Spillovers and strategic commitment in R&D

Abstract

This paper considers a one-stage Cournot duopoly of R&D. We characterize the Nash equilibrium of the one-stage game and provide a comparison with the two-stage version of the same Cournot model of R&D/product market competition. We look at R&D expenditures, profits, output and welfare. Under perfect symmetry, the one-stage model always leads to higher profits when the spillover parameter is not equal to 1/2. Moreover, the one-stage model implies more R&D expenditure and higher welfare if and only if the spillover parameter is greater than 1/2. The insights are robust to an n -firm generalization, but the differences between the one-stage game and the two-stage game disappear as the market becomes perfectly competitive.

Appendix

Proof for Proposition 2

First, we need to replicate the four variable comparisons for R&D expenditure, consumer surplus, profit and welfare between the one-stage model and two-stage model.

The difference in R&D expenditure: $\bar{Y}_n-Y^*_n$ can be simplified as

$$\begin{aligned} \frac{b (a-c)^2 (-1+n) (1+n) (-1+2 \beta ) \gamma ^2 (2 (n+\beta -n \beta )+b (1+n) (-1-3 n+2 (-1+n) \beta ) \gamma )}{ (1-2 b (1+n) \gamma )^2 \left( n+\beta -n \beta -b (1+n)^2 \gamma \right) ^2}. \end{aligned}$$

We claim that $(2 (n+\beta -n \beta )+b (1+n) (-1-3 n+2 (-1+n) \beta ) \gamma )<0$, so that $\bar{Y}_n-Y^*_n$ has the same sign as $(1-2 \beta )$, which leads to the conclusion as in Proposition 1. The inequality can be simplified to be $\frac{2 (n-(n-1)\beta )}{(1+n) (1+3 n-2 (n-1) \beta ) }<b \gamma$. By Assumptions (A2) (first inequality) and (A1) (second inequality), $b \gamma>\frac{a n}{c (1+n)^2}>\frac{n^2}{ (1+n)^2}> \frac{2 (n-(n-1)\beta )}{(1+n)(1+3 n-2 (n-1) \beta ) }$, the last inequality can be verified to hold for all $n\ge 2$ and $0<\beta <1$.

Since consumer surplus $\int _{0}^{2q^{*}}(a-bt)dt$ increases in the firm’s output, while the output increases in effective cost reduction, $q^*= \frac{(a-c+ \text {effective cost reduction})}{b(n+1)}$, and effective cost reduction increases in R&D expenditure. So CS has the same sign comparison as the R&D expenditure. The same logic has been employed in the duopoly scenario.

Note that ${\bar{Y}}_n=\frac{(a-c)^2 \gamma }{\left( -1+\frac{b (1+n)^2 \gamma }{n (1-\beta )+\beta }\right) ^2}\rightarrow 0$ when $n\rightarrow \infty$. Also $Y^*_n=\frac{\gamma (a-c)^2}{(2b(n+1)\gamma -1)^2}\rightarrow 0$ when $n\rightarrow \infty$. So ${\bar{Y}}_n-Y^*_n\rightarrow 0$. So the differences in R&D expenditure and CS disappear as $n\rightarrow \infty$.

The difference in profit, $\bar{\pi }_n-\pi ^*_n$ can be simplified as

$$\begin{aligned} -\frac{b (a-c)^2 (-1+n)^2 \gamma ^2 (1-2 \beta )^2 \left( -1-2 n-\beta +n \beta +3 b (1+n)^2 \gamma \right) }{(1+(-1+n) \beta ) (1-2 b (1+n)\gamma )^2 \left( n+\beta -n \beta -b (1+n)^2 \gamma \right) ^2}. \end{aligned}$$

Except for the negative sign in the front, all other terms are non-negative. Particularly, the term $\left( -1-2 n-\beta +n \beta +3 b (1+n)^2 \gamma \right) >0$ requires $3 b (1+n)^2 \gamma >1+2 n+\beta -n \beta$, which requires $b \gamma >\frac{1+2n+\beta -n \beta }{3 (1+n)^2 }$. We have shown right before, in proving for the R&D expenditure, that $b \gamma >\frac{n^2 }{ (1+n)^2}$, and one can verify that $\frac{n^2}{ (1+n)^2}>\frac{1+2 n+\beta -n \beta }{3 (1+n)^2 }$ always holds for all $n\ge 2$ and $0<\beta <1$. So $b \gamma >\frac{1+2n+\beta -n \beta }{3 (1+n)^2 }$ is proved. So $\bar{\pi }_n-\pi ^*_n\le 0$, while the equality holds only when $\beta =\frac{1}{2}$.

