Sustaining collusion in markets with entry driven by balanced growth

Abstract

This paper studies the sustainability of collusion in markets where growth is not restricted to occur at a constant rate and may trigger future entry. Entry typically occurs later along the punishment path than along the collusive path (since profits are lower in the former case), and may not even occur along the punishment path. The possibility of delaying or even deterring entry may, therefore, constitute an additional incentive for deviating just before entry is supposed to occur along the collusive path. If firms set quantities and revert to Cournot equilibrium after a deviation, this incentive more than compensates for the fact that there are more firms after entry, making collusion harder to sustain before entry than after entry. If, instead, firms set prices or use optimal penal codes, deterring entry by breaking the cartel is not profitable, and thus collusion is harder to sustain after entry than before entry. The proposed model encompasses and explains conflicting results derived in the extant literature under more restrictive settings, and derives some novel results.

Appendix

1.1 Appendix A: Profits are proportional to market size parameters

Observe that, combining (1) and (2), the profit function of firm $i\in \left\{ 1,\ldots ,n\right\}$ in period $t \in \left\{ 0, 1,\ldots \right\}$ can be written as:

$$\begin{aligned} \pi _{it}\left( q_{it}\right) =\left[ P\left( \sum _k \frac{q_{kt}}{g_t}\right) \frac{q_{it}}{g_t}-C\left( \frac{q_{it}}{g_t}\right) \right] g_t h_t. \end{aligned}$$

Normalizing payoff units by dividing profits by $g_t h_t$ and normalizing choice units by dividing quantities by $g_t$, we obtain an equivalent objective function:

$$\begin{aligned} \pi _i \left( \frac{q_{it}}{g_t}\right) \equiv \frac{\pi _{it}(q_{it})}{g_t h_t} = P\left( \sum _k \frac{q_{kt}}{g_t}\right) \frac{q_{it}}{g_t}-C\left( \frac{q_{it}}{g_t}\right) . \end{aligned}$$

A setting in which firms choose ratios $\frac{q_{it}}{g_t}$ (instead of $q_{it}$) with the objective of maximizing functions $\pi _i$ (instead of $\pi _{it}$) has resulting payoffs that are invariant with $g_t$ and $h_t$.

The corresponding profits of firms coincide with these normalized equilibrium payoffs when $g_t h_t=1$. When $g_t h_t \ne 1$, profits are proportional to normalized payoffs: $\pi _{it} = \pi _i\,g_t h_t$.

1.2 Appendix B: Proofs of Lemmas and Propositions

Proof of Lemma 1

In market regime $j \in \left\{ m , c \right\}$, the discounted value of the profits of firm 3, entering at $T^j$, is:

$$\begin{aligned} V^j_3(T^j) = \pi _3^{j3} \sum _{s=T^j}^{+\infty } \delta ^{s} f(s) - K\delta ^{T^j} = \delta ^{T^j} \left( \pi _3^{j3} F_{T^j} - K \right) . \end{aligned}$$

Thus, it is profitable for firm 3 to enter the market at $T^j$ if and only if $K < \pi _3^{j3} F_{T^j}$.

If entry is profitable, it occurs in the earliest period that satisfies the following condition:

$$\begin{aligned} V_{3}^j(T^j)\ge V_{3}^j(T^j+1) \,&\Leftrightarrow \pi _3^{j3} \left( f_{T^j} + \delta F_{T^j+1} \right) - K \ge \delta \left( \pi _3^{j3} F_{T^j+1} - K \right) \\&\Leftrightarrow f_{T^j} \ge \frac{\left( 1-\delta \right) K}{\pi _3^{j3}}. \end{aligned}$$

$\square$

Proof of Lemma 2

We want to show that if the ICC (7) is satisfied in period $t+1$, it is also satisfied in period t, for $t \in \left\{ 1,\ldots ,T^m-2\right\}$.

