Problem IV. Firefighters (30 points) John and Maria are the only two residents of Econvil- lage. Both get utility on their private consumption good, c, and the total number of firefighters 2 hired for Econvillage, F = f1 + fn. Specifically, John's utility is u = 4 In(c) + In(F) and Maria's utility is um = 4 In(cm) + In(F). So their utilities are identical. Each of them has an income of $90. Price of the consumption good and price/wage of each firefighter are both equal to $1. 1. How many firefighters are hired if the government does not intervene? How many are paid for by John? By Maria? Be clear about the problem setup for both persons including the choice variables, objective function, and the constraints. (7 pts) 2. What is the socially optimal number of firefighters? Be clear about the social planner's problem setup including the planner's choices, objective function, and the constraint. (7 pts) 3. Compare the social optimum and the private optimum. How and why do they differ? What type of problem exists there? (4 pts) 4. How would the situation change if John and Marin were a family together i.e. they had perfect altruism towards each other? Just explain the intuition. Math is not required here. (5 pts) Problem IV. Firefighters (30 points) John and Maria are the only two residents of Econvil- lage. Both get utility on their private consumption good, c, and the total number of firefighters 2 hired for Econvillage, F = f1 + fn. Specifically, John's utility is u = 4 In(c) + In(F) and Maria's utility is um = 4 In(cm) + In(F). So their utilities are identical. Each of them has an income of $90. Price of the consumption good and price/wage of each firefighter are both equal to $1. 1. How many firefighters are hired if the government does not intervene? How many are paid for by John? By Maria? Be clear about the problem setup for both persons including the choice variables, objective function, and the constraints. (7 pts) 2. What is the socially optimal number of firefighters? Be clear about the social planner's problem setup including the planner's choices, objective function, and the constraint. (7 pts) 3. Compare the social optimum and the private optimum. How and why do they differ? What type of problem exists there? (4 pts) 4. How would the situation change if John and Marin were a family together i.e. they had perfect altruism towards each other? Just explain the intuition. Math is not required here. (5 pts)