Answer to Question #247233 in Microeconomics for umair

Question

2. A consumer has preferences characterized by the utility function u(x1, x2) = In 21 + x2. a) What type of preferences are these? Solve for an expression for this consumer's MRS. Sketch 3 different indifference curves for this consumer.

b) Suppose M = 15, P1 = 1, P2 = 3. Use the tangency condition MRS = - to solve for the optimal amount of good 1. Given this, determine the optimal amount of good 2. Sketch this optimal choice on a graph of the budget set. Include an indifference curve through your optimal point.

c) Now increase income to M = 21. Derive the new optimal choice and show it on a graph as in b)

d) Explain any difference between the points chosen in b) and c)

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.

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.

(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:

d)

 Income effect leads to substitution effect