Answer to Question #244978 in Microeconomics for Marley

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

Utility function:

$u(x_1,x_2)=lnx_1+x_2$

(a)

$MRS=\frac{MU_{x1}}{MU_{x2}}\\MU_{x1}=marginal\space utility \space of \space good \space x_1\\MU_{x2}=marginal\space utility \space of \space good \space x_2\\MU_{x1}=\frac{∂u}{∂_{x1}}\\MU_{x1}=\frac{1}{x_1}\\And,\\MU_{x2}=\frac{∂u}{∂_{x2}}\\MU_{x2}=1\\MRS=\frac{\frac{1}{x1}}{1}\\MRS=\frac{1}{x_1}$

The graphical presentation of three indifference curves is given below:

According to the above figure, the x-axis measures the units of good x1, and the y-axis measures the unit of good x2. The IC1 shows the indifference curve along which the utility level is 10. The IC2 shows the indifference curve along which the utility level is 20. The IC3 shows the indifference curve along which the utility level is 30. 

 

(b)

$M=15\\P_1=1\\P_2=3$

The budget constraint will be as given below

$M=x_1P_1+x_2P_2\\15=1x_1+3x_2$

At equilibrium,

$MRS=\frac{P_1}{P_2}\\\frac{1}{x_1}=\frac{1}{3}\\x_1=3$

Let us substitute x1=3 in the budget constraint

$15=1×3+3x_2\\3x_2=15-3\\3x_2=12\\x_2=\frac{12}{3}\\x_2=4$

The units of good x1 are 3 and the units of good x2 are 4.

Optimal point (x1, x2) = (3, 4)

Graphical presentation:

According to the above figure, point E shows the equilibrium point or optimal point. At equilibrium, the consumer consumes 3 units of good x1 and 4 units of good x2.

 

(c).

The budget constraint will be as given below:

$M=x_1P_1+x_2P_2\\21=1x_1+3x_2$

At equilibrium,

$MRS=\frac{P_1}{P_2}\\\frac{1}{x_1}=\frac{1}{3}\\x_1=3$

Let us substitute x1=3 in the budget constraint

$21=1×3+3x_2\\3x_2=21-3\\3x_2=18\\x_2=\frac{18}{3}\\x_2=6$

Optimal point (x1, x2) = (3, 6)

The units of good x1 are 3 and the units of good x2 are 6.

Graphical presentation:

According to the above figure, point E shows the equilibrium point or optimal point. At equilibrium, the consumer consumes 3 units of good x1 and 6 units of good x2.

d)

 Income effect leads to substitution effect