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Downstream new product development and upstream process innovation

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Research and development (R&D) in upstream and downstream markets influence each other. This is because, in assembly industries, when upstream input prices are low, downstream firms can easily introduce or develop new products. The introduction of a new product creates a new final good market, creating an increased demand for inputs. This greater demand for inputs provides an incentive for upstream firms to reduce their costs through R&D. In this study, we consider both downstream R&D for new product introduction and upstream R&D for cost reduction. We show that if upstream R&D is efficient (inefficient), the results of the downstream new product introduction race are strategic complements (substitutes). Furthermore, in terms of timing, the upstream firm determines its input price after observing the downstream firm’s investment decision, with the upstream firm extracting benefits from downstream R&D by raising the input price. It is well-known that this behavior by upstream firms impedes downstream investment (the hold-up problem). Despite this timing structure, we show that the more downstream firms invest, the lower the input price.

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  1. See, for example, the following site: (accessed 2022–07–23).

  2. In addition to these examples for the assembly industry, we consider an example of a fuel supplier (upstream) and the transportation industry (fuel user: downstream).

  3. Because downstream new product introduction increases the number of final products, it increases input demand. As empirically shown by Fontana and Guerzoni (2008), process innovation (i.e., cost-reducing R&D) becomes a more important innovative activity for firms when the market size is large. Thus, cost-reducing R&D by upstream firms is partly consistent with the empirical evidence.

  4. We consider price competition in Sect. 5.1.

  5. Additionally, Dawid et al. (2010) considered new product development in a duopoly setting. However, their model has no upstream sector, and the R&D types differ between the two firms: one firm engages in a new product development project and the other firm engages in cost-reducing R&D. Although Dobson and Waterson (1996) and Grossman (2007) also considered a similar scenario in which firms choose their number of differentiated goods, there is no upstream market in these models. Battaggion and Tedeschi (2021) considered both cost-reducing R&D and demand-enhancing R&D in a duopoly, whereas Lin and Zhang (2022) considered product innovation in which the new product entails a safety risk.

  6. This case is the same as the scenario in which “one is a multi-product firm and the other is a single-product firm.” Hence, the strategic substitute type of equilibrium in our new product introduction model includes this scenario. Inomata (2018) and Kawasaki, Lin and Matsushima (2014) also considered the coexistence of multi-product and single-product firms.

  7. Our main results do not alter in price competition. For more details, see Sect. 5.1.

  8. The other possible setting is that existing and new products are differentiated. The formula in such a case is \(p_{e,i}=1-(q_{e,i}+q_{e,j})- \gamma (q_{n,i}+q_{n,j})\), \(i,j=1,2\); \(i\not =j\). However, our main results do not alter; hence, we use a simpler form (1).

  9. Even if we use a Shubik and Levitan-type demand function, our main result does not alter.

  10. We implicitly assume that downstream firms use Leontief technology. That is, the existing product and the new product are produced using common inputs and inputs purchased from different competing fringe firms, respectively. Here, for simplicity, we normalize the price of inputs purchased from competing fringe firms to zero.

  11. Chowdhury (2005) defined \(\Phi _I\) as a non-strategic benefit of R&D and \(\Phi _N\) as a strategic benefit of R&D.

  12. We focus on pure-strategy equilibria, but if there is more than one pure-strategy equilibrium, there also exists a symmetric mixed-strategy equilibrium. Each downstream firm chooses I with probability \(\theta\). Then, solving \(\theta (\pi ^{II}_{Di}-f) +(1-\theta )(\pi ^{IN}_{D1}-f) = \theta \pi ^{NI}_{D1} +(1-\theta ) \pi ^{NN}_{Di}\) for \(\theta\), we obtain the symmetric mixed-strategy equilibrium \(\theta\).

  13. Banerjee and Lin (2003) showed that a fixed-price contract (i.e., input price is decided first in the process) that uses the input price resolves this hold-up problem. Conversely, Takauchi and Mizuno (2019) demonstrated that a fixed-price contract can harm upstream and downstream firms.

