Bonnie and Clyde are business partners. Whenever they work, they have to work together. Their only source of income is profit from their partnership. Their total profit per year is 50H, where H is the number of hours that they work per year. Since they must work together, they both must work the same number of hours, so the variable “hours of labor” is like a public “bad” for the two person community consisting of Bonnie and Clyde. Bonnie’s utility function is UB(CB,H) = CB −.02H2 and Clyde’s utility function is UC(CC,H) = CC −.005H2, where CB and CC are the annual amounts of money spent on consumption for Bonnie and for Clyde.
(a) If the number of hours that they both work is H, what is the ratio of Bonnie’s marginal utility of hours of work to her marginal utility of private goods? ___________ What is the ratio of Clyde’s marginal utility of hours of work to his marginal utility of private goods? __________
(b) If Bonnie and Clyde are both working H hours, then the total amount of money that would be needed to compensate them both for having to work an extra hour is the sum of what is needed to compensate Bonnie and the amount that is needed to compensate Clyde. This amount is approximately equal to the sum of the absolute values of their marginal rates of substitution between work and money. Write an expression for this amount as a function of H. __________ How much extra money will they make if they work an extra hour? _________
(c) Write an equation that can be solved for the Pareto optimal number of hours for Bonnie and Clyde to work.
Find the Pareto optimal H.