Suppose Rosario faces a two-period time-allocation problem. Rosario can allocate 10 hours of time in each period between two activities: work and family. The utility function for the first time period is

where x is the amount of time spent at work, 10 – x is the amount of time spent with family, and the subscript refers to time period. Thus the marginal utility of time at work in the first period is

In the second period, the utility function is given by

Marginal utility of time at work in the second period is given by

Thus, total utility for both periods is given by

Resulting in total marginal utility of work in period 1 of

and total marginal utility of work in period 2 of

The optimal allocation can be found by setting marginal utility of time at work in each period to zero. If Rosario brackets broadly, what will be the optimal allocation of time between work and family in both time periods? How will Rosario allocate her time if she displays melioration? What is the level of utility in each solution?

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