Management Turnover, Strategic Ambiguity and Supply Incentives

A.1 SPNE without Ambiguity

We look for the symmetric SPNE where the two firms accept their contracts. By backward induction, we find that firms accept contracts whenever their profit is positive: π _i (c) = (1 − q _i − γq _j )q _i − f _i . This happens whenever f _i ≤ (1 − q _i − γq _j )q _i , for all i ∈ {1, 2}. At the contracting stage, the monopolist anticipates this decision, and sets each fixed so as to capture all the firms’ rents because its profit, π _U = f _i + f _j , is increasing in f _i . In addition, since we focus on the symmetric equilibrium, we have f _i = f _j = f and thus q _i = q _j = q/2 which leads to f = (1 − (q/2) − γ(q/2))(q/2) = (1 − (1 + γ)(q/2))(q/2). The monopolist then maximizes its profit π _U = (1 − (1 + γ)(q/2))q. The First Order Condition gives dπ _U /dq = 1 − (1 + γ)q = 0 and therefore: q _i = q _j = 1/[2(1 + γ)].

Substituting these values into the fixed fees, we have f _i = f _j = 1/[4(1 + γ)] which directly gives the monopolist’s profit π _U = 1/[2(1 + γ)]. Symmetrically, the consumer surplus can be rewritten as: CS = (1 + γ)q ² which gives CS = 1/[4(1 + γ)].□

A.2 SPNE with Strategic Ambiguity and Sequential Contracting

We look for the SPNE in pure strategies of the game and thus solve the game using backward induction. We now refer to the new manager and D ₁ as the same entity to ease the explanation and notations. For the same reason we also do not write the history of strategies as functions of the history of past strategies.

When D ₂ observes its offer, it also observes the previous contract offer and D ₁’s decision. D ₂ accepts the contract whenever its profit is positive. Formally, this means:

where π ₂(a ₂ = 1|c, a ₁) = (1 − q ₂ − γa ₁ q ₁)q ₂ − f ₂. We focus on equilibria where the firms accept their offers, which means that the monopolist’s offer to D ₂ must satisfy f ₂ ≤ (1 − q ₂ − γa ₁ q ₁)q ₂.

When the monopolist makes the offer to D ₂, it anticipates D ₂’s decision, given its own previous decision and that of D ₁. The monopolist thus maximizes π _U = f ₁ + f ₂ where f ₂ ≤ (1 − q ₂ − γa ₁ q ₁)q ₂ and f ₁ is sunk (because already paid by D ₁ at this stage). The profit is increasing in f ₂ so the monopolist extracts all the rent and the profit rewrites π _U = (1 − q ₂ − γa ₁ q ₁)q ₂. The monopolist maximizes this profit for any contract offer ( q 2 * ( a 1 , c 1 ) , f 2 * ( a 1 , c 1 ) ) such that

where q 2 * ( c 1 , a 1 ) is simply the Cournot best response to a ₁ q ₁.

When D ₁ gets its offer c ₁, it has to anticipate the other firms’ future decisions. This anticipation is critical. D ₁ perfectly anticipates D ₂’s decision, a 2 * . However, since D ₁ is ambiguous towards the monopolist’s decision, it weighs up the best and worst outcome induced by all the strategies available to the monopolist at the next stage, c 2 ∈ C 2 .

More formally, and by applying Eq. (6), the result is that D ₁’s expected profit from accepting the offer is E c 2 π 1 a 1 = 1 , c 2 , a 2 * | c 1 = ( 1 − α ) max c 2 ∈ C 2 π 1 a 1 = 1 , c 2 , a 2 * | c 1 + α min c 2 ∈ C 2 π 1 a 1 = 1 , c 2 , a 2 * | c 1 . D ₁ accepts whenever its expected profit is positive and we now formally get:

where the expected profit simplifies to E c 2 π 1 a 1 = 1 , c 2 , a 2 * | c 1 = ( 1 − α ) ( 1 − q 1 ) q 1 + α ( 1 − γ − q 1 ) q 1 − f 1 if q ₁ ≤ 1 − γ and E c 2 π 1 a 1 = 1 , c 2 , a 2 * | c 1 = ( 1 − α ) ( 1 − q 1 ) q 1 + 0 − f 1 when q ₁ ≥ 1 − γ.

Intuitively, given contract offer c ₁ = (q ₁, f ₁), the best outcome appears in the event where the monopolist offers nothing to D ₂, q ₂ = 0, and requests nothing from them (so that D ₂ accepts this contract). The worst outcome appears in the scenario where the monopolist offers the maximum quantity to D ₂, i.e. q ₂ = 1, which potentially drives the market price to zero, and requests nothing in exchange for such a quantity (so that again D ₂ accepts). Under imperfect substitution, even if the monopolist gives the maximal quantity to the rival q ₂ = 1, but provides a sufficiently low quantity to D ₁, q ₁ ≤ 1 − γ, the latter can expect at worst a positive market price and hence positive profits.

