The Welfare Implications of the Meeting Design of a Cartel

Abstract

This paper investigates the welfare implications of introducing a delegation problem into a price-fixing collusion game. Within a model in which each cartel conspirator has an irreplaceable market expertise, I demonstrate that the delegation of decisions to representatives for concealment purposes can lead to inefficient decisions. In this context, while a more severe antitrust policy contributes to deterrence, it can also induce surviving cartels to maximize concealment through delegation, which creates inefficiencies that are not considered in standard models of collusion. Leniency programs can exacerbate this perverse effect of policy.

Appendix

$\blacklozenge$Lemma 1: For a Complete cartel, the probability of finding cartel evidence during an inspection to a firm is 1. Given $\rho _A = \rho _B= \rho$, the probability of conviction is: $s^{C} = \rho _A (1 - \rho _B) \ + \ \rho _B (1 - \rho _A) \ + \ \rho _B \ \rho _A \ = \ \rho (2 - \rho )$.

For a Representative cartel, an inspection ends in conviction if it is carried out in a division that is headed by a meeting manager; w.l.o.g., identify this with $j=1$. Hence, if we denote with $\rho _{ij}$ the probability of an inspection to division j of firm i, the probability of conviction is: $s^{R} = \rho _{A1} (1 - \rho _B) \ + \ \rho _{A1} \rho _{B2} \ + \ \rho _{B1} (1 - \rho _A) \ + \ \rho _{B1} \rho _{A2} \ + \ \rho _{A1}\rho _{B1}$. Since (1) $\rho _{A} = \rho _{B} = \rho$, and (2) within a firm, divisions have the same probability of inspection, $\rho _{i1} = \rho _{i2} = \frac{1}{2} \rho _i$, the above equation yields: $s^{R} = \rho - \frac{1}{4} \rho ^{2}$.

Finally, given (1) $s^{C}(0) = s^{R}(0) =0$ , (2) $\frac{\partial s^{C}}{\partial \rho } , \frac{\partial s^{R}}{\partial \rho } >0$, and (3) $s^{C}(1) = 1 > \frac{3}{4} = s^{R}(1)$, it holds that $s^{C}$ is higher than $s^{R}$ for any value of $\rho \in [0 , 1 ]$.

$\blacklozenge$Proposition 1(equilibrium in pure strategies): Given that profits increase monotonically in the number of managers on the board, it is optimal to maximize managers’ attendance, $n_i =N = 2$. Proposition 1 is the standard result of Nash Equilibrium (NE) in a duopoly competition Bertrand with firms with equal and constant marginal costs. A brief proof of it requires at least three steps:

Let us first prove that there is no NE with $p_i \ne p_j$. Assume $p_i< p_j$. Since i has the lowest price, it serves all demand. Firms’payoffs are $\Pi _i(p_i,p_j) = 2(p_i - c)> 0$ (I have assumed $p_i > c$, as $p_i<0$ yields negative profits), and $\Pi _j(p_i,p_j) = 0$. In this context, choosing $p_i$ is a profitable deviation for firm j. Therefore, $p_i< p_j$ is not a NE; and, with the same logic, $p_j< p_i$ is not either. If there is a NE, it must be at $p_i = p_j$.

Let us prove that there is no NE with $p_i = p_j = p^{*} \ne c$. Assume $p_i=p_j = p^{*} < c$. In this context, firms split the demand in half ($q_i=q_j=1$), but they get negative profits. Hence, a price $p^{*} < c$ would never be chosen. Assume, instead, that $p_i=p_j = p^{*} > c$. Again, firms split the demand in half; but note that firm i can increase profits by reducing its price. In this way, i serves all of the demand, and its profits are $\Pi _i (p^{*}- \epsilon , p^{*}) = (p^{*}_i - \epsilon - c) 2$, which are close to $(p^{*} - c) 2$ for $\epsilon \rightarrow 0$. With the same logic, firm j also finds in a price reduction a profitable deviation. Hence $p_i = p_j = p^{*} > c$ is not a NE either.

Finally, let us show that $p_i = p_j =p^{*} = c$ is a NE. For $p_i = p_j =p^{*} = c$, firms split demand and obtain zero profits each: $\Pi _i(p^{*} , p^{*}) = 0$. In this context, if a firm reduces its price, it obtains negative profits. If, instead, it increases its price, the rival charges $p^{*}$ and serves all demand. As there is no profitable deviation from the candidate outcome, this is a NE.

$\blacklozenge$Proposition 2: In the main text.

$\blacklozenge$Corollary 1: Given $\alpha \in [0; 1]$, the payoff distribution rule that equals the managers’ expected profits from collusion must verify: $\Pi ^{R}_{i1}(p^{c} | q_i =1) = \Pi ^{R}_{i2}(p^{c} | q_i =1)$, for $i=A,B$. Since: $\Pi ^{R}_{i1}(p^{c} | q_i =1) = \left( p^{c} - c \right) \left( 1 - \gamma \phi _{1} \right) \alpha - h^{R} \left( f^{m} + \frac{f}{2} \right) \$ and $\Pi ^{R}_{i2}(p^{c} | q_i =1) = \left( p^{c} - c \right) \left( 1 - \gamma \phi _{1} \right) \left( 1 - \alpha \right) - h^{R} \frac{f}{2}$, the optimal $\alpha$ is $\alpha = \frac{1}{2} + \frac{h^{R} f^{m} }{2 \left( p^{c} - c \right) \left( 1 - \gamma \phi _{1} \right) }> \frac{1}{2}$.

$\blacklozenge$Proposition 3: In the main text.

$\blacklozenge$Lemma 2: Holds from Propositions 2–3, and Corollary 2.

