Advertising in an oligopoly with differentiated goods under general demand and cost functions: A differential game approach

TL;DR

This paper analyzes advertising strategies in a dynamic oligopoly with differentiated goods using differential game theory, highlighting how solution types depend on strategic interactions and spillover effects.

Contribution

It introduces a differential game framework to compare open-loop and closed-loop advertising strategies under general demand and cost functions.

Findings

Open-loop and closed-loop solutions differ based on strategic substitutes or complements.

Memoryless closed-loop and feedback solutions are equivalent without spillover effects.

The analysis provides insights into optimal advertising in differentiated goods markets.

Abstract

We present an analysis of advertising activities in a dynamic oligopoly with differentiated goods by differential game approach under general demand and cost functions. Mainly we show the following results. The comparison of the open-loop solution and that of the closed-loop solution depends on whether the outputs of the firms are strategic substitutes or strategic complements, and the memoryless closed-loop solution and the feedback solution are equivalent when there is no spillover effect of advertising activities.

Equations

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

$$\frac{\partial p_i(A_i(t),q_1(t),q_2(t),\dots ,q_n(t))}{\partial q_i(t)}<0,\frac{\partial p_i(A_i(t),q_1(t),q_2(t),\dots ,q_n(t))}{\partial q_j(t)}<0,j\ne i,$$

$$\frac{\partial p_i(A_i(t),q_1(t),q_2(t),\dots ,q_n(t))}{\partial A_i(t)}>0,$$

$$\frac{\partial p_i}{\partial A_i(t)}+\frac{{\partial }^2{p}_i}{\partial A_i(t)\partial q_i(t)}{q}_i(t)>0.$$

$$\frac{\partial p_j}{\partial q_i(t)}+\frac{{\partial }^2{p}_j}{\partial q_i(t)\partial q_j(t)}{q}_j(t)<0,j\ne i,$$

$$\frac{\partial p_j}{\partial q_i(t)}+\frac{{\partial }^2{p}_j}{\partial q_i(t)\partial q_j(t)}{q}_j(t)>0,j\ne i.$$

$$c(q_i(t)),i\in \{1,2,\dots ,n\},c(q_i(t))>0.$$

$$\frac{d{A}_i(t)}{dt}=\Gamma (k_i(t),K_{-i}(t))-\delta A_i(t),i\in \{1,2,\dots ,n\},\Gamma (k_i(t),K_{-i}(t))>0,$$

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

$$\{\begin{array}{rl}& \frac{\partial \Gamma (k_i(t),K_{-i}(t))}{\partial k_i(t)}>0,\frac{\partial \Gamma (k_i(t),K_{-i}(t))}{\partial K_{-i}(t)}>0,\frac{{\partial }^2\Gamma (k_i(t),K_{-i}(t))}{\partial k_i(t{)}^2}\le 0,\\ & \frac{{\partial }^2\Gamma (k_i(t),K_{-i}(t))}{\partial k_i(t{)}^2}+(n-1)\frac{{\partial }^2\Gamma (k_i(t),K_{-i}(t))}{\partial k_i(t)\partial K_{-i}(t)}\le 0,\\ & \left|\frac{\partial \Gamma (k_i(t),K_{-i}(t))}{\partial k_i(t)}\right|>\left|\frac{\partial \Gamma (k_i(t),K_{-i}(t))}{\partial k_i(t)}\right|.\end{array}$$

$$\gamma (k_i(t)),i\in \{1,2,\dots ,n\},\gamma (k_i(t))>0.$$

$${\gamma }^{\prime }(k_i(t))>0,{\gamma }^{\prime \prime }(k_i(t))>0.$$

$$p_i(A_i(t),q_1(t),q_2),\dots ,q_n(t))q_i(t)-c(q_i(t))-\gamma (k_i(t)).$$

$$\underset{q_i(t),k_i(t)}{\max }{\int }_0^{\infty }{e}^{-\rho t}[p_i(A_i(t),q_1(t),q_2),\dots ,q_n(t))q_i(t)-c(q_i(t))-\gamma (k_i(t))]dt,$$

