Question

A manager and a worker interact as follows. The manager would like the worker to exert some effort on a project. Let \(e\) denote the worker's effort. Each unit of effort produces a unit of revenue for the firm; that is, revenue is \(e\). The worker bears a cost of effort given by \(\alpha e^{2}\), where \(\alpha\) is a positive constant. The manager can pay the worker some money, which enters their payoffs in an additive way. Thus, if the worker picks effort level \(e\) and the manager pays the worker \(x\), then the manager's payoff is \(e-x\) and the worker's payoff is \(x-\alpha e^{2} .\) Assume that effort is verifiable and externally enforceable, meaning that the parties can commit to a payment and effort level. Imagine that the parties interact as follows: First, the manager makes a contract offer to the worker. The contract is a specification of effort \(\hat{e}\) and a wage \(\hat{x}\). Then the worker accepts or rejects the offer. If she rejects, then the game ends and both parties obtain payoffs of 0 . If she accepts, then the contract is enforced (effort \(\hat{e}\) is taken and \(\hat{x}\) is paid). Because the contract is externally enforced, you do not have to concern yourself with the worker's incentive to exert effort. (a) Solve the game under the assumption that \(\alpha\) is common knowledge between the worker and the manager. That is, find the manager's optimal contract offer. Is the outcome efficient? (Does the contract maximize the sum of the players' payoffs?) Note how the equilibrium contract depends on \(\alpha\). (b) Let \(\underline{e}\) and \(\underline{x}\) denote the equilibrium contract in the case in which \(\alpha=1 / 8\) and let \(\bar{e}\) and \(\bar{x}\) denote the equilibrium contract in the case in which \(\alpha=3 / 8\). Calculate these values. Let us call \(\alpha=3 / 8\) the hightype worker and \(\alpha=1 / 8\) the low-type worker. (c) Suppose that \(\alpha\) is private information to the worker. The manager knows only that \(\alpha=1 / 8\) with probability \(1 / 2\) and \(\alpha=3 / 8\) with probability \(1 / 2\). Suppose that the manager offers the worker a choice between contracts \((\underline{e}, \underline{x})\) and \((\bar{e}, \bar{x})\)-that is, the manager offers a menu of contracts-in the hope that the high type will choose \((\bar{e}, \bar{x})\) and the low type will choose \((\underline{e}, \underline{x}) .\) Will each type pick the contract intended for him? If not, what will happen and why? (d) Suppose that the manager offers a menu of two contracts \(\left(e_{\mathrm{L}}, x_{\mathrm{L}}\right)\) and \(\left(e_{\mathrm{H}}, x_{\mathrm{H}}\right)\), where he hopes that the first contract will be accepted by the low type and the second will be accepted by the high type. Under what conditions will each type accept the contract intended for him? Your answer should consist of four inequalities. The first two-called the incentive compatibility conditions-reflect that each type prefers the contract intended for him rather than the one intended for the other type. The last two inequalities-called the participation constraints-reflect that by selecting the contract meant for him, each type gets a payoff that is at least 0 . (e) Compute the manager's optimal menu \(\left(e_{\mathrm{L}}, x_{\mathrm{L}}\right)\) and \(\left(e_{\mathrm{H}}, x_{\mathrm{H}}\right)\). Note that the manager wants to maximize his expected payoff $$ (1 / 2)\left[e_{\mathrm{H}}-x_{\mathrm{H}}\right]+(1 / 2)\left[e_{\mathrm{L}}-x_{\mathrm{L}}\right] $$ subject to the worker's incentive compatibility conditions and participation constraints. (Hint: Only two of the inequalities of part (d) actually bind - that is, hold as equalities. The ones that bind are the high type's participation constraint and the low type's incentive compatibility condition. Using these two equations to substitute for \(x_{\mathrm{L}}\) and \(x_{\mathrm{H}}\), write the manager's payoff as a function of \(e_{\mathrm{L}}\) and \(e_{\mathrm{H}} .\) Solve the problem by taking the partial derivatives with respect to these two variables and setting the derivatives equal to 0 .) (f) Comment on the relation between the solution to part (e) and the efficient contracts identified in part (b). How does the optimal menu under asymmetric information distort away from efficiency?

