Collusion, Shading, and Optimal Organization Design

Proof

The coefficient of the marginal information rent 1 + p k 1 − p increases as the parameter p increases. Hence, the virtual marginal information rent (and so the marginal virtual cost) h 1 − h 1 + p k 1 − p ∂ C X L − θ − ∂ X L − ∂ C X L − θ ̄ ∂ X L increases as p increases. This brings about the decrease in the optimal output X L CP . Similarly, 1 + p k 1 − p increases as k increases. Hence, the virtual marginal information rent (and so the marginal virtual cost) increases as k increases. This brings about the decrease in the optimal output X L CP . □

Proof

In the following formulations of marginal virtual surplus of the two regimes: Collusion-proof “without shading” regime (CP) and “with shading” regime (CPS),

The optimal solution X L CP satisfies the first-order condition ∂ J CP X L CP , θ − ∂ X L = 0 . Then,

Therefore X L CP cannot be optimal for the behavioral, “with shading” regime (CPS). A marginal decrease in X _L from X L CP would increase the virtual surplus J CPS X L , θ − of “with shading” regime (CPS). Thus, we have X L CPS < X L CP for β > 0.

Comparative statics is straight forward. From (25), the derivative J X L CPS X L , θ is decreasing in β (shading strength by the agent). That is,

Hence, the optimal solution with ex-post shading X L CPS is decreasing in β. □

Proof

First, the expected virtual surplus in the Collusion-proof “without shading” regime (CP) is h p X H FB − C X H FB − θ ̄ − k U CP θ ̄ + 1 − p X H CP − C X H CP − θ ̄ − U CP θ ̄ + ( 1 − h ) p X L FB − C X L FB − θ − + 1 − p X L CP − C X L CP − θ − where U CP θ ̄ = C X L CP − θ − − C X L CP − θ ̄ i s t h e i n f o r m a t i o n r e n t i n t h i s r e g i m e ( C P ) .

The maximized expected virtual surplus in the Collusion-proof “without shading” regime (CP) is

Next, the expected virtual surplus in the collusion-proof “with shading” regime (CPS) is

where U CPS θ ̄ = C X L CPS − θ − − C X L CPS − θ ̄ is the information rent in this regime (CPS), and X H CPS and X L CPS are determined by (15) and (16). By reminding the modified coalition-proof constraint W S CPS θ ̄ − β U CPS θ ̄ ≥ k U CPS θ ̄ , and substituting X H CPS = X H FB , the maximized expected virtual surplus in the collusion-proof “with shading” regime (CPS) is as follows.

This is transformed as follows.

Taking the difference of the above two maximized expected virtual surpluses, the condition for the expected virtual surplus in the collusion-proof “with shading” regime (CPS) to be smaller than the one in the collusion-proof “without shading” regime (CP) is as follows.

The RHS of the inequality is negative from the following “Revealed Preference” relation

The LHS of the inequality is positive for β > 0. Therefore, VS^CPS < VS^CP always holds, which means that the maximized expected virtual surplus is always smaller in the behavioral regime (CPS) than in the no behavioral regime (CP), and implies the existence of “Haggling Cost” when the shading behavior occurs. □

Proof

Substituting the optimal solution X H CPS = X H FB for H-type, the expected virtual profit for the principal in the Collusion-Proof regime with shading (CPS) is written as follows.

where U θ ̄ = C X L − θ − − C X L − θ ̄ is the information rent in this regime.

Similarly, substituting the optimal solution X H EC = X H FB for H-type, the expected virtual profit for the principal in Equilibrium Collusion regime (EC) is written as follows.

Hence, which regime can achieve higher efficiency depends on the comparison of the following two optimal values.

By applying the optimization and envelope theorem, we find that

Putting this together with the above lemma, when the shading strength β ≤ 1 − k, we have X L CPS ≥ X L EC ⇔ VS CPS ≥ VS EC , and so the principal optimally chooses the Collusion-Proof regime with shading (CPS). Similarly, when the shading strength β ≥ 1 − k, we have X L CPS ≤ X L EC ⇔ VS CPS ≤ VS EC , and so the principal optimally chooses the Equilibrium Collusion regime (EC). □

Proof

EC implies the partial equilibrium collusion between H-type and the supervisor, θ ̄ , S , and the supervisor does not collude with L-type. That is, on the one hand, s = θ ̄ and r = ϕ, and on the other hand, s = θ − and r = θ − , with probability p.