We focus on equilibria where the firms accept the contract, so the monopolist’s offer to D ₁ must satisfy f ₁ ≤ (1 − α)(1 − q ₁)q ₁ + α max{(1 − γ − q ₁)q ₁, 0}. When the monopolist decides on the offer for D ₁, it anticipates the other firms’ strategies. The monopolist thus maximizes π _U = f ₁ + f ₂ where f 2 ≤ ( 1 − γ q 1 ) 2 4 and f ₁ ≤ (1 − α)(1 − q ₁)q ₁ + α max{(1 − γ − q ₁)q ₁, 0}. The profit is increasing in the fees so, for a given level of pessimism α, the monopolist extracts all the rent.

A.2.2 Imperfect Substitutes

At the time the monopolist enters into contract with the first firm, it maximizes

π U = 1 − γ q 1 2 2 + ( 1 − α ) ( 1 − q 1 ) q 1 + α max { ( 1 − γ − q 1 ) q 1 , 0 }

We have two cases to consider, depending on if q ₁ ≤ 1 − γ or q ₁ ≥ 1 − γ.

Case 1. q ₁ ≤ 1 − γ.

We have max{(1 − γ − q ₁)q ₁, 0} = (1 − γ − q ₁)q ₁, and the first order condition and the second order condition respectively are:

(15) ∂ π U ∂ q 1 = 0 ⇔ 1 2 − ( 2 α + 1 ) γ + γ 2 − 4 q 1 + 2 = 0

(16) ∂ 2 π U ∂ 2 q 1 ≤ 0 ⇔ 1 2 γ 2 − 4 ≤ 0

The FOC is satisfied when evaluated at q 1 ( α , γ ) = 2 − γ ( 1 + 2 α ) 4 − γ 2 . Let us suppose q 1 L ( α , γ ) ≡ 2 − γ ( 1 + 2 α ) 4 − γ 2 . Equation (16) holds whenever (α, γ) ∈ [0,1]². Therefore, q 1 L ( α , γ ) is always a maximum for π _U . When α ≤ α ̄ ̄ ( γ ) ≡ 1 γ − 1 2 , we have q 1 L ( α , γ ) ≥ 0 . q 1 L ( α , γ ) is thus the maximum in this region. When α > α ̄ ̄ ( γ ) ≡ 1 γ − 1 2 , we find q 1 L ( α , γ ) < 0 . Since a quantity must be positive, the maximum on this region is 0. Last, q ₁(α, γ) ≤ 1 − γ is satisfied as long as α ≤ 1 2 − γ 2 + γ − 2 γ + 3 and henceforth q ₁(α, γ) = 1 − γ is the maximum on the region where α > 1 2 − γ 2 + γ − 2 γ + 3 . Figure 4 summarizes our findings. The red line refers to α ̄ ̄ ( γ ) and the blue line refers to 1 2 − γ 2 + γ − 2 γ + 3 .

Figure 4:

The solution for q ₁ < 1 − γ.

Case 2. q ₁ ≥ 1 − γ.

We have max{(1 − γ − q ₁)q ₁, 0} = 0, and the first order condition and the second order condition respectively are:

