Collusion and Market Power
 19 minutes read  3927 wordsContext
 Markets with intense competition tend to reduce the competing firms' market power. Firms could form cartels to control the market and extract a greater share of the economic surplus if left unregulated. For this reason, collusion and cartel formation are illegal practices in most market economies today.
 However, legally binding contracts are not necessary for firms to coordinate. Firms can use dynamic strategies to collude tacitly.
 What kind of strategies can lead to tacit collusion?
 What is the role of the means of competition in tacit collusion?
 How can competition authorities measure market power?
Course Structure Overview
Lecture Structure and Learning Objectives
Structure
 Our Customers are Our Enemies (Case Study)
 Tacit Collusion
 Trigger Strategies with Quantity Competition
 Trigger Strategies with Price Competition
 Market Power
 Current Field Developments
Learning Objectives
 Illustrate that in dynamic settings, credible promises and threats can be used to induce tacit collusion.
 Illustrate that tacit collusion is achievable both when firms compete in quantities and prices
 Describe tacit collusion under price competition.
 Describe tacit collusion under quantity competition.
 Explain how market power can be statistically measured and estimated.
Our Customers are Our Enemies
 Lysine is an amino acid that speeds the development of lean muscle tissue in humans and animals.
 It is essential for humans, but we cannot synthesize it.
 It has to be obtained from the diet.
The Lysine Industry
 At the end of the 1980s, the world lysine industry consisted of three significant sellers:
 Ajinomoto,
 Kyowa, and
 Sewon.
 The three largest consumption regions were Japan, Europe, and North America.
 Most production took place in Japan, but it was based on imports of US dextrose.
 Ajinomoto had the largest share of the world market.
The ADM Entry
 In February 1991, Archer Daniel Midland Co. (ADM) entered the market and built by far the world’s largest lysine plant in the US.
 ADM hired biochemist Mark Whitacre, Ph.D., as head of the new division.
 ADM’s plant was three times the size of Ajinomoto’s largest plant.
 ADM gave Ajinomoto and Kyowa executives an unrestricted tour to show its production capacity.
 Companies engaged in a price war.
 Three months before ADM’s entry, the average US lysine price was \($1.22\) per pound.
 After an 18month price war, the US price averaged \($0.68\) per pound.
 ADM’s share of the US market reached \(80\%\).
The Lysine Association
 After the price war, ADM was willing to soften competition.
 In 1992, Mark Whitacre and his boss Terrance Wilson met with top Ajinomoto and Kyowa managers.
 Wilson proposed forming a world lysine association that would regularly meet.
 The new association would collect and distribute market information.
 Wilson also suggested that the new association could provide a convenient cover for illegal pricefixing discussions! (Connor 2001)
 After a year, the lysine association was founded, met quarterly, and performed the two functions that Wilson proposed.
Price Fixing
 There were \(25\) price fixing meetings in total.
 The first one took place in the Nikko Hotel in Mexico on June 23, 1992.
 The average Lysine price immediately jumped by more than \(12\%\).
 Consensus was not always easy to reach. The companies distrusted each other!
 There was a breakdown of the cartel during the spring and summer of 1993, and the lysine price plummeted.
 The crisis was resolved at a meeting in Irvine, California in October 1993 between ADM’s and Ajinomoto Executives. But…
 This meeting and many others were caught on video by the FBI.
Frenemies
WILSON: The only thing we need to talk here because we are gonna get manipulated by these God damn buyers, they’re sh, they can be smarter than us if we let them be smarter.
MIMOTO: (Laughs).
WILSON: Okay?
MIMOTO: (ui).
WILSON: They are not your friend. They are not my friend. And we gotta have ‘em. Thank God we gotta have ‘em, but they are not my friends. You’re my friend. I wanna be closer to you than I am to any customer. ‘Cause you can make us, I can make money, I can’t make money. At least in this kind of a market. And all I wanna is ta tell you again is let’slet’s put the prices on the board.
Coconspirator explains how endofyear compensation scheme eliminates incentives to cheat on cartel
The Whistleblower
 In 1992, Mark Whitacre became an FBI whistleblower. He is the highestlevel corporate executive to ever have done so.
 Whitacre’s wife pressured him into becoming a whistleblower. She threatened to inform the FBI herself if Whitacre wouldn’t do it.
 Whitacre informed the FBI that he and other ADM executives performed illegal pricefixing operations.
 Over the next three years, Whitacre collected information and recorded conversations with ADM executives and competitors.
 The ADM settlement involved the greatest US federal charges at the time.
 Whitacre was convicted for embezzling and money laundering, which he performed while cooperating with the FBI in the price fixing case.
What can we Learn?
