Asymmetric Collusion with Growing Demand

Abstract

We characterize collusion sustainability in markets where demand growth triggers the entry of a new firm whose efficiency may be different from the efficiency of the incumbents. We find that the profit-sharing rule that firms adopt to divide the cartel profit after entry is a key determinant of the incentives for collusion (before and after entry). In particular, if the incumbents and the entrant are very asymmetric, collusion without side-payments cannot be sustained. However, if firms divide joint profits through bargaining and are sufficiently patient, collusion is sustainable even if firms are very asymmetric.

Appendices

Appendix A: Model Simulations

In this Appendix, we study whether the distribution of the industry capital across firms, k, and the magnitude of the fixed entry cost, F, qualitatively affect the sustainability of collusion before and after entry.

Looking at ICC (9), we conclude that the magnitude of F has no direct impact on collusion sustainability after entry (regardless of the profit-sharing rule). Moreover, the magnitude of F only affects the incentives for collusion before entry if it changes the ability of incumbents to delay entry by disrupting the collusive agreement, i.e., to change t͂ _c − t͂ _m , with m ∈ {N, p}. Using Eq. 31, we conclude that, if time were a continuous variable, the value of F would have no impact on incentives for collusion before entry (since t _c − t _m does not depend on F). However, the magnitude of F may be relevant because time is discrete. Table 1 shows how many periods the entrant delays its entry if it expects to compete (rather than collude) with incumbents after entry.

Table 1 Difference between entry periods under competition and collusion with the Nash bargaining rule and without side-payments, $(\tilde {t}_c-\tilde {t}_{N},\tilde {t}_c-\tilde {t}_p)$, for k = 0.4

Regarding the value of k, there is no doubt that it affects conditions for collusion sustainability before and after entry. Analyzing Fig. 7, we conclude that: the more symmetric are the incumbents and the entrant, the more sustainable is the collusive agreement after entry. Furthermore, if firms are quite asymmetric, collusion is impossible in the absence of side-payments. It is not so straightforward to guess the impact of k on the ICC for collusion sustainability before entry, (22), since a change on k affects expressions for the profits and the entry periods. Using Eq. 31, we can show that the more efficient are the incumbents (i.e., the higher is the value of k), the higher is t _c − t _m . Typically, the entry delay, t͂ _c − t͂ _m , will also increase in k, which makes collusion more difficult to sustain.

1.1 A.1 Collusion with the Nash Bargaining Rule

If firms adopt the Nash bargaining rule to share their joint profits after entry, collusion after entry is sustainable for all values of k ∈, as long as the discount factor is sufficiently high.

Below, we plot the ICCs for collusion sustainability (before and after entry) when k = 0. 1, $k=\frac {1}{3}$ and k = 0. 45, in order to cover three different scenarios: (i) the incumbents are significantly more efficient than the entrant; (ii) the three firms are symmetric;⁵⁷ and (iii) the entrant is significantly more efficient than the incumbents.

Analyzing in Fig. 14, we conclude that a significant increase in the fixed entry cost, F, has a small impact on conditions for collusion sustainability before and after entry. An increase in F only accentuates the erratic behavior of the critical discount factor for collusion sustainability before entry. In particular, there are some pairs (δ, μ), for which collusion is sustainable before entry with F = 10 but not sustainable with F = 1 (more evident when k = 0. 1).

Fig. 14

Collusion sustainability when firm adopt the Nash bargaining rule after entry

We also find that a change of the distribution on industry capital, k, has a non-monotonic effect on collusion sustainability after entry, since the dashed line (corresponding to the ICC for collusion sustainability after entry) moves: (i) to the left, when k increases from 0. 1 to $\frac {1}{3}$; and (ii) to the right, when k increases from $\frac {1}{3}$ to 0. 45. This finding supports the evolution of the critical discount factor observed in Fig. 7. For parameters in regions E of Fig. 14a, b, collusion could be sustainable before entry, but the cartel would break down after entry. However, this only seems to happen when the entrant is significantly more efficient than the incumbents.

1.2 A.2 Collusion Without Side-Payments

Recall that collusion without side-payments is only possible after entry if k ∈ (0. 199, 0. 436). As, in Section 6, we considered k = 0. 4, we will only plot conditions for collusion sustainability when k = 0. 25 (Fig. 15).

Fig. 15

Collusion sustainability when firms make no side-payments after entry (k = 0.25)

Simulations in this Appendix allow us to conclude that, as long as firms are not too asymmetric, collusion is sustainable before and after entry with both the profit-sharing rules (after entry). In addition, it is not possible to say whether collusion is easier to sustain before or after entry: it depends on the demand growth rate, the discount factor, the value of k and the profit-sharing rule after entry.

Appendix B: Determination of Profits

1.1 B.1 Cournot Competition

1.1.1 B.1.1 Before Entry

Suppose that, in period t before entry, the two incumbents are competing in quantities. In this case, incumbent i ∈ {1, 2} chooses the quantity that maximizes its individual profit:

$$\pi_{it}^{c2}(q_{it}, q_{jt})=[\mu^{t}-(q_{it}+q_{jt})]q_{it}-\frac{q_{it}^{2}}{2k} ,$$

with j ≠ i and j ∈ {1, 2}. The corresponding first-oder condition is:

$$ -2 q_{it}-\frac{q_{it}}{k}-q_{jt}+\mu ^t=0.$$

(32)

Similarly, the first-order condition of the maximization problem of firm j is:

$$ -2 q_{jt}-\frac{q_{jt}}{k}-q_{it}+\mu ^t=0.$$

(33)

