Question

Suppose Richard has the following utility function over leisure and consumption:

U(C, L) = C × (L − 56)

where C is units of consumption and L is hours of leisure consumption per week. Richard receives $100 in Welfare benefits per week. The price of a unit is equal to 1. There are 168 hours in a week.

(a) Richard receives $100 in Welfare benefits per week. The price of a unit of consumption is equal to $1 per unit of consumption. Determine if the following statement is either True or False and provide supporting evidence. A wage of $.90 per hour is sufficient to induce Richard to supply a non-zero quantity of labor per week.

(b) Suppose the wage is $10 per hour. Determine the optimal bundle of consumption and leisure as well as the number of hours worked.

Answer 1

(a) The budget constraint would be as , for price times C is the expenditure on consumption and w times H is the income form working H hours for $w wage, and pi is the non labor income. For the given values, we have or for L<168, and C=100 for L>168.

The slope of the utility curve would be as or or or . The slope of the budget line would be or . When both slopes are equated, we have the optimal combination of bundle as or or . Putting this in the budget constraint, we have or or or or . Now, a non zero quantity of labor would mean that we must have or or , ie or or or . This means that wage more than about $0.8928 would induce a positive labor supply per week, and hence, a wage of $0.9 per week will induce to supply a non-zero quantity of labor. Hence, the statement is True.

(b) We have the combination of optimal bundle as where , and for the given wage, it is where or or . Putting this in the budget constraint or , we have or or hours, and since , we have or or dollars. These are the required optimal bundles of consumption and leisure per week, for the given wage of $10.