A waiter can provide either good or poor service. Providing good service costs the waiter of $12, while poor service has no effort cost ($0). The social custom is that waiters get tipped $15 for good service and $0 for poor service (but there is no legal obligation to pay the tip). There are two types of diners in the economy. Type 1 customers have a $20 disutility of not paying the tip. Type 2 customers have a $10 disutility of not paying the tip. The waiter cannot distinguish which type of customer someone is until after they have paid the bill. In this economy, what is the minimum proportion of Type 1 diners required in order to induce the waiter to provide good service in the subgame perfect equilibrium? Write your answer as a proportion (between 0 and 1) and round up to one decimal place.
Question
A waiter can provide either good or poor service. Providing good service costs the waiter of 0). The social custom is that waiters get tipped 0 for poor service (but there is no legal obligation to pay the tip). There are two types of diners in the economy. Type 1 customers have a 10 disutility of not paying the tip. The waiter cannot distinguish which type of customer someone is until after they have paid the bill.
In this economy, what is the minimum proportion of Type 1 diners required in order to induce the waiter to provide good service in the subgame perfect equilibrium? Write your answer as a proportion (between 0 and 1) and round up to one decimal place.
Solution
To solve this problem, we need to understand the waiter's decision-making process. The waiter will provide good service if the expected benefit of doing so is greater than the cost.
The cost of providing good service is $12.
The benefit of providing good service depends on the type of customer. If the customer is Type 1, the waiter will receive a tip of 10) is less than the tip itself ($15).
Let's denote the proportion of Type 1 customers as p. The expected benefit of providing good service is therefore $15*p.
The waiter will provide good service if 12. Solving this inequality for p gives p > 15 = 0.8.
Therefore, the minimum proportion of Type 1 customers required to induce the waiter to provide good service in the subgame perfect equilibrium is 0.8, or 80%.
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