 Why did ADM initially engage in a price war?
 Why did it initiate the price fixing discussions afterward?
 Why was there so much distrust among companies?
Tacit Collusion
 When there is a future, firms can use reputation and/or threats to coordinate.
 Coordination increases their profits.
 Explicit contracts are not needed.
 Trigger strategies can induce coordination.
 Trigger strategies work both when firms compete in quantities and prices.
Tacit Collusion in Quantity Competition
 When coordinating, firms produce at the monopolistic level and split the market.
 Alternatively, firms play their static Cournot strategies.
 On every date, there are two potential game states.
 Either both firms have always cooperated in the past.
 Or, at least one of them deviated.
 The trigger strategy is to produce the monopolistic quantity at state 1 and the Cournot quantity at state 2.
The Problem
 Suppose that both firms discount future profits by \(\delta \in (0,1)\).
 Firms aim to maximize the series of discounted flow profits, i.e.,
\[V_{i} = \max_{(q_{i,t})_{t}} \left\{ \sum_{t=0}^{\infty} \delta^{t} \pi_{i,t}(q_{1,t}, q_{2,t}) \right\}.\]
Non Coordinating Equilibrium
 There are many equilibria in the game.
 The non coordinating equilibrium is one in which the stage game Nash equilibrium is played at every date.
 All firms use the oligopolistic quantity competition production levels at every date, which leads to flow profits
\[\pi_{i,t} =  \frac{(c_{1}p_{0})^{2}}{9 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]
 Each firm’s expected future series of profits is
\[V_{i} =  \frac{1}{1\delta}\frac{(c_{1}p_{0})^{2}}{9 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]
Coordinating Equilibrium with Trigger Strategies
 Another equilibrium can be obtained when firms play the trigger strategies.
 Firms produce half the monopolistic quantity and split the monopolistic profits, i.e.,
\[\pi_{i,t} =  \frac{1}{2}\frac{(c_{1}p_{0})^{2}}{4 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]
 If firms coordinate with these profits as flow utilities, they get
\[V_{m, i} =  \frac{1}{1\delta}\frac{(c_{1}p_{0})^{2}}{8 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]
Deviating from Coordination
 Is there any incentive to deviate?
 Suppose that firm 1 deviates at the current date.
 The best response to firm 2’s choice of monopolistic profits is
\[q_{1,0} = \frac{3}{8}\frac{c_{1}p_{0}}{p_{1}}.\]
 Then, firm 1’s profit at the current date is
\[\pi_{1,t} =  \frac{9}{64}\frac{(c_{1}p_{0})^{2}}{p_{1}}.\]
 On future dates, the firms revert to the non coordinating quantities because they use trigger strategies.
 The total discounted profit of the deviating firm is
\[V_{d,1} =  \frac{9}{64}\frac{(c_{1}p_{0})^{2}}{p_{1}}  \frac{\delta}{1  \delta} \frac{(c_{1}p_{0})^{2}}{9 p_{1}}.\]
Can Coordination be Supported?
 The coordination outcome is sustainable if the deviation profit is less than the coordination profit, namely if
\begin{align*} V_{d,1} &=  \frac{9}{64}\frac{(c_{1}p_{0})^{2}}{p_{1}}  \frac{\delta}{1  \delta} \frac{(c_{1}p_{0})^{2}}{9 p_{1}} \\ &<  \frac{1}{1  \delta} \frac{(c_{1}p_{0})^{2}}{8 p_{1}} = V_{m,1}. \end{align*}
 This is equivalent to \(\delta > \frac{9}{17}\).
 As long as firms value future profits enough, there is room for cooperation.
Tacit Collusion in Price Competition
 When coordinating, firms produce at the monopolistic level and split the market.
 Alternatively, firms play their static Bertrand strategies.
 On every date, there are two potential states.
 Either both firms have always cooperated in the past.
 Or, at least one of them deviated.
 The trigger strategy is to use the monopoly price at state 1 and the Bertrand price at state 2.
The Problem
 Suppose again that both discount future profits by \(\delta \in (0,1)\).
 Firms aim to maximize the series of discounted flow profits, i.e.,
\[V_{i} = \max_{(p_{i,t})_{t}} \left\{ \sum_{t=0}^{\infty} \delta^{t} \pi_{i,t}(p_{1,t}, p_{2,t}) \right\}.\]
Non Coordinating Equilibrium
 Firms price at marginal cost when non coordinating.
 They make zero profits at each date.
 Therefore, \(V_{i}=0\).
Coordinating Equilibrium with Trigger Strategies
 As in the case of quantity competition, if firms produce half the monopolistic quantity and split the monopolistic profits, they get
\[\pi_{i,t} =  \frac{1}{2}\frac{(c_{1}p_{0})^{2}}{4 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]
 If firms coordinate, then
\[V_{m, i} =  \frac{1}{1\delta}\frac{(c_{1}p_{0})^{2}}{8 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]
Deviating from Coordination
 Suppose that firm 1 deviates at the current date.
 The best response to firm 2’s choice of monopolistic pricing is to slightly undercut the market price by \(\varepsilon\)
 Then, firm 1 takes all the market and makes almost the monopolistic profit
 For simplicity, drop \(\varepsilon\), so that
\[\pi_{1,t} =  \frac{(c_{1}p_{0})^{2}}{4 p_{1}}.\]
 On future dates, the firms have zero profits because they use trigger strategies.
 The total discounted profit of the deviating firm is
\[V_{d,1} =  \frac{(c_{1}p_{0})^{2}}{4 p_{1}}.\]
Can Coordination be Supported?

