Consider a restaurant that serves only one type of meal at a price of $???? > 0. The restaurant needs waiters. There are two types of risk-neutral waiters: good waiters and bad waiters. Good waiters can serve 10 customers a day while bad waiters can only serve 5 customers a day. The social norm for a tip is 15%. Thus, waiters, good or bad, get 15% of $???? per customer. Note that good waiters do not get better tips (i.e., higher percentage). However, they get more dollars in tips because they serve more customers. Assume that the cost of working as a waiter, regardless of ability, is zero. Furthermore, good or bad waiters, get a net benefit (payoff) $???????? > 0 per day in their next best alternative job. In each day, numerous customers come to the restaurant, so finding customers to serve is not an issue. Suppose that an applicant accepts a job as a waiter at this restaurant if her payoff is at least equal to her payoff in her next best alternative job. Otherwise, she won’t work for the restaurant. Suppose the manager wants to attract only good waiters. However, when applicants arrive at the restaurant for jobs as waiters, the manager is unable to identify the ones who are good waiters and the ones who are bad waiters. To solve this problem, the manager decides to ask applicants the wage they want to be paid per day. (i) Construct a separating equilibrium such that good waiters ask for a wage ????? where W^min ? ????^? < W^max, bad waiters ask for a wage higher than or equal to W^max, and the manager only hires applicants who ask for ????? because he believes (correctly) that only good waiters choose a wage of ????? ? [W^min, ????^max). You should state the manager’s equilibrium beliefs and out-of-equilibrium beliefs. (ii) What is the manager’s profit-maximizing wage? Will this necessarily be the equilibrium wage? (iii) Can the manager achieve his goal of attracting only good waiters if he does not allow tipping? Explain carefully. (iv) Recall that good waiters do not get better tips (i.e., higher percentage of the bill). However, they get more dollars in tips because they serve more customers. Suppose instead that, as is usually the case, that good waiters have a higher probability than bad waiters of being tipped. Suppose instead that good and bad waiters can serve the same number of customers, ????, but good waiters have a higher probability of being tipped or getting a good tip. The price of a meal is still $????. Let a good tip be 15% and bad tip be 5%. Let pg be a good waiter’s probability of getting a 15% tip and 1 ? pg be her probability of getting a bad tip. Let the corresponding probabilities for a bad waiter be pb and 1 ? pb, where 0 < pb < pg < 1. Repeat the exercise in (i).
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