Question

A bicycle manufacturer (the "buyer," abbreviated B) wishes to procure a new robotic system for the production of mountain-bike frames. The firm contracts with a supplier (S), who will design and construct the robot. The contractual relationship is modeled by the following game: The parties first negotiate a contract specifying an externally enforced price that the buyer must pay. The price is contingent on whether the buyer later accepts delivery of the robot (A) or rejects delivery (R), which is the only event that is verifiable to the court. Specifically, if the buyer accepts delivery, then he must pay \(p_{1}\); if he rejects delivery, then he pays \(p_{0}\). After the contract is made, the seller decides whether to invest at a high level (H) or at a low level (L). High investment indicates that the seller has worked diligently to create a high-quality robot-one that meets the buyer's specifications. High investment costs the seller 10. The buyer observes the seller's investment and then decides whether to accept delivery. If the seller selected \(\mathrm{H}\) and the buyer accepts delivery, then the robot is worth 20 units of revenue to the buyer. If the seller selected \(\mathrm{L}\) and the buyer accepts delivery, then the robot is only worth 5 to the buyer. If the buyer rejects delivery, then the robot gives him no value. (a) What is the efficient outcome of this game? (b) Suppose the parties wish to write a "specific-performance" contract, which mandates that the buyer accept delivery at price \(p_{1}\). How can \(p_{0}\) be set so that the buyer has the incentive to accept delivery regardless of the seller's investment? Would the seller choose H in this case? (c) Under what conditions of \(p_{0}\) and \(p_{1}\) would the buyer have the incentive to accept delivery if and only if the seller selects H? Show that the efficient outcome can be obtained through the use of such an "option contract." (d) Fully describe the negotiation equilibrium of the game, under the assumption that the parties have equal bargaining weights.

Short answer

(a) Both accept and invest high; (b) Set p_0 low, p_1 = any; (c) p_1 = 15; (d) Each party earns 5.

Step-by-step solution

Identify Payoffs for Different Investment and Acceptance Combinations

To solve this problem, let's first list the payoffs for the buyer (B) and the supplier (S) for each possible combination of investment levels (H/L) and acceptance decisions (A/R). - If B accepts (A) and S invests high (H): Buyer's payoff = 20 - p_1, Seller's payoff = p_1 - 10. - If B accepts (A) and S invests low (L): Buyer's payoff = 5 - p_1, Seller's payoff = p_1. - If B rejects (R): Both payoff = 0 - p_0 (Buyer = -p_0, Seller = p_0).

Determine the Efficient Outcome

The efficient outcome occurs when the total surplus (the sum of Buyer and Seller payoffs) is maximized. This happens when B accepts delivery, and S invests at the high level (H). In this scenario, the buyer receives 20 in value, and the seller incurs a cost of 10, so net surplus = 20 - 10 = 10.

Analyze Specific-Performance Contract

To create a specific-performance contract for question b, set p_0 low enough that rejection is unfavorable to B regardless of H/L, meaning: -p_0 < max(20 - p_1, 5 - p_1). Thus, ensure that p_1 offers sufficient surplus for B such that the seller has the incentive to choose L due to being paid regardless.

Design an Option Contract to Align Incentives

To address question c, set conditions such that B accepts only when faced with a high-quality investment. We need: 20 - p_1 > 0 > 5 - p_1; while also ensuring B rejects when they see L: -p0 < 5 - p_1. Therefore seller chooses H because only this investment method will yield acceptance.

Describe the Negotiation Equilibrium

For question d, equal bargaining weights imply equal sharing of surplus. Set p_1 such that 20 - p_1 = p_1 - 10, giving p_1 = 15. This outcome maximally appropriates total surplus while balancing incentives, ensuring both accept on H and are refrained from renegotiating terms post-contract.

Key concepts

Contract Theory

In the game theory scenario outlined, contract theory plays a significant role in defining the relationship between the bicycle manufacturer and the supplier. Contract theory seeks to understand how contracts are structured to align the interests of involved parties under conditions of asymmetric information and divergent goals.