The coordination outcome is sustainable if the deviation profit is less than the coordination profit, namely if
\begin{align*} V_{1,d} &=  \frac{(c_{1}p_{0})^{2}}{4 p_{1}} \\ &<  \frac{1}{1  \delta} \frac{(c_{1}p_{0})^{2}}{8 p_{1}} = V_{1,m}. \end{align*}

This is equivalent to \(\delta > \frac{1}{2}\).

As long as firms value future profits enough, there is room for cooperation.

Why is the lower bound \(\delta\) in quantity competition greater than in price competition?
Market Power
 In legal cases of competition law, market power is a central element based on which decisions are drawn.
 Market power refers to the ability of the firm to raise prices above marginal cost (the perfectly competitive price level).
 Marginal cost is not always observable, so it is not always easy to assess market power.
The Lerner Index
 The Lerner index (usually denoted \(L\)) is defined by the markup, i.e., the difference between price and marginal cost, as a percentage of the price.
\[L = \frac{pc’}{p}\]
 Prices are observed, but marginal costs are typically not, and firms are not always eager to reveal this information.
Concentration Index
 A concentration index is a statistic that measures the degree of concentration of the market. A market is concentrated when only a few firms have a large share of the market.
 The concentration ratio is the sum of the market shares of a subset of firms in the market. For example, with \(n\) firms in the market, \(\alpha_{i} = q_{i} / Q\) denoting the share of firm \(i\), and a \(M\) denoting a subset firms, the Mconcentration ratio is given by
\[I_{M} = \sum_{i\in M} \alpha_{i}.\]
 If \(I_{m}\) is close to one, it means that most of the market is controlled by the \(m\) firms included in the calculation.
 If \(m\) is small, this suggests that the market is concentrated.
Herfindahl Index
 The Herfindahl Index is a statistic that measures the degree of concentration of the market by considering the full distribution of market shares. It is defined as the sum of squares (\(\mathcal{L}^2\) norm) of market shares of all active firms in the market.
 For \(n\) firms in the market, with \(\alpha_{i} = q_{i} / Q\) denoting the share of firm \(i\), the Herfindahl index is
\[I_{H} = \sum_{i=1}^{n} \alpha_{i}^{2}.\]
 The closer is the value of \(I_{H}\) to one, the more concentrated the market is.
Estimating Market Power
 Suppose that we know the inverse market demand \(p(q;x)\), where \(q\) is the market quantity and \(x\) is a vector of other exogenous characteristics affecting demand.
 One approach to estimating market power is to start from the equation
\[MR(\lambda) = p + \lambda p’(q;x) q\]
 If \(\lambda=0\), then \(MR(\lambda) = p\), i.e., the market is perfectly competitive.
 If \(\lambda=1\), then the market is a monopoly, i.e.,
\[MR(\lambda) = p + p’(q;x) q.\]
 If \(\lambda=1/n\), then the market is represented by a symmetric \(n\text{firm}\) Cournot model, i.e.,
\[MR(\lambda) = p + \frac{1}{n}p'(q;x) q.\]
Equilibrium
 Suppose that \(c'(q;w)\) is the marginal cost function, where \(w\) is a vector of exogenous characteristics affecting cost.
 In equilibrium, for all firms in the market, it holds
\[p = c'(q;w)  \lambda p'(q;x) q.\]
 From this expression, we can obtain estimates of \(\lambda\).
Current Field Developments
 Recent work further investigates methods for estimating market power.
 There is a plethora of methods for estimating market power (Perloff, Karp, and Golan 2007).
 Recent antitrust cases in the EU are those against Google for Goggle Shipping and Google AdSense.
 Google has been fined with over \(8\) billion Euro for these cases.
