Question

A consumer buys two goods, good X and a composite good Y. The utility function is given as U(X,Y) = In3XY. The price of X is Py, the price of Y is Py and Income is I. 1) Derive the demand equation for good X. ( 5 marks) 2) Are the two goods X and Y complements or substitutes? Why? ( 5 marks) 3) Suppose that I=$10 and suppose that initially the Px = $1 and subsequently Px falls and becomes $0.5 per unit. Calculate numerically the value the price effect as a result of the drop in Py and graph the results. (5 marks)

Answer 1

Те &#kh њим е дом ал- U(x7) = 3x Now, Mosaid иҒ0:1 % x, му е əv (Ху). ЗХу. Main uld 4 ФУ, У 3y 1 х Әх у ә) muy = Qulx, y) = 3х - 3xy Suppose price of x = px puce of y=py Income - I Budget constraint I= XPxt ypy и!ія малітв'я Condition is Р MUX Ру му PX Ру г) У Рx ус Ру ) ХР» - УРу - Ха уРу с ) 1C Scanned with CamScanner

are neither substitutes or complements. They are substanting X= .ypy in budget constraint we get? РЯ. I- XBx typy I 2) I - YPY PX + YPY PX I= 2y Py =) ye Zey 2PY I 2PX Py. 2PY X = YPY рх AS PX Hices demand equation for good x = I 2px: Now demand for x = I 2PX Now, 2X Ĉ0 apy Now, cross-puce elasticity of x, ax. Py O х exy = or by cross puce elasticity is Zno, hence, Xad y fue as indepedent goods. Now ini tally Px=1, I= $10, Pue 0 y = Py. Hill budget constant I=XPX typy e) 100 X + Y py Scanned with CamScanner

(Mucic, will Now horizontal infucest of the budget line hure horizontal intcept of the budget line , Untical into cept of the budget line: [put y=0 los xt 6 x 0) =) [put X=0) 10- öt YPY X210 =) 2) jo 2 у PY Joinig horizontal and rustical intrcept, we get the initial budget line AB. demad for X Now initially demand for y= 2pY 910-5 2pY 2X) 도 py T 10 apy to in ital budget line AB at initial consuiption 5 budle (x, y) = (5 py I=10 and Niew, Px falls and become py= $0.5, while Риги у= Ру. Now, budget line I=APY toy Py =) 1030.5 x + YPY [put y=0 10= 0.5X to 20 52 X= $S$canned with CamScanner cer73

AC. Hice, the budget line will (x, y) = (10, PY! 0 :) Which is tuqunt to the inital in diffrence u= lngxY intical intercept of the budget line [ put X=0] 10 Y? = 10 2PX 2X0.5 S 10- (0.5 xo) typy -) PY Joining horizontal and intical untecept we get the rostate out ward from AB to AC. 10 Now new dem and for x = I demand for Ya 2PY Ру Haces the new consumption Anqant to new budget line AC at budle 5 Now, we have to find the I = 10 2PY new in difference Cure I c2 will be decomposition bulle. ütlity cuve, ii ) tagant to a budget. line, with its slope equels to the slope of the new budget line. Now, to gut. on the origial indifference Cuve, we substitute the initial bulle (X,Y) (5 5 bicton. in the utility м» =) = ln (3x 5 x 3xy - 75 Y 25 у = xy2 = 25 = x 25 3 ху =) ci 2 CSScanned with CamScanner

condition : Now for Augency of slope of in diffance (muc equals the slope the new budget line. X 0.5 PY +x\h-|-|- 0.5 PY 0.5X =) =) YPY Xa 2 ypy YPY 0.5 . E) - E) Y = 2y py substititing x= 2ypy in equation () we gut? Xyz 25 E) 2 y Pyx Y?? 25 273 25 25 y3 = 12.5 2 py (1205) 2.32 3 PY lo =) Ру 1/ 1) c) Y ? Py 1/3 Py ½ 1/3 8Y AS X = 2 Y Py = 2/ 2:32 4.64 py/ tice the decomposition budle is (X,Y)= (4.64 Py 23 2.32) PY CSScanned with CamScanner

graph the usetal budles tite Now substitution effect for good X Now, we can de composition bulle as: final buelle and fiul bulle X = 10 y=s Y5 2/3 64 Py (4.6 Inital bulle Decom posidon bolle X = 5 X = 4.64 Py / PY PY Y= 2,32 Py% = (De composition bulle - Initial bulle; -5) Income effect for good x (fial bulle bulle - decomposition budle (10-4.64.py/ effect - fial bulle - inital bulle Ĉ (10-5) -5 Huce x falls, consueen 5 more uits of x, due to the price effect - Price bus pull of CS$canned with CamScanner

5 esla a I le M ICI Y LL 5. 4646,4% 10 208 SE L IE PE SE= substitutin effect IE= Income effect PE - effect puce. CSScanned with CamScanner