There are two goods, X and Y, available in arbitrary non-negative quan- tities (so the consumption set is R2).
Beenish has preferences over consumption bundles that can be represented by a differentiable, strictly increasing and strictly quasi-concave utility function u: RR+, given by
u(x, y) = xyl-a
where x is the quantity of good X, y is the quantity of good Y, and a € (0,1) is a preference parameter.
1. Suppose Beenish has 80 AED. In a nearby store (called "Store A"), the price of good X is p= 2 AED per unit of good X, and the price of good Y is q = 8 AED per unit of good Y. However, Beenish also has a time constraint. At Store A, it takes tx= 4 minutes per unit to purchase good X and ty = 1 minutes per unit to purchase good Y. That means, purchasing a consumption bundle (x, y) ER at Store A would take 4x + y minutes. Beenish has only 80 minutes to do her shopping.
(a) In an appropriate diagram, illustrate Beenish's constraint set, given that she has both a monetary constraint and time constraint. (b) Describe Beenish's demand for good X and her demand for good Y as a function of her preference parameter a € (0,1), and illustrate her demand for good X, i.e., (a), in an appropriate diagram. (e) For what values of a does Beenish not spend all of her wealth? For what values of a does Beenish spend all of her wealth and use up all of her available time?
2. Now suppose that there is another store (called "Store B"), which is further away, where Beenish could do her shopping instead. At Store B, the goods are half-price: good X only costs 1 AED per unit and good Y only costs 4 AED per unit. At Store B, it still takes tx = 4 minutes per unit to purchase good X and ty= 1 minutes per unit to purchase good Y. However, it also takes 40 minutes to drive to Store B and back, and so Beenish's total time for shopping is only 40 minutes if she travels to Store B.
(a) In an appropriate diagram, illustrate Beenish's constraint set, given that she has both a monetary constraint and a time constraint but now has the option between two stores.
(b) Describe Beenish's demand for good X and her demand for good Y as a function of her preference parameter a, and illustrate her demand for good X, i.e., x(a), in an appropriate diagram.
(c) For what values of a does Beenish shop in Store A, and for what values of a does she travel to Store B?
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