Combining Eqs. 32 and 33, we obtain the equilibrium output incumbent i in period t:

$$q_{it}^{c2}=\frac{k}{1+3k}\mu ^{t} .$$

The corresponding profit is:

$$\Pi_{it}^{c2}=\frac{k(1+2 k)}{2(1+3 k)^{2}}\mu^{2t}.$$

1.1.2 B.1.2 After Entry

Suppose now that the incumbents and the entrant compete in quantities in period t. Incumbent i ∈ {1, 2} chooses the quantity that maximizes:

$$\pi_{it}^{c3}\left(q_{it},q_{jt}, q_{lt}\right)= \left[\mu^{t}- (q_{it}+q_{jt}+q_{lt})\right]q_{it}-\frac{q_{it}^{2}}{2k}.$$

The corresponding first-order condition is:

$$ -2 q_{it}-\frac{q_{it}}{k}-q_{jt}-q_{lt}+\mu^t=0 .$$

(34)

The profit function of the entrant is given by:

$$\pi_{3t}^{c3}\left(q_{it},q_{jt}, q_{3t}\right)= \left[\mu^{t}- (q_{it}+q_{jt}+q_{3t})\right]q_{3t}-\frac{q_{3t}^{2}}{2(1-2k)} .$$

Thus, the corresponding first-order condition is:

$$ -q_{it}-q_{jt}-2q_{3t}-\frac{q_{3t}}{1-2 k}+\mu ^t=0 .$$

(35)

Combining the best-reply functions of the three firms, we obtain:

$$q_{1t}^{c3}=q_{2t}^{c3}=\frac{2 (1-k) k }{3+3 k-8 k^{2}}\mu ^{t}\quad \text{and} \quad q_{3t}^{c3}=\frac{(1+k) (1-2 k) }{3+3 k-8 k^{2}} \mu ^t.$$

Market shares of incumbent i ∈ {1, 2} and the entrant are constant over time and, respectively, given by:

$$\lambda_{i}^{c3}=\frac{q_{it}^{c3}}{2q_{it}^{c3}+q_{3t}^{c3}}=\frac{2 (1-k) k}{1+3 k-6 k^{2}} \quad \text{and} \quad \lambda_{3}^{c3}=\frac{(1+k) (1-2 k)}{1+3 k-6 k^{2}}.$$

(36)

Finally, individual profits are:

$$ \Pi_{it}^{c3}=\frac{2k(1+2k)(1-k)^{2}}{\left(3+3k-8k^{2}\right)^{2}}\mu^{2 t} \quad \text{and} \quad \Pi_{3t}^{c3}=\frac{(1+k)^{2} \left(3-10k+8k^{2}\right) }{2 \left(3+3k-8k^{2}\right)^{2}}\mu^{2t} .$$

(37)

1.2 B.2 Collusion

1.2.1 B.2.1 Before Entry

If the incumbents collude in period t before entry, they produce the quantities that maximize:

$$ \pi_{t}^{m2}\left(q_{1t},q_{2t}\right)=\left[\mu^t-\left(q_{1t}+q_{2t}\right)\right](q_{1t}+q_{2t})-\left(\frac{{q_{1t}}^{2}}{2 k}+\frac{{q_{2t}}^{2}}{2k}\right) .$$

(38)

Solving the corresponding first-order conditions, we obtain:

$$\left\{ \begin{array}{l} \mu^t-2q_{1t}-2q_{2t}-\frac{q_{1t}}{k}=0 \\ \mu^t-2q_{1t}-2q_{2t}-\frac{q_{2t}}{k}=0 \end{array} \right . \quad \Leftrightarrow \quad q_{1t}^{m2}=q_{2t}^{m2}=\frac{k}{1+4k}\mu ^{t} .$$

Substituting these quantities in Eq. 38 and dividing it by two, we obtain the collusive profit of incumbent i:

$$\Pi_{it}^{m2}=\frac{k}{2(1+4k)} \mu^{2t}.$$

1.2.2 B.2.2 After Entry

B.2.2.1 Full collusion without side-payments

If the three firms maximize joint profits in period t, they produce quantities that maximize:

$$\pi_{t}^{m3}\left(q_{1t}, q_{2t}, q_{3t}\right)\,=\,\left[\mu^{t}\,-\,\left(q_{1t}\,+\,q_{2t}\,+\,q_{3t}\right)\right]\left(q_{1t}\,+\,q_{2t}\,+\,q_{3t}\right)\,-\,\left[\frac{{q_{1t}}^{2}}{2 k}\,+\,\frac{{q_{2t}}^{2}}{2k}\,+\,\frac{{q_{3t}}^{2}}{2(1\,-\,2k)}\right].$$

(39)

Solving the corresponding first-order conditions, we find the collusive output levels:

$$\left\{ \begin{array}{l} -2 q_{1t}-\frac{q_{1t}}{k}-2q_{2t}-2q_{3t}+\mu^t=0 \\ -2q_{1t}-2 q_{2t}-\frac{q_{2t}}{k}-2q_{3t}+\mu^t=0 \\ -2q_{1t}-2q_{2t}-2q_{3t}-\frac{q_{3t}}{1-2k}+\mu^t=0 \end{array} \right . \quad \Leftrightarrow \left\{ \begin{array}{l} q_{1t}^{m3}=q_{2t}^{m3}=\frac{k}{3}\mu ^{t} \\ q_{3t}^{m3}= \frac{1-2k}{3}\mu ^t\end{array} \right. . $$

Substituting them in Eq. 39, we obtain the profit of the cartel $\Pi _{t}^{m3}=\frac {\mu ^{2t}}{6}$.

