Question

A firm and worker interact as follows. First, the firm offers the worker a wage \(w\) and a job \(z ; z=0\) denotes the "safe" job and \(z=1\) denotes the "risky" job. After observing the firm's contract offer \((w, z)\), the worker accepts or rejects it. These are the only decisions made by the firm and worker in this game. If the worker rejects the contract, then he gets a payoff of 100 , which corresponds to his outside opportunities. If he accepts the job, then the worker cares about two things: his wage and his status. The worker's status depends on how he is rated by his peers, which is influenced by characteristics of his job as well as by random events. Specifically, his rating is given by \(x\), which is either 1 (poor), 2 (good), or 3 (excellent). If the worker has the safe job, then \(x=2\) for sure. In contrast, if the worker has the risky job, then \(x=3\) with probability \(q\) and \(x=1\) with probability \(1-q\). That is, with probability \(q\), the worker's peers think of him as excellent. When employed by this firm, the worker's payoff is \(w+v(x)\), where \(v(x)\) is the value of status \(x\). Assume that \(v(1)=0\) and \(v(3)=100\), and let \(y=v(2)\). The worker maximizes his expected payoff. The firm obtains a return of \(180-w\) when the worker is employed in the safe job. The firm gets a return of \(200-w\) when the worker has the risky job. If the worker rejects the firm's offer, then the firm obtains 0 . Compute the subgame perfect equilibrium of this game by answering the following questions.(a) How large must the wage offer be in order for the worker rationally to accept the safe job? What is the firm's maximal payoff in this case? The parameter \(y\) should be featured in your answer. (b) How large must the wage offer be in order for the worker rationally to accept the risky job? What is the firm's maximal payoff in this case? The parameter \(q\) should be featured in your answer. (c) What is the firm's optimal contract offer for the case in which \(q=1 / 2\) ? Your answer should include an inequality describing conditions under which \(z=1\) is optimal.

Short answer

(a) The wage must be \(100-y\) with firm payoff \(80+y\). (b) The wage must be \(100(1-q)\) with firm payoff \(100+100q\). (c) Risky job is optimal if \(y < 70\).

Step-by-step solution

Determine the Worker Payoff for Safe Job

If the worker accepts the safe job, his rating is always 2. Therefore, his payoff is determined by his wage offer plus the value of the status. The worker's payoff for accepting the safe job becomes: \(w + y\). For the worker to choose the safe job, this payoff must be at least equal to rejecting the offer, which earns him a payoff of 100. So, \(w + y \geq 100\). Thus, the wage offer \(w\) must be at least \(100 - y\).

Calculate the Firm's Payoff for Safe Job

If the firm successfully offers the safe job and the worker accepts, the firm receives a payoff of \(180 - w\). The firm's goal is to maximize this payoff while ensuring the worker accepts the offer with the least wage possible, which is \(w = 100 - y\). Therefore, the firm's maximal payoff is \(180 - (100 - y) = 80 + y\).

Determine the Worker Payoff for Risky Job

For the risky job, the worker's rating can be either 3 with probability \(q\) or 1 with probability \(1-q\). The expected payoff becomes \(w + q(100)\) because \(v(3) = 100\) and \(v(1) = 0\). The worker will accept the risky job if this expected payoff is at least 100. Therefore, \(w + 100q \geq 100\). Solving for \(w\), we find \(w \geq 100(1 - q)\).

Calculate the Firm's Payoff for Risky Job

The firm's payoff for the risky job is \(200 - w\). To offer the job at the least wage, set \(w = 100(1 - q)\). Thus, the firm's maximal payoff is \(200 - 100(1 - q) = 100 + 100q\).

Optimal Contract Offer for the Firm When \(q = \frac{1}{2}\)

Given \(q = \frac{1}{2}\), we compare the firm’s payoffs. The payoff for the safe job is \(80 + y\), and for the risky job, it is \(100 + 100 \times \frac{1}{2} = 150\). The firm will prefer the risky job if \(80 + y < 150\), which simplifies to \(y < 70\). Therefore, under this condition, offering the risky job is optimal.

Key concepts

Subgame Perfect Equilibrium

The concept of a subgame perfect equilibrium (SPE) is crucial in game theory.
It ensures that players make optimal decisions not just for the game as a whole, but for every part or subgame within it. In simpler terms, it's about taking the best possible actions at every step.
Imagine a game as a series of smaller games (subgames). Each subgame itself is a decision point. For a strategy to be a subgame perfect equilibrium, it must be optimal for each subgame.
In our exercise, the firm and the worker make decisions based on wages and jobs. To find the subgame perfect equilibrium, we address the game by "backward induction":

• First, decide the best response in the later stages of the game.
• Then work backwards to ensure every decision leads to optimal payoff.

For the firm, offering jobs with wages is the main decision. Solving to find which job to offer and at what wage can achieve the equilibrium where the worker's choice aligns with the firm's best interest.

Expected Payoff

Expected payoff is a critical concept when it comes to decision-making under uncertainty. It refers to the average value a player can expect to receive based on all possible outcomes of a game or decision.
In our exercise, the term arises when evaluating the worker's potential earnings from either the safe or risky job.For the worker:

• The safe job leads to a certain and straightforward payoff: wage plus status value.
• The risky job, however, involves uncertainty: a chance to be rated either 'excellent' or 'poor'.

Thus, expected payoff is calculated by evaluating the probabilities of these outcomes.The formula used: \[ \text{Expected Payoff (Risky)} = w + q \times v(3) + (1-q) \times v(1) \]Simplifying this based on given values results in:\[ \text{Expected Payoff (Risky)} = w + 100q \]The worker will select the option that maximizes his expected payoff, aligning his choice with a rational decision-making framework.

Contract Theory

Contract theory examines how individuals or entities manage relationships or agreements under conditions of incomplete information.
In the given exercise, this theory dictates the interactions between the firm and the worker. The objective here is crafting a contract—a job offer with a specific wage—that the firm presents to a worker:

• The worker evaluates the offer based on the provided wage and their acceptance maximizes their payoff.
• The firm is motivated to maximize profits while ensuring the worker accepts the offer.

The theory involves anticipating the worker's response to different wages and job types. Pricing the job accurately relies on understanding the worker's utility and outside options. By analyzing the conditions where a worker accepts a job (like wage ≥ outside opportunity value), contract theory supports designing optimal incentives for both parties.

Risk and Decision Making

Risk and decision making are intricately connected in the world of economics and game theory.
They revolve around making a choice that navigates uncertain outcomes effectively. In the exercise, the worker must decide between a safe or risky job offer.

• The safe job provides a guaranteed outcome with minimal risk.
• Conversely, the risky job carries potential for higher status, yet comes with uncertainty.

Evaluating risk involves considering probability and its impact on the worker's status and payoff. The key is to weigh the potential benefits against the possible downsides. Risk preference plays a crucial role:

• A risk-averse worker might shy away from potential failure, favoring the predictable safe job.
• In contrast, a risk-seeker might gamble on the chance of excellent status.

Effective decision making in such scenarios involves understanding one's risk tolerance and assessing expected payoffs to make informed, optimal choices.