4. Assume police officers are a pure public good, that hiring officers has a total cost of Q, and that the opportunity cost of a dollar spent on police is one unit of a private consumption good r. Further, assume there are two consumers (a and d) who have different preferences but the same income of 1. Consumer a has the advantage of choosing the total number of police officers but must pay the entire cost. They have quasi-linear utility Ug = 1 + 1, and their individual optimization problem is max V+* s.t. 1 = Q.10 2 Consumer d is disadvantaged and does not have a say in policing levels but is exempt from contributing towards policing costs. They have utility Ud = 6 +&d and their individual optimization problem is max Id +xdst. 1 = rd (a) (2) How many police officers Q will consumer a hire? (b) (2) At a's choice of Q (found in part a) if consumer d could change the total 3 number of officers, would they prefer to increase Q, decrease Q, or leave Q unchanged? Explain why in terms of their preferences. (c) (2) State the problem of a utilitarian social planner choosing Q, 1a, and 3d to maximize U+Ud subject to the resource constraint that the total income of 2 equals the sum of police officer costs and private good consumption. Then take first order conditions and derive an equation for the optimal choice of Q. Hint: you won't be able to explicitly solve for Q, but Q should be the only unknown in your equation. (d) (1) Will the social planner's optimum have more or fewer police officers than the situation where consumer a chooses? Hint: isolate consumer a's marginal utility of Q and compare to the first order condition associated with their privately optimal choice. 4. Assume police officers are a pure public good, that hiring officers has a total cost of Q, and that the opportunity cost of a dollar spent on police is one unit of a private consumption good r. Further, assume there are two consumers (a and d) who have different preferences but the same income of 1. Consumer a has the advantage of choosing the total number of police officers but must pay the entire cost. They have quasi-linear utility Ug = 1 + 1, and their individual optimization problem is max V+* s.t. 1 = Q.10 2 Consumer d is disadvantaged and does not have a say in policing levels but is exempt from contributing towards policing costs. They have utility Ud = 6 +&d and their individual optimization problem is max Id +xdst. 1 = rd (a) (2) How many police officers Q will consumer a hire? (b) (2) At a's choice of Q (found in part a) if consumer d could change the total 3 number of officers, would they prefer to increase Q, decrease Q, or leave Q unchanged? Explain why in terms of their preferences. (c) (2) State the problem of a utilitarian social planner choosing Q, 1a, and 3d to maximize U+Ud subject to the resource constraint that the total income of 2 equals the sum of police officer costs and private good consumption. Then take first order conditions and derive an equation for the optimal choice of Q. Hint: you won't be able to explicitly solve for Q, but Q should be the only unknown in your equation. (d) (1) Will the social planner's optimum have more or fewer police officers than the situation where consumer a chooses? Hint: isolate consumer a's marginal utility of Q and compare to the first order condition associated with their privately optimal choice
