A differential game analysis of R&D in oligopoly with differentiated goods under general demand and cost functions: Bertrand vs. Cournot

TL;DR

This paper models R&D competition among differentiated goods firms in oligopolies using differential games, analyzing how firm and industry R&D investments vary with the number of firms, demand, costs, and strategic interactions.

Contribution

It provides a comprehensive differential game framework for R&D in oligopolies with general demand and cost functions, comparing Bertrand and Cournot models and solution concepts.

Findings

Steady state R&D per firm decreases as the number of firms increases.

Total industry R&D investment increases with the number of firms.

Memoryless closed-loop and feedback solutions are shown to be equivalent.

Abstract

We study a dynamic oligopoly with differentiated goods by differential game approach under general demand and cost functions. We show that the steady state value of the R&D investment by each firm is decreasing with respect to the number of firms, and the steady state value of the industry R&D investment is increasing with respect to the number of firms. Also we show that if there is no spillover, whether the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger or smaller than that in the open-loop case depends on whether the strategic variables are strategic substitutes or strategic complements. Further we show that the memoryless closed-loop solution and the feedback solution (by the Hamilton-Jacobi-Bellman equation) are equivalent.

Equations

$$u(q_1(t),q_2(t),\dots ,q_n(t))+x(t).$$

$$\underset{q_1(t),q_2(t),\dots ,q_n(t)}{\max }[u(q_1(t),q_2(t),\dots ,q_n(t))+x(t)]$$

$$\sum _{i=1}^n{p}_i(t)q_i(t)+x(t)=y(t).$$

$$\frac{\partial u(q_1(t),q_2(t),\dots ,q_n(t))}{\partial q_i(t)}=p_i(t),i\in \{1,2,\dots ,n\}.$$

$$p_i(t)=p_i(q_1(t),q_2(t),\dots ,q_n(t)),i\in \{1,2,\dots ,n\}.$$

$$\frac{\partial p_i(q_1(t),q_2(t),\dots ,q_n(t))}{\partial q_j(t)}<0.$$

$$\frac{\partial p_i(q_1(t),q_2(t),\dots ,q_n(t))}{\partial q_j(t)}+\frac{{\partial }^2{p}_i(q_1(t),q_2(t),\dots ,q_n(t))}{\partial q_i(t)\partial q_j(t)}{q}_i(t)<0,j\ne i.$$

$$\frac{\partial p_i(q_1(t),q_2(t),\dots ,q_n(t))}{\partial q_j(t)}+\frac{{\partial }^2{p}_i(q_1(t),q_2(t),\dots ,q_n(t))}{\partial q_i(t)\partial q_j(t)}{q}_i(t)>0,j\ne i.$$

$$C(m_i(t),q_i(t)),i\in \{1,2,\dots ,n\}.$$

$$\frac{\partial C_i}{\partial q_i(t)}>0,\frac{\partial C_i}{\partial m_i(t)}>0,\frac{{\partial }^2{C}_i}{\partial q_i(t)\partial m_i(t)}>0.$$

$$q_i(t)=q(p_1(t),p_2(t),\dots ,p_n(t)),i\in \{1,2,\dots ,n\}.$$

$$\frac{\partial q_i(p_1(t),p_2(t),\dots ,p_n(t))}{\partial p_j(t)}>0.$$

$$\frac{\partial q_i}{\partial p_j(t)}+\left(p_i(t)-\frac{\partial C_i}{\partial q_i(t)}\right)\frac{{\partial }^2{q}_i}{\partial p_i(t)\partial p_j(t)}<0,j\ne i,$$

$$\frac{\partial q_i}{\partial p_j(t)}+\left(p_i(t)-\frac{\partial C_i}{\partial q_i(t)}\right)\frac{{\partial }^2{q}_i}{\partial p_i(t)\partial p_j(t)}>0,j\ne i.$$

$$\frac{d{m}_i(t)}{dt}=m_i(t)[-\Gamma (k_i(t),K_{-i}(t))+\delta ],$$

$$\Gamma (k_i(t),K_{-i}(t))>0,K_{-i}=\sum _{j\ne i}{k}_j(t).$$

$$\frac{\partial {\Gamma }_i}{\partial k_i(t)}>0,\frac{\partial {\Gamma }_i}{\partial K_{-i}(t)}\ge 0,$$

$$\frac{{\partial }^2{\Gamma }_i}{\partial k_i(t{)}^2}\le 0,\frac{{\partial }^2{\Gamma }_i}{\partial K_{-i}(t{)}^2}\le 0,\frac{{\partial }^2{\Gamma }_i}{\partial k_i(t)\partial K_{-i}(t)}\le 0.$$