  14. Because the figure for equilibrium in the differentiated price competition case is almost the same as that in the quantity competition case, we omit it.

  15. These thresholds are given by \(\psi _I \equiv \frac{2 (1-c)^2 (\gamma ^5-\gamma ^4-8 \gamma ^2+8 ) k^2}{(\gamma +2)^2 [ \gamma + (-4 \gamma ^2+8 \gamma +8 ) k-3 ]^2}\), \(\psi _N \equiv \frac{2 (1-c)^2 (\gamma ^5-3 \gamma ^4-4 \gamma ^3+4 \gamma +2 ) k^2}{(2 \gamma +1)^2 [ \gamma + (-4 \gamma ^2+8 \gamma +8 ) k-3 ]^2}\), \(\phi _I \equiv \frac{(1-c)^2 (\gamma ^5-\gamma ^4-8 \gamma ^2+8) k^2}{2 (\gamma ^2-2 \gamma -2)^2 (1-2 (\gamma +2) k)^2}\), and \(\phi _N \equiv \frac{(1-c)^2 (\gamma ^5-3 \gamma ^4-4 \gamma ^3+4 \gamma +2)k^2}{2 (\gamma ^2-2 \gamma -2)^2 ((4 \gamma +2) k-1)^2}\).


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The authors would like to thank the editor and two anonymous referees for their constructive and helpful comments. This work was supported by JSPS KAKENHI [grant numbers 18K01613, 20K01646, 22H00043, 20K01678, 19H01483]. We thank Maxine Garcia, PhD, from Edanz ( for editing a draft of this manuscript. All errors are our own.

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Appendix A. Proofs

Proof of Proposition 1

(i) \({x}^{II} - {x}^{IN} = \frac{ 2k (1-c)(1-\gamma ) }{ {L}_{N} } >0\) and \({x}^{IN}-{x}^{NN} = \frac{ 2k (1-c)(1-\gamma )(2-\gamma ) }{ {L}_{I} } >0\), where \(L_{N} \equiv [(4\gamma +2)k -1] [ 4(2+2\gamma -\gamma ^2)k -(3-\gamma )]\) and \(L_{I} \equiv [2(\gamma +2)k -1] [4(2+2\gamma -\gamma ^2)k -(3-\gamma )]\). (ii) The partial derivative of \(x^r\) with respect to k yields \(\partial {x}^{II}/\partial k = - \frac{2 (1-c) (2 \gamma +1)}{ [ (4 \gamma +2)k-1 ]^2}<0\), \(\partial {x}^{IN}/\partial k = - \frac{4 (1-c) (3-\gamma ) ( 2+2\gamma -\gamma ^2) }{ [ 4(2+2\gamma -\gamma ^2)k -(3-\gamma ) ]^2}<0\), and \(\partial {x}^{NN}/\partial k = - \frac{2 (1-c) (\gamma +2)}{ [ 2(\gamma +2)k -1 ]^2 }<0\). The partial derivative of \(x^r\) with respect to \(\gamma\) yields \(\partial {x}^{II}/\partial \gamma = -\frac{4 (1-c) k}{ [ (4 \gamma +2) k-1 ]^2}<0\), \(\partial {x}^{IN}/\partial \gamma = -\frac{4 (1-c) (\gamma ^2-6 \gamma +8) k}{ [ 4(2+2\gamma -\gamma ^2)k -(3-\gamma ) ]^2}<0\), and \(\partial {x}^{NN}/\partial \gamma = -\frac{2 (1-c) k}{ [ 2 (\gamma +2) k -1 ]^2} <0\). \(\square\)

