Question

Consider the following two-player team production problem. Each player \(i\) chooses a level of effort \(a_{i} \geq 0\) at a personal cost of \(a_{i}^{2}\). The players select their effort levels simultaneously and independently. Efforts \(a_{1}\) and \(a_{2}\) generate revenue of \(r=4\left(a_{1}+a_{2}\right)\). There is limited verifiability in that the external enforcer (court) can verify only the revenue generated by the players, not the players' individual effort levels. Therefore, the players are limited to revenue-sharing contracts, which can be represented by two functions \(f_{1}:[0, \infty) \rightarrow[0, \infty)\) and \(f_{2}:[0, \infty) \rightarrow[0, \infty)\). For each player \(i, f_{i}(r)\) is the monetary amount given to player \(i\) when the revenue is \(r\). We require \(f_{1}(r)+f_{2}(r) \leq r\) for every \(r .\) Call a contract balanced if, for every revenue level \(r\), it is the case that \(f_{1}(r)+f_{2}(r)=r\). That is, the revenue is completely allocated between the players. A contract is unbalanced if \(f_{1}(r)+f_{2}(r)

Short answer

Efficient efforts are 2 and 2; Nash eq. under sharing is 1 and 1; no balanced contract induces efficiency; effort verifiability would solve issues.

Step-by-step solution

Determine Efficient Effort Levels

To find the efficient effort levels, we maximize the joint value function: \[ 4(a_1 + a_2) - a_1^2 - a_2^2. \]Take the derivative of this function with respect to both \( a_1 \) and \( a_2 \), set them equal to zero, and solve for the critical points:\[ \frac{\partial}{\partial a_1} (4(a_1 + a_2) - a_1^2 - a_2^2) = 4 - 2a_1 = 0, \]\[ \frac{\partial}{\partial a_2} (4(a_1 + a_2) - a_1^2 - a_2^2) = 4 - 2a_2 = 0. \]Solving both equations, we find that the efficient effort levels for both players are:\[ a_1 = 2, \quad a_2 = 2. \]

Determine Nash Equilibrium Under Revenue Sharing Contract

Under the given contract, each player receives half of the revenue minus their personal cost:\[ 2(a_1 + a_2) - a_i^2. \]Each player chooses their effort level to maximize their payoff:For Player 1: \[ 2(a_1 + a_2) - a_1^2. \]Taking the derivative with respect to \( a_1 \) and setting to zero:\[ \frac{\partial}{\partial a_1} [2(a_1 + a_2) - a_1^2] = 2 - 2a_1 = 0. \]For Player 2:\[ 2(a_1 + a_2) - a_2^2. \]Taking the derivative with respect to \( a_2 \) and setting to zero:\[ \frac{\partial}{\partial a_2} [2(a_1 + a_2) - a_2^2] = 2 - 2a_2 = 0. \]Both equations yield \( a_1 = 1 \) and \( a_2 = 1 \). Hence, the Nash equilibrium under this contract is \( a_1 = 1 \), \( a_2 = 1 \).

Explore Balanced Contract for Efficient Nash Equilibrium

A balanced contract requires the total revenue to be allocated completely to the players: \[ f_1(r) + f_2(r) = r. \]We need to verify if a balanced contract can incentivize efficient efforts (\( a_1 = 2 \), \( a_2 = 2 \)).Assume a contract of the form where each player receives a different share:\[ f_1(r) = \alpha r, \quad f_2(r) = (1-\alpha) r. \]To incentivize \( a_1 = 2 \) and \( a_2 = 2 \), set up the derivative of each player's maximization problem and match with efficient levels insights.Through trial or numerical methods, it can generally be found that such specific alpha values do not allow symmetric & simultaneous equilibrium effort decisions that correspond to both playing optimal values concurrently.Thus, no balanced contract will achieve this solution given the problem setup.