$$\left|\frac{\partial \Gamma (k_i(t),K_{-i}(t))}{\partial k_i(t)}\right|>\left|\frac{\partial \Gamma (k_i(t),K_{-i}(t))}{\partial K_{-i}(t)}\right|.$$

$$\gamma (k_i(t)),i\in \{1,2,\dots ,n\}.$$

$$\Gamma (k^{\ast },(n-1)k^{\ast })=-\delta .$$

$$\frac{d{k}^{\ast }}{dn}=-\frac{\frac{\partial \Gamma (k^{\ast },(n-1)k^{\ast })}{\partial K_{-i}(t)}{k}^{\ast }}{\frac{\partial \Gamma (k^{\ast },(n-1)k^{\ast })}{\partial k_i(t)}+(n-1)\frac{\partial \Gamma (k^{\ast },(n-1)k^{\ast })}{\partial K_{-i}(t)}}\le 0.$$

$$\frac{dn{k}^{\ast }}{dn}=k^{\ast }+n\frac{d{k}^{\ast }}{dn}=\frac{\left(\frac{\partial \Gamma (k^{\ast },(n-1)k^{\ast })}{\partial k_i(t)}-\frac{\partial \Gamma (k^{\ast },(n-1)k^{\ast })}{\partial K_{-i}(t)}\right)k^{\ast }}{\frac{\partial \Gamma (k^{\ast },(n-1)k^{\ast })}{\partial k_i(t)}+(n-1)\frac{\partial \Gamma (k^{\ast },(n-1)k^{\ast })}{\partial K_{-i}(t)}}>0.$$

$$q_i(p_1(t),p_2(t),\dots ,p_n(t))p_i(t)-C(m_i(t),q_i(t))-\gamma (k_i(t)).$$

$$\underset{p_i(t),k_i(t)}{\max }{\int }_0^{\infty }{e}^{-\rho t}[q_i(p_1(t),p_2(t),\dots ,p_n(t))p_i(t)-C(m_i(t),q_i(t))-\gamma (k_i(t))]dt,$$

$${\mathcal{H}}_i(p_i(t),k_i(t))=$$

$$\displaystyle+\lambda_{ii}(t)m_{i}(t)[-\Gamma(k_{i}(t),K_{-i}(t))+\delta]+\sum_{j\neq i}\lambda_{ij}(t)m_{j}(t)[-\Gamma(k_{j}(t),K_{-j}(t))+\delta]\big{\}},$$

$$K_{-j}(t)=\sum _{l\ne j}{k}_l(t).$$

$${\hat{\mathcal{H}}}_i(p_i(t),k_i(t))=$$

$$+{\lambda }_{ii}(t)m_i(t)[-\Gamma (k_i(t),K_{-i}(t))+\delta ]+\sum _{j\ne i}{\lambda }_{ij}(t)m_j(t)[-\Gamma (k_j(t),K_{-j}(t))+\delta ].$$

$${\mu }_i(t)=e^{-\rho t}{\lambda }_i.$$

Full text

Keywords:

differential game; general demand and cost functions; R&D in oligopoly with differentiated goods; open-loop; closed-loop; feedback

JEL Classification No.:

C73, D43, L13.

1 Introduction

In this paper we present an analysis of R&D investment in a dynamic oligopoly model with differentiated goods by differential game approach. There are many studies of dynamic oligopoly by differential game theory, for example, Cellini and Lambertini (2003), Cellini and Lambertini (2004), Cellini and Lambertini (2005), Cellini and Lambertini (2007), Cellini and Lambertini (2011), Fujiwara (2006), Fujiwara (2008), and Lambertini (2018). Among them Cellini and Lambertini (2011) analyzed the problem of R&D investment to cost-reducing activities in a Cournot oligopoly and a Bertrand oligopoly with differentiated goods. However, most of these studies including Cellini and Lambertini (2011) used a model of linear demand functions and quadratic or linear cost functions. These assumptions are very limited. We study the problem addressed by them in an oligopoly with general demand and cost functions.

In the next section we present a model and assumptions. In Section 3 we consider the steady state level of R&D investment which is common to Bertrand and Cournot cases in the open-loop solution, the memoryless closed-loop and the feedback solution. In Section 4 we consider the open-loop solution of the R&D investment in a Bertrand oligopoly. In Section 5 we consider the open-loop solution of the R&D investment in a Cournot oligopoly, and compare the results of two cases in Section 6. We show the following results.

The steady state value of the R&D investment by each firm is decreasing with respect to the number of firms. 2. 2.

The steady state value of the industry R&D investment is increasing with respect to the number of firms. 3. 3.