In our example, the manufacturer (buyer) and the supplier are involved in a contractual negotiation over the supply and delivery of a robotic system. They need a contract that clearly specifies terms to ensure that the supplier provides a high-quality product and is incentivized to meet those requirements. The contract must balance payment structures to accommodate both parties' interests effectively.

• The price payable (\(p_{1}\) and \(p_{0}\)) is contingent on the manufacturer's decision to accept or reject the product based on its quality.
• A well-crafted contract should ensure the supplier feels compensated adequately for their investment level while encouraging timely and high-quality delivery.
• The contract should include provisions for enforcement, clearly outlining what happens if terms are not met, thus preventing renegotiation and ensuring compliance.

As the game unfolds, contract theory helps ensure that the supplier is incentivized to produce a product meeting specifications, while the buyer remains motivated to honor the contract when these specifications are met.

Negotiation Equilibrium

Negotiation equilibrium in this context refers to the state at which both buyer and supplier reach an agreement that neither party desires to deviate from. This is crucial in ensuring that the game resolves without the need for ongoing negotiation or dispute.

In achieving negotiation equilibrium, the concept of equal bargaining weights is essential. Here, both parties have identical negotiating power, meaning they stand to gain equally from the cooperation. Let's understand this with the following points:

• The buyer and supplier must agree on a price \(p_{1}\) that equates the benefit each gain from the transaction. This price reflects the shared surplus from a properly executed transaction when the robot is accepted.
• If the buyer's payoff from accepting the robot exactly offsets the supplier's cost plus bargaining benefit, equilibrium is achieved. In mathematical terms for this case, that means a \(p_{1}\) value where the buyer's value (20) minus \(p_{1}\) equals the supplier's cost (10) plus their payoff from \(p_{1}\).
• At equilibrium, both parties retain no incentive to deviate, maintaining focus on producing and accepting high-value work.

By careful calculation and aligning on equal payoff through the agreed contracts, negotiation equilibrium ensures cooperation is sustained, and both parties profit from successful deliveries.

Investment Incentives

Investment incentives are mechanisms put in place to encourage the supplier to invest in producing a high-quality product. Without these, the supplier might choose a low investment, thereby reducing the product’s value and disrupting the desirable outcome.

Here are key considerations in creating investment incentives throughout the contract game:

• For the supplier, investing at a high level (H) incurs a cost of 10 but ultimately results in a product that provides 20 in value to the buyer. The contract must make sure that this extra cost incentivization comes from the expectation of a high payoff (\(p_{1}\)) under acceptance conditions.
• If the buyer is set to reject low-quality products, the supplier realizes their only path to receive payment is through high investment, making H the optimal choice.
• Offering a higher discrepancy in acceptance payoffs between high and low-quality products can help shift the supplier’s choice decisively toward investing in H.

By conditioning delivery acceptance on observed investment and structuring payments to reflect level distinctions, both the buyer and the supplier are motivated towards maintaining high standards and achieving the desired product quality.

Efficient Outcome

Achieving an efficient outcome means structuring actions and incentives such that the total surplus—the combined benefits minus costs—is maximized across all parties.

In our scenario, this desired state hinges on aligning the outcomes where the supplier invests at a high level, thus producing a valuable robot accepted by the buyer:

• The total surplus in the game is the difference between the value to the buyer for a high-quality robot (20) and the supplier’s investment cost (10), resulting in a surplus of 10.
• To ensure this efficiency, contracts need to reflect an acceptance price (\(p_{1}\)) that makes the action of accepting a genuinely valuable robot beneficial solely under high investment conditions.
• Any outcome where the supplier chooses low investment results in reduced value (only 5) and thus should fall short of the buyer’s acceptance criteria, preventing a decrement in total potential surplus.

By precisely managing payoff differentials and ensuring deliverables meet expectations, an efficient outcome is maintained, benefiting both parties and maximizing the shared value derived from their contract.