A draw the budget set budget c l 100 l 9 1000 10 l b

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a ) Draw the budget set. Answer: 0 50 100 Budget Set 0 50 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c l 100 - l/ 9 1000 - 10 l b ) Does the consumer’s problem have a solution? That is, can utility be maximized over the budget set? Explain. Answer: Yes, the consumer’s problem has a solution. It suffices to show the budget set is compact since the Weierstrass Theorem will imply there is a solution. The budget set is the interesection of four sets: { ( c, l ) : l 0 } , { ( c, l ) : c 0 } , { ( c, l ) : c 1000 - 10 l } , and { ( c, l ) : c 100 - l/ 9 } . Each of these sets is the inverse image of a closed interval under a continuous function, and so each is closed. As the intersection of closed sets, the budget set is closed.

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MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 8, 2001 Page 3 Since l 0, c 100 - l/ 9 100. Since c 0, 10 l 1000 - c 1000, so l 100. The budget set is contained in [0 , 100] × [0 , 100] (see the diagram), and thus is bounded. Since it is both closed and bounded, it is compact. By the Weierstrass Theorem, the consumer’s problem has a solution. Note: Polygonal budget sets of this type arise when consumers face progressive taxation, where the number of sides depends on the number of tax brackets. The phase-out of welfare benefits as income increases has a similar effect.
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