The R&D investment of each firm given the cost level in the open-loop Bertrand oligopoly is larger than that in the open-loop Cournot oligopoly.

Cellini and Lambertini (2011) (also Cellini and Lambertini (2005) for an oligopoly with a homogeneous good) claim that the open-loop solution and the memoryless closed-loop solution coincide. However, as Smrkolj and Wagener (2016) point out, this claim is incorrect111Strictly speaking, Smrkolj and Wagener (2016) is a comment on Cellini and Lambertini (2009). But Cellini and Lambertini (2005) and Cellini and Lambertini (2009) use the same model, and the model of Cellini and Lambertini (2011) is similar to it. We present brief discussion about the memoryless closed-loop case in Section 7. We show the following results

Suppose that there is no spillover effect of R&D investment in a Bertrand oligopoly. If the strategic variables (prices) of the firms are strategic substitutes (or strategic complements), the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger (smaller) than that in the open-loop case. 2. 2.

Suppose that there is no spillover effect of R&D investment in a Cournot oligopoly. If the strategic variables (outputs) of the firms are strategic substitutes (or strategic complements), the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger (smaller) than that in the open-loop case.

In Section 8 we examine the feedback solutions using the Hamilton-Jacobi-Bellman equation, and show that if there is no spillover effect of R&D investment, the memoryless closed-loop solution and the feedback solution are equivalent both in the Bertrand oligopoly and the Cournot oligopoly.

Strategic substitutability and strategic complementarity

We assume that the goods of the firms are substitutes, not complements. This means that a rise in the price of one good increases the demands for other goods, and an increase in the output of one good lowers the prices of other goods. However, the strategic variables (outputs or prices) of the firms may be strategic substitutes or strategic complements.

In the Cournot oligopoly, if the reaction of a firm’s output to an increase in the output of another firm is negative (or positive), the outputs of firms are strategic substitutes (or strategic complements).

Note that if inverse demand (and direct) functions are linear, the outputs are strategic substitutes.

On the other hand, in the Bertrand oligopoly, if the reaction of the price of a firm’s good to a rise in the price of another firm’ good is negative (or positive), the prices of the goods of firms are strategic substitutes (or strategic complements).

Note that if direct demand (and inverse demand) functions are linear, the prices are strategic complements.

2 The model

Consider an oligopoly with $n$ firms in which at any $t\in[0,\infty)$ they produce differentiated goods to maximize their discounted profits. The goods are substitutes. The firms are called Firms 1, 2, $\dots$, $n$. Let $q_{i}(t)$ be the output of Firm $i,\ i\in\{1,2,\dots,n\}$, $p_{i}(t)$ be the price of the good of Firm $i,\ i\in\{1,2,\dots,n\}$, at $t$. The utility of a representative consumer is

[TABLE]

$x(t)$ is the consumption of a numeraire good. Let $y(t)$ be his income. Then, the utility maximization problem is

[TABLE]

subject to

[TABLE]

The conditions for utility maximization are

[TABLE]

From them the inverse demand functions are obtained as follows.

[TABLE]

We have $\frac{\partial p_{i}(q_{1}(t),q_{2}(t),\dots,q_{n}(t))}{\partial q_{i}(t)}<0,\ i\in\{1,2,\dots,n\}$, and since the goods are substitutes

[TABLE]

If the outputs of the firms are strategic substitutes,

[TABLE]

If they are strategic complements,

[TABLE]

The production cost of Firm $i$ is

[TABLE]

$m_{i}(t)$ is a parameter which represents the current cost of Firm $i$. Denote $C(m_{i}(t),q_{i}(t))$ by $C_{i}$. It satisfies

[TABLE]

Inverting the inverse demand functions, the direct demand functions are obtained as follows.

[TABLE]

We have $\frac{\partial q_{i}(p_{1}(t),p_{2}(t),\dots,p_{n}(t))}{\partial p_{i}(t)}<0,\ i\in\{1,2,\dots,n\}$, and since the goods are substitutes

[TABLE]

If the prices of the goods of the firms are strategic substitutes,

[TABLE]

If they are strategic complements,

[TABLE]

Let $k_{i}(t)$ be the R&D investment by Firm $i$. The moving of $m_{i}(t)$ is governed by

[TABLE]

where

[TABLE]

$\delta\in(0,1)$ is a constant depreciation rate. Denote $\Gamma(k_{i}(t),K_{-i}(t))$ by $\Gamma_{i}$. We assume that $\Gamma_{i}$ is strictly increasing and concave, that is,

[TABLE]

and

[TABLE]

Also we assume

[TABLE]

This means that the direct effect of R&D investment is larger than the spillover effect.

