Exercises PT4 - Exercises PT4[ Ch30:Exchange , quantities

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Exercises PT 4 [Please do not distribute]
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Ch 30: Exchange Some problems ask you to find a competitive equilibrium.  For an economy with two goods, the following procedure is  often a good way to calculate equilibrium prices and  quantities. Since demand for either good depends only on the ratio of  prices of good 1 to good 2, it is convenient to set the  price of good 1 equal to 1 (i.e. the  numeraire ) and let  p 2   be the price of good 2. With the price of good 1 held at 1, calculate each  consumer’s demand for good 2 as a function of  p 2 . Write an equation that sets the total amount of good 2  demanded by all consumers equal to the total of all  participants’ initial endowments of good 2. Solve this equation for the value of  p 2  that makes the  demand for good 2 equal to the supply of good 2. (When the  supply of good 2 equals the demand of good 2, it must also  be true that the supply of good 1 equals the demand for  good 1.) Plug this price into the demand functions to determine  quantities.
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30.1 Morris and Philip consume wine and books. Morris has an  initial endowment of 60 books and 10 bottles of wine.  Philip has an initial endowment of 20 books and 30 bottles  of wine. They have no other assets and make no trades with  anyone other than each other. For Morris, a book and a bottle of wine are perfect  substitutes. His utility function is  U ( b,w ) =  w , where  is the number of books he consumes and  is the number  of bottles of wine he consumes. Philip’s preferences are more subtle and convex. He has a  Cobb-Douglas utility function,  U ( b,w ) =  bw . (a)  In the Edgeworth box with Morris’s consumption  measured from the lower left and Philip’s measured from  the upper right corner of the box, mark the initial  endowment and label it  E . Use red ink to draw Morris’s  indifference curve that passes through his initial  endowment. Use blue ink to draw in Philip’s indifference  curve that passes through his initial endowment. (Remember  that quantities for Philip are measured from the upper  right corner, so his indifference curves are “Phlipped  over.”)
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(b)  At any Pareto optimum, where both  people consume some of each good, it must  be that their marginal rates of  substitution are equal. What is Morris’s  marginal rate of substitution? What is  Philip’s MRS when he consumes the bundle,  ( b P ,w )? What does this tell you about the  Pareto equilibrium? Use black ink to draw  the locus of Pareto optimal allocations.
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