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

$$+{\lambda }_{ii}(t)({\Gamma }_i(k_i(t),K_{-i}(t))-\delta A_i(t))+\sum _{j\ne i}{\lambda }_{ij}(t)({\Gamma }_j(k_j(t),K_{-j}(t))-\delta A_j(t))\}.$$

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

$$+{\lambda }_{ii}(t)({\Gamma }_i(k_i(t),K_{-i}(t))-\delta A_i(t))+\sum _{j\ne i}{\lambda }_{ij}(t)({\Gamma }_j(k_j(t),K_{-j}(t))-\delta A_j(t)).$$

$${\mu }_{ii}(t)=e^{-\rho t}{\lambda }_{ii}(t),{\mu }_{ij}(t)=e^{-\rho t}{\lambda }_{ij}(t).$$

$$\frac{\partial {\hat{\mathcal{H}}}_i}{\partial q_i(t)}=p_i+\frac{\partial p_i}{\partial q_i(t)}{q}_i(t)-c^{\prime }(q_i(t))=0,$$

$$\frac{\partial {\hat{\mathcal{H}}}_i}{\partial k_i(t)}=-{\gamma }^{\prime }(k_i(t))+{\lambda }_{ii}(t)\frac{\partial {\Gamma }_i}{\partial k_i(t)}+\sum _{j\ne i}{\lambda }_{ij}(t)\frac{\partial {\Gamma }_j}{\partial K_{-j}(t)}=0.$$

$$\frac{\partial {\hat{\mathcal{H}}}_i^2}{\partial q_i(t{)}^2}=2\frac{\partial p_i}{\partial q_i(t)}+\frac{{\partial }^2{p}_i}{\partial q_i(t{)}^2}{q}_i(t)-c^{\prime \prime }(q_i(t))<0.$$

$$\frac{{\partial }^2{\hat{\mathcal{H}}}_i}{\partial k_i(t{)}^2}=-{\gamma }^{\prime \prime }(k_i(t))+{\lambda }_{ii}(t)\frac{{\partial }^2{\Gamma }_i}{\partial k_i(t{)}^2}+\sum _{j\ne i}{\lambda }_{ij}(t)\frac{{\partial }^2{\Gamma }_j}{\partial K_{-j}(t{)}^2}<0,j\ne i.$$

$$-\frac{\partial {\hat{\mathcal{H}}}_i}{\partial A_i(t)}=\frac{\partial {\lambda }_{ii}(t)}{\partial t}-\rho {\lambda }_{ii}(t),i\in \{1,2,\dots ,n\},$$

$$-\frac{\partial {\hat{\mathcal{H}}}_i}{\partial A_j(t)}=\frac{\partial {\lambda }_{ij}(t)}{\partial t}-\rho {\lambda }_{ij}(t),j\ne i.$$

$$\frac{\partial {\hat{\mathcal{H}}}_i}{\partial A_i(t)}=\frac{\partial p_i}{\partial A_i(t)}{q}_i(t)-\delta {\lambda }_{ii}(t),$$

$$\frac{\partial {\hat{\mathcal{H}}}_i}{\partial A_j(t)}=-\delta {\lambda }_{ij}(t),j\ne i.$$

$$\frac{d{A}_i(t)}{dt}={\Gamma }_i-\delta A_i(t)=0,$$

$$\frac{\partial {\lambda }_{ii}(t)}{\partial t}=0,\frac{\partial {\lambda }_{ij}(t)}{\partial t}=0,i\in \{1,2,\dots ,n\},j\ne i.$$

$${\lambda }_{other}=0.$$

$$(\rho +\delta ){\lambda }_{own}=\frac{\partial p_i}{\partial A_i(t)}{q}^{\ast }.$$

Full text

Keywords:

oligopoly; advertising; differential game; general demand and cost functions

JEL Classification No.:

C73, D43, L13.

1 Introduction

In this paper we present an analysis of advertising activities 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 (2003a), Cellini and Lambertini (2003b), 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 (2003b) analyzed the advertising activities in a Cournot oligopoly with differentiated goods. However, most of these studies including Cellini and Lambertini (2003b) 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 under general demand and cost functions.

