Marital bliss

Emily and Derek are married but have no children. Emily makes all the decisions about how to spend their (exogenous) family wealth (Y), specifically how purchases will be divided between the two. There are, however, many commodities that she may choose to buy for herself and Derek. Emily chooses purchases in such a way that Derek's utility is at least as great as ud. (Her best friend Shelly has promised Derek that he would attain that level of utility if he left Emily for her.) You should further assume that all household purchases are rival and excludable, so that Derek does not benefit from Emily's purchases on herself and conversely.

a. Show that a family version of Roy's identity holds for Emily's indirect utility function.

Suppose that Emily also decides how many hours she will work at the wage rate w, and her preferences can be represented by the utility function u = E•L, where E is her monetary expenditure on herself and L is the number of hours of her leisure time. That is, Emily no longer cares about the composition of purchases on herself but only on the total level. Assume that Derek does not value Emily's leisure.

b. Prove mathematically that, so long as she is at an interior solution, Emily will choose to work harder if her friend promises to make Derek happier should he leave Emily.