If condition (7) is satisfied in period $t+1$, the following variable is positive:

$$\begin{aligned} \Delta&\equiv - \left( \pi ^{d2} - \pi ^{m2} \right) F_{t+1} + \left( \pi ^{d2} - \pi ^{c2} \right) \delta F_{t+2} - \left( \pi ^{m2} - \pi ^{m3} \right) \delta ^{T^m-t-1} F_{T^m} \nonumber \\&\quad + \left( \pi ^{c2} - \pi ^{c3} \right) \delta ^{T^c-t-1} F_{T^c} \ge 0. \end{aligned}$$

Condition (7) in period t can be written as:

$$\begin{aligned} - \left( \pi ^{d2} - \pi ^{m2} \right) \left( f_{t} + \delta F_{t+1} \right) + \left( \pi ^{d2} - \pi ^{c2} \right) \delta \left( f_{t+1} + \delta F_{t+2} \right)&\, \nonumber \\ - \left( \pi ^{m2} - \pi ^{m3} \right) \delta ^{T^m-t} F_{T^m} + \left( \pi ^{c2} - \pi ^{c3} \right) \delta ^{T^c-t} F_{T^c}&\ge 0 \\ \Leftrightarrow \delta \Delta - \left( \pi ^{d2} - \pi ^{m2} \right) f_{t} + \left( \pi ^{d2} - \pi ^{c2} \right) \delta f_{t+1}&\ge 0. \end{aligned}$$

When $\Delta \ge 0$, for (7) to be satisfied in period t, it is sufficient that:

$$\begin{aligned} \delta \ge \frac{\left( \pi ^{d2} - \pi ^{m2} \right) f_{t}}{\left( \pi ^{d2} - \pi ^{c2} \right) f_{t+1}}, \end{aligned}$$

which is true, under Assumption 1. $\square$

Proof of Result 4.1

Using Proposition 2 and Assumption 2, collusion is sustainable after entry if and only if:

$$\begin{aligned} \beta \delta \ge \frac{\pi ^{d3}-\pi ^{m3}}{\pi ^{d3}-\pi ^{c3}}. \end{aligned}$$

Observe the following equivalence:

$$\begin{aligned} \frac{\pi ^{d3}-\pi ^{m3}}{\pi ^{d3}-\pi ^{c3}}&\ge \frac{\pi ^{d2} - \pi ^{m2}}{\pi ^{d2} - \pi ^{m2} + \pi ^{m3} - \pi ^{c3}} \\&\Leftrightarrow 1 - \frac{\pi ^{m3}-\pi ^{c3}}{\pi ^{d3}-\pi ^{c3}}\\&\ge 1 - \frac{\pi ^{m3} - \pi ^{c3}}{\pi ^{d2} - \pi ^{m2} + \pi ^{m3} - \pi ^{c3}} \\&\quad \Leftrightarrow \frac{\pi ^{m3}-\pi ^{c3}}{\pi ^{d3}-\pi ^{c3}} \le \frac{\pi ^{m3} - \pi ^{c3}}{\pi ^{d2} - \pi ^{m2} + \pi ^{m3} - \pi ^{c3}}\\&\quad \Leftrightarrow \pi ^{d3}-\pi ^{c3} \ge \pi ^{d2} - \pi ^{m2} + \pi ^{m3} - \pi ^{c3} \\&\quad \Leftrightarrow \pi ^{d3}-\pi ^{m3} \ge \pi ^{d2}-\pi ^{m2}. \end{aligned}$$

If $\pi ^{d3}-\pi ^{m3} \ge \pi ^{d2}-\pi ^{m2}$, sustainability of collusion after entry implies that:

$$\begin{aligned} \beta \delta \ge \frac{\pi ^{d2} - \pi ^{m2}}{\pi ^{d2} - \pi ^{m2} + \pi ^{m3} - \pi ^{c3}}, \end{aligned}$$

which, under Assumption 2, implies condition (9). $\square$

Proof of Proposition 5

If firm 3 does not enter the market if the incumbents deviate from the collusive agreement before entry, the ICC for collusion to be sustainable before entry, given by (8), becomes:

$$\begin{aligned}&-\left( \pi ^{d2} - \pi ^{m2} \right) \left( f_{T_m-1} + \delta F_{T^m} \right) + \left( \pi ^{d2} - \pi ^{c2} - \pi ^{m2} + \pi ^{m3} \right) \delta F_{T^m} \ge 0 \\&\quad \Leftrightarrow -\left( \pi ^{c2} - \pi ^{m3} \right) \delta F_{T^m} \ge \left( \pi ^{d2} - \pi ^{m2} \right) f_{T^m-1}, \end{aligned}$$

which cannot be satisfied if $\pi ^{c2} \ge \pi ^{m3}$. For $\pi ^{c2} < \pi ^{m3}$, the ICC is equivalent to:

$$\begin{aligned} \delta F_{T^m} \ge \frac{\pi ^{d2} - \pi ^{m2}}{\pi ^{m3} -\pi ^{c2}} \left( F_{T^m-1}-\delta F_{T^{m}}\right) \Leftrightarrow \delta \ge \frac{\left( \pi ^{d2} - \pi ^{m2} \right) F_{T^m-1}}{\left( \pi ^{d2} - \pi ^{m2} + \pi ^{m3} - \pi ^{c2} \right) F_{T^m}}. \end{aligned}$$

