Answers_to_HW_3

Answers_to_HW_3 - Fall 2009 Truman Bewiey Economics 359a...

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Unformatted text preview: Fall 2009 Truman Bewiey Economics 359a Answers to Homework #3 (Due Thursday, October 1) Problem: 3) For each of the two following Edgeworth box examples, catculate and draw an accurate picture of the set of ' _ N N : V—{(uA(xA),uB(xB))|x ER+,X eR+,x +xB eA+eB)} A B A and of “it: {(v , v) 1 there is a feasibie altocation (x , x ) such thatv s u (x ) and A B A B A A A VB 3 uB(xB) }. a)e =(1,0),e =(O,1),u(x,x)=3x +x,u(x,x)=x +3x. A B A! 2 1 2 B12 1 2 b) eA = (1, 0), e8 =(0, 1), uA(x1, x2) = min(x1, 2x2), uB(x1, x2) = min(2x1, x2). 0) eA= (1, 0), eI3 =(0, 1), uA(x1, x2) = x1x2, uB(x1, x2) = x1+ 3x2. Answer: a) . ‘ A 62L KL“v\Mrlwwmm...u..—.Ilfl<7W\;7<.,:y vwrmmwafirzmfirmxmwmtmaw mmmzmwwsmmmmwmmwwr; :1 s i; t 3 w m w wwwwwmwmwmmummzmcw In the Edgeworth box to the left above, the heavy lines mark the set of Pareto optimal allocations. The box maps into the dotted area V at the right in utility space. The point 0A in the box maps to the point (0, 4) in utility space, the point 0 maps to the point (1, 1), the point OB maps to the point (4, O), and the point D in the box maps to the point (3, 3). Roughly speaking, the utility vector mapping rotates the Edgeworth box counter clockwise by 90 degrees, squeezes it into the diamond shape and places it on the diamond to the right. The set “it is V together with everything to the southwest of it. Answer: b) 0 2i3 Iiiijiliiiii..(2r3.2/3) wwwwammmmwmw» g i. 1' w«wmm~mmmmm ww—wflnmmmmm In the Edgeworth box above, the regions shaded with dots form the set of Pareto optimal allocations and the dashed tines are indifference curves. The dotted area in the bottom figure is the image of the box in utility space, the set V. The point E maps to the point (2/3, 2/3) in utility space. The point 0A maps to the point (0, 1). The point 0I3 maps to the point (1, 0). The points C and D map to the point (0, 0) in utility space. The left and bottom edges of the box map to the vertical axis from (O, 0) to (O, 1). The right and top edges map to the horizontal axis from (O, 0) to the point (1, O). The shaded region from GA to E maps to the line from (0, i) to (2/3, 2/3) in utility space. The shaded region from E to OB maps to the line from (2/3, 2/3) to (1, 0) in utility space. The set 9L is V together with everything to the southwest of it. Answer: 0) In the Edgeworth box above, the heavy lines mark the set of Pareto optimai allocations andthe parallel straight lines are indifference curves for person B. One curved indifference curve is shown for person A. The point C has coordinates (t, 1/3) from the point of view of the origin for person A. E: E i t.A9am..u.MWmWWWmmmwammmmmmmmwwywmwwm. PW...»WWW...mmmtmmxmaWWnWlmmmmwmwwwwmmnmmmmmmmm % C Wy/fl‘ fl)“ The image of the Edgeworth box is the dotted area V in the above diagram. The point 0 A in the Edgeworth box maps to the point (0, 4) in utility space. The point C maps to the point ( 5/3, 2) in utility space, which is Iabeted again as C. The line from 0A to C in the box maps to the straight tine from (O, 4) to (43/3, 2) or C in utility space. The right-hand side of the box maps to the curve vB =3 — 3v: in utility space going from the point (0, 3) through the point ( 1,313, 2) or C to the point (1, O). The top edge of the box maps to the curve VB = 1 — v: going from the point (1, 0) to the point (0, 1) in utility space. The bottom edge of the box maps to the straight line going from the point (0, 4) to the point (0, 3) along the vertical axis in utility space. The left-hand edge of the Edgeworth box maps to the straight line from the point (0, 4) to the point (0, 1) along the vertical axis in utility space. The utility possibitity set 61L is V together with everything to the southwest of V. In order to calcutate the slanted straight line of Pareto optima in the Edgeworth box, use the equations x au lax au lax A2 = _—_ A 1 = B 1 = l, x at: lax au 16): 3 A1 A 2 B 2 so that x 2 ix . A2 3 A1 In order to calculate the curve vB = 3 — 3v} notice that along the line from D to 0B in the box, the utility of person A is v = ix”, and the utility of person 8 is A v : 3x = 3(1 —x ) = 3 —3v2. Similarly to calculate the curve vB=1 —v§, notice that B 82 A2 A along the top edge of the box the utility of person A VA ll IX” and the utility of person B is v =x =1-x =1—v2. B B1 A1 A In seeing what happens to points on the left-hand and bottom edges of the box, notice that person A has zero utility at every such point, so that it is only necessary to calculate the interval over which the utitity of person B varies. ’Z 1,; E if 9:; g E t g E E i WWW mm -vWWWMmm“mewmmmmwnmumtmmwmmm ...
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This note was uploaded on 09/12/2011 for the course ECON 350 at Yale.

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Answers_to_HW_3 - Fall 2009 Truman Bewiey Economics 359a...

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