Question

1.a) Suppose u(x1,x2) = x1^a, x2^(1-a) . Given M, P1, and P2 derive the demands for the two goods: Solve for MU1, MU2 and the MRS. Now use the tangency condition MRS =-p1/p2

together with the budget line to solve for X1 (M, P1, P2) and X2 (M,P1, P2). b) Now suppose a = 1. Further, suppose M 12, P1 = 2 and P2 = 2. Draw the budget set and show the optimal point chosen by this consumer (using your demands in a)). Include a reasonable sketch of an indifference curve through the optimal point. c) Keep all parameters as in b) the same except now raise Pi to 4. Draw the new budget set and show the new optimal point chosen by this consumer. Include a reasonable sketch of an indifference curve through this optimal point. d) Now set a = 1/3 but go pack to the original prices and income of b). Draw the budget set and show the optimal point chosen by this consumer. Include a reasonable sketch of an indifference curve through this optimal point.

Answer 1

1a)

Given information

Utility function

$U=X_1^aX_2^{1-a}$

For maximization of utility

$MRS=\frac{Px_1}{Px_2}\\ MRS=\frac{MUx_1}{MUx_2}$

$MUx_1=\frac{∂U} {∂X_1} =aX_1 ^{a−1} X_2^{ 1−a}\\ MUx_2=\frac{∂U} {∂X_2} =(1−a)X_1 ^a X_2 ^{1−a−1}\\ MRS=\frac{MUx_1} {MUx_2} =\frac{aX_1 ^{a−1} X_2^{ 1−a}}{ (1−a)X_1^ a X_2^{ 1−a−1}}\\ MRS=\frac{aX_2}{ (1−a)X_1}\\$

For equilibrium 

$MRS=−\frac{P_1}{ P_2}$ (negative sign shows the downward slopping budget line)

$\frac{aX_2}{ (1−a)X_1} =\frac{P_1}{ P_2}\\ X_1=\frac{aP_2X_2}{ (1−a)P_1}$

By substituting X₁ in budget constraint

$M=P_1x_1+P_2x_2\\ M=P_1\times \frac{aP_2X_2}{ (1−a)P_1} +P2X2$

By solving above equation

$X_2=\frac{1−a}{ P_2} \times M$ −−−−−− ordinary demand function for X₂

By using X₂ value we can find X₁ value

$X_1=\frac{aP_2X_2} {(1−a)P_1}\\ X_1=\frac{aP_2}{ (1−a)P_1} \times M\times \frac{1−a }{ P_2}\\ X_1=\frac{a}{ P_1} \times M$ −−−−−− Ordinary demand function for X₁

2a)

When Income=12

P₁=2

P₂=2

Budget constraint for the consumer

$M=P_1X_1+P_2X_2\\ 12=2X_1+2X_2$

When X₁=0----- Consumption of $X_2=\frac{12}{2}=6$

When X₂=0----- Consumption of $X_1=\frac{12}{2}=6$

We know the demand curve 

$X_1=\frac{a}{P_1}\times M\\ X_2=\frac{(1-a)}{P_2}\times M$

When a=1

X₁=6

X₂=0

Here corner solution would be applicable

Consumer will consume only X1 product

Optimum choice 

b)

Price of X₁ changes and P₁=4

Budget constraint for the consumer

$M=P_1X_1+P_2X_2\\ 12=4X_1+2X_2$

When X₁=0----- Consumption of $X_1=\frac{12}{2}=6$

When X₂=0----- Consumption of $X_1=\frac{12}{4}=3$

Demand function

$X_1=\frac{a}{P_1}\times M\\ X_2=\frac{(1-a)}{P_2}\times M$

X₁=3

X₂=0