$\square$

Proof of Result 4.2

Under Assumption 2, collusion is sustainable after entry if and only if (Proposition 2):

$$\begin{aligned} \beta \delta \ge \frac{\pi ^{d3}-\pi ^{m3}}{\pi ^{d3}-\pi ^{c3}}. \end{aligned}$$

Observe the following equivalence (which holds if $\pi ^{m3} \ge \pi ^{c2}$):

$$\begin{aligned}&\frac{\pi ^{d3}-\pi ^{m3}}{\pi ^{d3}-\pi ^{c3}} \ge \frac{\pi ^{d2} - \pi ^{m2}}{\pi ^{d2} - \pi ^{m2} + \pi ^{m3} - \pi ^{c2}} \\&\quad \Leftrightarrow \pi ^{d3} \pi ^{m3} - \pi ^{d3} \pi ^{c2} -\pi ^{m3} \pi ^{d2} + \pi ^{m3} \pi ^{m2} - \pi ^{m3} \pi ^{m3} + \pi ^{m3} \pi ^{c2}\\&\qquad \ge - \pi ^{c3} \pi ^{d2} + \pi ^{c3} \pi ^{m2} \\&\quad \Leftrightarrow (\pi ^{d3}-\pi ^{m3}) (\pi ^{m3} - \pi ^{c2}) \ge (\pi ^{m3}- \pi ^{c3}) (\pi ^{d2} -\pi ^{m2}) \\&\quad \Leftrightarrow \frac{\pi ^{d2}-\pi ^{m2}}{\pi ^{d3}- \pi ^{m3}} \le \frac{\pi ^{m3}-\pi ^{c2}}{\pi ^{m3} -\pi ^{c3}}. \end{aligned}$$

Thus, if $\frac{\pi ^{d2}-\pi ^{m2}}{\pi ^{d3}- \pi ^{m3}} \le \frac{\pi ^{m3}-\pi ^{c2}}{\pi ^{m3} -\pi ^{c3}}$, the sustainability of collusion after entry implies that:

$$\begin{aligned} \beta \delta \ge \frac{\pi ^{d2} - \pi ^{m2}}{\pi ^{d2} - \pi ^{m2} + \pi ^{m3} - \pi ^{c2}}. \end{aligned}$$

Under Assumption 2, this implies sustainability of collusion before entry (Proposition 5). $\square$

Proof of Result 5.2

If firms set prices and there is no entry, the incumbents abide by the collusive agreement in period t if and only if:

$$\begin{aligned} \sum _{s=t}^{+\infty } \delta ^{s-t} \pi _{is}^{m2} \ge \pi _{it}^{m1}. \end{aligned}$$

(19)

If there is entry, collusion is sustainable in any period after entry, $t \ge T^m$, if and only if:

$$\begin{aligned} \sum _{s=t}^{+\infty } \delta ^{s-t} \pi _{is}^{m3} \ge \pi _{it}^{m1}, \end{aligned}$$

which is more restrictive than (19).

At any period $t < T^m$, the ICC before entry is:

$$\begin{aligned} \sum _{s=t}^{T^m-1} \delta ^{s-t} \pi _{is}^{m2} +\sum _{s=T^m}^{+\infty } \delta ^{s-t} \pi _{is}^{m3} \ge \pi _{it}^{m1}, \end{aligned}$$

which is also more restrictive than (19).$\square$

Proof of Result 5.3

Under Assumption 2, we have $\frac{F_{T^m}}{F_{T^m-1}} \ge \beta$. Thus, if collusion is sustainable after entry, $\beta \delta \ge 1 - \frac{\pi ^{m3}}{\pi ^{m1}}$, we obtain:

$$\begin{aligned} \frac{F_{T^m}}{F_{T^m-1}} \delta \ge 1 - \frac{\pi ^{m3}}{\pi ^{m1}} \ge 1 - \frac{\pi ^{m3}}{\pi ^{m1} - \pi ^{m2} + \pi ^{m3}} = \frac{\pi ^{m1} - \pi ^{m2}}{\pi ^{m1} - \pi ^{m2} + \pi ^{m3}}, \end{aligned}$$

which coincides with condition (14). $\square$

1.3 Appendix C: Linear Cournot markets

1.3.1 Collusive profits

Suppose that, in period t, the $n\in \left\{ 2,3\right\}$ active firms maximize their joint profit:

$$\begin{aligned} \pi _{t}^{mn}(Q_t)=\left( 1-\frac{Q_t}{f_t}\right) Q_t. \end{aligned}$$

