Please answer parts a,b,c,d only for the Leontief utility function and show all work!!(30 pts) Continuing from Midterm: Families of Utility Functions
Answer the following questions for each of the following utility functions.
x_(i) are quantity of good i consumed, and a,b,c,d,e,s are parameters.
i. Cobb-Douglas type utility function:
u(x_(1),x_(2))=x_(1)^(a)x_(2)^(b)
where u(x_(1),x_(2))=af(x_(1))+bx_(2)f(x_(1))=ln(x_(1)),a>0b>0u(x_(1),x_(2))=[sum_(i=1)^2 a_(i)^((1)/(s))x_(i)^((s-1)/(s))]^((s)/(s-1))a_(i)>0s>0u(x_(1),x_(2))=min{ax_(1),bx_(2)}a>0b>0{:epsi lon_(x_(2)^(**))^(x_(1)^(**)))a=1,b=2,a_(1)=a_(2)=1x_(1)-x_(2)
1. (30 pts) Continuing from Midterm: Families of Utility Functions Answer the following questions for each of the following utility functions. T; are quantity of good i consumed, and a, b, c, d, e, s are parameters.
i. Cobb-Douglas type utility function:
u(x1,x2)=xxb
where0<a<1 and0<b<1
ti. Quasi-linear utility function:
u(x1,x2)=af(x1)+bx2
where fx)=ln),a>0 and b>0
iii. Constant Elasticity of Substitution utility function:
1s1s aixis
where a; > 0 and s > 0.
iv. Leontief utility function:
u(x1,x2)=min{ax1,bx2}
where a > 0 and b>0
(a) Check if the implicit function theorem is applicable to find the marginal rate of substitution (MRS) (hint: to all or just part of the consump tion bundles). Explain.
(b) Use the implicit function theorem to find the MRS for applicable consumption bundles for each utility function.
(c) For ii. and iii., solve the utility maximization problem to find the optimal consumption bundle - demand of each good.
(d) For ii. and iii., find the price elasticity of demand and cross price elasticity of demand for all the goods.
Extra Credit For iii., find the elasticity of substitution between goods 1 and 2 (i.e ). Extra Credit With a = 1,b = 2,a1 = a2 = 1, draw each utility function and the corresponding budget constraint on a - 2 plane. Indicate the optimal consumption bundle and optimal MRS.