Proof of Proposition 2

(i) \(w^{NN} - w^{IN} = \frac{ k(1-c) (2-\gamma )(1-\gamma ) }{ {L}_{I} } >0\) and \(w^{IN} - w^{II} = \frac{ k(1-c)(1-\gamma ) }{ {L}_{N} } >0\). (ii) The partial derivative of \(w^r\) with respect to k yields \(\partial w^{II}/\partial k = \frac{ (1-c)(2\gamma +1) }{ [(4\gamma +2)k -1]^2 } >0\), \(\partial w^{IN}/\partial k = \frac{ 2(1-c)(3-\gamma )(2+2\gamma -\gamma ^2) }{ [ 4(2+2\gamma -\gamma ^2)k -(3-\gamma ) ]^2 }>0\), and \(\partial w^{NN}/\partial k = \frac{ (1-c)(\gamma +2) }{ [2(\gamma +2)k -1]^2 }>0\). The partial derivative of \(w^r\) with respect to \(\gamma\) yields \(\partial w^{II}/\partial \gamma = \frac{ 2(1-c)k }{ [(4\gamma +2)k -1]^2 }>0\), \(\partial w^{IN}/\partial \gamma = \frac{ 2(1-c)k (4-\gamma )(2-\gamma ) }{ [ 4(2+2\gamma -\gamma ^2)k -(3-\gamma ) ]^2 }>0\), and \(\partial w^{NN}/\partial \gamma = \frac{ (1-c)k }{ [ 2(\gamma +2)k -1 ]^2 }>0\). \(\square\)

Proof of Proposition 3

By comparing \(\Phi _N\) with \(\Phi _I\), we obtain

$$\begin{aligned} \Phi _N -\Phi _I&= \frac{ (1-c)^2 (1-\gamma )^2 k^2 (1 -4 \gamma k) \ g(k,\gamma ) }{2 [1-2 (\gamma +2) k]^2 [(4 \gamma +2)k-1]^2 \left[ \gamma +\left( -4 \gamma ^2+8 \gamma +8\right) k-3\right] ^2} \nonumber \\ \text {and}~~ g(k,\gamma )&\equiv 16 ( 3 \gamma ^4 + 5 \gamma ^3 + 16 \gamma ^2 + 22 \gamma + 8 ) k^3 - 12 ( 5\gamma ^3 + 5\gamma ^2 + 8\gamma + 6 ) k^2 + 24\gamma ^2 k - 3\gamma + 3. \end{aligned}$$

We show that \(g(k,\gamma ) > 0\) and \(\textrm{sign}\{ \Phi _N -\Phi _I \}\) depend only on \(1 -4 \gamma k\). To prove \(g(k,\gamma ) > 0\), it is sufficient to show that \(g(k,\gamma )\) has its minimum value at \(k = k_0\) and \(c=1\), and that its value is positive.

First, we show that \(g(k, \gamma )\) is an increasing function of k; that is, \(g(k, \gamma )\) is smallest at \(k=k_0\). The first derivative \(g(k, \gamma )\) with respect to k is \(\partial g(k, \gamma ) \big / \partial k = 24 [ 2 (3 \gamma ^4+5 \gamma ^3+16 \gamma ^2+22 \gamma +8) k^2 -(5 \gamma ^3+5 \gamma ^2+8 \gamma +6 ) k +\gamma ^2 ]\). \(\partial g(k,\gamma )/ \partial k\) is a quadratic function of k and the coefficient of \(k^2\) is positive. Hence, by solving \(\partial g(k,\gamma )/ \partial k > 0\) for k, we obtain \(k < k_1\) and \(k>k_2\), where \(k_1\) and \(k_2\) are roots of \(g(k, \gamma )=0\) for k and \(k_1 < k_2\).

Fig. 4
figure 4

\(k_0\) at \(c=1\) and the two roots of \(g(k, \gamma )=0\)

As \(k_0 = 1/[2 c (2 \gamma +1)]\) decreases with c, \(k_0\) has its minimum value at \(c = 1\). We illustrate \(k_1\), \(k_2\), and \(k_0\) at \(c=1\) in Fig. 4. Using numerical calculation, we find that for \(\gamma \in [0, 1]\), the unique root of \(k_2 - k_0|_{c=1} = 0\) is \(\gamma = 1\). Hence, \(\partial g(k,\gamma )/ \partial k > 0\) for any \(k > k_0\). Therefore, \(g(k, \gamma )\) has its minimum value at \(k = k_0\).

