We focus on the link between relative-performance scheme and sustainability of upstream collusion. Apply backward induction to solve for the subgame perfect Nash equilibrium (SPNE).

In the third stage, each manager chooses his quantity to maximize his reward:

\(\mathop {Max}\limits_{{{q_i}}}\) ,\(i,j=\text{1,2}\),\(i\ne j\), (3)

where the superscript \(C\) and the subscript \(RP\) represent quantity competition in the downstream market and the relative-performance scheme, respectively.

Substituting (2) into (3) and solving the first-order conditions (FOCs) from (3), we have:

\({q}_{i}=\frac{2(a-{w}_{i})+{\gamma }({\lambda }_{i}-1)(a-{w}_{j})}{4-{{\gamma }}^{2}({\lambda }_{i}-1)({\lambda }_{j}-1)}\) , \(i,j=\text{1,2}\), \(i\ne j\). (4)

Differentiating with respect to \({\lambda }_{i}\) and \({\lambda }_{j}\), we have:

$$\frac{\partial {q}_{i}}{\partial {\lambda }_{i}}=\frac{2\gamma (-2{w}_{j}+\gamma {w}_{i}(1-{\lambda }_{j})+a(2-\gamma (1-{\lambda }_{j})\left)\right)}{{(4-{\gamma }^{2}-{\gamma }^{2}({\lambda }_{i}\left(1-{\lambda }_{j}\right)+{\lambda }_{j}\left)\right)}^{2}}>0$$

$$\frac{\partial {q}_{i}}{\partial {\lambda }_{j}}=\frac{{\gamma }^{2}\left(2\right(a-{w}_{1})+\gamma (a-{w}_{2}\left)\right(1-{\lambda }_{1}\left)\right)(1-{\lambda }_{1})}{{(4-{\gamma }^{2}(1-{\lambda }_{1}\left)\right(1-{\lambda }_{2}\left)\right)}^{2}}<0$$

The incentive variables will increase its own output and reduce the rival output.

In the second stage, owner \(i\) sets the incentive parameter simultaneously to maximize his own profit. So, owner \(i\)’s maximization problem is given by:

\(\underset{{\lambda }_{i}}{Max}\) \({{\varPi }_{i}}_{RP}^{C}=({p}_{i}-{w}_{i}){q}_{i}\), \(i=\text{1,2}\). (5)

Substituting (2) and (4) into (5) and solving the FOCs from (5) yield:

\({\lambda }_{i}=\frac{{\gamma }(a-{w}_{i}-a{\gamma }+{w}_{j}{\gamma })}{2a-2{w}_{j}-{\gamma }(a-{w}_{i}+a{\gamma }-{w}_{j}{\gamma })}\) , \(i,j=\text{1,2}\), \(i\ne j\). (6)

In the first stage, there are two choices for manufacturer \(i\): (i) to set the wholesale price \({w}_{i}\) non-cooperatively to maximize its own profit \({\pi }_{i}\); (ii) to collude in the upstream market so that they choose the wholesale price together to maximize the total profits \({w}_{1}{q}_{1}+{w}_{2}{q}_{2}\).

In the first case, there is a non-cooperative game, and manufacturer \(i\)’s problem is:

$$\underset{{w}_{i}}{Max}{\pi }_{i}={w}_{i}{q}_{i}$$

7

Plugging (4) and (6) back into (7) and solving the FOCs, we have:

$${{w}_{i}}_{RP}^{{C}^{N}}=\frac{2a-a{\gamma }-a{{\gamma }}^{2}}{4-{\gamma }-2{{\gamma }}^{2}}$$

$${{\pi }_{i}}_{RP}^{{C}^{N}}=\frac{{a}^{2}(1-{\gamma })(2+{\gamma }{)}^{2}(2-{\gamma })}{4(1+{\gamma })(4-{\gamma }-2{{\gamma }}^{2}{)}^{2}}$$

8

where the superscript \(N\) denotes Nash equilibrium.

Given \({\lambda }_{i}={\lambda }_{j}=0\), we have the input price without delegation \({w}_{i}=a-\frac{2a}{4-\gamma }\). Compared with the input price with delegation, we find that

$${{w}_{i}}_{RP}^{{C}^{N}}-{w}_{i}=-\frac{a{\gamma }^{3}}{\left(4-\gamma \right)\left(4-\gamma -2{\gamma }^{2}\right)}<0$$

Relative-performance delegation will increase the total output of the downstream firms and lower the input price of the upstream firm.

In the second case, two manufacturers form a cartel to set wholesale price together for the maximization of the joint profit:

\(\underset{{w}_{1},{w}_{2}}{Max}\) \({\pi }_{1}+{\pi }_{2}={w}_{1}{q}_{1}+{w}_{2}{q}_{2}\) (9)

Substituting. (4) and (6) into (9) and taking differentiation, we obtain:

$${{w}_{i}}_{RP}^{{C}^{C}}=\frac{a}{2}$$

$${{\pi }_{i}}_{RP}^{{C}^{C}}=\frac{{a}^{2}(2+{\gamma })}{16(1+{\gamma })}$$

10

where the superscript \(C\) denotes collusion occurring in the upstream market.

However, another situation should also be taken into consideration, where manufacturer \(i\) deviates privately from the cartel in some period and the other manufacturer does not notice this behavior until the next period. Under this circumstance, manufacturer \(i\) will maximize its own profit \({\pi }_{i}\) while the other still follows the agreement stipulated in the cartel:

\(\underset{{w}_{i}}{Max}\)
\({\pi }_{i}={w}_{i}{q}_{i}\) \(s.t.\) \({w}_{j}=\frac{a}{2}\), \(i,j=\text{1,2}\) \(i\ne j\). (11)

Substituting (4) and (6) into (11) leads to the following results:

$${{w}_{i}}_{RP}^{{C}^{D}}=\frac{4a-a{\gamma }-2a{{\gamma }}^{2}}{4\left(2-{{\gamma }}^{2}\right)}$$

$${{\pi }_{i}}_{RP}^{{C}^{D}}=\frac{{a}^{2}(4-{\gamma }-2{{\gamma }}^{2}{)}^{2}}{64(2-3{{\gamma }}^{2}+{{\gamma }}^{4})}$$

12

where the superscript \(D\) represents the deviation from collusion.

Manufacturer \(i\) is faced with a trigger strategy: it chooses collusion (i.e., \({w}_{i}=\frac{a}{2}\)) at period 1, and at the \({t}^{th}(t>1)\) period still sticks to collusion if both manufacturers set wholesale price \({w}_{i}=\frac{a}{2}\) in all the previous periods; otherwise, it uses the Nash equilibrium outcome (i.e., \({w}_{i}=\frac{2a-a{\gamma }-a{{\gamma }}^{2}}{4-{\gamma }-2{{\gamma }}^{2}}\)) as punishment in the \({t}^{th}\) and all the subsequent periods. Let \(\delta\) be the discount factor of each manufacturer, which measures the manufacturers’ patience or how much importance they attach to the future. Both manufacturers will then apply the trigger strategy for higher profits, i.e., they will not deviate from the cartel and upstream collusion will inevitably appear if and only if the following inequality holds:

$$\delta \ge \frac{{\pi }_{D}-{\pi }_{C}}{{\pi }_{D}-{\pi }_{N}}$$

13

where\({{\pi }}_{\text{D}}\): deviating profit; \({\pi }_{C}\): cooperative profit; \({\pi }_{N}\): Nash-punishment profit.

We then compare three payoffs (deviating, cooperative and punishment) in the model with delegation to those in the model without delegation, to see how delegation affects the cutoff \(\frac{{\pi }_{D}-{\pi }_{C}}{{\pi }_{D}-{\pi }_{N}}\).

We want to compare the model with delegation to the model with forward ownership in the case of partial collusion. In a model with forward ownership, Reisinger and Thomes (2017) show that for the case of non-binding retail, contract offers by the upstream firm that are observable to the rival retailer are not necessarily beneficial for collusive purposes which act as relative-performance delegation that upstream firm has incentive to deviate from collusion.

$${\sum }_{t=0}^{\infty }\frac{{a}^{2}(2+{\gamma })}{16(1+{\gamma })}{\delta }^{t}\ge \frac{{a}^{2}(4-{\gamma }-2{{\gamma }}^{2}{)}^{2}}{64(2-3{{\gamma }}^{2}+{{\gamma }}^{4})}+{\sum }_{t=1}^{\infty }\frac{{a}^{2}(1-{\gamma })(2+{\gamma }{)}^{2}(2-{{\gamma }}^{2})}{4(1+{\gamma })(4-{\gamma }-2{{\gamma }}^{2}{)}^{2}}{\delta }^{t}$$

14

where the left hand of Ineq. (14) represents the present value of the manufacturer if it always sticks to collusion and the right hand denotes the manufacturer’s present value if it deviates from the agreement reached in the cartel.