B.2.2.2 Partial collusion with simultaneous moves

Consider that the incumbents do not include the entrant in their agreement. Assume further that the cartel and the entrant simultaneously choose quantities. In this case, the incumbents set quantities such to maximize:

$$\pi_{t}^{pc3}\left(q_{1t},q_{2t}\right)=\left(\mu ^t-q_{1t}-q_{2t}-q_{3t}\right)\left(q_{1t}+q_{2t}\right)-\frac{q_{1t}^{2}}{2k}-\frac{q_{2t}^{2}}{2k},$$

while the entrant chooses the quantity that maximizes its individual profit:

$$\pi_{3t}^{pc3}(q_{3t})=\left(\mu ^t-q_{1t}-q_{2t}-q_{3t}\right)q_{3t}-\frac{q_{3t}^{2}}{2(1-2k)}.$$

Combining the best-response functions, we obtain:

$$\left\{ \begin{array}{l} -2 q_{1t}-\frac{q_{1t}}{k}-2 q_{2t}-q_{3t}+\mu^{t} =0 \\ -2 q_{1t}-2 q_{2t}-\frac{q_{2t}}{k}-q_{3t}+\mu ^t=0 \\ -q_{1t}-q_{2t}-2 q_{3t}-\frac{q_{3t}}{1-2 k}+\mu ^t=0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} q_{1t}=q_{2t}=\frac{2k(1-k) }{3 \left(1+2 k-4 k^{2}\right)}\mu ^{t} \\ q_{3t}=\frac{(1-2 k) (1+2 k)}{3 \left(1+2 k-4 k^{2}\right)}\mu ^{t} . \end{array} \right.$$

The corresponding profits of incumbent i ∈ {1, 2} and the entrant in period t are:

$$\Pi_{it}^{pc3}=\frac{2 k(1-k)^{2} (1+4 k)}{9 \left(1+2 k-4 k^{2}\right)^{2}} \mu^{2 t} \quad \text{and} \quad \Pi_{3t}^{pc3}=\frac{(1+2 k)^{2} \left(3-10 k+8 k^{2}\right)}{18 \left(1+2 k-4 k^{2}\right)^{2}} \mu^{2 t}.$$

(40)

B.2.2.3 Partial collusion with sequential moves

Consider that the incumbents do not include the entrant in the collusive agreement and the cartel behaves as a Stackelberg leader while the entrant is a follower. The entrant produces the quantity that maximizes:

$$\pi_{3t}^{ps3}\left(q_{1t}, q_{2t}, q_{3t}\right)=\left[\mu^t-\left(q_{1t}+q_{2t}+q_{3t}\right)\right]q_{3t}-\frac{q_{3t}^{2}}{2(1-2k)} .$$

Its best-response function to the output of the cartel (formed by the two incumbents) is:

$$q_{3t}\left(q_{1t},q_{2t}\right)=\frac{1-2 k}{3-4 k}\left(\mu^t-q_{1t}-q_{2t}\right) .$$

(41)

Incumbents choose quantities that maximize their joint profits, given the best-response function of firm 3:

$$\begin{array}{@{}rcl@{}} \pi_{t}^{ps3}\left(q_{1t},q_{2t}\right) &=& \frac{2(1-k)}{3-4k}\left(\mu^t-q_{1t}-q_{2t}\right)\left(q_{1t}+q_{2t}\right)-\left(\frac{q_{1t}^{2}}{2k}+\frac{q_{2t}^{2}}{2k}\right). \end{array}$$

The corresponding first-order conditions are:

$$\left\{ \begin{array}{l} -q_{1t}-\frac{q_{1t}}{k}-q_{2t}+\left(-1+\frac{1-2 k}{3-4 k}\right) \left(q_{1t}+q_{2t}\right)+\mu ^t-\frac{1-2 k}{3-4 k}\left(-q_{1t}-q_{2t}+\mu ^{t}\right)=0 \\ -q_{1t}-q_{2t}-\frac{q_{2t}}{k}+\left(-1+\frac{1-2 k}{3-4 k}\right) \left(q_{1t}+q_{2t}\right)+\mu ^t-\frac{1-2k}{3-4k}\left(-q_{1t}-q_{2t}+\mu ^{t}\right)=0 , \end{array} \right .$$

whose solution is:

$$q_{1t}^{ps3}=q_{2t}^{ps3}=\frac{2k(1-k)}{3+4 k-8 k^{2}}\mu ^t.$$

(42)

Substituting these quantities in the best-response function of firm 3, we obtain:

$$q_{3t}^{ps3}=\frac{(1-2 k)\left(3-4 k^{2}\right)}{(3-4 k)\left(3+4 k-8 k^{2}\right)}\mu ^t.$$

As a result, the profit of the incumbent i ∈ {1, 2} and of the entrant in period t are, respectively:

$$\Pi_{it}^{ps3}=\frac{2k(1-k)^{2}}{9-40 k^2+32 k^{3}}\mu^{2t} \quad \text{and} \quad \Pi_{3t}^{ps3}=\frac{(1-2 k) \left(3-4 k^{2}\right)^{2}}{2 (3-4 k) \left(3+4 k-8 k^{2}\right)^{2}}\mu^{2 t} .$$

1.3 B.3 Deviation

1.3.1 B.3.1 Before Entry

If incumbent i ∈ {1, 2} deviates from the collusive agreement in period t, it produces the quantity that maximizes:

$$\pi_{it}^{d2}\left(q_{it}\right)=\left[\mu^{t}-\left(q_{it}+\frac{k}{1+4k}\mu ^{t} \right)\right]q_{it}-\frac{q_{it}^{2}}{2k} .$$

