Consider a city. There are two types of houses in the city. Houses of type R where rich people live in, and houses of type p where poorer people live. There are many burglars in the city. In case of burglary, the owners of rich houses lose the value V = 100 while the owners of poor houses lose the value V = 40. We suppose burglars are unable to tell from the outside whether a house is rich or poor. Therefore, houses have the same probability p of being robbed. With probability p, the owner's net income is - Vif she is rich or-V if she is poor. With probability 1-p, net income is o whether the owner is rich or poor. Individuals have the same utility function, u(x) =3 (200+x) 1/2, where x denotes net income. House owners can install an alarm system at a cost c=30. With an alarm system, the burglar is arrested half of the time. In other words, with an alarm system, the owner's net income is - V -c if she is rich (or - V-c if she is poor) with probability p/2, and -c with probability 1 - p/2. R R P a. Write the expected utility of the rich and poor owners, with and without an alarm system. Show that rich owners install the alarm if p > 0.18. Show that poor owners install the alarm if p 20.5. Explain the difference between the two answers. Alarm systems are visible from the outside. Burglars can use this information to guess the type of house! b. Suppose that only rich houses have an alarm system. Consider a risk-neutral burglar who has to pay K if she is caught. If she burgles a poor house, she is certain to obtain V p If she burgles a rich house, she has a probability 1/2 of getting V, and a probability 1/2 of being arrested and getting - K Find the condition on K for burglars to prefer to target poor houses if all rich houses have an alarm system. Explain the trade-off between stealing rich people and being caught. The probability of burglary is p for both types of owners when nobody installs an alarm system, or when everyone installs an alarm system. When rich and poor owners do not make the same decisions, burgla