In the next section we present a model and assumptions. In Section 3 we consider the open-loop solution of advertising activities. In Section 4 we present the memoryless closed-loop solution of advertising activities. Cellini and Lambertini (2003b) claim that the open-loop solution and the memoryless closed-loop solution coincide. Cellini and Lambertini (2003b) said

the optimal values of the control variables of each player are not affected by state variables different from its own market size.

However, it is incorrect. The market size (or the accumulated advertising effect) of one good affects its price, then the demands of other goods are affected. We show that the comparison of the open-loop solution and the closed-loop solution depends on whether the outputs of the firms are strategic substitutes or strategic complements. If the outputs of the firms are strategic substitutes, the steady state value of the accumulated effects of advertising in the closed-loop solution is smaller than that in the open-loop solution. If the outputs of the firms are strategic complements, the steady state value of the accumulated effects of advertising in the closed-loop solution is larger than that in the open-loop solution. In Section 5 we analyze the feedback solution using Hamilton-Jacobi-Bellman equation, and show the equivalence of the memoryless closed-loop solution and the feedback solution when there is no spillover effect of advertising activities. In Section 6 we consider advertising activities in a cartel. We show that the adversing investment in the cartel may be larger than that in the open-loop solution. This is due to the spillover of advertising activities.

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 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 at $t$. The inverse demand function is written as

[TABLE]

$A_{i}(t)$ represents the market size of the good of Firm $i$, or the accumulated effects of advertising for the good of Firm $i$. It is a state variable. Denote $p_{i}(A_{i}(t),q_{1}(t),q_{2}(t),\dots,q_{n}(t))$ by $p_{i}$. The inverse demand functions for all firms are symmetric. We assume

[TABLE]

[TABLE]

and

[TABLE]

If the outputs of the firms are strategic substitutes

[TABLE]

and if they are strategic complements

[TABLE]

The production cost of Firm $i$ is

[TABLE]

This is common to all firms. $c(q_{i}(t))$ is strictly increasing and convex, that is, $c^{\prime}_{i}(q_{i}(t))>0$ and $c^{\prime\prime}_{i}(q_{i}(t))\geq 0$.

Let $k_{i}(t)$ be the advertising investment by Firm $i$. The moving of $A_{i}(t)$ is governed by

[TABLE]

where

[TABLE]

$\delta>0$ is the constant depreciation rate. We assume that $\Gamma(k_{i}(t),K_{-i}(t))$ is strictly increasing and concave, that is,

[TABLE]

The last condition means that the direct effect of advertising is larger than the spillover effect. $\Gamma$ is common to all firms. Denote $\Gamma(k_{i}(t),K_{-i}(t))$ by $\Gamma_{i}$.

The advertising investment cost of Firm $i$ is

[TABLE]

We assume that it is strictly increasing and strictly convex, that is

[TABLE]

$\gamma$ is common to all firms.

The instantaneous profit of Firm $i$ is written as

[TABLE]

The objective of Firm $i$ is

[TABLE]

subject to (2). $\rho>0$ is the discount rate.

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

[TABLE]

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

[TABLE]

Let

[TABLE]

$\mu_{ii}(t)$ and $\mu_{ij}(t)$ are the costate variables. Denote $\hat{\mathcal{H}}_{i}(q_{i}(t),k_{i}(t))$ by $\hat{\mathcal{H}}_{i}$.

3 Advertising in dynamic oligopoly: Open-loop solution

We seek to the solution of the open-loop approach.