Second, we show \(\partial g(k_0, \gamma )/\partial c <0\). The derivation yields

$$\begin{aligned} \frac{ \partial g(k_0, \gamma ) }{ \partial c } = \frac{ \partial g(k_0, \gamma ) }{ \partial k } \times \frac{ \partial k_0 }{ \partial c } = \frac{ \partial g(k_0, \gamma ) }{ \partial k } \left[ \frac{ - 1}{2 c^2 (2 \gamma +1)} \right] <0. \end{aligned}$$

The last inequality is satisfied because \(\partial g(k,\gamma )/ \partial k > 0\). Hence, \(g(k_0, \gamma )\) is a decreasing function for c and it has its minimum value when \(c = 1\).

From the above discussion, \(g(k_0, \gamma )\) has the following minimum value at \(k=k_0\) and \(c=1\): \(g(k_0, \gamma ) \big |_{c=1} = \frac{(1-\gamma )^2 (\gamma +1)}{(2 \gamma +1)^3} >0\). Because \(g(k_0, \gamma ) \big |_{c=1}\) is positive, \(\forall k>k_0\), \(g(k, \gamma ) > 0\). This result implies that  \(\textrm{sign}\{ \Phi _N -\Phi _I \}\) depends only on “\(1 -4 \gamma k\).” Hence, \(\Phi _N > \Phi _I\) iff \(k < 1/(4\gamma )\). \(\square\)

Appendix B. Derivation of equilibrium outcomes

In this appendix, we derive the equilibrium outcomes in each regime.

No one invests regime: \(NN\). First, we derive the equilibrium quantity of the final product. Because each firm decides the quantity of the final product that maximizes profit, the first-order condition (FOC) to maximize profit is

$$\begin{aligned} \frac{\partial \pi _{Di}}{\partial q_{e,i}} =1-w-2 q_{e,i}-\gamma q_{e,j} =0. \end{aligned}$$

Consequently, we obtain  \(q_{e,i}=\dfrac{1}{2}(1-w-q_{e,j})\). From this reaction function, we obtain

$$\begin{aligned} q_{e,i}=q_{e,j}=\frac{1-w}{2+\gamma }. \end{aligned}$$

Substituting Eq. (7) into the upstream firm’s profit function, we obtain

$$\begin{aligned} \pi _U =\frac{2 (1-w) (w-(c-x))}{\gamma +2}-k x^2. \end{aligned}$$

Because the upstream firm decides w and x to maximize its profit, the FOCs are

$$\begin{aligned} \frac{\partial \pi _U}{\partial w}&=\frac{2 (1-2 w+(c-x))}{\gamma +2} =0, \\ \frac{\partial \pi _U}{\partial x}&=\frac{2(1- w)}{2+\gamma }-2 k x =0. \end{aligned}$$

Thus, we obtain the equilibrium outcomes listed in the paper.

Mixed regime: \(IN\) or \(NI\). Without loss of generality, we assume that only \(Di\) invests. First, we derive the equilibrium quantity of final products. Because each firm decides the quantity of final products that maximizes the profit, the FOCs to maximize the profit are

$$\begin{aligned} \frac{\partial \pi _i}{\partial q_{e,i}}&=1-w-2q_{e,i}-2 \gamma q_{n,i}-\gamma q_{e,j}=0, \\ \frac{\partial \pi _i}{\partial q_{n,i}}&=1-w-2q_{n,i}-2 \gamma q_{e,i}-\gamma q_{e,j}=0, \\ \frac{\partial \pi _j}{\partial q_{e,j}}&=1-w-2q_{e,j}-\gamma (q_{e,i}+q_{n,i}) =0. \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} q_{e,i} =q_{n,i}&=\frac{1}{2(1+\gamma )}(1-w-q_{e,j}), \\ q_{e,j}&=\frac{1}{2}(1-w-\gamma (q_{e,i}+q_{n,i})). \end{aligned}$$

From the above reaction functions, we obtain

$$\begin{aligned} q_{e,i}=q_{n,i}=\frac{(1-w)(2-\gamma )}{2(2+2\gamma -\gamma ^2)};~~ q_{e,j} =\frac{1-w}{2+2\gamma -\gamma ^2}. \end{aligned}$$

Substituting Eq. (8) into the upstream firm’s profit function, we obtain

$$\begin{aligned} \pi _U =\frac{(1-w)(2-\gamma )(w-(c-x))}{2+2\gamma -\gamma ^2}-kx^2. \end{aligned}$$

Because the upstream firm decides w and x to maximize its profit, the FOCs are

$$\begin{aligned} \frac{\partial \pi _U}{\partial w}&=\frac{(3-\gamma )(-2w+1+(c-x))}{2+2\gamma -\gamma ^2} =0, \\ \frac{\partial \pi _U}{\partial x}&=\frac{(1-w)(3-\gamma )}{2+2\gamma -\gamma ^2}-2kx =0. \end{aligned}$$

Thus, we obtain the equilibrium outcomes listed in the paper.

All product-developers regime: \(II\). First, we derive the equilibrium quantity of the final product. Because each firm decides the quantity of final products that maximizes the profit, the FOCs to maximize the profit are

$$\begin{aligned} \frac{\partial \pi _{Di}}{\partial q_{e,i}}&=1-w-2q_{e,i}-2 \gamma q_{n,i}-(q_{e,j}+q_{n,j}) \gamma =0, \\ \frac{\partial \pi _{Di}}{\partial q_{n,i}}&=1-w-2q_{n,i}-2 \gamma q_{e,i}-(q_{e,j}+q_{n,j}) \gamma =0. \end{aligned}$$

Consequently, we obtain  \(q_{e,i}=q_{n,i}=\dfrac{1-w-(q_{e,j}+q_{n,j})\gamma }{2(1+\gamma )}\). From this, we obtain

$$\begin{aligned} q_{e,i}=q_{n,i}=q_{e,j}=q_{n,j} =\frac{1-w}{2 (2 \gamma +1)}. \end{aligned}$$

Substituting Eq. (9) into the upstream firm’s profit function, we obtain

$$\begin{aligned} \pi _U =\frac{2(1-w)(w-(c-x))}{2\gamma +1}-kx^2. \end{aligned}$$

Because the upstream firm decides w and x to maximize its profit, the FOCs are

$$\begin{aligned} \frac{\partial \pi _U}{\partial w}&=\frac{2 (-2 w+1+(c-x))}{2 \gamma +1} =0, \\ \frac{\partial \pi _U}{\partial x}&=\frac{2(1-w)}{(1 + 2 \gamma )}-2 k x =0. \end{aligned}$$

Thus, we obtain the equilibrium outcomes listed in the paper.

Appendix C. Other equilibrium outcomes

No one invests regime: \(NN\)

$$\begin{aligned}{} & {} {\pi }_U^{NN} = \dfrac{ (1-c)^2 k}{ 2 (\gamma +2) k-1 };\quad q_{e,i}^{NN} = \dfrac{ (1-c)k }{2 (\gamma +2) k-1};\quad {\pi }_{Di}^{NN} = \bigl ( q_{e,i}^{NN} \bigr )^2 \quad \textrm{for} ~ i=1,2, \\{} & {} CS^{NN} = \frac{(1-c)^2 (\gamma +1) k^2}{[1-2 (\gamma +2) k]^2};\quad TS^{NN} = \frac{(1-c)^2 k [(3 \gamma +7) k-1] }{[1-2 (\gamma +2) k]^2}. \end{aligned}$$

Mixed regime: \(IN\) or \(NI\)

$$\begin{aligned} {\pi }_U^{IN}&= \dfrac{ (1-c)^2 (3 - \gamma ) k }{ 4(2 +2 \gamma - \gamma ^2) k -(3 - \gamma ) }, \\ q_{e,1}^{IN}&= q_{n,1}^{IN} = \dfrac{ (1-c)(2 -\gamma ) k }{ 4(2 +2\gamma -\gamma ^2) k -(3 -\gamma ) };\quad q_{e,2}^{IN} = \dfrac{ 2 (1-c) k }{ 4(2 +2 \gamma - \gamma ^2) k -(3 - \gamma ) }, \\ {\pi }_{D1}^{IN}&= \dfrac{ 2 (1-c)^2 (2- \gamma )^2 (1 + \gamma ) k^2 }{ {\left[ 4(2 +2\gamma -\gamma ^2) k -(3 -\gamma ) \right] }^2 };\quad {\pi }_{D2}^{IN} = \left( {q}_{e,2}^{IN} \right) ^2,\\ CS^{IN}&= \frac{(1-c)^2 (\gamma ^3-7 \gamma ^2+8 \gamma +6 ) k^2}{ [\gamma + (-4 \gamma ^2+8 \gamma +8 ) k-3 ]^2}, \\ TS^{IN}&= \frac{(1-c)^2 k [ (7 \gamma ^3-33 \gamma ^2+24 \gamma +42 ) k-(\gamma -3)^2 ] }{ [\gamma +(-4 \gamma ^2+8 \gamma +8 ) k-3 ]^2}. \end{aligned}$$

Note that \(q_{e,2}^{NI}= q_{n,2}^{NI}= q_{e,1}^{IN}= q_{n,1}^{IN}\), \(q_{e,2}^{IN}= q_{e,1}^{NI}\), \({\pi }_{D2}^{NI}= {\pi }_{D1}^{IN}\), \({\pi }_{D1}^{NI}= {\pi }_{D2}^{IN}\), \(CS^{IN} = CS^{NI}\), and \(TS^{IN} = TS^{NI}\).

All product-developers regime: \(II\)

$$\begin{aligned}{} & {} {\pi }_{U}^{II} = \dfrac{(1-c)^2 k}{(4\gamma +2) k-1}, \\{} & {} q_{e, i}^{II} = q_{n, i}^{II} = \dfrac{ (1-c)k }{ 2 \left[ (4 \gamma +2) k -1 \right] };\quad {\pi }_{Di}^{II} = \dfrac{(1-c)^2 (\gamma +1) k^2}{ {2 \left[ (4 \gamma +2) k -1 \right] }^2 } \quad \textrm{for} ~ i=1,2,\\{} & {} CS^{II} =\frac{(1-c)^2 (3 \gamma +1) k^2}{2 [(4 \gamma +2) k-1]^2};\quad TS^{II} = \frac{(1-c)^2 k [(13 \gamma +7) k-2] }{2 [(4 \gamma +2) k-1]^2}. \end{aligned}$$

Appendix D. SPNE outcomes in downstream price competition.

To identify Bertrand rivalry, we attach “\(\hat{ }\)” to the variables of the equilibrium solutions.

No one invests regime: \(NN\) \(\hat{w}^{NN} =\tfrac{1-(c+1) (2-\gamma ) (\gamma +1) k}{1-2 (2-\gamma ) (\gamma +1) k}\), \(\hat{x}^{NN}=\tfrac{1-c}{2 (2-\gamma ) (\gamma +1) k-1}\), \(\hat{\pi }_{U}^{NN} =\tfrac{(1-c)^2 k}{2 (2-\gamma ) (\gamma +1) k-1}\), \(\hat{p}_{e,i}^{NN} =\tfrac{1-(\gamma +1) k (c-2 \gamma +3)}{2 \left( \gamma ^2-\gamma -2\right) k+1}\), and \(\hat{\pi }_{Di}^{NN} =\tfrac{(1-c)^2 (1-\gamma ) (\gamma +1) k^2}{\left( 2 \left( \gamma ^2-\gamma -2\right) k+1\right) ^2}\).

Mixed regime: \(IN\) or \(NI\) \(\hat{w}^{IN} =\tfrac{2 (c+1) (2 \gamma +1) ((2-\gamma ) \gamma +2) k-(\gamma (\gamma +5)+3)}{4 (2 \gamma +1) ((2-\gamma ) \gamma +2) k-(\gamma (\gamma +5)+3)}\), \(\hat{x}^{IN} =\tfrac{(1-c) (\gamma (\gamma +5)+3)}{4 (2 \gamma +1) ((2-\gamma ) \gamma +2) k-(\gamma (\gamma +5)+3)}\), \(\hat{\pi }_U^{IN} =\tfrac{(1-c)^2 (\gamma (\gamma +5)+3) k}{4 (2 \gamma +1) ((2-\gamma ) \gamma +2) k-(\gamma (\gamma +5)+3)}\), \(\hat{p}_{e,i}^{IN} =\tfrac{-(2 \gamma +1) k (c (\gamma +1) (\gamma +2)+5 (1-\gamma ) \gamma +6)+\gamma (\gamma +5)+3}{\gamma (\gamma +5)-4 (2 \gamma +1) ((2-\gamma ) \gamma +2) k+3}\), \(\hat{p}_{n,i}^{IN} =\tfrac{-(2 \gamma +1) k (c (\gamma +1) (\gamma +2)+5 (1-\gamma ) \gamma +6)+\gamma (\gamma +5)+3}{\gamma (\gamma +5)-4 (2 \gamma +1) ((2-\gamma ) \gamma +2) k+3}\), \(\hat{p}_{e,j}^{IN} =\tfrac{-2 (2 \gamma +1) k (2 \gamma c+c+2 (1-\gamma ) \gamma +3)+\gamma (\gamma +5)+3}{\gamma (\gamma +5)-4 (2 \gamma +1) ((2-\gamma ) \gamma +2) k+3}\), \(\hat{\pi }_{Di}^{IN} =\tfrac{2 (1-c)^2 (1-\gamma ) (2 \gamma +1) (3 \gamma +2)^2 k^2}{(4 (2 \gamma +1) ((2-\gamma ) \gamma +2) k-(\gamma (\gamma +5)+3))^2}\), and \(\hat{\pi }_{Dj}^{IN} =\tfrac{4 (1-c)^2 (1-\gamma ) (\gamma +1)^3 (2 \gamma +1) k^2}{(4 (2 \gamma +1) ((2-\gamma ) \gamma +2) k-(\gamma (\gamma +5)+3))^2}\).

All product-developers regime: \(II\) \(\hat{w}^{II} =\tfrac{(c+1) (3 \gamma +1) k-(\gamma +1)}{(6 \gamma +2) k-(\gamma +1)}\), \(\hat{x}^{II} =\tfrac{(1-c) (\gamma +1)}{(6 \gamma +2) k-(\gamma +1)}\), \(\hat{\pi }_U^{II} =\tfrac{(1-c)^2 (\gamma +1) k}{(6 \gamma +2) k-(\gamma +1)}\), \(\hat{q}_{e,i}^{II} = \tfrac{(3 \gamma +1) k (\gamma c+c-\gamma +3)-2 (\gamma +1)}{4 (3 \gamma +1) k-2 (\gamma +1)}\), and \(\hat{\pi }_{Di}^{II} = \tfrac{(1-c)^2 (1-\gamma ) (\gamma +1) (3 \gamma +1) k^2}{2 [2 (3 \gamma +1) k-(\gamma +1)]^2}\).

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Kawasaki, A., Mizuno, T. & Takauchi, K. Downstream new product development and upstream process innovation. J Econ 140, 209–231 (2023).

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