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.

1Marshallian demands are also known as Walrasian demands, ordinary demands, and

uncompensated demands.

2Marshallian demands are also known as Walrasian demands, ordinary demands, and

uncompensated demands.

3The constant per unit price of commodity one is p1 (0, ), the constant

per unit price of commodity two is p2

(0,), and the constant per unit

price of commodity three is p3 (0,

). The consumer has an income of

y (0,).

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)?