(43)

Solving the corresponding first-order condition, we obtain:

$$ -2 q_{it}-\frac{q_{it}}{k}+\mu ^t-\frac{k}{1+4 k}\mu ^t=0 \ \Leftrightarrow \ q_{it}^{d2}=\frac{k (1+3 k)}{1+6 k+8 k^{2}} \mu ^{t} .$$

Substituting this quantity in Eq. 43, we obtain:

$$\Pi_{it}^{d2}=\frac{k (1+3 k)^{2}}{2 (1+2 k) (1+4 k)^{2}} \mu^{2 t}.$$

1.3.2 B.3.2 After Entry

B.3.2.1 Full collusion

Consider that the three firms are colluding and firm i ∈ {1, 2, 3} defects in period t. It produces the quantity that maximizes:

$$\pi_{it}^{d3}(q_{it})=\left[\mu^{t}-\left(q_{it}+\frac{k_{j}}{3} \mu ^t+\frac{1-k_i-k_{j}}{3} \mu ^{t} \right)\right]q_{it}- \frac{q_{it}^{2}}{2 k_{i}} ,$$

with j ≠ i and j ∈ {1, 2, 3}. Solving the corresponding first-order condition, we find the deviating output:

$$-2 q_{it}+\mu ^t-\frac{q_{it}}{k_{i}}-\frac{1-k_i-k_{j}}{3} \mu ^{t} -\frac{k_{j}}{3} \mu ^{t} =0 \quad \Leftrightarrow \quad q_{it}^{d3}=\frac{ k_{i} \left(2+k_{i}\right)}{3 \left(1+2 k_{i}\right)} \mu ^t.$$

Thus, the deviating profit is:

$$\Pi_{it}^{d3}=\frac{k_{i}(2+k_i)^{2}}{18(1+2k_i)}\mu^{2t}.$$

B.3.2.2 Partial collusion with simultaneous moves

If incumbent i ∈ {1, 2} deviates in period t from the partial collusive agreement (under Cournot competition with the entrant), it produces the quantity that maximizes:

$$\pi_{it}^{pdc}(q_{it})= \left[\mu^t-q_{it}-\frac{2 (1-k) k}{3 \left(1+2 k-4 k^{2}\right)} \mu ^t-\frac{(1-2 k) (1+2 k)}{3 \left(1+2 k-4 k^{2}\right)}\mu ^{t}\right]q_{it}-\frac{q_{it}^{2}}{2 k}.$$

Solving the corresponding first-order condition, we obtain:

$$-2 q_{it}\,-\,\frac{q_{it}}{k}\,+\,\mu^{t}\,-\,\frac{2 (1\,-\,k) k}{3 \left(1\,+\,2 k\,-\,4 k^{2}\right)}\mu ^{t}\,-\,\frac{(1\,-\,2 k) (1\,+\,2 k) }{3 \left(1\,+\,2 k\,-\,4 k^{2}\right)}\mu ^{t}\,=\,0 \Leftrightarrow q_{it}\,=\,\frac{2 k \left(1\,+\,2 k\,-\,3 k^{2}\right)}{3 \left(1\,+\,4 k\,-\,8 k^{3}\right)} \mu ^t.$$

Therefore, the one-shot deviation profit is:

$$\Pi_{it}^{pdc3}=\frac{2 k \left(1+2 k-3 k^{2}\right)^{2} }{9 (1+2 k) \left(1+2 k-4 k^{2}\right)^{2}} \mu^{2 t}.$$

(44)

B.3.2.3 Partial collusion with sequential moves

Suppose that, in period t after entry, the incumbent i ∈ {1, 2} deviates. This firm makes its choice taking into account that the other incumbent will produce the collusive output level, but the entrant (playing after the incumbents) will play its best-response to the new level jointly produced by the incumbents. More precisely, if the incumbent i produces $q_{it}^{ds3}$ and the other incumbent produces $q_{jt}^{ps3}$, given by Eq. 42, the best-response function of the entrant is:

$$q_{3t}(q_{it})=\frac{1-2 k}{3-4k}\left(q_{it}+\frac{3+2k-6k^{2}}{3+4k-8 k^{2}} \mu^{t}\right).$$

The deviating incumbent maximizes its individual profit, given the best-reply function of the entrant and that the other incumbent produces $q_{jt}^{ps3}$:

$$\pi_{it}^{dps3}(q_{it})=\left[\mu^t-q_{it}-\frac{2k(1-k)}{3+4 k-8 k^{2}}\mu ^t-\frac{1-2 k}{3-4k}\left(q_{it}+\frac{3+2k-6k^{2}}{3+4k-8 k^{2}} \mu^{t}\right)\right] q_{it}-\frac{q_{it}^{2}}{2k}.$$

Solving the corresponding first-order condition, we obtain:

$$q_{it}^{dps3}=\frac{2k(1-k) \left(3+2 k-6 k^{2}\right)\mu^{t}}{\left(3-4 k^{2}\right) \left(3+4 k-8 k^{2}\right)},$$

and the corresponding deviation profit is:

$$\Pi_{it}^{dps3}=\frac{2k(1-k)^{2}\left(3+2k-6 k^{2}\right)^{2}}{(3-4 k) \left(3-4k^{2}\right)\left(3+4 k-8k^{2}\right)^{2}}\mu^{2t}.$$