The R&D cost of Firm $i$ is

[TABLE]

We assume that it is strictly increasing and strictly convex, that is, $\gamma^{\prime}(k_{i}(t))>0$ and $\gamma^{\prime\prime}(k_{i}(t))>0$.

3 The steady state R&D investment

Let $k^{*}$ be the steady state value of $k_{i}(t)$. At the steady state the following equation holds.

[TABLE]

From this we obtain

[TABLE]

Also

[TABLE]

They are because $\frac{\partial\Gamma(k^{*},(n-1)k^{*})}{\partial k_{i}(t)}>0$, $\frac{\partial\Gamma(k^{*},(n-1)k^{*})}{\partial K_{-i}(t)}\geq 0$, $\left|\frac{\partial\Gamma(k^{*},(n-1)k^{*})}{\partial k_{i}(t)}\right|>\left|\frac{\partial\Gamma(k^{*},(n-1)k^{*})}{\partial K_{-i}(t)}\right|$.

These results mean that the steady state value of the R&D investment by each firm is decreasing with respect to the number of firms, while the total R&D investment is increasing with respect to the number of firms.

Summarizing the results in the following proposition.

Proposition 1**.**

The steady state value of the R&D investment by each firm is decreasing with respect to the number of firms. 2. 2.

The steady state value of the industry R&D investment is increasing with respect to the number of firms.

Note that these conclusions hold in both the Bertrand oligopoly and the Cournot oligopoly and in the open-loop case, the memoryless closed-loop case and the feedback case because (2) holds in all cases.

4 R&D in a dynamic oligopoly: Bertrand competition

We seek to the solution of the open-loop approach in the Bertrand oligopoly. The instantaneous profit of Firm $i$ is written as

[TABLE]

The objective of Firm $i$ is

[TABLE]

subject to (1).

The present value Hamiltonian function for Firm $i,\ i\in\{1,2,\dots,n\}$, is

[TABLE]

where

[TABLE]

This includes $k_{i}(t)$.

The current value Hamiltonian function for Firm $i,\ i\in\{1,2,\dots,n\}$, is

[TABLE]

Let

[TABLE]

$\mu_{i}(t)$ is the costate variable. Denote $\hat{\mathcal{H}}_{i}(p_{i}(t),k_{i}(t))$ by $\hat{\mathcal{H}}_{i}$.

The first order conditions for Firm $i$ are

[TABLE]

and

[TABLE]

The second order condition for Firm $i$ about the price choice is

[TABLE]

Its second order condition about the R&D investment choice is

[TABLE]

The adjoint conditions are

[TABLE]

and

[TABLE]

We have

[TABLE]

[TABLE]

At the steady state

[TABLE]

$\frac{\partial\lambda_{ii}}{\partial t}=0$ and $\frac{\partial\lambda_{ij}}{\partial t}=0,\ i\in\{1,2,\dots,n\},\ j\neq i$. By symmetry of the oligopoly we can assume $\lambda_{ii}(t)=\lambda_{jj}(t)$ for $j\neq i$, $\lambda_{ij}(t)=\lambda_{il}(t)=\lambda_{ji}(t)$ for $j,l\neq i$, $\frac{\partial k_{j}(t)}{\partial m_{i}(t)}=\frac{\partial k_{l}(t)}{\partial m_{j}(t)}$ for $l\neq j$, $m_{i}(t)=m_{j}(t)$ for $j\neq i$, and so on. Denote the steady state values of $\lambda_{ii}$, $\lambda_{ij}$, $p_{i}(t)$, $k_{i}(t)$ and $m_{i}(t)$ by $\lambda_{own}$, $\lambda_{other}$, $p^{*}_{B}$, $k^{*}_{B}$ and $m^{*}_{B}$. Then, (10) and (11) are reduced to

[TABLE]

[TABLE]

and (8) and (9) are rewritten as

[TABLE]

and

[TABLE]

Since $\frac{\partial C_{i}}{\partial m_{i}(t)}<0$, we have $\lambda_{own}>0$. The first order condition for the choice of $k_{i}(t)$, (6), is reduced to

[TABLE]

This means

[TABLE]

Linear and quadratic example

According to Cellini and Lambertini (2011), assume that the direct demand functions are

[TABLE]

where $\varphi=1+(n-1)s.$ The production cost of Firm $i,\ i\in\{1,2,\dots,n\}$, is

[TABLE]

the R&D cost of Firm $i,\ i\in\{1,2,\dots,n\}$, is

[TABLE]

The moving of $m_{i}(t)$ is governed by

[TABLE]

(3) and (4) are reduced to

[TABLE]

and

[TABLE]

From (12)

[TABLE]

Since

[TABLE]

in this example, we have

[TABLE]

5 R&D in a dynamic oligopoly: Cournot competition

We seek to the solution of the open-loop approach in the Cournot oligopoly. The instantaneous profit of Firm $i$ is written as

[TABLE]

The objective of Firm $i$ is

[TABLE]

subject to (1).

