2) = ({x}^{{*}}-{x}^{{**}}){W}/36({k}+1)^2({k}+2), \end{aligned}$$
(B1)

where

$$\begin{aligned}&{W}\equiv -9\delta {kc}_{{M}}-\{18\gamma ({k}+1)^2({k}+2)- \delta ^2{k}(8{k}^2+16{k}+9)\}{x}^{{**}}\\&\quad -\,\delta ^2{k}(4{k}^2+8{k})({x}^{{*}}+{x}^{{**}}). \end{aligned}$$

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)

Using Lemmas 1 and 2, we have

$$\begin{aligned} {E}[ {q}_{{i}}^{{*}}({C}_{{i}})]= & {} \int _0^{{c}_{{M}}-\delta {x}^{{*}}}{q}_{{i}}^{ {*}}({c}_{{i}})~{f}^{{*}}_{{i}}({c}_{{i}}; {x}_{{i}})~{dc}_{{i}}\\= & {} \int _0^{{c}_{{M}}-\delta {x}^{{*}}}\left\{ (2{a}-3{c}_{ {i}})/6+{k}({c}_{{M}}-\delta {x}^{{*}})/6({k}+1)\right\} \\&\qquad \left\{ {kc}^{ {k}-1}_{{i}}/({c}_{{M}}-\delta {x}^{{*}})^{{k}}\right\} {dc}_{{i}}\\= & {} {a}/3 + {k}({c}_{{M}}-\delta {x}^{{*}})/6({k}+1) - \int _0^{{c}_{{M}}-\delta {x}^{{*}}}\left\{ {kc}^{{k}}_{ {i}}/2({c}_{{M}}-\delta {x}^{{*}})^{{k}}\right\} {dc}_{{i}}\\= & {} {a}/3 - {k}({c}_{{M}}-\delta {x}^{{*}})/3({k}+1), \end{aligned}$$

and

$$\begin{aligned} {E}[ {q}_{{i}}^{{**}}({C}_{{i}})]= & {} \int _0^{{c}_{{M}}-\delta {x}^{{**}}}{q}_{{i}}^{ {**}}({c}_{{i}}) {f}^{{**}}_{{i}}({c}_{{i}}; {x}_{{i}}) {dc}_{{i}}\\= & {} \int _0^{{c}_{{M}}-\delta {x}^{{**}}}\left\{ (2{a}-3{c}_{ {i}})/6+{k}({c}_{{M}}-\delta {x}^{{**}})/6({k}+1)\right\} \\&\qquad \left\{ {kc}^{ {k}-1}_{{i}}/({c}_{{M}}-\delta {x}^{{**}})^{{k}}\right\} {dc}_{{i}}\\= & {} {a}/3 + {k}({c}_{{M}}-\delta {x}^{{**}})/6({k}+1) - \int _0^{{c}_{{M}}-\delta {x}^{{**}}}\left\{ {kc}^{{k}}_{ {i}}/2({c}_{{M}}-\delta {x}^{{**}})^{{k}}\right\} {dc}_{{i}}\\= & {} {a}/3 - {k}({c}_{{M}}-\delta {x}^{{**}})/3({k}+1). \end{aligned}$$

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)

From Lemmas 1 and 2, we have

$$\begin{aligned} {CS}^{{*}}= & {} ({c}_{{M}}-\delta {x}^{{*}})^2\left[ 32{a}^2({k}+1)^2-({c}_{{M}}-\,\delta {x}^{{*}}) \left\{ 16a({k}+1)({k}+3)-(5{k}^2+18{k}+21)\right. \right. \\&\left. \left. \times \,({c}_{{M}}-\delta {x}^{{*}})\right\} \right] /144({k}+1)^2, \end{aligned}$$

and

$$\begin{aligned} {CS}^{{**}}= & {} ({c}_{{M}}-\delta {x}^{{**}})^2\left[ 32{a}^2({k}+1)^2-({c}_{{M}}-\delta {x}^{{**}}) \left\{ 16{a}({k}+1)({k}+3)-\,(5{k}^2+18{k}+21)\right. \right. \\&\left. \left. \times \,({c}_{{M}}-\delta {x}^{{**}})\right\} \right] /144({k}+1)^2. \end{aligned}$$

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 \({CS}^{{*}}<{CS}^{{**}}\).

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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

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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