Assume police officers are a pure public good, that hiring Q officers has a total cost of 1/8 Q, and that the opportunity cost of a dollar spent on police is one unit of a private consumption good x. 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 Q but must pay the entire cost. They have quasi-linear utility

U_a = Q + x_a and their individual optimization problem is

max_Q+xa Q +x_a s.t. 1 = 1/8 Q +x_a

Consumer d is disadvantage and does not have a say in policing levels but is exempt from contributing towards policing costs. They have utility U_d = 1/Q+ x_d and their individual optimization problem is

max_xd 1/Q +x_d s.t. 1=x_d

(a) How many police officers Q will consumer a hire?

(b) At a's choice of Q (found in part a) if consumer d could change the total number of officers, would they prefer to increase Q, decrease Q, or leave Q unchanged? Explain why in terms of their preferences.

(c) State the problem of a utilitarian social planner choosing Q, x_a, and x_d to maximize U_a + U_d 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) 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.

Transcribed image text