Appendix C: Proofs

1.1 Lower Bound for the Entry Cost (F)

Start by noticing that: if the entrant does not enter at t = 0 when incumbents are colluding, it does not enter at t = 0 if incumbents are competing. As a result, we only need to ensure that the present discounted value of the profits if entry occurs (under collusion) at period t = 1 exceeds the corresponding value at period t = 0. Using Eq. 17, we find that:

$$V^{m}(1)>V^{m}(0) \Leftrightarrow \beta_{3} \frac{\mu^{2} \delta}{1-\mu^{2} \delta} -\delta F > \beta_{3} \frac{1}{1-\mu^{2} \delta} - F \Leftrightarrow F> \frac{\beta_{3}}{1-\delta},$$

where β ₃ denotes the share of the entrant in the monopoly profit. Looking at Fig. 6b, we conclude that β ₃ > 0.9 −2k, ∀k. Substituting in the inequality above, we obtain:

$$F> \frac{0.9-2k}{1-\delta} = \frac{9-20k}{10(1-\delta)}.$$

Proof of Proposition 1

Firm 3 enters the market in the period that maximizes V ^c. If t was continuous, the optimal entry period would satisfy the following first-order condition:

$$\alpha_{33} \frac{\mu^{2t} \delta^{t}}{1-\mu^{2} \delta} ln {\left(\mu^{2} \delta\right)} -F \delta^{t} ln(\delta) =0 \ \Leftrightarrow \ t_c= \frac{1}{2 ln (\mu)} ln \left[ \frac{ln(\delta)}{ln\left(\mu^{2} \delta\right)} \frac{F\left(1-\mu^{2} \delta\right)}{\alpha_{33}}\right] .$$

As this expression may not be integer, the entrant must compare the value of V ^c in the two integers closest to t _c . □

Proof of Proposition 2

The critical (adjusted) discount factor for incumbent i ∈ {1, 2} to abide by the collusive agreement is greater than one if:

$$\begin{array}{@{}rcl@{}} \mu^{2}\tilde{\delta}_{ip}(k)>1 \Leftrightarrow \frac{\gamma_{3i}-\frac{k}{6}}{\gamma_{3i}-\alpha_{3i}} >1 \Leftrightarrow \frac{k \left(-3+18 k-3 k^2-72 k^3+64 k^{4}\right)}{6 \left(3+3k-8k^{2}\right)^{2}} <0 . \end{array}$$

As $k \in \left (0, \frac {1}{2}\right )$, the last inequality is equivalent to f(k) ≡ − 3 + 18k − 3k ² − 72k ³ + 64k ⁴ < 0. The roots of the first-order derivative of f are:

$$k_1=\frac{3-\sqrt{393}}{64}<0 \quad \vee \quad k_2= \frac{3+\sqrt{393}}{64} \quad \vee \quad k_3= \frac{3}{4}>\frac{1}{2} .$$

Thus, f ’ is positive for k ∈ (0, k ₂) and negative for k ∈ (k ₂, 1 / 2). This implies that f is strictly increasing in (0, k ₂) and it (strictly) decreasing in (k ₂, 1 / 2). Moreover,

$$f(0)<0, \quad f(k_2)>0 \quad \text{and} \quad f\left(\frac{1}{2}\right) > 0 .$$

As f is continuous, by the intermediate value theorem, there exists k ^∗ ∈ (0, k ₂) such that f(k ^∗) = 0. As f is increasing in this domain, k ^∗ is its unique root in (0, k ₂). As f(k ₂) > f(1 / 2) > 0 and f is strictly decreasing in (k ₂, 1 / 2), f has no roots in this interval. Thus,

$$f(k)<0 \ \wedge \ k \in \left(0, \frac{1}{2}\right) \Leftrightarrow k \in (0, k^{\ast}) .$$

Using, for example, the bisection method we can find that k ^∗ ≈ 0. 199.

Now, we make a similar analysis for the entrant:

$$\begin{array}{@{}rcl@{}} \mu^{2}\tilde{\delta}_{3p}(k)>1 \Leftrightarrow \frac{\gamma_{33}-\frac{1-2k}{6}}{\gamma_{33}-\alpha_{33}} >1 \Leftrightarrow \frac{2 k (1-2 k) (3-6 k-9 k^{2}+16 k^{3})}{3 (3+3k-8k^{2})^{2}}<0 . \end{array}$$

As $k \in \left (0, \frac {1}{2}\right )$, the last inequality is equivalent to g(k) ≡ 3 − 6k − 9k ² + 16k ³ < 0. As g is continuous, g(0) > 0 and g(1 / 2) < 0, there is k ^{∗ ∗} ∈ (0, 1 / 2), such that g(k ^{∗ ∗}) = 0. Let us now prove that k ^{∗ ∗}is unique. The roots of the first-order derivative of g are:

$$k_4=\frac{3-\sqrt{41}}{16} <0 \quad \vee \quad k_5=\frac{3+\sqrt{41}}{16} >\frac{1}{2}.$$

Therefore, g′ is negative for $k \in \left (0,\frac {1}{2}\right )$, which implies that g is strictly decreasing in this domain. As a result, k ^{∗ ∗} is unique and

$$g(k)<0 \ \wedge \ k \in \left(0, \frac{1}{2}\right) \Leftrightarrow k \in \left(k^{\ast\ast}, \frac{1}{2}\right) .$$

