Question

Consider a single parent with one child who has preferences for housing (X) and other items (Y) represented by the utility function U(x,y) =x1/2y1/2. Currently, the parent works 40 hours a week and earns a monthly income of M=$2,000 and lives in a city where the unit price of housing is Px =$4 per square foot per month. The unit price of other items is Py=$1. 1. Find the parent's optimal bundle and label it (x0​;y0​). 2. Compute the parent's indirect utility at the optimal bundle V0​=U(x0​;y0​). The parent would like to reduce her number of hours to have more time to supervise her son when he is out from school. She is wondering whether by moving to another town where housing costs P'x =$2 per square foot per month she could afford to work parttime. We are interested in the minimum budget the parent would need in the new town to enjoy the same welfare (standard of living) she currently enjoys. 3. Write the parent's expenditure minimization problem. 4. Write the Lagrangian for this problem. 5. Compute the first order conditions for the Lagrangian. 6. Solve the first order conditions and find the cheapest bundle of housing and other items that in the new town would give the parent the same utility she currently enjoys (the results here might not be round numbers. Round up to the closest integer). 7. Evaluate the cost of this bundle. 8. If she moved to the new town could the parent afford to cut her hours to 30a week and earn only M′=1,200 per month without lowering her standard of living?

Answer 1

To find the single parent's optimal bundle, we utilize her utility function and budget constraint, and then use Lagrange multipliers to find the bundle and corresponding indirect utility. The next step is to minimize expenditures while maintaining the same utility level in a new town with lower housing costs. Finally, we analyze if the parent can work part-time while maintaining the same standard of living.

Step-by-step explanation:

The single parent's optimal bundle can be determined using her utility function and budget constraint. With a monthly income (M) of $2,000, the cost of housing (Px) at $4 per square foot, and other items (Py) at $1, she would aim to optimize U(x,y) = x1/2y1/2. We first set up the budget constraint 4x + y = 2000, then use the method of Lagrange multipliers to find the optimal bundle (x0, y0). The indirect utility at the optimal bundle, V0 = U(x0, y0), will indicate the level of utility achieved.

For the expenditure minimization problem in the new town with a housing cost (P'x) of $2 per square foot, we need to minimize the expenditure while keeping the utility at the level of V0. The Lagrangian for this problem is set up with the new budget constraint and the same utility level. The first order conditions derived from the Lagrangian enable us to solve for the cheapest bundle that yields the same utility as the original town. Finally, by evaluating the cost of the cheapest bundle and comparing it to a part-time income (M') of $1,200, we can determine if the parent can afford to work less and still maintain her standard of living.