Jim and Tammy are partners in Business and in Life. As is all too common in this imperfect world, each has a little habit that annoys the other. Jim’s habit, we will call activity X, and Tammy’s habit, activity Y. Let x be the amount of activity X that Jim pursues and y be the amount of activity Y that Tammy pursues. Due to a series of unfortunate reverses, Jim and Tammy have a total of only $1,000,000 a year to spend. Jim’s utility function is UJ = cJ + 500 ln x − 10y, where cJ is the money he spends per year on goods other than his habit, x is the number of units of activity X that he consumes per year, and y is the number of units of activity Y that Tammy consumes per year. Tammy’s utility function is UT = cT + 500 ln y − 10x, where cT is the amount of money she spends on goods other than activity Y , y is the number of units of activity Y that she consumes, and x is the number of units of activity X that Jim consumes. Activity X costs $20 per unit. Activity Y costs $100 per unit.
(a) Suppose that Jim has a right to half their joint income and Tammy has a right to the other half. Suppose further that they make no bargains with each other about how much activity X and Y they will consume. How much of activity X will Jim choose to consume? __________ How much of activity Y will Tammy consume? ___________
(b) Because Jim and Tammy have quasilinear utility functions, their utility possibility frontier includes a straight line segment. Furthermore, this segment can be found by maximizing the sum of their utilities. Notice that
UJ (cJ, x, y) + UT (cT , x, y)
= cJ + 500 ln x − 20y + cT + 500 ln y − 10x
= cJ + cT + 500 ln x − 10x + 500 ln y − 10y.
But we know from the family budget constraint that cJ + cT = 1,000,000 − 20x − 100y. Therefore we can write
UJ (cJ, x, y) + UT (cT , x, y) = 1,000,000 − 20x − 100y + 500 ln x − 10x + 500 ln y − 10y
= 1, 000, 000 + 500 ln x + 500 ln y − 30x − 110y.
Let us now choose x and y so as to maximize UJ (cJ, x, y) + UT (cT , x, y). Setting the partial derivatives with respect to x and y equal to zero, we find the maximum where x = ___________ and y = __________ If we plug these numbers into the equation UJ (cJ, x, y) + UT (cT, x, y) = 1,000,000 + 500 ln x + 500 ln y − 30x − 110y, we find that the utility possibility frontier is described by the equation UJ + UC = ______________ (You need a calculator or a log table to find this answer.) Along this frontier, the total expenditure on the annoying habits X and Y by Jim and Tammy is _______________. The rest of the $1,000,000 is spent on cJ and cT. Each possible way of dividing this expenditure corresponds to a different point on the utility possibility frontier. The slope of the utility possibility frontier constructed in this way is ___________