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Unformatted text preview: Fall 2008 Truman Bewiey Economics 155a
Answers to Homework #3 (Due Thursday, October 2) Problem: 3) For each of the two following Edgeworth box examples, calculate and draw an
accurate picture of the set of _ N N __
V~{(uA(xA),uB(xB)) x eR+,xBeR+,x +xB—e +eB)} A A A 3% r
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:3 and of CM: {(v , v ) i there is a feasible allocation (x , x ) such that v s u (x ) and
A B A B A A A warmers mylRKMK rmmxmmmv VBS uB(xB) }. a) eA= (1, 0),eB= (0, 1), uA(x1, x2) =3X1 + x2,uB(x1,x2) =x1+3x2. b) eA = (1, 0), e13 =(0,1),uA(x1,x2) = min(x1, 2x2), uB(x1, x2) = min(2x1, x2). wm‘wmmwwmw' H
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x v H c) e =(1,0),eB=(O,1),uA(x,x) x+3x.
A 12 12 B1 1 Answer: a) ammuwm“rmmamWm* 'r{wmrwmazn‘rmerrwéawAz7.1mm»; mnwv «filmvﬂmiwmuﬁvwwolw'vw’a 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 C maps to the point (1, 1), the point 08 maps to the point (4, 0), and the point D in the box maps to the point (3, 3). Roughly speaking,
the utility vector mapping rotates the Edgewonh box counter clockwise by 90 degrees, squeezes it into the diamond shape and places it on the diamond to the right. The set GIL is V together with
everything to the southwest of it. Answer: b) 0 2/3 ..... (2/3. 2/3) in the Edgeworth box above, the regions shaded with dots form the set of Pareto optimal
allocations and the dashed lines 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 t)A maps to the point (0, 1). The point 0B 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 verticai axis from (0, 0) to (0, 1). The right and top edges map to the horizontal
axis from (0, O) to the point (1, 0). The shaded region from GA to E maps to the line from (O, 1) to (2/3, 2/3) in utility space. The shaded region from E to 0B maps to the line from (2/3,
2/3) to (t, 0) in utility space. The set GM is V together with everything to the southwest of it. Answer: 0) 0
A In the Edgeworth box above, the heavy lines mark the set of Pareto optimal allocations
and the parallel straight lines are indifference curves for person B. One curved indifference
curve is shown for person A. The point C has coordinates (1, 1/3) from the point of view of the
origin for person A. 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 labeled 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 utiiity space. The righthand side of the box maps to the curve VB = 3 — 3v: in utility space going from the point (0, 3) through the point ( 43/3, 2) or C to the point (1, 0). 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 utiiity 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 tefthand 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 utiiity space. The utitity possibility set 62L
is V together with everything to the southwest of V. in order to calculate the slanted straight line of Pareto optima in the Edgeworth box, use
the equations E:
a;
a;
1: meurart In order to catculate the curve vB = 3 — 3v: notice that along the line from D to GB in the box, the utility of person A is v = [x , and the utility of person B is
A2 A v = 3x = 3(1 —x ) = 3 —3v2. Similariy to calculate the curve v :1 —v2, notice that
B 92 A2 A B A along the top edge of the box the utility of person A vA = [XM 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 lefthand and bottom edges of the box, notice that
person A has zero utility at every such point, so that it is onty necessary to calculate the
interval over whEch the utility of person B varies. :35
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