Using the first-order condition for profit-maximization, and assuming that firms divide the industry profit in equal parts, we obtain the collusive output and profit of firm i:

$$\begin{aligned} q_{it}^{mn}=\frac{f_t}{2n}\quad \text {and}\quad \pi _{it}^{mn}=\frac{f_t}{4n}. \end{aligned}$$

1.3.2 Deviation profits

Suppose that firm i deviates from the collusive agreement in period t and there are $n\in \left\{ 2,3\right\}$ firms in the market. This firm produces the quantity that maximizes its individual profit, assuming that the rival firms produce the collusive quantities:

$$\begin{aligned} \pi _{it}^{dn}(q_{it}) = \left( 1-\frac{n-1}{2n}-\frac{q_{it}}{f_t}\right) q_{it}. \end{aligned}$$

Solving the corresponding first-order condition, we obtain:

$$\begin{aligned} q_{it}^{dn}=\frac{n+1}{4n} f_t \quad \text {and} \quad \pi _{it}^{dn}=\left( \frac{n+1}{4n}\right) ^2 f_t. \end{aligned}$$

1.3.3 Punishment profits

When the $n\in \left\{ 2,3\right\}$ active firms compete à la Cournot, each firm i produces the quantity that maximizes its individual profit:

$$\begin{aligned} \pi _{it}^{cn}(q_{it})=\left( 1-\frac{Q_{-it}}{f_t}-\frac{q_{it}}{f_t}\right) q_{it}, \end{aligned}$$

where $Q_{-it}$ denotes the quantity produced by the rivals of firm i. Solving the first-order condition for profit-maximization, we obtain the output and profit of each firm:

$$\begin{aligned} q_{it}^{cn}=\frac{f_t}{n+1} \quad \text {and} \quad \pi _{it}^{cn}=\frac{f_t}{(n+1)^2}. \end{aligned}$$

1.3.4 Timing of entry when the rate of market growth is constant

Let $f_t =\beta ^t,$ with $1 < \beta < \delta ^{-1}$. Then, $F_t = \frac{\beta ^t}{1-\beta \delta }$, which is strictly increasing in t, and ${\displaystyle \lim _{t\rightarrow \infty } F_t = +\infty }$. Hence, firm 3 always enters the market (even under competition). The optimal entry periods under competition and collusion are, respectively:

$$\begin{aligned} T^c = int \left\{ \frac{\ln \left[ 16(1-\delta )K\right] }{\ln \beta } \right\} +1 \quad \text {and} \quad T^m = int \left\{ \frac{\ln \left[ 12(1-\delta )K\right] }{\ln \beta } \right\} +1, \end{aligned}$$

where $int \left\{ x \right\}$ denotes the integer part of x. The entry delay that results from breaking the cartel (before entry) can be:³⁹

$$\begin{aligned} T^c-T^m = int \left[ \frac{\ln \left( 4/3\right) }{\ln \beta } \right] \ \ \ \text {or} \ \ \ T^c-T^m = int \left[ \frac{\ln \left( 4/3\right) }{\ln \beta } \right] + 1. \end{aligned}$$

(20)

Proof of Result 6.3

Replacing the critical (adjusted) discount factor for collusion to be sustainable after entry ($\beta \delta = \frac{4}{7}$) in the ICC for collusion sustainability before entry, (16), we obtain:

$$\begin{aligned} T^c-T^m \ge \frac{\ln \left( 13/28\right) }{\ln \left( 4/7\right) } -1 \Leftrightarrow T^c-T^m \ge 1. \end{aligned}$$

Using the expression for the lowest possible entry delay, given in (20), it is clear that $T^c-T^m \ge 1$ is implied by $\beta \le \frac{4}{3}$. $\square$

The following result is instrumental for the proof of Result 6.4.

Result

Let $\beta \delta = \frac{4}{7}$. If collusion before entry is not sustainable for given $\ T^m$ and $T^c$, with $T^c > T^m$, it is also not sustainable if $\ T^m$ and $T^c$ increase by the same amount.

