Question

Recall Example \(15.6,\) which covers tacit collusion. Suppose (as in the example) that a medical device is produced at constant average and marginal cost of \(\$ 10\) and that the demand for the device is given by \\[ Q=5,000-100 P \\] The market meets each period for an infinite number of periods. The discount factor is \(\delta\). a. Suppose that \(n\) firms engage in Bertrand competition each period. Suppose it takes two periods to discover a deviation because it takes two periods to observe rivals' prices. Compute the discount factor needed to sustain collusion in a subgame-perfect equilibrium using grim strategies. b. Now restore the assumption that, as in Example \(15.7,\) deviations are detected after just one period. Next, assume that \(n\) is not given but rather is determined by the number of firms that choose to enter the market in an initial stage in which entrants must sink a one-time cost \(K\) to participate in the market. Find an upper bound on \(n\). Hint: Two conditions are involved.

Short answer

Question: Compute the discount factor needed to sustain collusion in a subgame-perfect equilibrium using grim strategies. Also, restore the assumption that deviations are detected after just one period, and find an upper bound on the number of firms entering the market.

Step-by-step solution

Part a: Computing the Discount Factor Needed to Sustain Collusion

We have the demand function: \[ Q = 5000 - 100P \] We know that the constant average and marginal cost is \(10, so \)MC = AC = 10$. In each period, firms are involved in Bertrand competition, so if there is no collusion, firms will set the price equal to \(MC\) (\(P = 10\)). In this case, monopoly profits will be zero. However, if firms collude and act as a monopoly, they will set the price to maximize profits. First, let's find the monopoly price and quantity: 1. Set \(MC = MR\), where \(MR\) is the marginal revenue. 2. Find the corresponding quantity and price. Now, in order to sustain collusion, firms need to have a higher present value of profits when they collude than when they don't. Therefore, we need to find the discount factor needed for this to occur using grim strategies. We know that if a firm deviates, it will be detected after two periods. Thus, we have to compare the present value of profits when the firm colludes and when it does not: 1. Compute the present value of profits when the firm colludes, which is the monopoly profits discounted to the present time. 2. Compute the present value of profits when the firm does not collude, which is the discounted value of future profits from deviation and possible punishments. 3. These present values must be equal in order for collusion to be sustained. Solve for the discount factor.

Part b: Finding an Upper Bound on n

According to the hint, we need to consider two conditions in order to find an upper bound on n: 1. Deviations are detected after just one period. 2. The number of firms (n) is determined by the number of firms that choose to enter the market in an initial stage, in which entrants must sink a one-time cost (K) to participate in the market. In this case, calculate the present value of the profits when the firm colludes, and the present value of the profits when the firm does not collude for each firm. A firm will choose to enter the market if the present value of profits from collusion is higher than the one-time sunk cost (K). 1. Since deviations are detected after just one period, the present value expression for no collusion must be updated. 2. Calculate the present value of the profits when the firm colludes, and the present value of the profits when the firm does not collude for each firm in this new condition. 3. A firm will enter the market if the present value of profits from collusion is higher than the one-time sunk cost (K). So, we need to find the maximum number n such that the present value of monopoly profits is larger than K. 4. Solve for the upper bound on n.

Key concepts

Bertrand Competition

In microeconomic theory, Bertrand competition is a model that describes interactions between firms that compete on price. Imagine a marketplace where a few companies sell identical products and consumers opt for the cheapest offering. Under this scenario, if firms are rational and aim to maximize their profits, the competition will drive the prices down to the level of the marginal cost, which is the additional cost of producing one more unit. This is because if any firm tries to set a higher price, consumers will buy from the one with the lower price.

In a perfect Bertrand model with identical products and no collusion, the end result is zero economic profit for all firms as the price falls to equal the marginal cost. However, the real world is more nuanced, and firms often find ways to avoid such fierce competition, which can lead to tacit collusion—an unspoken agreement to keep prices at a higher level to the benefit of all firms involved.

Subgame-Perfect Equilibrium

The concept of subgame-perfect equilibrium comes from game theory and refers to a scenario where players choose strategies that constitute a Nash equilibrium within every subgame of a larger game. To put it simply, a subgame-perfect equilibrium is a strategy that does not only consider the immediate benefits but looks ahead at the implications of each action within every possible future situation that can unfold from the current point in the game.

This approach is crucial when analyzing situations of tacit collusion in the context of repeated games. In the case of Bertrand competition, firms might potentially sustain higher prices through collusion if the outcome is a subgame-perfect equilibrium; this maintains higher profits every period, factoring in the threat of punishment strategies if any firm deviates from the collusive agreement.

Grim Strategies

Within the strategy playbook of game theory is a category known as grim strategies. Applied within the scope of repeated games, a grim strategy essentially means playing nice until someone else deviates; after a deviation occurs, the grim strategy dictates a permanent shift to a punishment mode. This can be a powerful tool in deterring firms from breaking a collusive agreement.

For instance, in a repeated Bertrand competition scenario, firms might tacitly agree to keep prices high. If a grim strategy is employed, this pact is upheld only as long as no firm undercuts the agreed-upon price. Should any firm break ranks, the rest retaliate by reverting to competitive pricing for an indefinite period, making the defector incur substantial losses. The fear induced by grim strategies can foster an environment where tacit collusion is more sustainable, as deviation becomes less attractive.

Discount Factor

At the heart of understanding the intertemporal choices firms face is the concept of a discount factor, denoted by \( \delta \) in economic models. This factor measures how much future profits are valued today; it's a number between 0 and 1. The closer \( \delta \) is to 1, the more future profits are worth in present terms, indicating a lower rate of time preference.

When considering collusion in a repeating Bertrand competition, the discount factor is key. If firms are very patient (\( \delta \) is high), they value future profits almost as much as present profits and therefore are more likely to maintain a collusive agreement for the long-term benefits it brings. In contrast, if they're less patient (\( \delta \) is lower), the temptation to undercut and gain immediate profits might outweigh the less-valued future benefits of collusion. Calculating the discount factor that supports a subgame-perfect equilibrium with grim strategies offers insights into the sustainability of tacit collusion.