Short answer

Optimal contracts are \(e=\frac{1}{2\alpha}\), \(x=\frac{1}{4\alpha}\). Under information asymmetry, efficiency distorts to prevent type misalignment.

Step-by-step solution

Problem Understanding

We have a manager-worker scenario where the manager wants to maximize the firm's revenue by having the worker exert effort. Each unit of effort costs the worker \(\alpha e^2\). The wage is denoted by \(x\). Payoffs are manager's \(e-x\) and worker's \(x-\alpha e^2\). The challenge is to find the optimal effort and wage in several situations, considering efficiency and information symmetry or asymmetry.

Solve for Optimal Contracts (a)

For the manager to maximize her payoff, she needs to set the worker's payoff to zero, making the worker indifferent to accepting the contract or not. Set \(x - \alpha e^2 = 0\) which implies \(x = \alpha e^2\). The manager's problem is to maximize \(e - \alpha e^2\). Taking the derivative and setting it to zero:\[1 - 2\alpha e = 0 \Rightarrow e = \frac{1}{2\alpha}\]Substitute \(e\) into \(x\): \[x = \alpha \left(\frac{1}{2\alpha}\right)^2 = \frac{1}{4\alpha}\]

Calculate Equilibrium Contracts (b)

Substitute \(\alpha = \frac{1}{8}\) into our optimal solutions:\[\underline{e} = \frac{1}{2 \times \frac{1}{8}} = 4, \quad \underline{x} = \frac{1}{4 \times \frac{1}{8}} = 2\]Substitute \(\alpha = \frac{3}{8}\):\[\overline{e} = \frac{1}{2 \times \frac{3}{8}} = \frac{4}{3}, \quad \overline{x} = \frac{1}{4 \times \frac{3}{8}} = \frac{2}{3}\]

Incentive Compatibility with Private Information (c)

Given a menu of contracts \((\underline{e}, \underline{x})\) and \((\bar{e}, \bar{x})\), check if each worker type chooses the intended contract. Worker will select contract that maximizes payoff:- For low-type \(\alpha=\frac{1}{8}\), compare \(\underline{x} - \left(\frac{1}{8}\right)\underline{e}^2 \text{ vs. } \bar{x} - \left(\frac{1}{8}\right)\bar{e}^2\)- For high-type \(\alpha=\frac{3}{8}\), compare \(\bar{x} - \left(\frac{3}{8}\right)\bar{e}^2 \text{ vs. } \underline{x} - \left(\frac{3}{8}\right)\underline{e}^2\)This ensures each worker picks their respective contracts.

Incentive and Participation Constraints (d)

Derive conditions for each type to choose and participate:- Incentive Compatibility (IC) for low-type:\(\underline{x} - \frac{1}{8}\underline{e}^2 \geq \bar{x} - \frac{1}{8}\bar{e}^2\)- IC for high-type:\(\bar{x} - \frac{3}{8}\bar{e}^2 \geq \underline{x} - \frac{3}{8}\underline{e}^2\)- Participation Constraint (PC) for low-type:\(\underline{x} - \frac{1}{8}\underline{e}^2 \geq 0\)- PC for high-type:\(\bar{x} - \frac{3}{8}\bar{e}^2 \geq 0\)

Optimal Menu of Contracts (e)

Maximize manager's expected payoff:\[(1/2)(e_H - x_H) + (1/2)(e_L - x_L)\]Subject to IC and PC constraints. Use two binding conditions to solve for \(x_L\) and \(x_H\), then express payoff as a function of \(e_L\) and \(e_H\). Differentiate and solve:\[\text{\(x_L = \frac{1}{8}e_L^2\), (from low type PC) } \]\[\text{Substitute obtained \(x_H\), solve: } \]\[e_L = 2, e_H = \frac{5}{6}\]Express manager's payoff and find \(x_L\) and \(x_H\) using IC bindings.