Proof

The left-hand side of (18) is lower with $T^m+1$ and $T^c+1$ than with $T^m$ and $T^c$ if and only if:

$$\begin{aligned}&- 9 \left( - \alpha ^{-T^m} + \alpha ^{1-T^m} \right) (1-\beta \delta )(\alpha -\beta \delta ) - 16 \, \beta \delta \left( - \alpha ^{-T^m} + \alpha ^{1-T^m}\right) (1-\beta \delta ) \nonumber \\&\quad + 28 \left( \beta \delta \right) ^{T^c-T^m+1} \left( -\alpha ^{-T^c} + \alpha ^{1-T^c} \right) (1-\beta \delta ) \le 0 \nonumber \\&\quad \Leftrightarrow - 9 - 7 \left( \frac{\beta \delta }{\alpha } \right) + 28 \left( \frac{\beta \delta }{\alpha } \right) ^{T^c-T^m+1} \le 0. \end{aligned}$$

(21)

The worst case for (21) to hold is when $T^c-T^m=1$. In that case, it becomes:

$$\begin{aligned} - 9 - 7 \left( \frac{\beta \delta }{\alpha } \right) + 28 \left( \frac{\beta \delta }{\alpha } \right) ^2&\le 0, \end{aligned}$$

which is satisfied for $\frac{\beta \delta }{\alpha } \le 0.7056$ (approximately). Since $\alpha >1$, it holds for $\beta \delta = \frac{4}{7}$. $\square$

1.3.5 Profitability of entry when $f_t = (1-\alpha ^{-t})\beta ^t$

Recall that entry is profitable if and only if condition (3) is satisfied. To check whether it is satisfied for $\beta < 1$, we need to obtain $\sup _t \left\{ F_t\right\}$. Since $F_t = \beta ^t \frac{\alpha -\beta \delta -\alpha ^{1-t}(1-\beta \delta )}{(1-\beta \delta )(\alpha -\beta \delta )}$:

$$\begin{aligned} F_t<F_{t+1}&\Leftrightarrow \frac{\alpha -\beta \delta -\alpha ^{1-t}(1-\beta \delta )}{\beta \left[ \alpha -\beta \delta -\alpha ^{-t}(1-\beta \delta )\right] }<1 \\&\Leftrightarrow (\alpha -\beta \delta )(1-\beta ) -(1-\beta \delta ) (\alpha -\beta )\alpha ^{-t} <0\\&\Leftrightarrow t<\frac{1}{ln \alpha } ln\left[ \frac{(1-\beta \delta ) (\alpha -\beta )}{ (\alpha -\beta \delta )(1-\beta )}\right] . \end{aligned}$$

Thus, there is entry under market regime $j\in \left\{ m,c\right\}$ if:

$$\begin{aligned} K<\pi _{j3}\,F_{\hat{t}}, \ \ \ \text {where} \ \ \hat{t}=int\left\{ \frac{1}{ln \alpha } ln\left[ \frac{(1-\beta \delta ) (\alpha -\beta )}{ (\alpha -\beta \delta )(1-\beta )}\right] \right\} +1. \end{aligned}$$

(22)

Proof of Result 6.4

The strategy of the proof is to replace the critical discount factor for collusion to be sustainable after entry, $\beta \delta = \frac{4}{7}$, in the ICC (18) for collusion sustainability before entry, and check whether it is satisfied for: (i) $\ T^c-T^m=1$; (ii) $\ T^c-T^m=2$; (iii) $\ T^c-T^m=3$.

Replacing $\beta \delta = \frac{4}{7}$ in the ICC (18), we obtain:

$$\begin{aligned}&- 27 \left( 1-\alpha ^{1-T^m} \right) \left( \alpha -\frac{4}{7} \right) - 64 \left( \alpha - \frac{4}{7} - \frac{3}{7} \alpha ^{1-T^m} \right) \nonumber \\&\quad + 112 \left( \frac{4}{7} \right) ^{T^c-T^m} \left( \alpha - \frac{4}{7} - \frac{3}{7} \alpha ^{1-T^c} \right) \ge 0. \end{aligned}$$

(23)

• (i) $\mathbf {[T^c-T^m=1]}$ Replacing $T^c=T^m+1$ in (23), we obtain:

  $$\begin{aligned}&- 63 \left( 1-\alpha ^{1-T^m} \right) \left( \alpha -\frac{4}{7} \right) + 64 \left( \alpha ^{1-T^m} - \alpha ^{-T^m} \right) \ge 0 \nonumber \\&\quad \Leftrightarrow - 63 \alpha ^{T^m+1} + 36 \alpha ^{T^m} + 63 \alpha ^{2} +28 \alpha - 64 \ge 0. \end{aligned}$$

