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

06/05/2019 ∙ by Masahiko Hattori, et al. ∙ Doshisha University 0

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.

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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 firms in which at any they produce differentiated goods to maximize their discounted profits. The firms are called Firms 1, 2, , . Let be the output of Firm , be the price at . The inverse demand function is written as

represents the market size of the good of Firm , or the accumulated effects of advertising for the good of Firm . It is a state variable. Denote by . The inverse demand functions for all firms are symmetric. We assume

and

(1)

If the outputs of the firms are strategic substitutes

and if they are strategic complements

The production cost of Firm is

This is common to all firms. is strictly increasing and convex, that is, and .

Let be the advertising investment by Firm . The moving of is governed by

(2)

where

is the constant depreciation rate. We assume that is strictly increasing and concave, that is,

(3)

The last condition means that the direct effect of advertising is larger than the spillover effect. is common to all firms. Denote by .

The advertising investment cost of Firm is

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

(4)

is common to all firms.

The instantaneous profit of Firm is written as

The objective of Firm is

subject to (2). is the discount rate.

The present value Hamiltonian function for Firm , is

The current value Hamiltonian function for Firm , is

Let

and are the costate variables. Denote by .

3 Advertising in dynamic oligopoly: Open-loop solution

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

The first order conditions for Firm are

(5)

and

(6)

The second order condition about production is

The second order condition about advertising investment is

The adjoint conditions are

(7)

and

(8)

We have

(9)

and

(10)

At the steady state

(11)

By symmetry of the oligopoly we can assume , , , , for , and so on. Denote the steady state values of , , , and by , ,, and .

From (8) and (10) we have

From (7) and (9)

(6) is reduced to

Therefore, we get

(12)

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

(13)

and

(14)

From (6) with and , under the symmetry condition,

Thus, we have

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

The Jacobian matrix has the following trace and determinant

For the steady state to be a saddle point we need

If , the steady state is unstable. From , we have

Thus,

Let be the left-hand side of (12). Differentiating with respect to at the steady state yields

From the assumptions (3) and (4)

Therefore, means , and we have shown the following result.

Lemma 1.

The left-hand side of (12), , is decreasing with respect to .

Linear and quadratic example

Assume that the inverse demand function is

The advertising cost is

The production cost is

and the accumulation of advertising effects is written as

At the steady state, (5) is reduced to

Thus,

(12) is reduced to

Thus,

(15)

From (11)

(16)

Solving (15) and (16) yields

and

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

(17)

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

(18)
(19)

and

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

(20)

and

(21)

The terms in (20)

and the terms in (21)

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

We have

(22)
(23)

and

is obtained by (43) in Appendix. If the outputs of the firms are strategic substitutes , , and if they are strategic complements , .

At the steady state we have

and

By symmetry of the oligopoly we can assume for , for , for , , , for , and so on. Denote the steady state values of , , , and by , , , and . Then, with (22) and (23), (20) is rewritten as

(24)

The first order condition for the choice of , (19), is reduced to

From (24) this means

(25)

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

(26)

and