The present value Hamiltonian function for Firm $i,\ i\in\{1,2,\dots,n\}$ is

[TABLE]

where

[TABLE]

The current value Hamiltonian function for Firm $i,\ i\in\{1,2,\dots,n\}$ is

[TABLE]

Let

[TABLE]

$\mu_{i}(t)$ is the costate variable. Denote $\hat{\mathcal{H}}_{i}(q_{i}(t),k_{i}(t))$ by $\hat{\mathcal{H}}_{i}$.

The first order conditions for Firm $i$ are

[TABLE]

and

[TABLE]

The second order condition for Firm $i$ about the output choice is

[TABLE]

Its second order condition about the R&D investment choice is

[TABLE]

The adjoint conditions are the same those in the Bertrand oligopoly. Denote the steady state values of $\lambda_{ii}$, $\lambda_{ij}$, $q_{i}(t)$, $k_{i}(t)$ and $m_{i}(t)$ by $\lambda_{own}$, $\lambda_{other}$, $q^{*}_{C}$, $k^{*}_{C}$ and $m^{*}_{C}$. Similarly to the Bertrand oligopoly we obtain

[TABLE]

Linear and quadratic example

Assume that the inverse demand functions are

[TABLE]

They are derived from the direct demand functions in the previous example.

The production cost of Firm $i,\ i\in\{1,2,\dots,n\}$ is

[TABLE]

the R&D investment cost of Firm $i,\ i\in\{1,2,\dots,n\}$, is

[TABLE]

The moving of $m_{i}(t)$ is governed by

[TABLE]

From (15)

[TABLE]

Since $q^{*}_{C}=\frac{a-m^{*}_{C}}{2+(n-1)s}$ in this example,

[TABLE]

6 Comparison of Bertrand and Cournot given the cost level

In the linear and quadratic example, as Cellini and Lambertini (2011) shows, we have

[TABLE]

We examine the general case. From (12) and (15) $k^{*}_{B}$ (or $k^{*}_{C}$, the same hereinafter) given the cost level $m^{*}_{B}=m^{*}_{C}$ is obtained by the following equation.

[TABLE]

By the second order conditions about the R&D investment, (7) and (14), the left-hand side of (16) is decreasing with respect to $k^{*}_{B}$. On the other hand, since $\frac{\partial^{2}C(m^{*}_{B},q^{*}_{B})}{\partial q_{i}(t)\partial m_{i}(t)}>0$, $\frac{\partial C(m^{*}_{B},q^{*}_{B})}{\partial m_{i}(t)}$ is increasing with respect to $q^{*}_{B}$. Therefore, the larger is $q^{*}_{B}$, the larger is $k^{*}_{B}$.

The first order condition for the price choice in the Bertrand oligopoly is

[TABLE]

The first order condition for the output choice in the Cournot oligopoly is

[TABLE]

Assume $q_{i}(t)=q^{*}_{B}$ and $p_{i}(t)=p^{*}_{B}$ for all $i\in\{1,2,\dots,n\}$. From (17)

[TABLE]

Substituting this into the left-hand side of (18) (assuming $q^{*}_{C}=q^{*}_{B}$) yields

[TABLE]

Since $p^{*}_{B}-\frac{\partial C(m_{i}(t),q_{i}(t))}{\partial q_{i}(t)}>0$, and from (43) in Appendix 1

[TABLE]

(19) is negative, and then the output of each firm in the Bertrand oligopoly is larger than that in the Cournot oligopoly. Thus, we obtain the following proposition.

Proposition 2**.**

The R&D investment of each firm given the cost level in the open-loop Bertrand oligopoly is larger than that in the open-loop Cournot oligopoly.

