Question

Utility maximization with more than two goods Suppose that there four goods Q, R, X and Y , available in arbitrary non-negative quantities (so the the consumption set is R 4 +). A typical consumption bundle is therefore a vector (q, r, x, y), where q ≥ 0 is the quantity of good Q, r ≥ 0 is the quantity of good R, x ≥ 0 is the quantity of good X, and y ≥ 0 is the quantity of good Y . The consumer has a total wealth of w = 12, and the each of the goods has a strictly positive price. In particular, let pQ = 1 be the price of good Q, pR = 1 be the price of good R, pX = 2 be the price of good X, and pY = 2 be the price of good Y . In the following three questions, you are asked to study the utility maximization problem for the consumer for three different utility functions. Your final solution for each part should indicate the quantity of goods Q, R, X, and Y that the consumer will purchase at the given prices and wealth when the consumer’s preferences can be represented by the respective utility function (i.e., utility function u1 for part 1, u2 for part 2, or u3 for part 3). Please carefully show the reasoning behind all of your answers.

(a) Suppose that the consumer’s preferences over consumption bundles can be represented by the following utility function: u1(q, r, x, y) = min{qr, xy}. Formulate and solve the consumer’s utility maximization problem.

(b) Instead, now suppose that the consumer’s preferences over consumption bundles can be represented by the following utility function: u2(q, r, x, y) = qr + xy. Formulate and solve the consumer’s utility maximization problem.

Answer 1

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