Once again, using the bisection method, we find that k ^{∗ ∗} ≈ 0. 436. □

Proof of Proposition 3

Using β _3t = 1 − β _1t − β _2t , the first-order conditions corresponding to the maximization of the Nash product are:

$$\left\{ \begin{array}{l} \left[\left(1-\beta_{1t}-\beta_{2t}\right)\Pi_{t}^{m3}-\Pi_{3t}^{c3}\right]-\left[\beta_{1t} \Pi_{t}^{m3}-\Pi_{1t}^{c3}\right] =0\\ \left[\left(1-\beta_{1t}-\beta_{2t}\right)\Pi_{t}^{m3}-\Pi_{3t}^{c3}\right]-\left[\beta_{2t} \Pi_{t}^{m3}-\Pi_{2t}^{c3}\right] =0 \end{array} \right..$$

As$\Pi _{1t}^{c3}$, it follows that:

$$\beta_{1t}=\beta_{2t}=\frac{\Pi_{t}^{m3}+\Pi_{1t}^{c3}-\Pi_{3t}^{c3}}{3\Pi_{t}^{m3}} \equiv \beta_t.$$

Substituting the expressions for profits, we find that β _t is constant over time. The collusive profit of the incumbent i ∈ {1, 2} is:

$$\Pi_{it}^{N3} =\frac{1+6\alpha_{3i}-6\alpha_{33}}{18} \mu^{2t} = \frac{k \left(21-6 k-51 k^2+32 k^{3}\right) }{9 \left(3+3 k-8 k^{2}\right)^{2}}\mu^{2 t},$$

while:

$$\Pi_{3t}^{N3} =\frac{1-12\alpha_{3i}+12\alpha_{33}}{18} \mu^{2t} =\frac{27-30 k-93 k^2+60 k^3+64 k^{4} }{18 \left(3+3 k-8 k^{2}\right)^{2}}\mu^{2 t}.$$

Finally, the shares of incumbent i and entrant of the cartel profit are:

$$\begin{array}{@{}rcl@{}} \lambda_{i}^{N3}&=&\frac{\Pi_{it}^{N3}}{2\Pi_{it}^{N3}+\Pi_{3t}^{N3}}=\frac{2 k \left(21-6 k-51 k^2+32 k^{3}\right)}{3 \left(3+3 k-8 k^{2}\right)^{2}} \notag \\ \lambda_{3}^{N3}&=&\frac{27-30 k-93 k^2+60 k^3+64 k^{4}}{3 \left(3+3 k-8 k^{2}\right)^{2}}. \end{array}$$

(45)

□

Proof of Lemma 1

Incumbent i ∈ {1, 2} abides by the collusive agreement in a period t < t͂ _m before entry of firm 3 if the following inequality is satisfied:

$$\begin{array}{l} \beta_{2} \sum\limits_{s=t}^{\tilde{t}_m {\kern-1.5pt} -{\kern-1.5pt}1} (\mu^2\delta)^s {\kern-1.5pt}+{\kern-1.5pt} \beta_{3i} \sum\limits_{s=\tilde{t}_m}^{\infty} (\mu^2\delta)^s{\kern-1.5pt}\geq{\kern-1.5pt}\gamma_{2} (\mu^2\delta)^{t}{\kern-1.5pt} +{\kern-1.5pt} \alpha_{2} \sum\limits_{s=t{\kern-1.5pt}+{\kern-1.5pt}1}^{\tilde{t}_c{\kern-1.5pt}-{\kern-1.5pt}1} (\mu^2\delta)^s {\kern-1.5pt}+{\kern-1.5pt}\alpha_{3i} \sum\limits_{s=\tilde{t}_c}^{\infty} (\mu^2\delta)^s\ {\kern-1.5pt}\Leftrightarrow{\kern-1.5pt} \\ \beta_{2} \frac{(\mu^2\delta)^t{\kern-1.5pt}-{\kern-1.5pt}(\mu^2\delta)^{\tilde{t}_m}}{1{\kern-1.5pt}-{\kern-1.5pt}\mu^2\delta}{\kern-1.5pt}+{\kern-1.5pt}\frac{(\mu^2\delta)^{\tilde{t}_m}}{1{\kern-1.5pt}-{\kern-1.5pt}\mu^2\delta}{\kern-1.5pt}\geq{\kern-1.5pt}\gamma_{2} (\mu^2\delta)^{t} {\kern-1.5pt}+{\kern-1.5pt} \alpha_{2} \frac{ (\mu^2\delta)^{t{\kern-1.5pt}+{\kern-1.5pt}1}{\kern-1.5pt}-{\kern-1.5pt}(\mu^2\delta)^{\tilde{t}_c}}{1{\kern-1.5pt}-{\kern-1.5pt}\mu^2\delta} {\kern-1.5pt}+{\kern-1.5pt}\alpha_{3i} \frac{(\mu^2\delta)^{\tilde{t}_c}}{1{\kern-1.5pt}-{\kern-1.5pt}\mu^2\delta}. \end{array}$$

(46)

Multiplying both sides of the inequality by 1 − μ ² δ, we obtain:

$$\begin{array}{@{}rcl@{}} \beta_{2} \left[\left(\mu^{2}\delta\right)^t-\left(\mu^{2}\delta\right)^{\tilde{t}_{m}}\right]+\beta_{3i}\left[(\mu^{2}\delta)^{\tilde{t}_{m}}\right] \geq \notag && \gamma_{2} (\mu^{2}\delta)^{t} (1-\mu^{2}\delta) +\\ &&+ \alpha_{2} \left[ (\mu^{2}\delta)^{t+1}-(\mu^{2}\delta)^{\tilde{t}_{c}}\right] +\alpha_{3i} \left[(\mu^{2}\delta)^{\tilde{t}_{c}}\right] . \end{array}$$