(17) ∂ π U ∂ q 1 = 0 ⇔ 1 2 − 2 α − γ + q 1 4 α + γ 2 − 4 + 2 = 0

(18) ∂ 2 π U ∂ 2 q 1 ≤ 0 ⇔ 1 2 4 α + γ 2 − 4 ≤ 0

The FOC is satisfied when evaluated at q 1 ( α , γ ) = 2 ( 1 − α ) − γ 4 ( 1 − α ) − γ 2 . Let us suppose q 1 H ( α , γ ) ≡ 2 ( 1 − α ) − γ 4 ( 1 − α ) − γ 2 . Equation (18) holds whenever γ ∈ [0, 1] and 0 ≤ α ≤ 1 4 4 − γ 2 . Therefore, q 1 H ( α , γ ) is a maximum for π _U when 0 < γ < 1 and 0 ≤ α ≤ 1 4 4 − γ 2 , and a minimum otherwise. When 0 ≤ α ≤ 1 4 4 − γ 2 , we find q 1 H ( α , γ ) ≥ 1 − γ only if α ≤ ( γ − 2 ) γ 2 + γ − 1 2 − 4 γ . It implies that (i) q 1 H ( α , γ ) is the maximum in the area where α ≤ ( γ − 2 ) γ 2 + γ − 1 2 − 4 γ and (ii) q ₁(α, γ) = 1 − γ is the maximum in the area where ( γ − 2 ) γ 2 + γ − 1 2 − 4 γ < α < 1 4 4 − γ 2 . When α > 1 4 4 − γ 2 , q 1 H ( α , γ ) is a minimum, we find that the value is higher than 1 − γ meaning that the maximum on this part is either in q ₁(α, γ) = 1 or q ₁(α, γ) = 1 − γ. It can be shown that the profit at q ₁(α, γ) = 1 − γ is higher than that at q ₁(α, γ) = 1 so that q ₁(α, γ) = 1 − γ is the maximum on this area. Figure 5 summarizes our findings. The black line refers to (1/4)(4 − γ ²) and the blue line refers to ( γ − 2 ) γ 2 + γ − 1 2 − 4 γ .

Figure 5:

The solution for q ₁ ≥ 1 − γ.

We now derive the best solution for the monopolist in each region.

A.3 Maximum Profit at Optimum Solutions

Let us denote by π U L the profit function when q ₁ ≤ 1 − γ and π U H the profit function when q ₁ ≥ 1 − γ. Note that the profits are the same at q ₁ = 1 − γ. Several cases arise:

1. In the area where α ≤ 1 2 − γ 2 + γ − 2 γ + 3 , i.e. below the blue line of Figure 4, we find that

Therefore, q 1 H is the solution in this region.

1. In the area where 1 2 − γ 2 + γ − 2 γ + 3 < α < ( γ − 2 ) γ 2 + γ − 1 2 − 4 γ , i.e. between the blue lines of Figures 4 and 5, we find that

This is positive whenever α > α ̄ ( γ ) ≡ 1 8 − ( γ − 2 ) 2 ( γ − 1 ) 2 ( γ + 2 ) ( γ ( γ ( γ + 4 ) − 3 ) + 2 ) γ 4 − γ 2 − 4 γ 2 + γ + 8 γ and negative otherwise. The solution in this region is thus q 1 L ( α , γ ) when α > α ̄ ( γ ) and q 1 H ( α , γ ) , otherwise.

1. In the area where α > α ̄ ̄ ( γ ) , above the red line of Figure 4, we find that

The solution in this region is q ₁(α, γ) = 0.

1. In the last area, we find that

The solution in this region is q 1 L ( α , γ ) .

These thresholds are summarized in Figure 6 below, where q 1 T A ′ = q 1 H ( α , γ ) , q 1 T B ′ = 0 and q 1 T C ′ = q 1 L ( α , γ ) .

Figure 6:

(Equiv. Figure 1) Graph of solution partition.

With management turnover and imperfect substitutes, the SPNE strategy of U jointly is:

where Area A′ denotes the subset { ( α , γ ) ∈ [ 0,1 ] × ( 0,1 ) : α < α ̄ ( γ )  and  1 > γ > 1 2 ( 5 − 1 ) } such that α ̄ ( γ ) = ( 1 / 8 ) − ( γ − 2 ) 2 ( γ − 1 ) 2 ( γ + 2 ) ( γ ( γ ( γ + 4 ) − 3 ) + 2 ) / γ 4 − γ 2 − 4 / γ 2 + ( γ + 8 ) / γ , Area B′ denotes the subset { ( α , γ ) ∈ [ 0,1 ] × ( 0,1 ) : α ̄ ̄ ( γ ) < α < 1  and  1 > γ > 2 / 3 } such that α ̄ ̄ ( γ ) = 1 / γ − 1 / 2 and Area C′ denotes the rest of the set of parameters.□

B.1 Proof of Proposition 1

Denote by π _U (α) the profit of the monopolist under strategic ambiguity. By the results under perfect substitutes obtained in the above proof and displayed in Eq. (8) (or in Lemma 2 in the proof of Proposition 4), we have:

as long as α ≤ 1/2. For higher values of α, π _U is equal to 1/4. The monopoly profit with perfect substitutes is π U M = 1 / 4 . We then get:

which is strictly positive as long as α < 1/2 and null otherwise.□

B.2 Proof of Proposition 3

Take for example α = 1 and a sufficiently high γ, say γ > γ′, so that we are in area B. We get f 1 T + f 2 T = 1 / 4 while the monopoly profit is π M = π U B = 1 / ( 2 + 2 γ ) . With imperfect substitutes, we have γ′ < γ < 1, which implies that f 1 T + f 2 T = 1 / 4 < 1 / ( 2 + 2 γ ) = π M .□

B.3 Proof of Proposition 4

Remember that π U B = 1 / ( 2 ( 1 + γ ) ) is the benchmark profit without management turnover. Let us now denote by π U T the profit of the monopolist with management turnover. Lemma 2 summarizes the monopolist’s SPNE strategies according to the parameter values.

