^{1}

^{2}

^{3}

^{3}

^{1}

^{2}

^{3}

This paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation. The dynamics of the economy is characterised by a differentiable two-dimensional discrete time system. The paper stresses the importance of complementarity between products as a source of synchronisation in the long term, in contrast to the case of their substitutability. This means that when products are complements, players may coordinate their behaviour even if initial conditions are different. In addition, there exist multiple attractors so that even starting with similar conditions may end up generating very different dynamic patterns.

Strategic delegation is a relevant topic in both oligopoly theory and industrial organisation, and several papers have contributed to clarify questions related to the differences between the behaviour of profit-maximising firms and managerial firms (e.g., [

The present paper studies a nonlinear duopoly game with price competition and market share delegation and extends the study carried out by Fanti et al. [

The rest of the paper is organised as follows. Section

Consider a duopoly game with price competition, horizontal differentiation, and market share delegation contracts (see [

Both the firms have the same marginal cost

We now assume a discrete time (

Assume

It is of importance to observe that system (

Let

The feasible set of system

(a), (b). The feasible set

With regard to the study of the structure of the feasible set, a numerical procedure based on the study of the properties of the critical curves can be used (see, e.g., [

We denote the critical curve of rank-

The study of the structure of set

We now fix all parameter values but

Parameter values:

Notice that in the simulations presented in Figure

Let

If

If

(i) If

(ii) This statement can be proved simply considering the limits

According to Proposition

By taking into account the above-mentioned arguments, in what follows we will focus on the study of the dynamics produced by

Let

Let

If

The origin

If

Consider the system

Assume

Since

Assume

The question of the existence of an interior fixed point in the general case

In order to study the evolution of system

Since map

Let

The eigenvalues evaluated at the fixed point

Different from the case in which products are substitutes, the following Proposition can easily be verified.

If

If

From Proposition

The following condition for the local stability of

Let system

By taking into account Propositions

A feasible trajectory

With regards to the dynamics of synchronised trajectories, we first recall the result proved in Proposition

An attractor

Parameter values:

By taking into account the previous results and looking at the one-dimensional bifurcation diagrams in Figures

Let us consider now a duopoly with identical players that start from different feasible initial conditions and let

Let

Let

According to Proposition

If

Consider now a more complex situation; that is,

If

In order to investigate the existence of a Milnor attractor

Parameter values:

The study of the geometrical properties of the critical lines may be used to estimate the maximum amplitude of the bursts by obtaining the boundary of a compact trapping region of the phase plane in which the on-off intermittency phenomena are confined. Following Mira et al. [

If

Parameter values:

If we compute

Finally, we consider the case in which products are independent from each other and each manager behaves as a monopolist (

We now consider the case in which managers’ bonuses are evaluated differently; that is,

With regard to the existence of the Nash equilibrium, we recall that Proposition

By considering the analytical properties of

If the Nash equilibrium exists, then it is unique such that the equilibrium price is higher for the variety associated with a lower market share bonus (see Figure

If the Nash equilibrium is locally stable in the symmetric case, then it is also locally stable in the asymmetric case if and only if the perturbation on

In the case of heterogeneity, the Nash equilibrium loses stability via a flip bifurcation at which it becomes a saddle point and a stable

Synchronised trajectories do not emerge and synchronisation cannot occur, while multistability still emerges (compare Figures

(a) Parameter values:

To better describe point (iv) above, we recall that a situation in which synchronisation may occur is depicted in Figure

Similarly, observe that, with independent products and homogeneous managers, three coexisting attractors are owned (see Figure

Although the managers’ behaviours are no longer coordinated, it is interesting to stress that, different from the case of substitutability between products, the structure of the basin of attraction seems to become simpler than under homogeneous delegation contracts.

This paper has studied the mathematical properties of a nonlinear duopoly game with price competition and market share delegation contracts. The main aim was to extend the analysis carried out by Fanti et al. [

From an economic point of view, synchronisation is relevant because it implies coordination between players. Then, in a model with managerial firms and market share delegation contracts, coordination can (resp., cannot) hold when products are complements (resp., substitutes). In addition, we have also shown that multiple attractors may exist so that initial conditions matter.

The authors declare that there is no conflict of interests regarding the publication of this paper.