Consider the following model regarding hospital waiting times.

There are two public hospitalsi{A,B}. There is a continuum of patients with volume 1. A patient, indexed byj, seeks a treatment that can be provided by either hospital. Hospital A's quality of treatment isqh, and Hospital B's isql, withqh>ql>0. Assume that a proportion of patients know the hospitals' qualities, while the other 1- of patients are uninformed about the hospital quality. The uninformed patients hold the belief that each hospital's treatment quality is equally likely to be ofqhand ofql.

Each hospital's service rate is fixed ats>0.

A patientj's utility from receiving treatment at hospitalidepends on the treatment quality and her waiting timewij, with

Uj(qi,wij)=qi- wij.

Patients choose the hospitals simultaneously. Denote bynithe volume of patients at hospitali. After patients choose the hospitals, each patient's waiting timewijis a random draw from a uniform distribution on [0,ni/si]. We assume that patients choose hospitals only once, that is, they do not switch hospitals after knowing their waiting times.

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(1) In equilibrium, will all the uninformed patients choose Hospital A? If so, under what condition(s)? If not, why?

(2) In equilibrium, will all the uninformed patients choose Hospital B? If so, under what condition(s)? If not, why?

(3) In equilibrium, will all the informed patients choose Hospital A? If so, under what condition(s)? If not, why?

(4) In equilibrium, will all the informed patients choose Hospital B? If so, under what condition(s)? If not, why?