Managerial Collusive Behavior under Asymmetric Incentive Schemes

2.1 One-Shot Game

Each firm is run by a manager who receives a compensation based on absolute and relative profits (RP). Following Jansen, van Lier, and van Witteloostuijn (2009), manager i receives a compensation proportional to:

where πi=(1−Q)qi is the profit of firm i and θi is the weight put on the profit of (the rival) firm j. We assume that θi∈0,1. A high θi makes manager i more aggressive. On the other hand, if θi is low, then manager i behaves more like a pure profit maximizer. Let θ=(θ1,θ2) denote the distribution of managers’ incentives in the economy. We assume that θ is revealed and becomes common knowledge before managers interact on the output market.

It is important to note that θ is not endogenized at any stage of our model. Our focus is to investigate if asymmetry constitutes a factor supporting collusion.

2.2 Repeated Game

The repeated game consists of an infinite repetition of the game described above. We assume that the game is stationary in each period. In particular, the pair of incentive weights (or the distribution managers’ types) is the same in each period. Information is complete, time is discrete and δ is the common discount factor.

Managers maximize the discounted sum of their payoffs given by ∑t=0∞δtWit where Wit=πit−θitπjt represents manager i’s payoff at time t.

For simplicity and tractability, we restrict our attention to the case where stationary collusive agreements are enforced by grim trigger strategies. Intuitively, each manager chooses her specified collusive output in each period, as long as both managers continue do to so. If, however, one of the managers defects, the punishment is triggered: both managers revert to the static Cournot Nash outputs forever.

Let qiM and qiN denote the outputs of manager i in the collusive agreement (M) and the Cournot Nash equilibrium (N) respectively.

We denote by WiM=Wi(qiM,qjM) manager i’s payoff in the stationary collusive agreement. Let WiD=maxqiWi(qi,qjM) denote the one-shot payoff obtained by manager i when she optimally deviates from the collusive agreement, while manager j sticks to M. Finally, let WiN=Wi(qiN,qjN) denote manager i’s per-period payoff during the punishment phase. Given the common discount factor δ, the collusive agreement is a subgame-perfect Nash equilibrium (SPNE) if and only if the following incentive (no-deviation) constraints are satisfied:

where δ_i is the minimum level of the discount factor at which manager i can support the collusive agreement.^[2] Consequently, collusion is sustainable if and only if δ≥minδ_1,δ_2.

3.1 Collusive Stage

To simplify the demonstration, we define a (collusive) outcome by the pair (Q,s), where Q is the collusive output and s is the associated market share (or output quota) of, say, manager 1. This notation is equivalent to (q1,q2) since q1=sQ and q2=(1−s)Q. Using this notation, it results that managers’ payoffs can be written as W1=s−θ1+sθ1Q−Q2 and W2=1−s−sθ2Q−Q2. Since Q∈0,1, manager 1’s and manager 2’s payoffs are non-negative if s∈θ11+θ1,11+θ2.

Note that Q=1/2 is the aggregate monopoly output. In words, if negative payoffs are ruled out, then the total output which maximizes both managers’ payoffs simultaneously is unique and coincides with the monopoly output.

From Proposition 1, we argue that the monopoly output is a ‘‘natural’’ focal point on which managers are very likely to tacitly coordinate. It means that, for any value of s belonging to θ11+θ1,11+θ2, managers would agree to produce a total of Q=1/2.

Consequently, the monopoly (collusive) outputs are given by q1M=s/2 and q2M=(1−s)/2 and the monopoly (collusive) payoffs by:

It must be noted that the monopoly (collusive) output and payoff of each manager depend on the size of s: the larger s is, the higher q1M and W1M (the lower q2M and W2M), and vice versa.