Efficiency and Asymmetric Information (f)

The simulated contracts (e) deviate from efficient allocation (b) due to asymmetric information. The low-type effort \(e_L\) is efficient, but high-type \(e_H\) is reduced to prevent low-type from choosing the wrong contract, leading to lower total payoff compared to symmetric information.

Key concepts

Contract Theory

Contract theory is a branch of economics that studies how economic actors create contractual arrangements, often in scenarios where there are conflicting interests or incomplete information. In our scenario of manager-worker interaction, the essence of contract theory is seen in the agreement on an effort level \( \hat{e} \) and a wage \( \hat{x} \). Here, the manager's objective is to design a contract that maximizes her payoff while ensuring the worker's participation and motivation.

Contracts are crafted to align the incentives of both parties. This alignment becomes crucial, especially when there is verifiable effort and externally enforceable agreements. In practice, contract theory often explores how different contracts influence behavior, considering factors like the worker’s effort cost, given by \( \alpha e^2 \).

• Optimal contracts align worker effort with manager payoffs.
• Complex contracts might include contingencies based on worker performance.
• Contracts need to consider cost structures like effort cost \( \alpha e^2 \).

Manager-Worker Interaction

In the dynamics of a manager-worker interaction, the manager aims to incentivize the worker to exert effort on a project, while the worker assesses whether the offered compensation justifies their effort expenditure. The exercise demonstrates this by letting the worker's effort yield equivalent revenue, \( e \), to the firm, while the effort costs \( \alpha e^2 \) to the worker.

Such interactions are typically formalized through contracts, where the manager’s payoff is \( e-x \) and the worker's is \( x-\alpha e^2 \). The goal is to establish terms where the worker feels sufficiently incentivized to commit effort that aligns with the manager’s goals.

• The manager must balance payoff gains with incentive costs.
• Effective interaction requires transparency in cost and effort agreements.
• In the contract scenario, external enforceability ensures fair adherence to agreed terms.

Incentive Compatibility

Incentive compatibility focuses on ensuring that each party involved in a contract chooses actions that align with their best interests. In other words, the worker should naturally pick the contract designed for them without being swayed to choose an unsuitable contract. This becomes especially critical when dealing with different types of workers as seen with the high-type and low-type workers in our scenario.

The worker decides based on maximizing their payoff: the contract intended for their specific type should offer them greater utility than any alternative. For example, the low-type worker should find choosing their genuine contract more beneficial than selecting the high-type's contract. This is enforced using incentive compatibility conditions, such as:

• Low-type worker prefers \( (\underline{x}, \underline{e}) \) if \( \underline{x} - \frac{1}{8} \underline{e}^2 \geq \bar{x} - \frac{1}{8} \bar{e}^2 \).
• High-type worker prefers \( (\bar{x}, \bar{e}) \) if \( \bar{x} - \frac{3}{8} \bar{e}^2 \geq \underline{x} - \frac{3}{8} \underline{e}^2 \).

A well-designed incentive-compatible contract ensures that even when private information might skew decisions, each worker type acts in accordance with the contract meant for them.

Asymmetric Information

Asymmetric information occurs when one party in a transaction has more or better information than the other. In our manager-worker scenario, the uncertainty around the worker’s type - whether high-type or low-type - represents a classic case of asymmetric information. The manager only knows the probabilities of workers being one type or the other, but not the specific \( \alpha \) value for an individual worker.

This necessitates the creation of a menu of contracts, each designed to appeal to specific worker types despite the manager’s limited information. Asymmetric information can distort contracts from their efficient forms because the manager aims to prevent workers from "gaming" the system by choosing the more favorable contract.

• Asymmetric information leads to inefficiency by requiring a cautious approach in contract design.
• It encourages strategies such as offering a menu of contracts to identify worker types.
• Efficiency loss occurs as the manager balances incentives with type differentiation.

This scenario highlights the challenges of decision-making under imperfect information, where safeguarding against adverse selection is critical.