Question

1. Consider a consumer who has income of m = 30 and preferences represented by the utility function u(x, y) = 2x + 3 ln y. Note that this is quasilinear utility.

(a) Initially, suppose that prices are given by px = 12 and py = 1. Find the optimal bundle. This is point “A.”

(b) Suppose that the price of y increases to 4. In the notation we have used, this means p 0 y = 2. Find the optimal bundle after this price change. This is point “B.”

(c) Calculate m0 , the compensated income for this price change.

(d) Find the optimal bundle on the intermediate budget line defined by m0 , px, and p 0 y . This is point “C.”

(e) What are the substitution and income effects on x?

(f) What are the substitution and income effects on y?

Answer 1

Given: U= 2x+3lny

m=30, Px= 12, Py= 1

From the given information, we can write the budget line as follows:

Px.X + Py.Y =m

12X + Y= 30 (putting values of m, Px and Py)

In order to find optimal bundle, we should compute MRS from utility function and slope of budget line:

Slope = Px/Py,

Hence slope of budget line = 12

Now, MRSxy = MUx/MUy

MUx= du/dx= 2

MUy = du/dy= 3/y

Therefore, MRSxy= 2y/3

Since, equilibrium exist where budget line is tangent to indifference curve. So, slope of Indifference curve and budget line should be the same.

So, MRSxy = slope of budget line (tangency condition)

Hence, 2y/3 = 12 --> y= 18

So, x = (30-18)/12= 1 (by putting value of y in budget line equation)

a) therefore, optimal bundle, A(x,y) = (1, 18)

b). Now, price of Y has increased to 4. So,

New budget line will be,

12X +4Y= 30

Slope of new budget line = 12/4= 3

MRSxy = slope of budget line

2y/3= 3 --> y= 9/2 =4.5 and x = (30- 4.5×4)/12 = 1

So new optimal bundle, B =(1, 4.5)

f) similary for y, substitution effect is -13.5. shown in picture.