Rearranging the terms of the inequality, we obtain:

$$\left(\mu^{2}\delta\right)^{t}\left(\beta_{2}-\gamma_{2}\right) +\left(\mu^{2}\delta\right)^{t+1}\left(\gamma_2-\alpha_{2}\right) +\left(\mu^{2}\delta\right)^{\tilde{t}_{m}}\left(\beta_{3i}-\beta_{2}\right) +\left(\mu^{2}\delta\right)^{\tilde{t}_{c}}\left(\alpha_2-\alpha_{3i}\right) \geq 0$$

(47)

Substituting t = t͂ _m − 1 in the inequality, we obtain:

$$\left(\mu^{2}\delta\right)^{\tilde{t}_m-1}\left[\beta_{2}-\gamma_2 +\left(\gamma_2-\alpha_2+\beta_{3i}-\beta_{2}\right) \mu^{2}\delta +\left(\alpha_2-\alpha_{3i}\right) \left(\mu^{2}\delta\right)^{\tilde{t}_c-\tilde{t}_m+1}\right]\geq 0 .$$

As t͂ _c − t͂ _m + 1 > 1 and μ ² δ < 1, a sufficient condition for the ICC to be satisfied is:

$$A(\mu,\delta,k) \equiv \beta_{2}-\gamma_2 +\left(\gamma_2-\alpha_2+\beta_{3i}-\beta_{2}\right) \mu^{2}\delta +\left(\alpha_2-\alpha_{3i}\right) (\mu^{2}\delta)^{\tilde{t}_c-\tilde{t}_m+1} \geq 0 .$$

(48)

Consider now an arbitrary period before entry: t = t͂ _m − τ, with 1 ≤ τ ≤ t͂ _m . Substituting t = t͂ _m − τ in Eq. 47, we obtain:

$$\begin{array}{@{}rcl@{}} &&\left(\mu^{2}\delta\right)^{\tilde{t}_{m}\,-\,\tau}(\beta_{2}\,-\,\gamma_{2}) \,+\,\left(\mu^{2}\delta\right)^{\tilde{t}_{m}\,-\,\tau\,+\,1}(\gamma_{2}\,-\,\alpha_2) \,+\,\left(\mu^{2}\delta\right)^{\tilde{t}_{m}}(\beta_{3i}\,-\,\beta_{2}) \,+\,\left(\mu^{2}\delta\right)^{\tilde{t}_{c}}(\alpha_{2}\,-\,\alpha_{3i}) \geq 0 {} \;\; \Leftrightarrow \quad \notag \\ &&\left(\mu^{2}\delta\right)^{\tilde{t}_{m}\,-\,\tau}\left[\beta_{2}\,-\,\gamma_{2} \,+\, (\gamma_{2}\,-\,\alpha_{2}) \mu^{2}\delta \,+\,(\beta_{3i}\,-\,\beta_{2}) \left(\mu^{2}\delta\right)^{\tau} \,+\,(\alpha_{2}\,-\,\alpha_{3i}) \left(\mu^{2}\delta\right)^{\tilde{t}_{c}\,-\,\tilde{t}_{m}\,+\,\tau}\right]\geq 0 {} \;\;\Leftrightarrow\quad \notag \\ &&\left(\mu^{2}\delta\right)^{\tilde{t}_{m}-\tau}\left[A(\mu,\delta,k)+(\beta_{2}-\beta_{3i})\mu^{2}\delta \left[1-\left(\mu^{2}\delta\right)^{\tau-1}\right]\right] \geq 0. \end{array}$$

(49)

Notice that if condition (48) holds and β ₂ − β _3i > 0, the inequality (49) is also satisfied, since 1 − (μ ² δ)^{τ − 1} > 1. Therefore, to conclude the proof it is only missing to show that β ₂ − β _3i > 0.

The expressions for β ₂ and for β _3i are given in Eqs. 20 and 13, respectively. Therefore,

$$\beta_2-\beta_{3i}=\frac{k \left(39+6 k-201 k^2-88 k^3+320 k^{4}\right)}{18 (1+4 k) \left(3+3 k-8 k^{2}\right)^{2}} .$$

As k > 0, the inequality β ₂ − β _3i > 0 is equivalent to:

$$p(k)\equiv 39+6 k-201 k^{2}-88 k^{3}+320 k^{4} > 0 .$$

The first-order derivative of p is: p ^′(k) = 6 − 402k − 264k ² + 1280k ³, which is globally continuous. As a result,

$$\begin{array}{@{}rcl@{}} &[p^{\prime}(-0.5) < \wedge p^{\prime}(-0.4) > 0]&\quad \Rightarrow \quad \exists k_{1} \in (-0.5,-0.4) : p^{\prime}(k_1)=0\\ &[p^{\prime}(0) < 0 \wedge p^{\prime}(0.1) > 0]&\quad \Rightarrow \quad \exists k_{2} \in (0,0.1) : p^{\prime}(k_2)=0 \\ &[p^{\prime}(0.6) < 0 \wedge p^{\prime}(0.7) > 0]&\quad \Rightarrow \quad \exists k_{3} \in (0.6,0.7) : p^{\prime}(k_3)=0 . \end{array}$$

As p ^′ is a polynomial of third degree, its only zeros are k ₁, k ₂ and k ₃. We conclude, therefore, that p is increasing in (0, k ₂) and decreasing in $\left (k_{2},\frac {1}{2}\right )$. Thus, the minimum of p in the interval $\left (0,\frac {1}{2}\right )$ must be achieved at one limit of this interval. As $p(0) > p\left (\frac {1}{2}\right ) > 0$, we conclude that $p(k) > 0 \ , \forall k \in \left (0,\frac {1}{2}\right )$. This ends the proof. □