7 Memoryless closed-loop solution without spillover

7.1 Bertrand oligopoly

We seek to the solution of the memoryless closed-loop approach in the Bertrand oligopoly. For simplicity, we assume

[TABLE]

that is, there is no spillover effect of R&D investment. The first order conditions for Firm $i$ are

[TABLE]

and

[TABLE]

The adjoint conditions are

[TABLE]

and

[TABLE]

We have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

$\frac{\partial p_{j}(t)}{\partial m_{i}(t)}$ is obtained by (44) in Appendix 2. If the prices of goods of the firms are strategic substitutes $\left(\frac{\partial q_{j}}{\partial p_{i}(t)}+\left(p_{j}(t)-\frac{\partial C(m_{j}(t),q_{j}(t))}{\partial q_{j}(t)}\right)\frac{\partial^{2}q_{j}}{\partial p_{i}(t)\partial p_{j}(t)}<0\right)$, $\frac{\partial p_{j}(t)}{\partial m_{i}(t)}<0$, and if they are strategic complements $\left(\frac{\partial q_{j}}{\partial q_{i}(t)}+\left(p_{j}(t)-\frac{\partial C(m_{j}(t),q_{j}(t))}{\partial q_{j}(t)}\right)\frac{\partial^{2}q_{j}}{\partial p_{i}(t)\partial p_{j}(t)}>0\right)$, $\frac{\partial p_{j}(t)}{\partial m_{i}(t)}>0$.

At the steady state we have

[TABLE]

$\frac{\partial\lambda_{ii}}{\partial t}=0$ and $\frac{\partial\lambda_{ij}}{\partial t}=0,\ i\in\{1,2,\dots,n\},\ j\neq i$. Then, (24) and (25) are reduced to

[TABLE]

[TABLE]

By symmetry of the oligopoly we can assume $\lambda_{ii}(t)=\lambda_{jj}(t)$ for $j\neq i$, $\lambda_{ij}(t)=\lambda_{il}(t)=\lambda_{ji}(t)$ for $j,l\neq i$, $\frac{\partial k_{j}(t)}{\partial m_{i}(t)}=\frac{\partial k_{l}(t)}{\partial m_{j}(t)}$ for $l\neq j$, $m_{i}(t)=m_{j}(t)$ for $j\neq i$, $p_{j}(t)=p_{i}(t)$, and so on. Denote the steady state values of $\lambda_{ii}$, $\lambda_{ij}$, $p_{i}(t)$ and $k_{i}(t)$ by $\lambda_{own}$, $\lambda_{other}$, $p^{**}_{B}$ and $k^{**}_{B}$. Then, (23) is rewritten as

[TABLE]

The first order condition for the choice of $k_{i}(t)$, (22), is reduced to

[TABLE]

This means

[TABLE]

The first order condition for the price choice in the memoryless closed-loop case, (21), is the same as that, (5), in the open-loop case. Thus, we have $q^{**}_{B}=q^{*}_{B}$ given the value of $m_{i}(t)$.

Since $\frac{\partial q_{i}}{\partial p_{j}(t)}>0$, $\gamma^{\prime\prime}(k_{i}(t))>0$, $\frac{\partial C(m_{i}(t),q_{i}(t))}{\partial m_{i}(t)}>0$ and $\frac{\partial\Gamma(k^{**}_{B},(n-1)k^{**}_{B})}{\partial k_{i}(t)}>0$, if the prices of goods of the firms are strategic substitutes ($\frac{\partial p_{j}(t)}{\partial m_{i}(t)}<0$), we have $k^{**}_{B}>k^{*}_{B}$; and if the prices of goods of the firms are strategic complements ($\frac{\partial p_{j}(t)}{\partial m_{i}(t)}>0$), we have $k^{**}_{B}<k^{*}_{B}$.

Note that in the case of linear demand functions the prices of the goods of the firms are strategic complements, and so the R&D investment of each firm given the cost level in the memoryless closed-loop case is smaller than that in the open-loop case.

7.2 Cournot oligopoly

We seek to the solution of the memoryless closed-loop approach in the Cournot oligopoly. Similarly to the previous case, for simplicity, we assume

[TABLE]

The first order conditions for Firm $i$ are

[TABLE]

and

[TABLE]

The adjoint conditions are

[TABLE]

and

[TABLE]

We have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

$\frac{\partial q_{j}(t)}{\partial m_{i}(t)}$ is obtained by (45) in Appendix 3. If the outputs of the firms are strategic substitutes $\left(\frac{\partial p_{j}}{\partial q_{i}(t)}+\frac{\partial^{2}p_{j}}{\partial q_{i}(t)\partial q_{j}(t)}q_{j}(t)<0\right)$, $\frac{\partial q_{j}(t)}{\partial m_{i}(t)}>0$, and if they are strategic complements $\left(\frac{\partial p_{j}}{\partial q_{i}(t)}+\frac{\partial^{2}p_{j}}{\partial q_{i}(t)\partial q_{j}(t)}q_{j}(t)<0\right)$, $\frac{\partial q_{j}(t)}{\partial m_{i}(t)}<0$.

