Alia and Bob have just terminated their marriage. They have agreed to let Carol raising their only child, Daniel. The two parents hold no animosity toward one another, and each is intensely concerned about little Daniel's welfare. Their preferences are described by the utility functions:

(, ) =

and (, ) =

where y_A and y_B denote thousands of dollars "consumed" directly by the respective parents in a year, and x denotes thousands of dollars per year consumed by Daniel. Daniel's consumption is simply the sum of the support contributions from his mother and father. These contributions will be voluntary: neither parent has sought a legal judgment against the other. Assume throughout that = 1/4 and = 1/3. The marginal cost of X is 1.

a) Suppose Daniel's mother is unable to contribute anything toward Daniel's support, so that Bob must provide, out of his RM40 thousands annual income, for both his own consumption, y_B, and Daniel's consumption, x. Express Bob's budget constraint both analytically, and diagrammatically. Determine Bob's marginal rate of substitution between x and y_B at the choice he will make, and draw a diagram representing his choice problem. What levels of x and y_B will Bob choose?

b) Determine the two equations (viz., the marginal condition and the "on-the-constraint" condition) that characterize the Pareto optimal allocations if Alia and Bob both contribute. Alia's income is RM48 thousands and Bob is RM40 thousands. Find x