Consider a consumer that faces the following budget-constrained utility maximisation problem: Maximise U (91, 92, 93)...

 

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Consider a consumer that faces the following budget-constrained utility maximisation problem: Maximise U (91, 92, 93) = 91 + ln (9293) subject to p1q1+P292 +P393 ≤y, 91 > 0, 92>0, and 93 > 0, where P1 is the price of commodity one, p2 is the price of commodity two, P3 is the price of commodity three, and y is the consumer's income. The choice variables in this problem are 91, 92, and 93. You may assume that this consumer will optimally choose to exhaust his or her budget throughout this question. You may also assume that 0 < p < 2p1 < y <∞, 0 < P2 <∞, and 0 <p3 <∞o. Note that these inequality restrictions on the economic parameters of this problem ensure that it has a unique solution that involves the consumer choosing to purchase strictly positive amounts of all three commodities. 1. What conditions characterise an optimal consumption bundle for this consumer? Justify your answer. 2. What are the consumer's Marshallian demand functions (or corre- spondences) for each of the three commodities?2 Justify your answer. 3. What is the consumer's indirect utility function? Justify your an- swer? Consider a consumer that faces the following budget-constrained utility maximisation problem: Maximise U (91, 92, 93) = 91 + ln (9293) subject to p1q1+P292 +P393 ≤y, 91 > 0, 92>0, and 93 > 0, where P1 is the price of commodity one, p2 is the price of commodity two, P3 is the price of commodity three, and y is the consumer's income. The choice variables in this problem are 91, 92, and 93. You may assume that this consumer will optimally choose to exhaust his or her budget throughout this question. You may also assume that 0 < p < 2p1 < y <∞, 0 < P2 <∞, and 0 <p3 <∞o. Note that these inequality restrictions on the economic parameters of this problem ensure that it has a unique solution that involves the consumer choosing to purchase strictly positive amounts of all three commodities. 1. What conditions characterise an optimal consumption bundle for this consumer? Justify your answer. 2. What are the consumer's Marshallian demand functions (or corre- spondences) for each of the three commodities?2 Justify your answer. 3. What is the consumer's indirect utility function? Justify your an- swer?

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To solve the consumers budgetconstrained utility maximization problem we can use the Lagrange multiplier method Lets denote the Lagrange multiplier associated with the budget constraint as 1 To find t... View the full answer