1. In the context of the usual utility maximization problem involving n(>2) goods, prove that not all goods can be net complements and that at least one good must be normal.
2. Prove the envelope theorem and use it to derive Hotelling's lemma.
(a) Why lexicographic preferences do not yield a continuous utility function.
(b) The difference between cardinal and ordinal utility.
(c) Why profit maximization is not an assumption in the competitive general equilibrium model.
4. Average costs fall as output increases if and only if the production function is homogeneous of degree p>1. True or false? Explain your answers.
5. Consider the problems of maximizing utility u(x) subject to px ≤ y and minimizing expenditure px subject to u(x) ≥ u. Assuming that the utility function is continuous and strictly monotonic increasing and that solutions to both problems exist. Prove that utility maximization implies and is implied by expenditure minimization.
Terms such as lexicographic, cardinal/ordinal utility,envelope, Hotelling's lemma, and utility maximization are exemplified.