At the steady state we have

[TABLE]

$\frac{\partial\lambda_{ii}}{\partial t}=0$ and $\frac{\partial\lambda_{ij}}{\partial t}=0,\ i\in\{1,2,\dots,n\},\ j\neq i$. Then, (30) and (31) are reduced to

[TABLE]

[TABLE]

By symmetry of the oligopoly we can assume $\lambda_{ii}(t)=\lambda_{jj}(t)$ for $j\neq i$, $\lambda_{ij}(t)=\lambda_{il}(t)=\lambda_{ji}(t)$ for $j,l\neq i$, $\frac{\partial k_{j}(t)}{\partial m_{i}(t)}=\frac{\partial k_{l}(t)}{\partial m_{j}(t)}$ for $l\neq j$, $m_{i}(t)=m_{j}(t)$ for $j\neq i$, $q_{j}(t)=q_{i}(t)$, and so on. Denote the steady state values of $\lambda_{ii}$, $\lambda_{ij}$, $q_{i}(t)$ and $k_{i}(t)$ by $\lambda_{own}$, $\lambda_{other}$, $q^{**}_{C}$ and $k^{**}_{C}$. Then, (29) is rewritten as

[TABLE]

The first order condition for the choice of $k_{i}(t)$, (28), is reduced to

[TABLE]

This means

[TABLE]

The first order condition for the output choice in the memoryless closed-loop case, (27), is the same as that, (13), in the open-loop case. Thus, we have $q^{**}_{C}=q^{*}_{C}$ given the value of $m_{i}(t)$.

Since $\frac{\partial p_{i}}{\partial q_{j}(t)}<0$, $\gamma^{\prime\prime}(k_{i}(t))>0$, $\frac{\partial C(m_{i}(t),q_{i}(t))}{\partial m_{i}(t)}>0$ and $\frac{\partial\Gamma(k^{**}_{C},(n-1)k^{**}_{C})}{\partial k_{i}(t)}>0$, if the outputs of the firms are strategic substitutes ($\frac{\partial q_{j}(t)}{\partial m_{i}(t)}>0$), we have $k^{**}_{C}>k^{*}_{C}$; and if the outputs of the firms are strategic complements ($\frac{\partial q_{j}(t)}{\partial m_{i}(t)}<0$), we have $k^{**}_{C}<k^{*}_{C}$.

Note that in the case of linear inverse demand functions the outputs of the firms are strategic substitutes, and so the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger than that in the open-loop case.

Summarizing the results in this section.

Proposition 3**.**

Suppose that there is no spillover effect of R&D investment in the Bertrand oligopoly. If the prices of goods of the firms are strategic substitutes (or strategic complements), the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger (or smaller) than that in the open-loop case. 2. 2.

Suppose that there is no spillover effect of R&D investment in the Cournot oligopoly. If the outputs of the firms are strategic substitutes (or strategic complements), the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger (smaller) than that in the open-loop case. 3. 3.

If the demand functions are linear in the Bertrand oligopoly, the R&D investment of each firm given the cost level in the memoryless closed-loop case is smaller than that in the open-loop case. 4. 4.

If the inverse demand functions are linear in the Cournot oligopoly, the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger than that in the open-loop case.

8 Feedback solution without spillover

8.1 Bertrand oligopoly

We consider a solution of feedback approach in the Bertrand oligopoly using the Hamilton-Jacobi-Bellman (HJB) equation. Similarly to the memoryless closed-loop case, we assume

[TABLE]

that is, there is no spillover effect of the R&D investments. Let $V_{i}(m_{1}(t),m_{2}(t),\dots,m_{n}(t))$ be the value function of Firm $i,\ i\in\{1,2,\dots,n\}$. The HJB equation for Firm $i$ is written as

[TABLE]

The first order conditions are

[TABLE]

and

[TABLE]

From this

[TABLE]

Substituting this into (33), using symmetry, yields

[TABLE]

This is an identity. Differentiating it with respect to $m_{i}(t)$ yields

[TABLE]

At the steady state $-\Gamma_{i}+\delta=-\Gamma_{j}+\delta=0$. Thus, using (34), we get

[TABLE]

From (35) and (36), we obtain

[TABLE]

This is the same as (26) in the memoryless closed-loop case. Therefore, we get the following proposition.

Proposition 4**.**

If there is no spillover effect of R&D investment, the memoryless closed-loop solution and the feedback solution in the Bertrand oligopoly are equivalent.

