Consider a consumer whose preferences over bundles of strictly positive amounts of each of three distinct commodities can be represented by a utility function U : R. -> R of the form U (91, 92, 93) = 91 + 19293. The constant per unit price of commodity one is pie (0, co), the constant per unit price of commodity two is p, e (0. co), and the constant per unit price of commodity three is p3 6 (0, co). The consumer has an income of ME (0.20). 1. What is the consumer's budget-constrained utility maximisation prob- lem? 2. What is the Lagrangean function for the consumer's budget-constrained utility maximisation problem? 3. (What are the first-order conditions for the consumer's budget-constrained utility maximisation problem (assuming that a strictly positive amount of each commodity will optimally be purchased)? 4. What are the consumer's Marshallian demands for each of the com- modities (assuming that a strictly positive amount of each commod- ity is optimally purchased)? In the case of each commodity, identify whether the Marshallian demand mapping is a function or a cor- respondence. (You may assume that appropriate second-order con- ditions for a maximum and constraint qualification conditions are satisfied.) 5. What is the consumer's indirect utility function (assuming that a