Question

Bnnas O al UI IImelioRO0d T0l DClla! Eplalli. 2. Suppose that a consumer has utility U(X, Y) goods X and Y a) The prices of X and Y are S1 and $2 per unit respectively. Use a Lagrangian to solve for the optimal basket of goods. b) Suppose that the price of X increases to $2 per unit. Use a Lagrangian to solve for the new optimal basket of goods. Find the total effect of the price change on the consumption of each good. c) Use a Lagrangian to find the substitution effect of the increase in the price of good X on the consumption of each good. What income would the consumer need to attain the original level of utility if the price of X increased to $2 per unit? d) Find the income effect of the increase in the price of good X on the consumption of each good. Are the goods normal or inferior? Explain. e) Show that the total effect of the increase in the price of X is equal to the sum of the substitution effect and the income effect. = 2XY +X and income of $499 to spend on

Answer 1

Smati'on M-499 pa1 N Budget comat ant X+2y-499 -2XG TX Subjeet to Xty-403 xY+Xtd32-X-2y7 2Y+1 +E-17 0 工十んで = 2x +t-21-0 2x-24 2 491-X-2yニ0 dlivtale X0 X=29+1 2y+1 2 2X Substiture x= 2y+1 in eg = 498 Y= 498 X+2y = 2y+1+2y=493ラ48 4.5 +1 X= 250/ 2498 4 basket of goods is CXiy- C250, 418). optimal New BudgelConatnai nt-2x+2y = 499 pm'=2 U2X47 X Man 2X+P4I49 Subjeet 2x X + A49-2X-2y 247- (4 2Y+1 = 2Y1 AE a7 0

2X t11-21 0 493-2-24-0 2X+24-499 2 5 Divide et 27 24+1 24+1- 2X 24+ 1-X 2 Suesribute x value in hatim 44-488 +24 149 4 498 124.5 4 +1 250 125 2 ex", y") - C125, 438 o) 0 0ma malle 4 X-X 125-250 =)125 Puice affeet of good 70 tau Paice effeet of good y X T0 tal 4'"tn 4nR-498-0 bo attain ouiginal utfty The hneone segubud compured X.(Apa) [2-1] L2501=250 Am So Budlget constnant mw mtAm 499+ 250=99, 2xt24 =749 =2XY+X t AL 749-2x- 2y7 0) 24+1 X +1-27 -0 21 2X --x

JA 749 -2X-2y D 2メ+2y=49 APvPole get 21 X 2 aubsti tute nalua of Xn uation G) 2Y+1 + 24 -4y+1 = 749 2 y'2 748 = 187 4 xi8)+1 =187,5 2 [ sutory So, optfmal = (1875, 187 bundle &ET ISt Substituti n ef feet fou X =x'-X=187o 5-250 = 62.5 62.5 effeet for Y = y'-Y=187-124.5 Substi'tutin X = ×リ-X' = 125-187-5-C)62-5 effect for goo0d d Ducoome - y"-y'-124.5-187 E) 62.5 Income epfeet Pore good = whose dlemndl falls are the one Jne nosunal g0ods Jhe puce of good X inerenses that sReal Pneome of Pallo. Andl for moumal a fth fal in inesme. the income consum eu that 心) fall in pule of good w effeet ngatire, ineroase the ueal income oureh inerease demand for the good L that s, Palling frte 2 iuereaing is nogative aelation Juus, for normal good the ueome effect a negabive i's demmd that the income effeet Por Good X Good y PNSM parnt (C

Nosatives So, Bo th Xty goed as freroese in prce of good Xdeereases the real ineome denarol Ae nanal Palls aoct normal goodl totat Rrice =come epfeet substftutn effeet OFgood X C Parit d t-62.51+ C-62.5 = C25 =Total u C Parut effect of good X of good X CPosut c oPfect Part 7 ometSubstitation epfect Totau Puice Prom ffeet Of Good f ffeet of good ef Good y R.ttoS neome effeet t Lubstitutin epfeet of Good y + 62.5 = 0 = LoHes C-62.5] uostitutim +Dreome effeet ttence, To tal Puice effeer effret