The first order conditions for Firm $i$ are

[TABLE]

and

[TABLE]

The second order condition about production is

[TABLE]

The second order condition about advertising investment is

[TABLE]

The adjoint conditions are

[TABLE]

and

[TABLE]

We have

[TABLE]

and

[TABLE]

At the steady state

[TABLE]

[TABLE]

By symmetry of the oligopoly we can assume $\lambda_{ii}(t)=\lambda_{jj}(t)$, $\lambda_{ij}(t)=\lambda_{jl}(t)=\lambda_{ji}(t)$, $j\neq i,\ l\neq i,j$, $k_{i}(t)=k_{j}(t)$, $q_{i}(t)=q_{j}(t)$, $A_{i}(t)=A_{j}(t)$ for $j\neq i$, and so on. Denote the steady state values of $q_{i}(t)$, $k_{i}(t)$, $A_{i}(t)$, $\lambda_{ii}(t)$ and $\lambda_{ij}(t)$ by $q^{*}$, $k^{*}$,$A^{*}$, $\lambda_{own}$ and $\lambda_{other}$.

From (8) and (10) we have

[TABLE]

From (7) and (9)

[TABLE]

(6) is reduced to

[TABLE]

Therefore, we get

[TABLE]

From (2) and (7), under the symmetry condition,

[TABLE]

and

[TABLE]

From (6) with $\lambda_{ij}(t)=0$ and $\lambda_{ii}(t)=\frac{\gamma^{\prime}(k_{i}(t))}{\frac{\partial\Gamma_{i}}{\partial k_{i}(t)}}$, under the symmetry condition,

[TABLE]

Thus, we have

[TABLE]

Consider a system of dynamic equations (13) and (14). Linearization of them around the steady state is as follows.

[TABLE]

[TABLE]

The Jacobian matrix $\Omega$ has the following trace and determinant

[TABLE]

[TABLE]

For the steady state to be a saddle point we need

[TABLE]

If $\det{\Omega}>0$, the steady state is unstable. From $\frac{dA_{i}(t)}{dt}=0$, we have

[TABLE]

Thus,

[TABLE]

Let $\Phi$ be the left-hand side of (12). Differentiating $\Phi$ with respect to $A_{i}(t)$ at the steady state yields

[TABLE]

From the assumptions (3) and (4)

[TABLE]

Therefore, $\det{\Omega}<0$ means $\frac{\partial\Phi}{\partial A_{i}(t)}<0$, and we have shown the following result.

Lemma 1**.**

The left-hand side of (12), $\Phi$, is decreasing with respect to $A_{i}(t)$.

Linear and quadratic example

Assume that the inverse demand function is

[TABLE]

The advertising cost is

[TABLE]

The production cost is

[TABLE]

and the accumulation of advertising effects is written as

[TABLE]

At the steady state, (5) is reduced to

[TABLE]

Thus,

[TABLE]

(12) is reduced to

[TABLE]

Thus,

[TABLE]

From (11)

[TABLE]

Solving (15) and (16) yields

[TABLE]

and

[TABLE]

These are the results in Cellini and Lambertini (2003b).

4 Advertising in a dynamic oligopoly: Memoryless closed-loop solution without spillover

We seek to the solution of the memoryless closed-loop approach. As discussed in Introduction memoryless closed-loop and open-loop are not equivalent. For simplicity, we assume

[TABLE]

that is, there is no spillover effect of advertising investment. The first order conditions and the second order conditions for Firm $i$ are the same as those in the open-loop case. Using (17), they are

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

The adjoint conditions are different from those in the open-loop case. They are

[TABLE]

and

[TABLE]

The terms in (20)

[TABLE]

and the terms in (21)

[TABLE]

take into account the interaction between the control variables of the firms other than Firm $i$ and the current levels of the state variables.

We have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

$\frac{\partial q_{j}(t)}{\partial A_{i}(t)}$ is obtained by (43) in Appendix. If the outputs of the firms are strategic substitutes $\left(\frac{\partial p_{j}}{\partial q_{i}(t)}+\frac{\partial^{2}p_{j}}{\partial q_{i}(t)\partial q_{j}(t)}q_{j}(t)<0\right)$, $\frac{\partial q_{j}(t)}{\partial A_{i}(t)}>0$, and if they are strategic complements $\left(\frac{\partial p_{j}}{\partial q_{i}(t)}+\frac{\partial^{2}p_{j}}{\partial q_{i}(t)\partial q_{j}(t)}q_{j}(t)>0\right)$, $\frac{\partial q_{j}(t)}{\partial A_{i}(t)}>0$.

