# Using utility function to solve for optimal choice

Suppose that the typical consumer has the following utility function:

U(N, Y) = NÃ—Y,

where Y = income or expenditures on goods, and N = leisure (non-work) hours. The wage rate is given by w = $10. The consumer is initially taxed at the proportional rate of t1 = .4. The consumer has no unearned income (Y* = 0). The time constraint is given by

24 = N + H,

where H is hours of work.

Solve for the optimal choice. Graph this solution. How many hours of work is the consumer working? What is her income?

Can you show me the steps needed to solve the this problem?

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#### Solution Preview

First, we need to know the marginal rate of substitution (MRS) between leisure and income for this consumer. The MRS describes how much income is worth in terms of leisure. It is the slope of the indifference curve.

We obtain the MRS of substitution by differentiating the utility function. If is U =NY then we differentiate with respect to leisure to get MU(N) = Y and differentiate with respect to income ...

#### Solution Summary

Given a utility function and time constraint, the ideal work/leisure combination is found.