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RelativeResourceManager - Mathematical Economics 2‘“i...

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Unformatted text preview: Mathematical Economics 2‘“i ed. Baldani, Bradfield, and Turner Chapter 5 Answers Page 1 Answers to Problems from Chapter 5 5.1 (a) f(x,y)=auc2 +bxy+cy2 fx=2ax+by flmbx+2cy fxx=2a fxy “f” m f =20 fm=fm=fm=fm=fm=fm=fm=fm=0 (b) f(x,y,z)max2 +by2 +czz+cixy+pcz+lgzz j; =2ax+afy+jz I fy =2by+dx+k2 f; 2202+jx+ky fn m ——.25x"'1‘5y'5 fxy : fyx : .25x"5y"5 f”, = “25325)?” fm = .375x"2‘5y'5 f)”, = 3751753245 fxxy = fW 2 fm m ——.1253c'1'5y“5 fm = fwl= fw =w—.125x“'5y“"1'5 Mathematical Economics 2"“ ed. Baldani, Bradfield, and Turner Chapter 5 Answers Page 2 (d) f(w,x,y,z) = (Wm—Y" fw m .ZSW‘3/4 (xysz )2 :1: 25.7fm (wyzf‘m fy =: .5353F3’r‘g (WA:sz fz = .2553"4 (Mixtny fww : mfg “7/4 (xyz )1]4 fwx = w = "Ram/xv“ (WY/4 fw = fw — 335' W“ (3:2)“ f,” = rm = 1}g(wz)'3l4 (xyy/‘i f“ z ‘ 1‘35 “’4 WY" fl? = 13.: = Yaw“ (my/4 fa m z” m TEE—(“W (“WY/4 f»; 3 1%qu (wxz W f» WW f” M fwww m %W_1V4(Jg/z)w fm m gwa1/4(wyz)ll4 fwwx = fwm = fm = mg; w-TM x~3l4 ( yz)114 fm = fm = fyww == wéw-fl‘iy—m (my/4 fwwz = fm = fm = —.(%wm7/4Z«3/4 (xyy/a fm = fm : fm a _%x—7/4w_3/4 (yam fiacy = fm = 1;”, == ~éx~7f4ywsr4 (WW4 fm = f,m = fm 96.3; {7742414 (WY/4 fww m fm m fwyy m ‘63; ~7/4w—314(xz):/4 fyyx m fw m fw m —%y-7/4x—314(w2)y4 fm = fw = fw = méy-7/4Z—314(wx)1/4 fzzw = fm = fm = wézqflwéfl (WW4 fm = fm = fm -ézwmxwm (MO/)1” f” =f =f =—————7""7/412"3/4{wv\1/4 Mathematical Economics 2'"1 ed. Baldani, Bradfield, and Turner Chapter 5 Answers Page 3 (6) f(x, y, z) = Axayflz" fx = (Judged—132527 fy = BAx“y wl21’ f2 = yAxc‘yfizy‘l fxx “‘4' 0(Gt “1)Axm'2yfi-Z? fyy : I303“ 014353942? fzz = My ~1)Ax°‘y"zw fly = f3“ = on Ax“”‘yfi”‘z" f“ = fax = onyAxa'lyng fyz m fay m B'yAxmyfi"127"i fm = a(a ~1)(a—2)Ax“-3yfiz* fm, m £3(B—1)(13«2)Ax°‘yfi'327 fzzz ”V(V“1)(7”2)Axayfizyn3 fm = fny = fm = If)”: : fiw : fzyx 3 053T mulJ/wzfi"1 fm = fm 2 fm m (18({1—1) Axm’zywzy fm = f,m = fm = on! (on - 1)Ax°‘"2 3232*"1 fm = fm = fm WNW1)Ax°“‘y?’"2z"r fw = fm m 1;”, m Bv(B—1)Ax“y5”gz"“l = fm = fm = aY(“x’”‘1)A-XHYBZV_2 fay m fzyz ~'= fyzz =E3Y(Y~1)Axayfl”127‘2 (f) f(x,y,z)=aln(x»~xo)+bln(y—y0)+cln(z~20) _~ a m b m c fx—x—xo flay—yo fanny:sz b 0 fax:- a 2 fwm“ 2 fzzzw 2 (x"xo) (y'yo) (2‘30) fxy=fmmfxmfmmfyz=fzy10 b 0‘ fan—:2 a 3 fmmz 3 fzzz=2 3 (96-360) (rm) (2“20‘) fwxfmwfmmfmmflxymfw=fm=fm=fm=fm=0 1/? (g) f(x,y)= A(ax" +(1-~a)y") J; :13“? +(1—a)y°)‘l”")/D(W"'l) 13 =§(dx" +(lwa)y")“""vp(9(1-a)y""‘) _ f3 m figgilw +(1m a)y”)“’2"”" (paxp”‘)2 JEFF—Rafi + (1~a)y"){l"”yp (9(P~I)ax"'2) a = +<1—a)yp)‘*'2°”" (pa—— aw )2 +(1—a)y9)“'“’”(mpmlxww—z) m 240- p) m}; Twp+(1~a)yp)“"””°(pax”*‘)((p—1)(1-a)yp"1) Mathematical Economics 2'“l ed. Baldani, Bradfield, and Turner Chapter 5 Answers Page 5 5.3 (a) g; m (Zax + by) gqumqggfw =(2a):+by)(-16t"5)+(bx+2cy)(3—10t) (‘0) (Be carefill to distinguish between d, the coefficient on xy, and the differential symbols.) .41- .292 d_y . .612. dt *(Zax+dy+}z) dt +(2by+dx+kz) dt +(2cz+}x+ky) d! =(Zax+dy+ jz)(—16r5)+(2by+dx+kz)(3—100+(2cz+ jx+ky)[m%r7f4] (c) i a: (,5x"5y'5 )gfxg—i—(fix‘sf's m (.Syc‘fiy'5 )(m16t'5 )+(.5x'5y“‘5 )(3 ~10t) (‘0 i?[-25w“3/“(xyz>‘/“)%?+(25x”3’4(wyz)1/4)§x: + (.IZSyWB/4 (WXZ)1/4 + [2524/4 bang/)1”4 (6) ~63:- = (OLAxMyflz-T + (BAJcmyfi‘IZ'T + (yAxayfiZV'l ) g:— = (anW‘yfiZT )(—~16rS ) + (ESAxay'HZT )(3 —10z)+ (yAxmy52V”‘)[-—:—t’7l4] (Di: a flair b. 921+ .C 33 dt 3cme dt y—y0 dt 2—29 dt Whig))(m16t"5)+[yfy0)(3~10t)+[z:20][—%t”7’4] Mathematical Economics 2“" ed. Baldani, Bradfield, and Turner Chapter 5 Answers Page 6 <g (My-(m Mathematical Economics 2'“1 ed. Baldani, Bradfield, and Turner Chapter 5 Answers Page 10 5.5 (a) M: J; J; 8;: 3y 2ax+by 'bx+2cy m flax? +0“ a)y”)“‘p)/p (WM) flax" +(1—“W’ )WP (PG— 60y“) = ((zax+by)(paxp—l)“(bx+ 20y)(P(1‘a)J/ph1))[w§(ax9 41—61))” )wyp] fr j; J: .Saxw‘fiy52‘5 .Saxjy—fizfi .Saxfiyfiz—fi (b) r: gx gy gz m QAxamlbec beayb—lzc chabeC—i hx ha» 112 a/(x"xo) b/(y“yo) C/(Z'Zo) .5 —.5 5 .5 —.5 fx f3, f2 .Sa'sx"'5y'sz'5 .5a'5x y z‘ .Sa'5x5y z (c) Vim-g): gy g2: 2ax+dy+jz 2by+dx+kz 2cz+jx+ky hr by h gmxaw%y%z% %x%ya-%Z% %x%y%z“'% Mathematical Economics 2"d ed. Baidani, Bradfield, and Turner Chapter 5 AnsWeTS Page 13 5.7 For each part, call the left-hand side of the equation f (x, y). Then the slope of the level curve of the function x* (y) is, by the implicit function rule, equal to ~ 1; /fx . In order to find how the slope changes when y increases slightly, differentiate —- fy [fx with respect to y, remembering that the value of x will change too. That is, evaluate 5H;(x*(y)»y)/f;(x*(y)=y)) 6y (a)f(x,y)= 2x2 — 6xy+3y2 :2 12 f, =4xw6y fy =~6x+6y so —j;/fx =-(6y—6x)/(4x—6y). Ifyml then 2x2 ~6x+3 mm or 2x2 ——6x—~9=0 so x: 6iJ36—4(2)(——9) z (mm : (mm/:9,- xiii]? 2(2) 4 4 2 2 (Note that, since there are two values of x for each value of y, x is not a function ofy, except in the vicinity of a particular (34:, y) combination. So the slope of the level curve is ~(6y—6x)/(4x—6y)==—[6~6@¢§J§D/[4@agwfi]m6) = “[méagfiJ/ifi = swim-£0 0. 2J3 -2 6y 5y __—6(4x—6y)—~6(6y~—6x)+6(4x~6y)+4(6ym6x)§fl: 12 [Jam a) m (4x’6yf (4x*6y)2 5? (4th«6;¢)"2 y 6y . 