3.2 Deviation Stage

When deviation occurs, the deviating manager i considers manager j’s output qjM as given, and optimally selects the output that maximizes her own payoff. Let qiD denote the optimal deviation output for manager i. Thus, qiD is obtained by maximizing Wi=(1−qi−qjM)(qi−θiqjM). Calculations show that q1D=(1+θ1+s−sθ1)/4 and q2D=(2−s+sθ2)/4 and the resulting deviation payoffs are:

3.3 Competitive Stage

In the competitive stage, managers independently and simultaneously choose their firm outputs qi that maximize their own payoffs Wi=(1−qi−qj)(qi−θiqj). Solving ∂Wi(qi,qj)/∂qi=0 for qi, one obtains manager i’s best response output function: Ri(qj)=(1−(1−θi)qj)/2. The Cournot Nash equilibrium is the pair of outputs (q1N,q2N) such that qiN=Ri(qjN) holds for both managers, i≠j=1,2. It is easy to check that qiN=(1+θi)/(3+θ1+θ2−θ1θ2), i=1,2. Substituting these output levels into (3) yields, for a pair of incentives (θ1,θ2), the Nash equilibrium payoff of manager i:

In the static Cournot Nash equilibrium, the manager with the higher incentive toward relative profits sets a larger quantity but does not earn a larger payoff than the manager with the lower incentive: if, say, θ1>θ2, then q1N>q2N but W1N=W2N.

3.4 Critical Discount Factors

Before looking at the critical discount rates, it is important to identify some restrictions on the market share. To guarantee the existence of a discount rate that sustains collusion, compensations under collusion must be higher that those under the competitive equilibrium. In other words, we must find conditions on s such that WiM≥WiN.

Lemma 1 is an application of the folk theorem. It is easy to show that the interval of Expression 6 is not empty for any value of θ1,θ2. Consequently, for any pair (θ1,θ2), collusion can be sustainable if managers are patient enough.

Now, we find the minimum discount rate that supports collusion. By appropriate substitution of eqs [3]–[5] into eq. [2], we obtain critical discount factors for both managers as a function of the market sharing rule s:

where ϕ≡3+θ1+θ2−θ1θ2 and ψ≡2ϕ+4(1−θ1θ2).

Figure 1 depicts the critical discount factors of the two managers as a function of s for the particular case where the pair of incentive weights (θ1,θ2)=13,13.

Figure 1:

Minimum discount factor for collusion when θ1=θ2=1/3.

Using δ_1(s) and δ_2(s), we can study the conditions that make collusion sustainable. By definition, if for a given market share s, δ is greater than δ_1(s) and δ_2(s), then collusion is sustainable. Looking at Figure 1, the area σδ represents the set of all combinations of s and δ such that δ≥maxδ_1(s),δ_2(s). Explicitly, all pairs s,δ belonging to the set σδ make collusion sustainable.

Since we study the sustainability of collusion and not the collusive equilibrium itself, our focus is now on finding the condition that guarantees the existence of market shares that sustain collusion. Let s1(δ) be the minimum market share that sustains collusion for manager 1. Similarly, let s2(δ) be the maximum market share that sustains collusion for manager 2. Consequently, collusion is sustainable for any market share s belonging to the interval Isδ=s1(δ),s2(δ). Figure 2 illustrates a case where, if the discount factor is high enough, the set of all market shares that support collusion as a subgame perfect Nash equilibrium outcome is non-empty.

Figure 2:

Non-empty collusive market share set.

The following proposition states the minimum condition on the discount factor for collusion to be sustainable.

If Proposition 3 provides the minimum discount factor sustaining collusion, the following proposition states the market share attached to it.

In Figure 2, we identify by δ∗ the minimum discount rate sustaining collusion. The discount factor δ∗ corresponds to the balanced temptation equilibrium discount factor proposed by Friedman (1971). This criterion refers to the idea that managers split the market in such a way that both have the same incentive to deviate (same discount factor threshold). For the remaining of the paper, we use δ∗ to analyze the effect of incentive asymmetry on collusion: an increase in incentive asymmetry supports collusion if δ∗ increases.

4.1 Iso-stability Curves

To illustrate the effect of incentive asymmetry on collusion, we develop the concept of an iso-stability curve. An iso-stability curve is the set of all pairs (θ1,θ2) for which managers have the same discount rate threshold while the market share is given by s∗.^[4] Explicitly, an iso-stability curve ISδ is the set of all pairs (θ1,θ2) such that δ∗θ1,θ2=δ.

To graph iso-stability curves, we isolate θ2 in the equation δ∗θ1,θ2=δ and we obtain θ2 as a function of θ1 and δ.

