Question

Suppose that there two goods, X and Y , available in arbitrary nonnegative quantities (so the the consumption set is R 2 +). The consumer has preferences over consumption bundles that are monotone, strictly convex, and represented by the following (differentiable) utility function: u(x, y) = α √ x + (1 − α) √ y, where x is the quantity of good X, y is the quantity of good Y , and α ≥ 0 is a utility parameter. The consumer faces three different constraints. (1) A monetary constraint. It costs money to purchase the goods. The consumer has strictly positive wealth of w = 10 units (e.g., Dirhams); the price of good X is pX = 2 (Dirhams per unit of good X); the price of good Y is pY = 1 (Dirhams per unit of good Y ). (2) A time constraint. It takes time to purchase the goods. The consumer has a total endowment of time T = 10 (e.g., hours); it takes tX = 1 unit of time (hours) to purchase a unit of good X; it takes tY = 2 units of time (hours) to purchase a unit of good Y . (3) A space constraint. The consumer must transport goods with their vehicle, and the vehicle has only a limited amount of space. The consumer has a total space availability of S = 6 units (e.g., cubic meters); a unit of good X takes up sX = 1 units of space; a unit of good Y takes up sY = 1 units of space.

(a) In an appropriate diagram, illustrate each of the consumers three constraints, and indicate the consumers overall constraint set. Note that the consumer can only purchase a consumption bundle (x, y) if it satisfies all three constraints (i.e., the monetary, time and space constraint). Also, provide a formal description of the overall constraint set. Make sure you label diagrams clearly, and include as part of your answer any calculations about the slopes and intercepts of the three constraint lines.

(b) In an appropriate diagram, illustrate the consumers map of indifference curves. Describe how the indifference curves change as α changes. Make sure you label the diagram clearly, and include as part of your answer any calculations about the slopes and intercepts of the indifference curves.

(c) Formulate and solve the consumer’s utility maximization problem. The only parameter in this problem is α, and your final answer should describe the consumer’s demand for goods X and Y as a function of α. Finally, for what values of α does the consumer (i) spend all of their income, (ii) use up all of their available time, and (iii) use up all of the available space in their vehicle?

Answer 1

ㄡ 2 오ls .vsn (x184%、em y/.),冶ぐla t 2ぃer.リ.) 44 enau