Question

Question Two Suppose that in a two-commodity world, the consumer's utility function takes the form (x)-la +. This utility function is known as the constant elasticity of substitution utility function a) Show that when p-1, indifference curves become linear Show that when p o. this utility function comes to represent the same preference as the cable Douglas utility function () c) Show that as , indifference curves become right angles" that is this utility function has the limit the indifference map of the Leontief utility function )

Answer 1

Que) The elasheity of subshi futon calculates how easy it is to suvshute one good for another 5= d (12/714) d MRS X MRS 12/717 here MRS : - Marginal Rale of Substitution. Case I:- linear car o when elasticity of substitution Col is given ab infinity than two goods are considered to be poyect substitutes of each other. Y, л un nz) = ani+brz Slope = -a

Case when et 6=0 When elasticity of subshlution is 6=0; two goods are perject complements to each other. Utility function will be of L-Shape depicting consumer would always Consumer at the cooner of an 150 quant ucu,m2) = min { an, bn2} Slopes a U2 - --- ---- - ---> When: bx2 <ani then u = buz :ani <bu2 then u = au ani = b^2 then both qoods will be consumed When this happens, 12 - Q and consumption takes place at ventex of so quant. Iso indifference curve. 711 b case when 6=1 uchi, 12) = n, a n a This is the mathematical form of cobb-douglas function In this case, indiference curves are of" normal convex" shape and o cel as ticity of suosh tution) = 1

Q+CI-a) si increasing retwins Q+CI-a)= constant sie wins SU2 Q+Cl-a) < de cue asma me turns to scale. # Constant Elasticity of substitution A functional foom that incorporate all 3 previous case of 5= 1,0 and Q and allows o Celasticity) to take on other values as well is the constant elasticity of substitu hon CCES) whity function. The function is given u[m, M3] = Camp+azu, ) Up for p < ,p to and y>o 6= MRS d (nalna) d MRS - 12/up P-1 Fighstly, find MRS. MRS --MUNE E-pap-. - - ai - MU nz pazn, por a2712 Muni= y Cam, P+ a2 220g //pcpainopoly MU42= 1 x Camp + O2 U2P /p/ cp az U2P-1) MRS -- aC)'

= Y 7PM e dMRS d712/21 -a 02 sportiva Cupy muzyka over the paymen t 6 = d 2/nui MRS d MRS 12/u = con a 2 nzini CI-PC) () x gay ( ) -B (menos en la 4.) If pat; 615 = IC becomes linear 6.) If p=061-0-1- IC become convex (Cobb Douglas) c.) If p=-a 6-10 itata IC become I shape (Minimum function)