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fin-401-05faa

# fin-401-05faa - 1 20th December 2005 Instructions Answer 7...

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1 20th December 2005 Instructions : Answer 7 of the following 9 questions. All questions are of equal weight. Indicate clearly on the fi rst page which questions you want marked. 1. Assume some consumer chooses bundle x j at prices p j , j = 1 , 2 , 3 , where p 1 = (1 , 1 , 2) ; x 1 = (5 , 19 , 9) p 2 = (1 , 1 , 1) ; x 2 = (12 , 12 , 12) p 3 = (1 , 2 , 1) ; x 3 = (27 , 11 , 1) . Show these data satisfy WARP. Is there an intransitivity in the revealed prefer- ences? Justify your answer. Cost of these bundles at these prices 1 2 3 Deductions 1 42 48 40 1 " 3 2 33 36 39 2 " 1 3 52 48 50 3 " 2 These preferences are consistent with WARP but clearly they violate transitivity. 2. Income and substitution e ff ects are an important part of the theory of consumer behaviour. Consider a price-taking consumer in a two-good world. Let m denote money income, ( p 1 , p 2 ) prices, x j ( p 1 , p 2 , m ) Marshallian demands, and h j ( p 1 , p 2 , u ) Hicksian demands. Write the Slutsky equation for the e ff ect of changing the price of good 2 on the demand for good 1 . Now suppose the person is endowed not with money income m but quantities of goods 1 and 2 ( e 1 , e 2 ) . Rewrite the Slutsky equation for the e ff ect of changing the price of good 2 on the demand for good 1 . x 1 ( p 1 , p 2 , m ) p 2 = h 1 ( p 1 , p 2 , u ) p 2 x 2 x 1 ( p 1 , p 2 , m ) m (1) With endowments x 1 ( p 1 , p 2 , p 1 e 1 + p 2 e 2 ) p 2 = x 1 ( p 1 , p 2 , m ) p 2 + e 2 x 1 ( p 1 , p 2 , m ) m . Using (1)

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2 x 1 ( p 1 , p 2 , p 1 e 1 + p 2 e 2 ) p 2 = h 1 ( p 1 , p 2 , u ) p 2 x 2 x 1 ( p 1 , p 2 , m ) m + e 2 x 1 ( p 1 , p 2 , m ) m = h 1 ( p 1 , p 2 , u ) p 2 + ( e 2 x 2 ) x 1 ( p 1 , p 2 , m ) m 3. Consider the two-state world of Andy and Brian. In the good state each has a wealth of 100 ; in the bad state each has a wealth of 50 , assuming they do not trade with each other. Let the probability of each state be 1 / 2 . The only way in which they di ff er is Andy is risk averse and Brian is risk neutral. Both are expected utility maximizers. Describe as precisely as you can the Pareto e cient allocations for these two people. Now suppose there are many Andys and an equal number Brians. Describe as precisely as you can the competitive equilibrium of this economy. Draw an Edgeworth rectangle with wealth in the good state along the horizontal axis and wealth in the bad state along the vertical axis. The dimensions of the rectangle are 200 by 100 and the endowment point is right in the middle. Put Andy’s origin in the lower left corner and Brian’s origin in the upper right corner. Andy is
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fin-401-05faa - 1 20th December 2005 Instructions Answer 7...

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