 In 2020, a case has been filed against Facebook in the US for suppressing competition from social media rivals.
Concise Summary
 Firms can employ trigger strategies to induce collusion in the market.
 Explicit use of contracts is not required.
 Collusion can be sustained both under quantity and price competition.
 Quantity competition requires firms to value future profits more than price competition.
 Market power can be measured by statistics such as the Herfindahl index.
 Market power can be estimated using a variation of the first order condition of the Cournot model.
Further Reading
 Watson (2008, chaps. 22, 23)
 Belleflamme and Peitz (2010, secs. 2.3, 3.5, 14.2.1)
 Varian (2010, secs. 28.10, 28.11)
Mathematical Details
Repeated simultaneous quantity competition with two firms
When there is a future, firms can use reputation and/or threats to coordinate without explicitly cooperating. This gives them the ability to increase their profits. A simple threat that can induce coordination is the trigger strategy.
The trigger strategy
The goal of the trigger strategy is that the firms produce at the monopolistic level and split the market. If any firm deviates from this strategy, the other firm reverts to producing the oligopolistic quantity for all future dates.
The problem
Suppose that there are two firms in the market. Both discount future profits by \(\delta \in (0,1)\). Let demand be affine and costs linear. Firms aim to maximize the series of flow profits, i.e., \[V_{i} = \max_{(q_{i,t})_{t}} \left\{ \sum_{t=0}^{\infty} \delta_{t} \pi_{i,t}(q_{1,t}, q_{2,t}) \right\}.\]
Non coordinating equilibrium
There are many equilibria in the game. A simple equilibrium is one that the stage game Nash equilibrium is played are every date. All firms use the oligopolistic quantity competition production levels at every date, which leads to flow profits \[\pi_{i,t} =  \frac{(c_{1}p_{0})^{2}}{9 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\] Therefore, each firm’s expected future series of profits is \[V_{i} =  \frac{1}{1\delta}\frac{(c_{1}p_{0})^{2}}{9 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]
Coordinating equilibrium with trigger strategies
Another equilibrium can be obtained when firms play the trigger strategies. Firms produce half the monopolistic quantity and split the monopolistic profits \[\pi_{i,t} =  \frac{1}{2}\frac{(c_{1}p_{0})^{2}}{4 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\] If firms coordinate with these profits as flow utilities, they get \[V_{m, i} =  \frac{1}{1\delta}\frac{(c_{1}p_{0})^{2}}{8 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]
Is there any incentive to deviate? Suppose that Firm 1 deviates at the current date. The best response to Firm 2’s choice of monopolistic profits is \[q_{1,0} = \frac{3}{8}\frac{c_{1}p_{0}}{p_{1}}.\] Then, firm 1’s profit at the current date is \[\pi_{1,t} =  \frac{9}{64}\frac{(c_{1}p_{0})^{2}}{p_{1}}.\] On future dates, the firms revert to the non coordinating quantities because they use trigger strategies. Therefore, the total discounted profit of the deviating firm is \[V_{d,1} =  \frac{9}{64}\frac{(c_{1}p_{0})^{2}}{p_{1}}  \frac{\delta}{1  \delta} \frac{(c_{1}p_{0})^{2}}{9 p_{1}}.\]
The coordination outcome is sustainable if the deviation profit is less than the coordination profit, namely if
\begin{align*} V_{1,d} &=  \frac{9}{64}\frac{(c_{1}p_{0})^{2}}{p_{1}}  \frac{\delta}{1  \delta} \frac{(c_{1}p_{0})^{2}}{9 p_{1}} \\ &<  \frac{1}{1  \delta} \frac{(c_{1}p_{0})^{2}}{8 p_{1}} = V_{1,m}, \end{align*}
which is equivalent to \(\delta > \frac{9}{17}\). Thus, as long as firms value future profits enough, there is room for cooperation.
Exercises
Group A