$\blacklozenge$Lemma 3:${\textit{Deterrence effect}:}$ A more severe antitrust policy improves deterrence if the thresholds $v^{C}$ and $v^{R}$ increase with f, $f^{m}$, and $\rho$. Since:

$$\begin{aligned} v^{C}= c +\ \frac{ \delta \ h^{C} \left( 2f^{m} + f \right) }{\left( 2\delta - 1 \right) } , v^{R}= c +\ \frac{ \delta \ h^{R} \left( f^{m} + f \right) }{\left( 2\delta - 1 - \gamma \phi _{1} \right) } \end{aligned}$$

for $\delta > \frac{1}{2}$ and $\gamma < \frac{2 \delta - 1}{\phi _{1}}$, the partial derivatives of $v^{C}$ and $v^{R}$ with respect to f, $f^{m}$ and $\rho$ are all positive. Hence, the above statement holds.

${\textit{Design-distortion }\text {effect}}$: Given threshold $v^{N} = \frac{ h^{C} F^{C} - h^{R} F^{R} \ }{\gamma \phi _{1}} + c \$, we need to prove the following three statements:

St.1: ‘More severe antitrust policies induce some cartels to switch design for sustainability purposes, from the Representative to the Complete one.’. If we define $\gamma _{n1}$ as the value of $\gamma$ at which $v^{N}$ and $v^{C}$ cross each other, and $\gamma _{n2}$ as the analogous value but for the case in which $v^{N}$ and $v^{R}$ cross each other, Statement 1 holds for $\gamma _{n2} < \gamma _{n1}$.

St.2: ‘Higher fines lead some surviving cartels to switch their Complete design to the Representative design’: $\frac{\partial v^{N}}{\partial f} , \frac{\partial v^{N}}{\partial f^{m}} >0$.

Given: $\frac{\partial v^{N}}{\partial f^{m}} = \frac{2h^{C} - h^{R}}{\gamma \phi _{1}} >0$ and $\frac{\partial v^{N}}{\partial f} = \frac{h^{C} - h^{R}}{\gamma \phi _{1}} >0$, statement 2 holds.

St.3: ‘More inspections have an ambiguous effect on the cartel’s design decision. There exist${{\hat{\rho }}}$and${{\hat{f}}}$, such that: for$\rho > {{\hat{\rho }}}$and$f > {{\hat{f}}}$, more inspections lead some surviving cartels to switch their Representative design to the Complete design. Otherwise, the opposite holds’. That is: $\frac{\partial v^{N}}{\partial \rho } <0$ for $\rho > {{\hat{\rho }}}$ and $f > {{\hat{f}}}$; while $\ \frac{\partial v^{N}}{\partial \rho }> 0$ otherwise.

In $\frac{\partial v^{N}}{\partial \rho } = \frac{\left( h'^{C} - h'^{R} \right) (f + f^{m}) + h'^{C} f^{m}}{\gamma \phi _{1}}$, the sign depends entirely on the numerator. In this, the first term in brackets is positive for $\rho < 2/3$, and negative the other way around, since $h'^{C} = \sigma (2 - 2 \rho ) > 0$ and $h'^{R}= \sigma (1 - 1/2 \rho ) >0$. Hence, defining ${{\hat{\rho }}} = 2/3$, if $\rho < {{\hat{\rho }}}$, $\frac{\partial v^{N}}{\partial \rho } > 0$. Otherwise, there exists ${{\hat{f}}} = f^{m} \sigma (3 - 7/2 \rho )$ such that $\frac{\partial v^{N}}{\partial \rho } >0$ for $f < {{\hat{f}}} \$ and $\ \frac{\partial v^{N}}{\partial \rho }< 0$ for $f > {{\hat{f}}}$.

$\blacklozenge$Lemma 4: Holds from Propositions 2–3, Corollary 2, and Lemmas 2–3.

$\blacklozenge$Proposition 4: To prove that the threshold $v^{R}$ is higher when the AA inspects an entire firm rather than a single division, substitute $h^{C}$ for $h^{R}$ in Eq. 7. Since $h^{R}<h^{C}$, the result is straightforward. Working analogously for equation 8, the reader can prove that the opposite holds for the threshold $v^{N}$.

$\blacklozenge$ Lemma 5: Holds from Proposition 4 and the fact that firm-wide inspections do not distort the deterrence effect of the policy that is observed in Lemma 3; this is true also for the main results that are related to the design-effect that is also described in that Lemma.

$\blacklozenge$Proposition 5: For the analysis of the threshold parameters ${{\hat{\theta }}}$ and ${{\hat{\theta }}}^{R}$ see the main text. The threshold ${{\hat{\theta }}}^{N}$ arises from the ICC of a Complete cartel that after the introduction of a LP finds that: (1) collusion under its current design is not sustainable anymore; and (2) conspiring with the Representative design yields higher profits than deviating with a leniency report. That is: $(v-c) - \theta F^{C} \le \ \frac{\delta }{1 - \delta }\ \left[ (v-c)(1-\gamma \phi _{1}) - h^{R} F^{R} \right]$. Solving for $\theta$: ${{\hat{\theta }}}^{N} = \frac{(v-c) - \frac{\delta }{(1 - \delta )}\ \left[ (v-c)(1-\gamma \phi _{1}) - h^{R} F^{R}\right] }{F^{C}}$. With a little bit of algebra, the reader can prove that $\ {{\hat{\theta }}}^{N} \in ({{\hat{\theta }}}^{R} , {{\hat{\theta }}})$.

As in the basic set-up, the welfare implications of the policy depend directly on its ability to prevent collusion under the Representative design.