Since on these SPNE, firms always accept the contract, we have our next lemma which displays the monopolist’s equilibrium profits with respect to the parameter values.

We then get:

which is positive whenever α < α ̃ a ( γ ) ≡ 1 2 − ( γ − 2 ) γ 3 + 1 ( γ + 1 ) 2 − γ − 2 γ + 1 + 3 and negative otherwise in area A, always negative in area B and positive whenever α < α ̃ c ( γ ) ≡ − γ 2 − γ 4 − 5 γ 2 + 4 + γ + 2 2 γ 2 + γ while negative otherwise in area C. Figure 2 summarizes our findings and is redisplayed below (Figure 7).

1. Let’s additionnally prove that the thresholds are inferior to 1/2 for γ ∈ (0, 1),

   1. Consider α ̃ c ( γ ) = − γ 2 − γ 4 − 5 γ 2 + 4 + γ + 2 2 γ 2 + γ ≤ 1 / 2 . It is equivalent to 2 + γ − γ 2 − 4 − 5 γ 2 + γ 4 ≤ ( 1 / 2 ) 2 ( γ + γ 2 ) ⇒ 2 − 2 γ 2 ≤ 4 − 5 γ 2 + γ 4 ⇒ 4 ( 1 − γ 2 ) 2 ≤ 4 − 5 γ 2 + γ 4 ⇒ 4 − 8 γ 2 − 4 γ 4 ≤ 4 − 5 γ 2 + γ 4 which is true for γ ∈ (0, 1).

   2. For α ̃ a ( γ ) = 1 2 − ( γ − 2 ) γ 3 + 1 ( γ + 1 ) 2 − γ − 2 γ + 1 + 3 ≤ 1 / 2 , we have − ( γ − 2 ) γ 3 + 1 ( γ + 1 ) 2 − γ − 2 γ + 1 + 3 ≤ 1 ⇒ − γ − 2 γ + 1 + 3 − 1 ≤ ( γ − 2 ) γ 3 + 1 ( γ + 1 ) 2 ⇒ γ − γ 2 γ + 1 ≤ ( γ − 2 ) γ 3 + 1 ( γ + 1 ) 2 ⇒ γ − γ 2 ≤ ( γ − 2 ) γ 3 + 1 ⇒ ( γ − γ 2 ) 2 ≤ 1 + γ 4 − 2 γ 3 ⇒ γ 2 − 2 γ 3 + γ 4 ≤ 1 − 2 γ 3 + γ 4 which is true for γ ∈ (0, 1).

Figure 7:

(Equiv. Figure 2) The monopolist’s profits.

B.4 Proof of Eq. 11

From π U T ( α ̃ ( γ ) , γ ) − π U B ( γ ) = 0 , we get:

We can then decompose the profits. First, using π U T ( q 2 * ( q 1 ) , q 1 ) at q 1 T ( α , γ ) , we have

At this point ∂ π U T / ∂ q 1 ( q 1 T ) = 0 , the expression simplifies to

Because f 1 q 1 T = ( 1 − α ) 1 − q 1 T q 1 T + 1 q 1 T < 1 − γ α 1 − γ − q 1 T q 1 T and f 2 q 1 T = ( 1 / 4 ) ( 1 − γ q 1 T ) 2 , we find

By the same process, we obtain

Finally, it is easy to see that π U B q 1 * , q 2 * implies the same derivative irrespective of the area considered. We get

From Eqs. (22)–(24), we find that:

This is Eq. (11) in our main text.

In area A, where q 1 T = q 1 T A we have:

And the sign is positive.

In area C, where q 1 T = q 1 T C we have:

And the sign can be positive or negative. However, evaluating the sign at α = α ̃ c ( γ ) we find that

which is negative provided 0 < γ < − 1 + 3 ≈ 0.730 . This threshold is lower than the threshold at which α ̃ intersects area A, γ ≈ 0.784. Therefore, the slope is slightly positive above γ = − 1 + 3 and negative below.□

B.5 Proof of Proposition 5

1. It is straightforward to see that the firms make no profit in the case without management turnover.

2. With management turnover, the monopolist takes advantage of ambiguity when the manager is sufficiently optimistic ( α ≤ α ̃ ( γ ) ) . This area encompasses a sub-part of Area A and Area C. From Eq. (12), we find that:

   1. in the intersection of Area A with the area where α ≤ α ̃ ( γ ) , the profit of Firm 1 is: