Relative-Performance Delegation destabilizes Upstream Collusion

doi:10.21203/rs.3.rs-1979811/v1

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Relative-Performance Delegation destabilizes Upstream Collusion

https://doi.org/10.21203/rs.3.rs-1979811/v1

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This paper analyzes upstream firms’ collusive sustainability when downstream firms adopt the relative-performance delegation in an infinitely repeated Cournot or Bertrand game. We find that relative-performance delegation makes managers acting more aggressive and that leads to more difficult for upstream collusion to be sustained comparing with sales-revenue delegation regardless of the competition modes. The driving force behind our result is that downstream relative-performance delegation decreases input price, while it makes more profit for deviated firm.

JEL classification: D21; D43; L13; L21

Relative-performance delegation

Upstream collusion

Vertically related market

Competition modes

The relative-performance delegation is widely used in practice, in which the managers’ incentive contract is based not only on own firms profits but also on the rival’s relative profits. Salas-Fumas (1992), and Miller and Pazgal (2001, 2002) have argued that relative-performance delegation is more popular delegation contract in business world. Miller and Pazgal (2001) demonstrate that if the owners are able to manipulate their managers’ behavior under the relative performance contracts, the equilibrium outcome is the same, regardless of the competition modes.

In a seminal paper, Deneckere (1983) in a repeated Cournot and Bertrand duopolies with linear demand and symmetric marginal costs, derives some basic results on the ability to maximize profits jointly.^{^[1]} Since then, there are more papers investigate collusive effects from different perspectives. Lambertini and Trombetta (2002), Spagnolo (2005), and Matsumura and Matsushima (2012) examine the implication of managerial delegation on collusive stability in horizontal market with or without differentiated products, but are not in the context of vertically related markets. Even though Nocke and White (2007), Normann (2009), Piccolo and Reisinger (2011), and Reisinger and Thomes (2017) focus on tacit collusion in the vertical case, but without considering managerial incentives. Bian et al. (2013) consider sales-revenue delegation in a vertical framework and examine the sustainability of upstream collusion under downstream Cournot and Bertrand competition. Wang and Wang (2021) extend Bian et al. (2013) in two directions: (1) confining to full collusion, they revisit the Bertrand result for close substitutes and find in contrast the possibility that downstream managerial delegation impedes upstream collusion; (2) the results in both Cournot and Bertrand models are sensitive to the case of partial collusion.

Considering cross ownership among firms in a vertical related market without downstream delegation, Charistos et al. (2022) show that symmetric passive forward ownership without input price discrimination hinders the sustainability of upstream collusion.[2] This paper analyzes upstream firms’ collusive sustainability when downstream firms adopt the relative-performance delegation. We need to clarify the difference between exogeneous cross ownership referring the investor’s shareholding interest and endogenous relative-performance delegation denoting market competition between owner’s concern of firm’s profit.

We show that without observable vertical arrangement have two identified effects on manufacturers' incentives to stay or deviate from tacit collusion. First, downstream firm without binding of upstream ownership will intensify product market competition in a one-shot game paying lower input price. Hence, punishment profits are higher, thereby disincentivizing deviation more profitable. Second, product substitutability and relative-performance delegation intensify market competition, and reduce rival’s profit that is favorable for the upstream firm to deviate from collusion. Hence, relative-performance delegation makes more difficult for upstream collusion comparing with sales-revenue delegation coupled with increasing product substitutability. Our results have different mechanisms for reaching destine.

This paper is organized as follows. Basic model is provided in Section 2. The analysis of upstream collusion under quantity and price competition are provided in Section 3 and 4. Section 5 concludes the paper.

[1] More works on the collusion behavior in the Cournot or/and Bertrand infinitely repeated game can be found in Abreu (1988), Chang (1991), Lambertini and Sasaki (2001), and Østerdal (2003), among many others.

[2] Sun and Wang (2022) analyze the impact of vertical cross‐ownership with input price discrimination on social welfare. Under forward cross‐ownership, a higher degree of product differentiation and a higher degree of cross‐ownership raise industry profit, consumer surplus, and social welfare.

We assume that there exist two manufacturers (indexed by 1 and 2) producing differentiated goods and two retailers (indexed by 1 and 2) in a vertically related market. Manufacturers choose retailers to wholesale their commodity. Moreover, retailer $i$ has a two-level governance structure consisting of an owner $i$ and a manager $i$ ($i=\text{1,2}$). For simplicity, we suppose that both managers’ opportunity cost and manufacturers’ marginal cost are zero. The representative consumer’s utility function is given by:

$$U({q}_{1},{q}_{2})=a({q}_{1}+{q}_{2})-\frac{1}{2}({q}_{1}^{2}+2{\gamma }{q}_{1}{q}_{2}+{q}_{2}^{2})+m$$

where ${q}_{1}$ and ${q}_{2}$ represent two manufacturers’ production quantity. The parameter ${\gamma }\in \left(\text{0,1}\right)$ measures the degree of the product differentiation, and $m$ denotes composite goods.

The inverse demand function is derived as follows:

${p}_{i}=a-{q}_{i}-{\gamma }{q}_{j}$ , $i,j=\text{1,2}$,$i\ne j$. (2)

We assume that both owners have two alternative incentive schemes: (i) the relative-performance scheme under which manager $i$’s reward is given by: $\left[\right(1-{\lambda }_{i}){\pi }_{i}+{\lambda }_{i}({\pi }_{i}-{\pi }_{j}\left)\right]=({\pi }_{i}-{\lambda }_{i}{\pi }_{j})$, where ${\lambda }_{i}\in \left[\text{0,1}\right]$ is the weight that owner $i$ puts on the relative performance, ${\pi }_{i}$ and ${\pi }_{j}$ respectively denote retailer $i$’s and retailer $j$’s profits, and ${\alpha }_{i}$ is an exogenous scaling parameter; (ii) the pure-profit scheme under which manager $i$’s salary is ${\pi }_{i}$.