For the welfare, ${\bar{W}}_n-W^*_n$ can be simplified as

$$\begin{aligned} \frac{(1-2 \beta ) b (a-c)^2 (-1+n) n (2+n) \gamma ^2 \left( -1-3 n-2 \beta +2 n \beta +4 b (1+n)^2 \gamma \right) }{2 (1-2 b (1+n) \gamma )^2 \left( n+\beta -n \beta -b (1+n)^2 \gamma \right) ^2} \end{aligned}$$

We claim that ${\bar{W}}_n-W^*_n$ has the same sign as $1-2\beta$, which is stated in the Proposition. To validate that, we need $-1-3 n-2 \beta +2 n \beta +4 b (1+n)^2 \gamma >0$, or $b \gamma >\frac{1+3 n+2\beta -2n \beta }{4 \text { }(1+n)^2 }$, which can be verified to be true, since $b \gamma >\frac{ n^2}{ (1+n)^2}$ and $\frac{n^2}{ (1+n)^2}>\frac{1+3 n+2\beta -2n \beta }{4 (1+n)^2 }$ for all $n\ge 2$ and $0<\beta <1$.

Now, we want to show that all these differences will disappear, moreover, in a quite monotonic manner, as n approaches infinity. First, let us show this for the R&D expenditure. Differentiating $\bar{Y}_n-Y^*_n$ with respect to n, it equals $2 b (a-c)^2 \gamma ^2 A$, where A is the following term:

$$\begin{aligned} \frac{2 \left( \frac{n+1}{2} \right) ^3}{\left( \frac{n+1}{2} (-1+2 b (1+n) \gamma ) \right) ^3}-\frac{(1+n) (1+n (-1+\beta )-3 \beta ) (n (-1+\beta )-\beta )}{\left( -n-\beta +n \beta +b (1+n)^2 \gamma \right) ^3} \end{aligned}$$

It can be verified that for the denominators, $\frac{n+1}{2} (-1+2 b (1+n) \gamma )>-n-\beta +n \beta +b (1+n)^2 \gamma >0$, and for the numerators, $\frac{(n+1)^2}{ 4}<(-1+n (1-\beta )+3 \beta ) (n (1-\beta )+\beta )$, two always hold for all $n\ge 3$ and $0<\beta <\frac{1}{2}$. So the sign of A should be negative. Thus $\bar{Y}_n-Y^*_n$ decreases in n for $0<\beta < \frac{1}{2}$, and since $\bar{Y}_n-Y^*_n\ge 0$ for $0<\beta <\frac{1}{2}$ and $\bar{Y}_n-Y^*_n\rightarrow 0$ when $n\rightarrow \infty$, we conclude that $\bar{Y}_n-Y^*_n$ decreases in n and approaches 0 when $n\rightarrow \infty$ . Similarly, one can verify that the two inequalities for the denominators and the numerators flip signs when $1>\beta >\frac{1}{2}$, so that $\bar{Y} _n-Y^*_n$ increases in n for all $n\ge 3$ and $1>\beta >\frac{1}{2}$. Combined with the fact that $\bar{Y}_n-Y^*_n\le 0$ for $1>\beta >\frac{1}{2}$ , it proves that $\bar{Y}_n-Y^*_n$ increases in n to 0 as $n\rightarrow \infty$.

Because of the relation between the R&D expenditure and the consumer surplus, as argued shortly above, the consumer surplus should follow the same pattern as the R&D expenditure when $n\rightarrow \infty$.

The monotonic tendency of disappearing difference between the one-stage and two-stage models for the profit and welfare can also be proved, by playing with the differentiation terms, the inequalities and the assumptions, but they are too bulky to be presented here, so an interested reader can request from the authors a formal proof. Here, we want to give a last account on the profit and welfare as to simply prove what is stated in Proposition 2, which, the readers may notice, contains no account of monotonicity. We have shown that $\bar{Y}_n-Y^*_n\rightarrow 0$, which implies $\bar{q} _n-q^*_n\rightarrow 0$, one discussed already when we dealt with consumer surplus. Now that the profit is only affected by quantity q and the effective cost reduction achieved by Y, one immediately gets $\bar{\pi } _n-\pi ^*_n\rightarrow 0$. Lastly, as the differences of both consumer surplus and profit disappear in perfect competition, it follows that $\bar{W} _n-W^*_n\rightarrow 0$. And we have proved Proposition 2. $\square$