At the steady state we have

[TABLE]

and

[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 $j,l\neq i$, $\frac{\partial k_{j}(t)}{\partial A_{i}(t)}=\frac{\partial k_{l}(t)}{\partial A_{j}(t)}$ for $l\neq j$, $A_{i}(t)=A_{j}(t)$, $k_{j}(t)=k_{i}(t)$, $q_{j}(t)=q_{i}(t)$ for $j\neq i$, and so on. Denote the steady state values of $\lambda_{ii}$, $\lambda_{ij}$, $A_{i}(t)$, $q_{i}(t)$ and $k_{i}(t)$ by $\lambda_{own}$, $\lambda_{other}$, $A^{**}$, $q^{**}$ and $k^{**}$. Then, with (22) and (23), (20) is rewritten as

[TABLE]

The first order condition for the choice of $k_{i}(t)$, (19), is reduced to

[TABLE]

From (24) this means

[TABLE]

From (2) and (20 with (22), (23)) we obtain

[TABLE]

and

[TABLE]

Under the assumption $\frac{\partial\Gamma_{i}}{\partial K_{-i}(t)}=0$, for the steady state to be a saddle point we need

[TABLE]

where

[TABLE]

We assume $\det{\Omega^{\prime}}<0$. This system of dynamics are obtained from (26) and (27).

The first order condition for the output choice in the memoryless closed-loop case, (18), is the same as that, (5), in the open-loop case. Thus, we have $q^{**}=q^{*}$ given the value of $A_{i}(t)$.

We show the following proposition.

Proposition 1**.**

Assume that there is no spillover effect of advertising investment.

If the outputs of the firms are strategic substitutes, the steady state value of $A_{i}(t)$ in the closed-loop solution is larger than that in the open-loop solution. 2. 2.

If the outputs of the firms are strategic complements, the steady state value of $A_{i}(t)$ in the closed-loop solution is smaller than that in the open-loop solution.

Proof.

Let us compare (25) with (12) in the open-loop case. Suppose that $A^{*}=A^{**}$. Then, $q^{*}=q^{**}$, $k^{*}=k^{**}$. They satisfy (25), and the left-hand side of (12), which is denoted by $\Phi$, is negative because $\frac{\partial p_{i}}{\partial q_{j}(t)}<0$ and $\frac{\partial q_{j}(t)}{\partial A_{i}(t)}<0$. Since $\Phi$ is decreasing with respect to $A_{i}(t)$, the steady state value of $A_{i}(t)$ in the open-loop solution is smaller than that in the closed-loop solution, or the steady state value of $A_{i}(t)$ in the closed-loop solution is larger than that in the open-loop solution. 2. 2.

If the outputs of the firms are strategic complements, $\frac{\partial q_{j}(t)}{\partial A_{i}(t)}<0$, and the left-hand side of (12) is positive. Therefore, the steady state value of $A_{i}(t)$ in the closed-loop solution is smaller than that in the open-loop solution.

∎∎

5 Advertising in a dynamic oligopoly: Feedback solution without spillover

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

[TABLE]

Let $V_{i}(A_{1}(t),A_{2}(t),\dots,A_{n}(t))$ 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]

(29) is the same as (18) in the memoryless closed-loop case. From (30)

[TABLE]

Substituting this into (28), using symmetry, yields

[TABLE]

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

[TABLE]

At the steady state $\Gamma_{i}-\delta A_{i}(t)=\Gamma_{j}-\delta A_{j}(t)=0$. Thus, using (29), we get

[TABLE]

From this and (31),

[TABLE]

This is the same as (25). Therefore,

Proposition 2**.**

If there is no spillover effect of the advertising investment, the memoryless closed-loop solution and the feedback solution are equivalent.