Consider two firms competing in quantities for an infinite number of dates. The discount factor is \(\delta \in (0, 1)\).The market’s inverse demand is \(p(q)=220q\), and both firms have a marginal cost equal to \(10\)
 Find the equilibrium under non cooperation (Cournot competition) at each date. Calculate the quantity that a monopolist would produce at each date. Calculate the profits in both cases and compare them.
 Write down the trigger strategies of the two firms that can lead to tacit collusion.
 What is the most profitable deviation from the trigger strategy at the initial date? What is the profit of the deviation at this date? How does the profit compare with the coordination profit of the trigger strategy?
 For which values of \(\delta\) is collusion sustainable?

In the stage game, each firm maximizes \[\max_{q_{i}} (220  q_{i}  q_{j}  10 ) q_{i},\] which implies that \(q_{i} = q_{j} = q_{c} = 70\). When firms do not cooperate, they choose to produce these quantities at all dates. In this case, the market price is \[p(2q_{c}) = 220  140 = 80,\] and each firm makes profit \[\pi_{c} = (80  10 ) 70 = 4900.\]
The monopolist would instead maximize \[\max_{q} (220  q  10 ) q,\] which gives \(q_{m} = 105\). The market price is then \[p(q_{m}) = 220  105 = 115,\] and the monopolist makes profit \[\pi_{m} = (115  10) 105 = 11025.\]

The trigger strategy that can lead to collusion is for each firm \(i\) to produce according to the rule
\begin{align*} q_{i,t} = \left\{ \begin{aligned} \frac{q_{m}}{2} & \quad\quad q_{j,s}=\frac{q_{m}}{2} \text{ for all } s < t \\ q_{c} & \quad\quad \text{ otherwise } \end{aligned}\right. . \end{align*}

Given that firm \(j\) plays the trigger strategy, firm \(i\) maximizes \[\max_{q_{i}} \left(220  q_{i}  \frac{105}{2}  10\right) q_{i},\] which implies that the best firm \(i\) can do, if it deviates from the trigger strategy, is to produce \(q_{d} = \frac{315}{4}\). The profit of firm \(i\) is then \[\pi_{d} = \frac{99225}{16},\] which is greater than \(pi_{m}/2\). Thus, the firm can make greater flow profit at the initial date if it deviates from coordination.