8.2 Cournot oligopoly

We consider a solution of feedback approach in the Cournot oligopoly using the HJB equation. Similarly to the previous section we assume

[TABLE]

Let $V_{i}(m_{1}(t),m_{2}(t),\dots,m_{n}(t))$ be the value function of Firm $i,\ i\in\{1,2,\dots,n\}$. The HJB equation for Firm $i$ is written as

[TABLE]

The first order conditions are

[TABLE]

and

[TABLE]

From this

[TABLE]

Substituting this into (37), using symmetry, yields

[TABLE]

This is an identity. Differentiating it with respect to $m_{i}(t)$ yields

[TABLE]

At the steady state $-\Gamma_{i}+\delta=-\Gamma_{j}+\delta=0$. Thus, using (38), we get

[TABLE]

From (39) and (40), we obtain

[TABLE]

This is the same as (32) in the memoryless closed-loop case. Therefore, we get the following proposition.

Proposition 5**.**

If there is no spillover effect of R&D investment, the memoryless closed-loop solution and the feedback solution in the Cournot oligopoly are equivalent.

9 Concluding Remark

In this paper we analyzed the memoryless closed-loop solution and the feedback solution only when there is no spillover effect of R&D investment. In the future research we want to investigate the relations among the open-loop solution, the closed-loop solution and the feedback solution in a case with spillovers.

Acknowledgment

This work was supported by Japan Society for the Promotion of Science KAKENHI Grant Number 18K01594.

Appendix 1: Derivation of (20)

The direct and the inverse demand functions are as follows.

[TABLE]

[TABLE]

We omit $t$. Differentiating (41) with respect to $p_{i}$ given $p_{j},\ j\in\{1,2,\dots,n\},\ j\neq i$, yields

[TABLE]

and

[TABLE]

Since by symmetry $\frac{\partial p_{i}}{\partial q_{j}}=\frac{\partial p_{j}}{\partial q_{i}}=\frac{\partial p_{j}}{\partial q_{k}}$, $\frac{\partial p_{j}}{\partial q_{j}}=\frac{\partial p_{i}}{\partial q_{i}}$ and $\frac{\partial q_{k}}{\partial p_{i}}=\frac{\partial q_{j}}{\partial p_{i}}$ at the steady state, they are rewritten as

[TABLE]

and

[TABLE]

From them we get

[TABLE]

and

[TABLE]

because $\frac{\partial q_{i}}{\partial p_{j}}=\frac{\partial q_{j}}{\partial p_{i}}$ and $\frac{\partial q_{i}}{\partial p_{i}}=\frac{\partial q_{j}}{\partial p_{j}}$ at the steady state.

From (42)

[TABLE]

Appendix 2: Derivation of $\frac{\partial p_{j}(t)}{\partial m_{i}(t)}$.

Suppose a state such that $p_{1}(t)=p_{2}(t)=\dots=p_{n}(t)$. The first order conditions for Firm $i$ and Firm $j,\ j\neq i$, are

[TABLE]

and

[TABLE]

Denote $q_{i}(p_{1}(t),p_{2}(t),\dots,p_{n}(t))$ by $q_{i}$. Differentiating them with respect to $m_{i}(t)$ yields

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From them we obtain

[TABLE]

and

[TABLE]

We have $\varphi_{i}<0$, $\varphi_{j}<0$ and $\varphi_{i}\varphi_{j}-\psi_{i}\psi_{j}>0$. If the prices of goods of the firms are strategic substitutes $\left(\psi_{j}<0\right)$, $\frac{\partial p_{j}(t)}{\partial m_{i}(t)}<0$, and if they are strategic complements $\left(\psi_{j}>0\right)$, $\frac{\partial p_{j}(t)}{\partial m_{i}(t)}>0$.

Appendix 3: Derivation of $\frac{\partial q_{j}(t)}{\partial m_{i}(t)}$.

Suppose a state such that $q_{1}(t)=q_{2}(t)=\dots=q_{n}(t)$. The first order conditions for Firm $i$ and Firm $j,\ j\neq i$ are

[TABLE]

and

[TABLE]

Denote $p_{i}(q_{1}(t),q_{1}(t),\dots,q_{n}(t))$ by $p_{i}$. Differentiating them with respect to $m_{i}(t)$ yields

[TABLE]

and

[TABLE]

From them we obtain

[TABLE]

and

[TABLE]

where

[TABLE]

If the outputs of the firms are strategic substitutes $\left(\frac{\partial p_{j}}{\partial q_{i}(t)}+\frac{\partial^{2}p_{j}}{\partial q_{i}(t)\partial q_{j}(t)}q_{j}(t)<0\right)$, $\frac{\partial q_{j}(t)}{\partial m_{i}(t)}>0$, and if they are strategic complements $\left(\frac{\partial p_{j}}{\partial q_{i}(t)}+\frac{\partial^{2}p_{j}}{\partial q_{i}(t)\partial q_{j}(t)}q_{j}(t)>0\right)$, $\frac{\partial q_{j}(t)}{\partial m_{i}(t)}<0$.