01 May Question: Please answer the following question. Please show all your work/explanation. This problem provide…
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.
- Write the maximization problem of the household assuming prices PN and PH and wealth W.
- Recognizing that this is a constrained optimization problem, write the Lagrangian and let λ denote the lagrange multiplier.
- Differentiate the Lagrangian with respect to N and H to obtain the first order conditions for an optimum with respect to N and H.
- 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.
- 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?
- 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?
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