Proof of Lemma 3

If the incumbents anticipate partial collusion with simultaneous moves after entry, they abide by the collusive agreement before entry if:⁵⁸

$$C+\zeta_{3i}\mu^{2} \delta +\left(\alpha_{2}-\alpha_{3i}\right) \left(\mu^{2}\delta\right)^{\tilde{t}_c-\tilde{t}_{pc}+1} \geq 0 ,$$

(50)

where $C=\beta _{2}-\gamma _{2}+\left (\gamma _{2}-\alpha _{2}-\beta _{2}\right )\mu ^{2} \delta$. Similarly, if the incumbents anticipate partial collusion with sequential moves after entry, the ICC for collusion sustainability before entry is:

$$C+\sigma_{3i}\mu^{2} \delta +(\alpha_{2}-\alpha_{3i}) \left(\mu^{2}\delta\right)^{\tilde{t}_c-\tilde{t}_{ps}+1} \geq 0,$$

(51)

where σ _3i is given by Eq. 27. Below, we prove that if inequality (50) holds, inequality (51) is also satisfied. In other words, we prove that:

$$\begin{array}{@{}rcl@{}} &&\sigma_{3i}\mu^{2} \delta +(\alpha_{2}-\alpha_{3i}) \left(\mu^{2}\delta\right)^{\tilde{t}_c-\tilde{t}_{ps}+1} > \zeta_{3i}\mu^{2} \delta +(\alpha_{2}-\alpha_{3i}) \left(\mu^{2}\delta\right)^{\tilde{t}_c-\tilde{t}_{pc}+1} \Leftrightarrow \\ && \left(\mu^{2}\delta\right)^{\tilde{t}_c-\tilde{t}_{ps}+1}-\left(\mu^{2}\delta\right)^{\tilde{t}_c-\tilde{t}_{pc}+1} > \frac{\zeta_{3i}-\sigma_{3i}}{\alpha_2-\alpha_{3i}}. \end{array}$$

Start by noticing that:

$$\sigma_{33}<\zeta_{33} \ \Leftrightarrow \ -\frac{8 (1-k)(1-2 k)^{2} k^{2} (9+18 k-44 k^2-36 k^3+56 k^4)}{9 (3-4 k) (1+2 k-4 k^2)^{2} (3+4 k-8 k^2)^{2}}<0,$$

which is satisfied for $k\in \left (0,\frac {1}{2}\right )$, since 9 + 18k − 44k ² − 36k ³ + 56k ⁴ > 6 in this domain. This means that the single-profit of the entrant is lower under sequential moves than under simultaneous moves. As a result, entry occurs later under sequential moves (t͂ _ps > t͂ _pc ), which implies that: $(\alpha _{2}-\alpha _{3i}) \left (\mu ^{2}\delta \right )^{\tilde {t}_c-\tilde {t}_{ps}+1}-(\alpha _{2}-\alpha _{3i}) \left (\mu ^{2}\delta \right )^{\tilde {t}_c-\tilde {t}_{pc}+1}>0.$.

Each incumbent profits more under sequential moves than under simultaneous moves, since:

$$\zeta_{3i}-\sigma_{3i}=-\frac{8 k^{3} (1-k)^{2} (1-2 k)^{2}}{9(3-4 k) (3+4 k-8 k^2)(1+2 k-4 k^2)^{2}}<0$$

This concludes the proof, since α ₂ − α _3i > 0. □

Proof of Remark 5

The incumbent i ∈ {1, 2} profits more than the entrant if:

$$\Pi_{it}^{ps3} \geq \Pi_{3t}^{ps3} \Leftrightarrow -\frac{9-30 k-16 k^2+100 k^3-64 k^{4}}{2 (3-4 k) (3+4 k-8 k^2)^{2}} \geq 0.$$

In the domain $\left (0,\frac {1}{2}\right )$, the last inequality is equivalent to p(k) ≡ 9 − 30k − 16k ² + 100k ³ − 64k ⁴ ≤ 0. The first-order derivative of p is: p ^′(k) = − 30 − 32k + 300k ² − 256k ³, which is globally continuous. Moreover:

$$\begin{array}{@{}rcl@{}} [ p^{\prime}(-0.3) > 0 \wedge p^{\prime}(-0.2)<0]&\quad \Rightarrow\quad \exists k_{1} \in (-0.3,-0.2) : p^{\prime}(k_1)=0,\\ \quad[p^{\prime}(0.5) < 0 \wedge p^{\prime}(0.6)>0]&\quad \Rightarrow \quad \exists k_{2} \in (0.5,0.6) : p^{\prime}(k_2)=0,\\ \quad[ p^{\prime}(0.8) > 0 \wedge p^{\prime}(0.9)<0 ]&\quad \Rightarrow \quad \exists k_{3} \in (0.8,0.9) : p^{\prime}(k_3)=0 . \end{array}$$

As p ^′ is a polynomial of third degree, its only roots are k ₁, k ₂ and k ₃. Thus, p ^′ has no roots in the interval $\left (0,\frac {1}{2}\right )$, which means that p is strictly decreasing in this interval. Therefore, the polynomial p can have, at most, one zero in the interval $\left (0,\frac {1}{2}\right )$. As:

$$p(0.3415)>0 \quad \text{and} \quad p(0.3416)<0 ,$$

there is k ^∗ ∈ (0. 3415, 0. 3416) such that p(k ^∗) = 0. Thus, if $k \in \left (k^{\ast },\frac {1}{2}\right )$, then p(k) ≤ 0. □