Solving Ineq. (14) yields the threshold \({\delta }_{RP}^{C}\) as follows. The upstream market will fall into collusion if and only if each manufacturer’s \(\delta\) satisfies \(\delta \ge {\delta }_{RP}^{C}\).

$${\delta }_{RP}^{C}=\frac{(4-{\gamma }-2{{\gamma }}^{2}{)}^{2}}{32-16{\gamma }-31{{\gamma }}^{2}+8{\gamma }^{3}+8{{\gamma }}^{4}}$$

15

$$\frac{d{\delta }_{RP}^{C}}{d{\gamma }}=\frac{4\gamma \left(2+{{\gamma }}^{2}\right)\left(4-{\gamma }-2{{\gamma }}^{2}\right)}{{\left(32-16{\gamma }-31{{\gamma }}^{2}+8{{\gamma }}^{3}+8{{\gamma }}^{4}\right)}^{2}}>0$$

16

When it comes to the pure-profit scheme, as shown in Bian et al. (2013), the critical \({\delta }_{P}^{C}\) is:

$${\delta }_{P}^{C}=\frac{(4-{\gamma }{)}^{2}}{32-16{\gamma }+{{\gamma }}^{2}}$$

17

$$\frac{d{\delta }_{P}^{C}}{d{\gamma }}=\frac{8\left(4-{\gamma }\right){\gamma }}{{\left(32-16{\gamma }+{{\gamma }}^{2}\right)}^{2}}>0$$

18

where the subscript \(P\) denotes the pure-profit scheme.

Furthermore, Bian et al. (2013) investigate downstream sales-revenue delegation, which is denoted by the subscript \(R\). They calculate the critical value \({\delta }_{R}^{C}\) and find \({\delta }_{R}^{C}>{\delta }_{P}^{C}\).

$${\delta }_{R}^{C}=\frac{{\left(4-{\gamma }-{{\gamma }}^{2}\right)}^{2}}{32-16{\gamma }-15{{\gamma }}^{2}+4{{\gamma }}^{3}+2{{\gamma }}^{4}}$$

19

$$\frac{d{\delta }_{R}^{C}}{d{\gamma }}=\frac{2\gamma \left(4+{{\gamma }}^{2}\right)\left(4-{\gamma }-{{\gamma }}^{2}\right)}{{\left(32-16{\gamma }-15{{\gamma }}^{2}+4{{\gamma }}^{3}+2{{\gamma }}^{4}\right)}^{2}}>0$$

20

We obtain three partial derivatives with respect of \(\gamma\) in Ineq. (16), (18) and (20), and find that higher product differentiation will hinder upstream collusion. The same reasoning was provided by Deneckere (1983) for downstream firm’s collusion without any types of managerial delegation: sustaining collusion becomes more difficult with an increasing degree of substitutability.

By comparing (15), (17) and (19), upstream firms are difficult to sustain collusion in relative-performance scheme than in downstream profit-and-revenue incentive schemes. (See Fig. 1 for depiction).

We have the following Proposition 1.

**Proposition 1**

*In a quantity competition market, the relative-performance delegation impedes upstream collusion, i.e., .*

The main reasoning is that when the owners symmetrically adopt relative-performance delegation, the managers will behave more aggressively when the firms compete in quantities. The upstream firm will then have more incentive to deviate by lowering input prices. An increase in the degree of product differentiation (the products becomes more homogenous), sustaining upstream collusion becomes more difficult. Same as Miller and Pazgal (2001), relative-performance delegation intensifies downstream quantity competition and has a quantity-enhancing effect, similar to Bian et al. 2013) and Wang and Wang (2021).

[3] Delbono and Lambertini (2020) show that managerial delegation based upon relative performance may generate collusive outcomes observationally equivalent to those typically associated with repeated games or cross ownership.

[4] Notice that .