Downstream new product development and upstream process innovation

Abstract

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.

Appendices

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

$k_0$ 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

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

(7)

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

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

(8)

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

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

(9)

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 $\hat{\pi }_{Di}^{II} = \tfrac{(1-c)^2 (1-\gamma ) (\gamma +1) (3 \gamma +1) k^2}{2 [2 (3 \gamma +1) k-(\gamma +1)]^2}$.