Question

Construct a game with joint decisions by augmenting the Cournot duopoly game in the following way: Suppose the two firms have a profit-sharing pact that is enforced by the government (legal collusion). First, the firms simultaneously select quantities, \(q_{1}\) and \(q_{2}\), at a marginal cost of 10 . The price is determined from the inverse demand curve, \(p=100-q_{1}-q_{2}\), and the total revenue \(p\left(q_{1}+q_{2}\right)\) is deposited into a joint bank account. Then the firms must negotiate over how to share the revenue. Model the negotiation as a joint decision over firm l's share of the revenue, \(m\), so that agreement yields a payoff of \(m-10 q_{1}\) to firm 1 and \(p\left(q_{1}+q_{2}\right)-m-10 q_{2}\) to firm 2 . If they disagree then neither firm obtains any revenue, so firm 1 gets \(-10 q_{1}\) and firm 2 gets \(-10 q_{2}\).

Short answer

Firms negotiate to share revenue equally using Nash Bargaining after choosing equilibrated quantities.

Step-by-step solution

Understand the Game Components

In this exercise, we're dealing with a Cournot duopoly where two firms, referred to as firm 1 and firm 2, are producing quantities \(q_1\) and \(q_2\) respectively. The marginal cost for producing each unit is 10 for both firms. The market price, \(p\), is determined by the inverse demand curve \(p = 100 - q_1 - q_2\). The objective of each firm is to maximize its own profit under a profit-sharing agreement, enforced legally, that involves joint decisions on quantities followed by negotiation over revenue division.

Formulate the Revenue Function

First, calculate the total revenue based on the inverse demand curve and production quantities. Total revenue comes from multiplying the price \(p\) by the sum of quantities \(q_1 + q_2\). Thus, the total revenue is described as: \[ TR = p(q_1 + q_2) = (100 - q_1 - q_2)(q_1 + q_2) \]

Determine the Profit-Sharing Agreement

The revenue \(TR\) is placed into a joint account. Each firm must negotiate how to split this total revenue using a share parameter \(m\) for firm 1. Firm 1's payoff is \(m - 10q_1\) and firm 2's payoff is \(TR - m - 10q_2\). Disagreement on the split results in each firm only bearing the cost, \(-10q_1\) for firm 1 and \(-10q_2\) for firm 2. Their payoffs depend on successfully agreeing on \(m\).

Set Up Negotiation Constraints

Both firms would like to agree on a share that maximizes their expected payoffs. The agreement constraints are structured so that firm 1's share \(m\) must satisfy \(m - 10q_1 > -10q_1\) and firm 2's share \(TR - m - 10q_2 > -10q_2\). This ensures both firms are better off agreeing on the share rather than not agreeing.

Equilibrium Analysis

Determine the equilibrium quantities \(q_1^*\) and \(q_2^*\) that maximize joint profit before negotiation. In a typical Cournot duopoly without negotiation, firms maximize their own profits, which might lead us to equilibrium strategies. However, here firms cooperate by sharing profits, so they need to come up with \(q_1^*\) and \(q_2^*\) that maximize \(TR = (100 - q_1 - q_2)(q_1 + q_2)\). Solving this joint profit maximization leads to symmetry \(q_1^* = q_2^* = q^*\).

Use Nash Bargaining Solution for Revenue Division

Utilize the Nash Bargaining Solution (NBS) to determine the point where both firms will split profits such that \[ m = \frac{TR}{2} + \frac{10(q_2 - q_1)}{2}\]. This solution assumes that firms share the surplus equally from the collaborative game perspective.

Key concepts

Profit-Sharing Agreement

A profit-sharing agreement is a collaborative arrangement whereby the profits generated by two or more parties are divided according to a predetermined formula. In the context of a Cournot duopoly game, profit-sharing plays a critical role in determining how firms distribute their joint earnings. The two firms collaborate on deciding the quantities to produce, and then negotiate their individual shares of the profits.

In the exercise, after firms determine their production quantities, the profits are pooled into a joint account. The distribution of these profits is crucial for ensuring mutual benefit. A share parameter, "m", is negotiated which dictates how much firm 1 will receive. Firm 1's payoff from this profit-sharing arrangement is calculated as:

• Firm 1's payoff: \[ m - 10q_1 \]

For firm 2, the payoff is the remaining profit after subtracting firm 1's share, minus their costs:

• Firm 2's payoff: \[ TR - m - 10q_2 \]

If the firms can't agree on the division, they bear only their own production costs, getting negative payoffs. Hence, the agreement must be attractive enough that both firms prefer it over no agreement. Ensuring this agreement takes place legally under government enforcement ensures that neither firm deviates from the agreed terms, making the deal credible.

Inverse Demand Curve

The inverse demand curve is a fundamental concept in economics that describes the relationship between the price of a good and the quantity demanded. In simpler terms, it tells us how much customers are willing to pay for a specific amount of goods.

In a Cournot duopoly, like the one in the exercise, the price of the product is determined by the inverse demand curve. This curve is given by the equation:\[ p = 100 - q_1 - q_2 \]This equation implies that the price decreases as the total quantity produced by the two firms (\(q_1 + q_2\)) increases. The inverse relationship means if firms produce more, the price they can charge is lower.

• Maximizing revenue involves understanding how each additional unit of product affects the market price.
• It's critical for firms to find the quantity that maximizes their benefit while keeping in mind others' impacts.

Therefore, strategic decision-making on production quantities is essential, as it directly influences both total revenue and individual profits.

Nash Bargaining Solution

The Nash Bargaining Solution is a concept used to find an equitable distribution of benefits in a negotiation between multiple parties. It is named after mathematician John Nash, who proposed that parties will reach a solution where they split payoffs fairly based on their alternative options. This method assumes some rationality and cooperation among the negotiating parties.

In the scenario where firms are engaged in legal collusion, the Nash Bargaining Solution helps determine how profits are split fairly. The goal is to find an outcome where neither firm could be better off without making the other worse off.

In the exercise, the solution is expressed as:\[ m = \frac{TR}{2} + \frac{10(q_2 - q_1)}{2} \]This formula represents a fair split where each firm gains equally from the cooperative game, taking into account their respective costs and outputs.

The Nash Bargaining Solution provides clarity and simplicity by allowing the firms to compute and agree on profit-sharing in a way that maximizes joint satisfaction. It also ensures that the outcome is balanced, meaning that both firms leave the negotiation with profits surpassing their initial costs.