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 : R3+ ?? R of the form

U (q1, q2, q3) = q1 + ?q2q3.

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 fifirst-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 demands3 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 qualifification conditions are

satisfified.)

5. What is the consumer's indirect utility function (assuming that a

strictly positive amount of each commodity is optimally purchased)?

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 : R1 > R of the form U (91: (1'2: 93) = 9'] + V 92%}- 1Marshallian demands are aLso known as W'alrasian demands, ordinary demands, and uncompensated demands. 2l't'Iarshallian demands are aLso known as \"Talrasian demands, ordinary demands, and uncompensated demands. The constant per unit price of commodity one is In E (0, 00), the constant per unit price of commodity two is p2 E (D, on), and the constant per unit price of commodity three is 333 E (0, oo). The consumer has an income of y E (0, on). 1. What is the consumer's budgetcoast rained utility maximisation prob lem'? 2. What is the Lagrangean function for the consumer's budgetconstrained utility maximisation problem? 3. (What are the rstorder conditions for the consumer's budgetconstrained utility maximisation problem (assuming; that a strictly positive amount of each commodity will optimally be purchased)? 4. 1What are the consumer's Marshallian demands3 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 It-'Iarshallian demand mapping is a. function or a cor respondence. (You may assume that appropriate seconderder con ditions for a maximum and constraint qualication conditions are satised.) 5. What is the consumer's indirect utility function (assuming; that a strictly positive amount of each commodity is optimally purchased)