Both demand and cost asymmetries are considered in oligopoly model with managerial delegation. It shows that (i) both efficient and inefficient firms with delegation have second move advantage under quantity setting and first move advantage under price competition; (ii) the extended games under both quantity and price competition have subgame equilibria. Lastly, the social welfare of all strategy combinations is considered to find that when the efficient firm moves first and the inefficient firm moves second under price competition, the social welfare can be higher than Bertrand case, if the efficiency gap between the two firms is huge.

Industrial organization analysis in oligopoly and duopoly usually assumes that firms move either simultaneously or sequentially. Technically speaking, it is Cournot (Bertrand) game or Stackelberg game. In a seminal paper, Singh and Vives [

There is literature which starts to research the strategy choice of timing: when do the duopolies move simultaneously and when do they move sequentially? Hamilton and Slutsky [

The strategic use of managers’ contracts in competition games was introduced and being widely used from the 1980s. Fershtman and Judd [

Though there is horizontal product differentiation in Hamilton and Slutsky [

To explore the question, asymmetry between the two firms with sales revenue delegation is considered in this paper while demand and cost asymmetry were not considered by Lambertini [

The conclusions of this paper are threefold: if products are substitute and the assumption of positive primary outputs holds, firstly, both efficient and inefficient firms with delegation have second move advantage under quantity setting and first move advantage under price competition; secondly, the extended games under quantity competition and under price competition have subgame equilibria; thirdly, the social welfare can be higher when the efficient firm moves first and the inefficient firm moves second under price competition than the Bertrand case, if the efficiency gap between the two firms is huge.

The rest of the paper is organized as follows. Section

Consider a linear duopoly model used in Singh and Vives [

On the supply side, these two firms are assumed to have zero fixed cost competing either in quantities or in prices with asymmetric constant marginal costs,

Recall that both demand asymmetry

For simplicity, we focus on the case of substitute goods and assume that both firms are active in the market. (It is known that the competition modes, the property of goods (substitutes), and the strategic relationship (strategic substitute or complement) in oligopolistic competition cause the difference of results. See Sklivas [

Therefore, when

Cournot case is investigated in this subsection. The payoff function of owner

If the owners of these two firms decide to move simultaneously, no matter

The incentive schemes:

If the owner of firm 1 chooses to move first and firm 2 opts to take action later, that is,

The incentive schemes:

Similarly, if the owner of firm 1 wants to move later and firm 2 wishes to move earlier, that is,

The incentive schemes:

It is worth mentioning that in quantity competition with delegation, both firms’ incentive schemes under four strategy combinations are less than or equal to 1, especially in the two sequential subgames; the leaders’ incentive schemes are equal to 1. This is interesting because when the owner decides to move first, it is already an aggressive strategy, which makes it needless to set managerial schemes to make the manager aggressive as well because the manager’s bonus which he maximizes is just the firm’s profit.

Followed with the Cournot case in the former subsection, Bertrand case is analyzed within this subsection. Similarly, the payoff function of owner

If the owners of these two firms decide to move simultaneously, no matter earlier or later, that is,

The incentive schemes:

If the owner of firm 1 prefers to move first and firm 2 chooses to move later, that is,

The incentive schemes:

Similarly, if the owner of firm 1 is in favor of moving later and firm 2 determines to move earlier, that is,

The incentive schemes:

Note that in price competition with delegation, both firms’ incentive schemes under four strategy combinations are more than or equal to 1, especially in two sequential subgames; the leaders’ incentive schemes are always equal to 1. The reason is similar to quantity competition. (In essence, an owner of a managerial firm hiring an aggressive manager may lead the firm to enjoying a dominant position replicating the performance of a Stackelberg-leader, provided that rivals remain entrepreneurial. One can refer to Basu [

In contrast with Singh and Vives [

When parameters

both efficient and inefficient firms charge higher prices under Cournot equilibrium than under Bertrand competition;

both firms produce less under Cournot than under Bertrand competition;

both firms make more profits under Cournot than under Bertrand competition.

See the appendix.

Conventional wisdom suggests that a decrease in the degree of product differentiation always reduces firms’ profits by increasing the intensity of product market competition, irrespective of the competition mode, that is, irrespective of the fact that firms compete à la Cournot or à la Bertrand in the product market. The theoretical reason behind this result can be understood by referring to the standard differentiated duopoly model, in which a decrease in the degree of product differentiation diminishes total demand and induces firms to compete more aggressively [

From Lemma

After realizing the results of the game, the endogenous timing problem is able to be resolved. First, the payoffs of simultaneous Nash subgame and two sequential subgames under quantity competition and price competition should be, respectively, compared.

When

in the quantity-setting extended game, each firm’s profit under simultaneous game is lower than the profit when being a follower in the sequential game, but higher than the leadership payoff; that is,

in the price-setting extended game, each firm makes more profit when being a follower under sequential game than under simultaneous game, but less when being a leader; that is,

Thus both owners’ preferences for the order of the roles their firms play are

See the appendix.

Realizing the preference order of the owners under the considered game structures makes it possible to figure out the owners’ timing choice. We have the following proposition.

(1) When firms compete in quantity, the extended game has unique pure-strategy subgame-perfect equilibrium.

(2) When firms compete in price, the extended game has two pure-strategy subgame-perfect equilibria, and mixed strategy equilibrium exists.

(1) From Proposition

(2) Proposition

As a result, both

Though the equilibrium of the two extended games is being found out here, the most Pareto-efficient outcome is not determined yet, which can be figured out through the comparison of the welfare under all strategy combinations.

The total social surplus is defined as follows and can be derived through the substitution of the equilibrium outcomes back to the model under each case:

Comparing the total social surplus obtained under different competition modes, we have the following proposition.

For any

it is more Pareto-efficient under price competition than all other cases under quantity competition; that is, for any

when both firms act as quantity-setters, the welfare ranking of simultaneous and sequential games is

when firms compete in price, the welfare ranking of the three cases with different move orders is

See the appendix.

Generally, price competition is more Pareto-efficient than quantity competition, no matter the firms are symmetric or not. Under quantity competition, the Nash equilibrium is

Many papers have discussed the endogenous timing game, but those papers usually assumed duopoly market with symmetric firms. Due to the specification they had, the roles of the firms in the games they concluded are often the same. Thus they did not really answer the question of endogenously determined move order. It is because when one firm chooses to act as a leader, it is not sufficient for the desirability of the other symmetric firm to act as a follower.

Therefore, to solve the question and figure out the reason, this paper extends the Hamilton and Slutsky [

When

Price:

Profit:

When

If

The sign of last formula is uncertain. For

The author declares that this paper is not supported by any funding institutions or any government’s divisions.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author wishes to thank the assistance provided by Zhigang Zhang, Tien-Der Han, and Tai-Liang Chen and constructive suggestions received from two reviewers and academic editor Henry X. Wang.