6 Cartel

We consider full cartelization in which the firms cooperatively determine their outputs and advertising investments to maximize the discounted total profits. By symmetry we assume $p_{i}(t)=p_{j}(t)$, $q_{i}(t)=q_{j}(t))$, $k_{i}(t)=k_{j}(t)$, $A_{i}(t)=A_{j}(t)$ for any $j\neq i$. Denote the values of them by $p(t)$, $q(t)$, $k(t)$, $A(t)$. The objective of the firms is

[TABLE]

subject to

[TABLE]

Let $\hat{\mathcal{H}}$ be the current value Hamiltonian function, then

[TABLE]

Let

[TABLE]

$\mu(t)$ is the costate variable. Denote $p(A(t),q(t),\dots,q(t))$ by $p$. The first order conditions are

[TABLE]

and

[TABLE]

The adjoint condition is

[TABLE]

At the steady state

[TABLE]

[TABLE]

Denote the steady state values of $A(t)$, $\lambda(t)$, $q(t)$ and $k(t)$ by $A^{c}$, $\lambda^{c}$, $q^{c}$ and $k^{c}$. From (35)

[TABLE]

Then, from (34) we have

[TABLE]

We show the following proposition.

Proposition 3**.**

If there is no spillover effect, the value of $A(t)$ in the cartel is smaller than the value of $A_{i}(t)$ in the open-loop solution.

Proof.

Compare (33) and the first order condition in the open-loop case (5). If $q_{i}(t)$ satisfies (33), the left-hand side of (5) is

[TABLE]

Therefore, under the second order condition for the output choice in the open-loop case, the output of each firm in the open-loop solution is larger than that in the cartel, or the output of each firm in the cartel is smaller than that in the open-loop solution, that is, $q^{c}<q^{*}$ given $A(t)$ . 2. 2.

Assume $\frac{\partial\Gamma_{i}}{\partial K_{-i}(t)}=0$. Then, (36) is reduced to

[TABLE]

Suppose that $A(t)$ and $q^{c}$, which is obtained from (33) with $A(t)$, satisfy (37). Then, the left-hand side of (12) in the open-loop case, which is denoted by $\Phi$, is negative because by (1) $\frac{\partial p_{i}}{\partial A(t)}q_{i}(t)$ is deceasing with respect to $q_{i}(t)$ and $q^{*}>q^{c}$. Since $\Phi$ is decreasing with respect to $A_{i}(t)$, the value of $A_{i}(t)$ in the open-loop solution is larger than the value of $A(t)$ in the cartel, or the value of $A(t)$ in the cartel is smaller than the value of $A_{i}(t)$ in the open-loop solution. ∎

∎

If there is spillover effect of advertising activities, the value of $A(t)$ in the cartel may be larger than $A_{i}(t)$ in the open-loop solution.

Linear and quadratic example

Assume that the inverse demand function is

[TABLE]

The advertising cost is

[TABLE]

The production cost is

[TABLE]

and the accumulation of advertising effects is written as

[TABLE]

At the steady state (33) and (34) are reduced to

[TABLE]

and

[TABLE]

From (35) with $\frac{\partial\lambda(t)}{\partial t}=0$,

[TABLE]

From (38)

[TABLE]

From $\frac{dA(t)}{dt}=k+(n-1)\beta k-\delta A=0$,

[TABLE]

Solving (39), (40), (41) and (42),

[TABLE]

[TABLE]

In Cellini and Lambertini (2003b) the current value Hamiltonian function in (32) is written as follows.

[TABLE]

Then, we obtain

[TABLE]

[TABLE]

These are the results in Cellini and Lambertini (2003b).

7 Concluding Remark

We have studied the problem of advertising activities in a dynamic oligopoly under general demand and cost functions by a differential game approach. We have shown that the results in Cellini and Lambertini (2003b) in a case of linear demand and quadratic cost functions can be generalized to a case of general demand and cost functions, but some results depend on the property of demand functions, strategic substitutability or strategic complementartity.

Acknowledgment

This work was supported by Japan Society for the Promotion of Science KAKENHI Grant Number 18K01594 and 18K12780.

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

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

[TABLE]

and

[TABLE]

Differentiating them with respect to $A_{i}(t)$ yields

[TABLE]

and

[TABLE]

Let

[TABLE]

From them we obtain

[TABLE]

and

[TABLE]

where

[TABLE]

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