The maximized expected virtual profit for the principal in Equilibrium Collusion regime (EC) is as follows.

where U EC θ ̄ = C X L EC − θ − − C X L EC − θ ̄ and X L EC is determined by (21).

The maximized expected virtual profit for the principal in Two-tier, No-supervisor regime (TW) is as follows.

where U TW θ ̄ = C X L TW − θ − − C X L TW − θ ̄ and X L TW is determined by (9).

Then, we have X L EC < X L TW .

For the payoff comparison between EC and TW, we can apply the argument in Suzuki (2018).

From the “Revealed Preference” argument, the following two inequalities hold.

Adding them up, we obtain the comparison result on EC and TW.

That is, TW (Two-tier structure with No Supervisor) is strictly dominated by EC (Three-tier hierarchy with partial equilibrium collusion) for all k ∈ 0,1 , β > 0 . A rationale is that having the contract for the L-type contingent on the supervisor’s report r ∈ θ − , ϕ (Three-tier hierarchy) X L FB r = θ − w · p p , X L EC r = ϕ w . p 1 − p is strictly better than the “pooling contract” (Two-tier hierarchy) X L TW w . p 1 . Figure 3 shows this point. □

Proof

From Proposition 5 and Lemma 3, when β + k ≤ 1, the Collusion-proof principle still holds: CPS ≥ EC > TW.

When β + k ≥ 1, we have already known that EC ≥ CPS, and have also checked that EC > TW always holds, when Z = 0. Therefore, the remaining one is the comparison between CPS and TW.

The expected virtual surplus in the collusion-proof “with shading” regime (CPS) is given by (29), where W S CPS θ ̄ = k + β U CPS θ ̄ , X H CPS = X H FB , and X L CPS is determined by (16).

The maximized expected virtual surplus in the collusion-proof “with shading” regime (CPS) is given by (30).

Then, the following two “Revealed Preference” relations hold.

Adding them up, we obtain the comparison result (Step 1) on CPS and TW,

If p k + β + 1 − p ≤ 1 ⇔ β + k ≤ 1 , we obtain Step 2.

Thus, we have

This means that CPS > TW when β + k ≤ 1. CPS ≥ EC when β + k ≤ 1 is also known.

Since EC > TW has already been proven, we have the comparison result CPS ≥ EC > TW.

Next, when β + k ≥ 1, we cannot have (45) (the above Step 2).

Then, as the shading parameter β becomes larger (as β → +∞), the optimal output X L CPS , which is determined by (16), goes to zero, X L CPS → 0 . The potential aggrievement (information rent) for the agent also goes to zero, U CPS θ ̄ → 0 . Hence, the equilibrium payoff of the Collusion-proof regime with shading (CPS) goes to p h X H FB − C X H FB − θ ̄ + 1 − h X L FB − C X L FB − θ − : The First Best Expected Total Surplus.

On the other hand, the payoff of the Two-tier, No Supervisor regime (TW) is independent of β, p

Hence, which payoff is greater between (CPS) and (TW) at β → +∞ depends on the relative size of

Case1

There exists a cutoff value of shading strength β* such that for β > β * ≥ 1 − k “Two-tier Hierarchy (TW)” Payoff dominates “Collusion-proof” regime with ex-post shading (CPS).

Case2

“Two-tier, No Supervisor” Hierarchy (TW) is not optimal even for β → +∞, but Collusion-proof regime with ex-post shading (CPS) is optimally chosen. The point is that Shut-down is endogenously chosen in the states of θ , ϕ , that is, the optimal output goes to zero, X L CPS → 0 , and the potential aggrievement (information rent) also goes to zero, U CPS θ ̄ → 0 .

As p becomes smaller, the states of θ , ϕ with probability 1 − p increase. Then the principal cannot neglect her decision X L CPS any more in the supervisory no information state ϕ, in the form of X L CPS → 0 . However, the cost of collusion-proof constraint, or the shading cost which the principal will eventually bear becomes very large. Since it is too costly, the principal switches to the Two-tier Regime (TW) which induces X L TW in both states. □