This practice, however, does not make the firm overall better off. Only shortsighted firms prefer this scenario. Firms that value future profits prefer coordination. Specifically, coordination can be sustained if and only if \[\frac{1}{1\delta} \frac{\pi_{m}}{2} \ge \pi_{d} + \frac{\delta}{1 \delta}\pi_{c}\] or equivalently \[\delta \ge \frac{\frac{\pi_{m}}{2}  \pi_{d}}{\pi_{c} \pi_{d}} = \frac{\frac{11025}{2}  \frac{99225}{16}}{4900  \frac{99225}{16}} = \frac{11025}{20825} = \frac{9}{17}.\]

Consider two firms competing in prices for a potentially infinite number of dates. The two firms have marginal costs equal to \(c\). If the two firms set the same price, they split the market so that firm \(1\) gets a share \(\mu\) and firm \(2\) gets a share \(1\mu\). The firms do not discount future profits, but there is a probability \(p > 0\) that the market closes down after each date. The firms use trigger strategies to coordinate and split the monopolistic profit.
 Find the equilibrium if \(p=1\).
 Suppose that \(\mu = \frac{1}{2}\). Find the values of \(p<1\) for which collusion is sustainable. How do you interpret the result?
 Find the values of \(p<1\) for which collusion is sustainable for any value of \(\mu\in(0,1)\). What is the equilibrium relationship between \(p\) and \(\mu\)? How do you interpret the result?
 If \(p=1\), the firms compete only for one date, and the game reduces to Bertrand competition. The firms set \(p_{1}=p_{2}=c\).
 The most profitable deviation of each firm at the initial date is to slightly undercut the monopoly price by \(\varepsilon\). By doing so, the firm can take all the market demand and achieve almost the monopolistic profit. For simplicity, assume that the deviating firm earns exactly the monopolistic profit. Then, the deviation is not profitable if and only if \[\frac{1}{p}\frac{\pi_{m}}{2} \ge \pi_{m} \iff p \le \frac{1}{2}.\] Collusion is sustainable if the probability that the market closes down is small and the firms have the potential to benefit from future profits.
 With arguments analogous to the \(\mu=\frac{1}{2}\) case, the condition for firm \(1\) is \[\frac{1}{p}\mu \pi_{m} \ge \pi_{m} \iff p \le \mu,\] and for firm \(2\) is \[\frac{1}{p}(1\mu) \pi_{m} \ge \pi_{m} \iff p \le 1\mu.\] Therefore, collusion is sustainable if and only if \[p \le \min\{\mu, 1\mu\}.\] If \(\mu < \frac{1}{2}\), the equilibrium condition becomes less stringent when \(\mu\) increases. In contrast, the equilibrium condition becomes more stringent when \(\mu\) increases for \(\mu>\frac{1}{2}\). The most lenient equilibrium condition is obtained for \(\mu=\frac{1}{2}\). The further is \(\mu\) from \(\frac{1}{2}\), the less are the flow profits that one of the two firms makes and, as a result, coordination based on future benefits becomes less profitable.
Group B

Consider two firms competing in quantities for an infinite number of dates. The discount factor is \(\delta \in (0, 1)\).The market’s inverse demand is \(p(q)=p_{0}+p_{1}q\) and both firms have marginal cost equal to \(c\), where \(p_{0}>c>0\) and \(p_{1}<0\). Suppose that firms use the following titfortat strategy. If the other firm has produced half the monopolistic quantity last date, then produce half the monopolistic quantity at this date. Otherwise, produce the Cournot quantity.
 Find the equilibrium under non cooperation (Cournot competition) at each date. Calculate the profits of the firms.
 Find the total discounted profits if each firm produces half the monopolistic quantity at each date.
 Find the most profitable deviation of firm \(i\) at the initial date. Calculate the profit of the deviating firm at the initial date.
 Suppose that at the second date, the deviating firm reverts to producing half the monopolistic quantity. Calculate the profit it makes at this date.
 For which values of \(\delta\) is collusion sustainable?

