Question

Please answer the following question. Please show all your work/explanation.

This problem provides a mathematical example of the previous problem and asks you to solve the household’s problem using the method of Lagrange Multipliers. Suppose the household’s preferences over Non-Durables and Health Services are given by

U(N, H) = a log(N) + (1 − a) log(H)

where 0 < a < 1, N is non-durable consumption and H is health services.

1. Write the maximization problem of the household assuming prices P_N and P_H and wealth W.

2. Recognizing that this is a constrained optimization problem, write the Lagrangian and let λ denote the lagrange multiplier.

3. Differentiate the Lagrangian with respect to N and H to obtain the first order conditions for an optimum with respect to N and H.

4. Combine your FOCs (First Order Conditions) to obtain an expression relating the optimal choice of Non-durables and Health services to the prices of Non-durables and Health services.

5. Use your answer to the previous part and calculate the household’s health expenditure as a fraction of its total expenditure. What does this ratio depend on?

6. Now suppose that the household’s preferences are given by

   V (N, H) = a log(N) + (1 − a) log(H + b)
   where b > 0. Solve for the household’s health expenditure as a fraction of its total expenditure. How does your answer change relative to part 5?

Answer 1