Consider Unbalanced Contract for Efficient Nash Equilibrium

Examine whether an unbalanced contract can support the efficient effort level:Consider hypothetical revenue compensation functions yielding \( f_1(r) + f_2(r) < r \) at critical, perhaps significantly lowering or destroying a portion of revenue post-division, but instead mixed specified returns without real numerical breakdown this hypothetical basis cannot be further leveraged without adjustments.The mechanism behind unbalancing typically alters outside-optimal operational stability, using tools like bonuses or adjustments might indirectly reach close to targets with additional modeling.

Discuss the Role of Verifiability of Efforts

If the court can verify each player's effort level directly, contracts would not be limited to revenue sharing and could depend explicitly on the efforts \( a_1 \) and \( a_2 \). This verification would facilitate perfect incentive alignment since players would be rewarded based on their effort rather than derived outcomes, ensuring true efficient efforts; thus balancing both effort representation and reward absolutely, diminishing coordination issues seen primarily in absence of procedural certainty.

Key concepts

Nash Equilibrium

The Nash Equilibrium refers to a situation in a game where each player makes the best decision they can, given the choices of the other players. In the context of our two-player team production problem, both players choose an effort level that maximizes their own payoff, considering the effort level chosen by the other player.

When each player receives half of the revenue generated by both players' efforts, they seek to optimize their gain after subtracting their individual effort cost. By solving their payoff function, each player independently decides an effort level where no one has anything to gain by changing only their own strategy if the other's effort level remains constant.

• For Player 1: Maximizes \(2(a_1 + a_2) - a_1^2\), leading to \(a_1 = 1\).
• For Player 2: Maximizes \(2(a_1 + a_2) - a_2^2\), leading to \(a_2 = 1\).

The resulting Nash Equilibrium in this scenario is when both players choose effort levels of 1.

Revenue Sharing

Revenue sharing in this context involves distributing the total revenue between the two players according to certain rules. Each player's share is determined without knowing the actual effort levels, due to limited verifiability by the enforcer.

In our problem, a contract specifying that each player receives half of the revenue, minus their effort cost, aligns their incentives only partially with total revenue maximization. This creates a situation where players naturally tend to exert lesser effort compared to optimal levels, i.e., the effort levels yielding the maximum joint benefit: \(4(a_1 + a_2) - a_1^2 - a_2^2\).

• The assigned revenue for each player based on this structure is inefficient as it does not drive the optimum contribution due to shared gains versus individual costs.
• This is where different contract designs may attempt to better align incentives.

Efficient Effort Levels

Efficient effort levels refer to those combinations of effort by the players that maximize the joint benefit from the task. The joint benefit function can be expressed as \(4(a_1 + a_2) - a_1^2 - a_2^2\).

To find these levels, we take the derivative of this joint value with respect to each player's effort and set them equal to zero. This results in optimal effort levels of \(a_1 = 2\) and \(a_2 = 2\), which means both players exert more effort in comparison to those chosen at the Nash Equilibrium given the current revenue-sharing contract.

• These efforts maximize the collective output, ensuring that the total value generated minus the total costs to players is the highest possible.
• Unfortunately, without changing the payoff structure, players are not incentivized to reach these effort levels.

Contract Theory

Contract Theory involves designing optimal contractual arrangements to align the interests of different parties involved in a transaction.

In the exercise, revenue-sharing contracts are the tools available under limited verifiability conditions. Players cannot be tied directly to specified effort levels, since these are unobservable. Therefore, contracts have to be structured based on the revenue output, which both players collectively influence.

• Balanced contracts share all revenue with no excess or deficit, matching payouts to the total generated.
• Unbalanced contracts might leave some revenue unallocated or wasted.
• Under current constraints, achieving efficient efforts as a Nash Equilibrium via such contracts remains challenging unless methods to verify and motivate individual efforts are included.

Expanding contract mechanisms with bonuses or other incentives would be a move toward efficient contract design, especially if effort levels could be verifiably enforced.