As in the grim trigger strategy case, if the firms do not coordinate, they produce \[q_{c} = \frac{c  p_{0}}{3 p_{1}},\] make flow profits \[\pi_{c} = \frac{(p_{0}  c)^{2}}{9 p_{1}},\] and total discounted profits \[V_{c} = \frac{1}{1\delta}\frac{(p_{0}  c)^{2}}{9 p_{1}}.\]

In this case, each firm produces \[\frac{q_{m}}{2} = \frac{c  p_{0}}{4 p_{1}},\] makes flow profit \[\frac{\pi_{m}}{2} = \frac{(p_{0}  c)^{2}}{8 p_{1}},\] and total discounted profit \[V_{m} = \frac{1}{1\delta}\frac{(p_{0}  c)^{2}}{8 p_{1}}.\]

The most profitable deviation of firm \(i\) is calculated by solving \[\max_{q_{i}} \left(p_{0} + p_{1} \left(q_{i} + \frac{q_{m}}{2}\right)  c\right) q_{i},\] which leads to \[q_{1} = \frac{3}{8}\frac{cp_{0}}{p_{1}},\] and flow profit \[\pi_{1} = \frac{9}{64}\frac{(cp_{0})^{2}}{p_{1}}.\]

Firm \(i\) is producing half the monopolistic quantity, and firm \(j\) is producing the Cournot quantity, punishing the initial date deviation. Thus, firm \(i\) makes profit
\begin{align*} \pi_{2} &= \left(p_{0} + p_{1} \left(\frac{q_{m}}{2} + q_{c}\right)  c\right) \frac{q_{m}}{2} \\ &= \frac{5}{48}\frac{(cp_{0})^{2}}{p_{1}}. \end{align*}

There are two cases we should consider. The deviating firm may revert back to producing half the monopolistic quantity after the deviation, or it can resort to producing the Cournot quantity. In the first case, the payoff is \[V_{r} = \pi_{1} + \delta \pi_{2} + \delta^{2} V_{m},\] while in the second case, it is \[V_{n} = \pi_{1} + \delta V_{c}.\] The second case is excluded if and only if
\begin{align*} V_{r} \ge V_{n} &\iff \pi_{2} + \delta V_{m} \ge V_{c} \\ &\iff \frac{5}{48}\frac{(cp_{0})^{2}}{p_{1}}  \frac{\delta}{1\delta} \frac{1}{8}\frac{(cp_{0})^{2}}{p_{1}} \ge  \frac{1}{1\delta} \frac{1}{9}\frac{(cp_{0})^{2}}{p_{1}} \\ &\iff \delta \ge \frac{1}{3}. \end{align*}
Moreover, for collusion to be sustainable, we need
\begin{align*} V_{m} \ge V_{r} &\iff V_{m} \ge \pi_{1} + \delta \pi_{2} + \delta^{2} V_{m} \\ &\iff \frac{1  \delta^{2}}{1\delta} \frac{1}{8} \frac{(cp_{0})^{2}}{p_{1}} \ge \frac{9}{64}\frac{(cp_{0})^{2}}{p_{1}}  \delta \frac{5}{48}\frac{(cp_{0})^{2}}{p_{1}} \\ &\iff (1 + \delta) \frac{1}{8} \ge \frac{9}{64} + \delta \frac{5}{48} \\ &\iff \delta \ge \frac{3}{4}. \end{align*}