12 3 3 l 3 12 8 WhICh, when :1, e uals ----~——-—-—- —¢m\/§]w[im+m))=m[imj y q (4x—6y)2[[2 2 2J3 2 (4x-6y)2 2,}; If x is positive, the slope of the level curve when y =1 gets more positive as y increases. If x is negative, the slope of the level curve when y :1 gets less positive as y increases. Mathematical Economics 2“d ed. Baldani, Bradfield, and Turner Chapter 5 Answers Page 14 (b)f(x,y)m 2x2 —6Jijy+3y2 = 36 fx m 4x~6y fy m *6x+6y so fify/fx = “(6yw6x)/(4x——6y). Ify=l then 2x2 ~6x+3 = 36 or 2x:Z ~6x—33 m 0 so x = 6ilf36—4(2)(—33) _ 6iJ300 a 6:106 M £35 2(2) 4 4 2 * 2 (Note that, since there are two values of x for each value of y, x is not a function of y, except in the vicinity of a particular (x, y) combination. So the slope of the level curve is --(6y~—6.x)/(4x—6y)=~{6—6[~3i§-J§D/[4[§tg~fi]w6] I 5 5 3 3 =— ~—-‘— 3 i~ 3mi +->0. [ 2+2fj/ 3f 10$? 2 al—MWJ/Ml/fic (x*(y)=y)) = 3(‘(6y*6x*(y))/(4x*(y)"53’» 3y 6y zw6(4x~—6y)—6(6y-6x)+6(4x-6y)+4(6y~6x)§g: 12 [JEN 9.3g] (4xw6y)2 (4x—6y)2 6y (4x—6y)2 y e» . 12 3 5 3 3 36 Wthh, wholly: 1, equals 106 : igfi So if x is positive, the slope gets more positive as y increases; ifx is negative, the slope gets less positive as y increases. (c)f(x,y) = J}; = 36 “is 23.35. m =1. finiagtmnmi, Ify :1 then J; = 36 orx = 362 m 1296. So the slope of the level curve is ~1296 < O. which, when y =1, equals 1296 +1296 > 0. So as y increases, the slope of the level curve becomes less negative. Mathematical Economics 2lid ed. Baldam’, Braclfield, and Turner Chapter 5 Answers P513615 1 y 1 x x fr“; 3: 13”; 5 5° “Ii/fare“;- Ify=l then J; =100 orx =1002 =10,000. So the slope of the level curve is m 10,000 < 0. a(*fir(x*(y)=y)) = 6(*x*(y)/y):_[_1]+i 6y 6y y yz which, when y = 1, equals 10, 000 + 10,000 > 0. So as y increases, the slope of the level curve becomes less negative. (e)f (x, y) = 5364312 = 66 fx 3 23c'fi'y'2 f3, zx‘éy‘é so =w-x/2y. lfy :21 then Sx‘4 m 66 orx m(66/5)1M w 633. So the slope of the level curve is w» 633/2 < 0. a(-fi(x*(y)ay)) =a(—x*(y)/2y),,[,_ ,1 ],___x__ 6y 6y . 2y 2322 which, when y = 1, equals about 633 > 0. So as y increases, the slope of the level curve becomes less negative. (Whey) = 5x'4y‘2 = 99 fx :29«7"6}2‘2 fy 2:64)!“ so =mx/2y. Ify m1 then 5x4 m 99 or x m(99/5)1"4 m 1745. So the slope of the level curve is w 1745/ 2 < 0. 2y 2y 3y 5y which, when y = 1, equals about 1745 > 0. So as y increases, the slope of the level curve becomes less negative ...
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RelativeResourceManager - Mathematical Economics 2‘“i...

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