While eq. [10] seems difficult to analyze, it is easy to show that, for any θ1∈0,1 and δ∈9/17,1,^[5]

1. IS curves are decreasing and convex;

2. IS curves are symmetric with respect to the line θ2=θ1;

3. when θ2(θ1,δ)=θ1, the slope of the IS curve is –1.

Figure 3:

Iso-stability curves.

We can see that δ∗ increases when (θ1,θ2) moves closer to (1,1). This result is consistent with Lemma 2: an increase in θi does not support collusion.

4.2 Incentive Asymmetry

The next step consists in the separation of the effect of an increase in the θi, i=1,2, from an increase in incentive asymmetry. Indeed, when (θ1,θ2) changes, the difference between θ1 and θ2 changes but the level of θi changes as well. As we have seen above, an increase in either θi increases δ∗. However, the effect of asymmetry (difference between θ1 and θ2) is, as yet, unclear.

To clarify this point, we divide the change in the pair θ1,θ2 in two components: a change in average and a change in asymmetry. First, let Θ be the average of θ1 and θ2. Second, we define Δ as the Euclidean distance between θ1 and θ2. Without loss of generality, we suppose θ1≥θ2 and therefore, Δ=θ1−θ2. Using these two notations, we are able to separate the effect of an increase in the level of the θi from that of a change in asymmetry.

Figure 4 illustrates the effects of an increase in Θ and Δ on δ∗. A change in Θ when Θ remains constant is represented by a movement along the line of slope 1 while a change in Δ when Θ remains unchanged by a movement along the line of slope –1.

Figure 4:

Effects on δ∗ of a change in Θ and Δ.

The formal proof of Proposition 4 is in the Appendix. Graphically, the result is clear: an increase in the incentive asymmetry permits to reach a lower iso-stability curve. We already know that δ∗ increases when an incentive parameter increases (∂δ∗/∂θi>0). However, when θ1 increases and θ2 decreases, the effect on collusion is mitigated and finally negative. It seems that the effect of the decision to use incentive compensations is relatively more important when the incentive parameter θ is low. In other words, the gain from the reallocation of production from firm 2 to firm one decreases as θ1 increases. This is due to the relative lower gain under deviation since firm 1 already produces a larger quantity.

Another way to look at this result is to see the impact of a change in asymmetry on the set of market shares supporting collusion.

As stated previously, our focus is on the link between incentive asymmetry and collusion and not the actual equilibrium market shares. From Corollary 2, collusion remains sustainable if it was sustainable before an increase in asymmetry. However, this does not mean that the initial set is a subset of the new set of market shares. In other words, if a market share supported collusion before a change in asymmetry, it does not mean it still supports collusion thereafter. When asymmetry increases while the average remains unchanged, δ1_ and δ2_ move in the same direction. Consequently, the interval Iδ moves as well. Some market shares that belonged to the interval of sustainable market shares do not belong to the interval anymore.

4.3 Collusion Rate of Substitution

If Proposition 4 is useful to isolate the effect of an increase in asymmetry from an increase in the average value of the incentive parameters, it does not cover all possible cases. Indeed, if asymmetry and average increase altogether, Proposition 4 cannot predict the final effect on δ∗. However, there exists another approach that covers more cases. This second approach involves the introduction of a new concept: the collusion rate of substitution.

The collusion rate of substitution (CRS) is given by the negative of the ratio of derivatives of δ∗ with respect to θ1 and θ2, i.e.

Since θ1 and θ2 belong to 0,1, then CRS varies between −1/4 and –4. Technically, CRS is the slope of the iso-stability curve passing through θ1,θ2 (see Figure 5).

Figure 5:

Collusion rate of substitution.

To study the effect of a change in θ1 and θ2 on δ∗, we compare the variation in θ1 and θ2 with CRS at the initial pair θ1,θ2. Let us still assume that θ1>θ2 and let Δθ1,Δθ2 be the variation in θ1 and θ2.

Figure 6 illustrates the three possibilities presented in Proposition 5.

Figure 6:

Collusion rate of substitution.

First, point A represents the case when θ1 and θ2 decrease. As discussed above, a decrease in θ1 and θ2 leads to a decrease of δ∗. The second case (Δθ1<0, Δθ2>p0) is illustrated by point B. In this case, the decrease in θ1 must be large enough to compensate the effect of an increase in θ2 in order to support collusion. The third case (point C) is analogous to the second case.