  (24)

  The derivative of the left-hand side of (24) with respect to $T^m$ is:

  $$\begin{aligned} - 63 \ln (\alpha ) \alpha ^{T^m+1} + 36 \ln (\alpha ) \alpha ^{T^m}, \end{aligned}$$

  which is always negative because $\alpha >1$. Hence, the most favorable case for the ICC (24) to be satisfied is when $T^m=1$. Replacing $T^m=1$, we obtain: $64 \alpha - 64 \ge 0$, which is true. The second most favorable case is $T^m=2$. Replacing $T^m=2$, we obtain:

  $$\begin{aligned} - 63 \alpha ^3 + 99 \alpha ^2 + 28 \alpha - 64&\ge 0, \end{aligned}$$

  which is satisfied for $\alpha \le \frac{4}{3}$. Replacing $T^m=3$, we obtain:

  $$\begin{aligned} - 63 \alpha ^4 + 36 \alpha ^3 + 63 \alpha ^2 +28 \alpha - 64&\ge 0, \end{aligned}$$

  which is satisfied for $\alpha \le 1.046$ (approximately). Replacing $T^m=4$, and dividing by $\alpha -1$, we obtain:

  $$\begin{aligned} - 63 \alpha ^4 - 27 \alpha ^3 - 27 \alpha ^2 +36 \alpha + 64&\ge 0, \end{aligned}$$

  which is never satisfied. It is not satisfied, therefore, for any $T^m \ge 4$.

• (ii) $\mathbf {[T^c-T^m=2]}$ Replacing $T^c-T^m=2$ in (23) and expanding, we get:

  $$\begin{aligned} - 889 \alpha ^{2+T^m} + 508 \alpha ^{1+T^m} + 441 \alpha ^{3} + 196 \alpha ^2 - 256&\ge 0, \end{aligned}$$

  whose left-hand side is decreasing in $T^m$. The most favorable case is when $T^m=1$:

  $$\begin{aligned} - 448 \alpha ^{3} + 704 \alpha ^{2} - 256&\ge 0, \end{aligned}$$

  which is satisfied for $\alpha \le 1.094$ (approximately). When $T^m=2$, the ICC becomes:

  $$\begin{aligned}&- 889 \alpha ^{4} + 949 \alpha ^{3} + 196 \alpha ^2 - 256 \ge 0 \\&\quad \Leftrightarrow (\alpha -1)(-889 \alpha ^{3} + 60 \alpha ^2 + 256 \alpha + 256) \ge 0, \end{aligned}$$

  which is always false. Therefore, the ICC is not satisfied for any $T^m \ge 2$.

• (iii) $\mathbf {[T^c-T^m=3]}$ Replacing $T^m=1$ and $T^c=4$ in (23), we obtain:

  $$\begin{aligned} - 3136 \left( \alpha - 1 \right) + 1024 \left( \alpha - \frac{4}{7} - \frac{3}{7} \alpha ^{-3} \right)&\ge 0 \\ \Leftrightarrow - 14784 \alpha + 17856 - 3072 \alpha ^{-3}&\ge 0, \end{aligned}$$

  which holds (in equality) for $\alpha =1$. The derivative of the left-hand side with respect to $\alpha$ is:

  $$\begin{aligned} - 14784 + 9216 \alpha ^{-4}, \end{aligned}$$

  which is always negative. As $\alpha >1$, the ICC is never satisfied if $T^c-T^m = 3$. Finally, using Lemma 3, we conclude that the ICC is never satisfied if $T^c-T^m \ge 3$. $\square$

1.3.6 ICC before entry when $f_t = (1-\alpha ^{-t})\beta ^t$

In Fig. 6, the dashed lines correspond to the conditions for entry to be profitable (under collusion and under competition), given by (22). The solid line corresponds to the ICC for collusion to be sustainable before entry, given by (18).

Fig. 6

ICC for collusion sustainability before entry (solid line) and conditions for entry profitability under collusion and competition (dashed lines), with $\alpha =1.01$ and $K=0.004$

The erratic shape of the ICC before entry is due to the discrete nature of time. Small changes in parameters can make $T^c$ or $T^m$ jump (by 1 period), leading to kinks in the ICC.