We consider an infinitely repeated game where a multi-stage game occurs in each period. The timing of multi-stage game is that manufacturer $i$ ($i=\text{1,2}$) in the upstream market simultaneously chooses its own wholesale price ${w}_{i}$ ($i=\text{1,2}$) in the first stage; owner $i$ sets its incentive variable ${\lambda }_{i}$ ($i=\text{1,2}$) in the second stage; in the last stage, manager $i$ competes in the downstream market by choosing the quantity${q}_{i}$ ($i=\text{1,2}$). For the pure-profit scheme, it degenerates into a two-stage game: in the first stage, the manufacturers set the wholesale price simultaneously, and the managers of both retailers choose the quantity or price in the last stage.

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:

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

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

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

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

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:

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

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

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:

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

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

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:

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

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.

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

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 $\delta \ge {\delta }_{RP}^{C}$.

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

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

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

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

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

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 ${\delta }_{R}^{C}>{\delta }_{P}^{C}$.

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

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

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 .

We want to make a comparison of collusive stability of upstream firms under Cournot and Bertrand model.

The demand function is expressed as follows:

$${q}_{1}=\frac{(1-{\gamma })a-{p}_{1}+{\gamma }*{p}_{2}}{1-{{\gamma }}^{2}}$$

$${q}_{2}=\frac{(1-{\gamma })a-{p}_{2}+{\gamma }*{p}_{1}}{1-{{\gamma }}^{2}}$$

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

From the FOC, we have

$${p}_{1}=\frac{{w}_{1}(-2+{{\gamma }}^{2}(-1+{\lambda }_{1}){\lambda }_{2}-{\gamma }{w}_{2}(1+{\lambda }_{1})+a(-2+{\gamma }+{{\gamma }}^{2}-(-1+{\gamma }){\gamma }{\lambda }_{1})}{-4+{{\gamma }}^{2}+{{\gamma }}^{2}\left({\lambda }_{1}\right(-1+{\lambda }_{2})-{\lambda }_{2})}$$

$${p}_{2}=\frac{{w}_{2}(-2+{{\gamma }}^{2}{\lambda }_{1}(-1+{\lambda }_{2}\left)\right)-{\gamma }{w}_{1}(1+{\lambda }_{2})+a(-2+{\gamma }+{{\gamma }}^{2}-(-1+{\gamma }\left){\gamma }{\lambda }_{2}\right)}{-4+{{\gamma }}^{2}+{{\gamma }}^{2}\left({\lambda }_{1}\right(-1+{\lambda }_{2})-{\lambda }_{2})}$$

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

$$\underset{{\lambda }_{i}}{Max}{{\varPi }_{i}}_{RP}^{B}=\left({p}_{i}-{w}_{i}\right){q}_{i} i=\text{1,2}$$

Solving the FOCs from (25) yields

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

where the superscript B denotes Bertrand competition.

In the next part, we adopt the same process as in the Cournot model to find the results of three cases.

$\text{I}\text{n} \text{t}\text{h}\text{e} \text{f}\text{i}\text{r}\text{s}\text{t} \text{c}\text{a}\text{s}\text{e},$ we have

$${w}_{1}=\frac{2a-a{\gamma }-a{{\gamma }}^{2}}{4-{\gamma }-2{{\gamma }}^{2}}$$

$${w}_{2}=\frac{2a-a{\gamma }-a{{\gamma }}^{2}}{4-{\gamma }-2{{\gamma }}^{2}}$$

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

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

Note that $\left(27\right)$are identical with (12), we do not need to compute the second and the third cases since all the computations are exactly the same as that obtained under Cournot competition. We then conclude that under Bertrand competition in the downstream market, the relative-performance delegation hinders upstream collusion, i.e.,${\delta }_{RP}^{B}>{\delta }_{R}^{B}>{\delta }_{P}^{B}$

We then have the Proposition 2.

Proposition 2

The relative-performance delegation in a downstream market impede upstream collusion at the same degree regardless of the competition mode in downstream market, i.e.,

Proposition 2

supports one important proposition put forth by Miller and Pazgal (2001, 2005) that if the owners are able to manipulate their managers’ behavior via relative-performance, the equilibrium outcome is the same regardless of the choice of quantity setting or price setting in a horizontal differentiated market. We further demonstrate that in a vertical related market with differentiated products, upstream firms’ collusion sustainability encounters the same obstacle regardless of the choice of quantity setting or price setting when downstream firms adopt the relative-performance delegation in an infinitely repeated Cournot or Bertrand game with trigger strategy punishment.

This paper analyzes upstream firms’ collusion sustainability when downstream firms adopt the relative-performance delegation in an infinitely repeated game. We demonstrate that the relative-performance delegation makes more difficult for upstream collusion comparing with sales-revenue delegation. We further demonstrate that in a vertical related market with differentiated products, upstream firms’ collusion sustainability encounters the same obstacle regardless of the choice of competition modes when downstream firms adopt the relative-performance delegation in an infinitely repeated Cournot or Bertrand game with trigger strategy punishment.

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No competing interests reported.

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Relative-Performance Delegation destabilizes Upstream Collusion

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Abstract

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1. Introduction

2. The Model

3. Upstream Collusion: Cournot Competition

4. Upstream Collusion: Bertrand Competition

5. Concluding Remarks

References

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