Answer to Question #273333 in Microeconomics for cookie

Question

A person’s utility function is of the form U(x,y) = 5xy. The prices of good x and y are P_x = $4 and P_y = $2, respectively. The person’s income is $1200.

 

(a) Show that these preferences are homothetic?

(b) What quantities of x and y should the consumer purchase to maximize his

utility?

(c) Determine the person’s income offer curve (IOC). Draw it.

(d) Explain whether each of the two goods is normal or inferior.

(e) Derive the Engel curve for x. Draw it.

Answer 1

Solution:

a.). Preference relation is homothetic if and only if it can be represented by a utility function that is homogeneous of degree one.

U(x,y) = 5xy

This function presents a homothetic function.

b.). To maximize utility:

U(x,y) = 5xy

Derive MRS:

MRS = $\frac{MU_{x} }{MU_{y} } = \frac{Px }{Py }$

MUx = 5y

MUy = 5x

$\frac{MU_{x} }{MU_{y} } = \frac{Px }{Py }$

$\frac{5y }{5x } = \frac{4}{2}$

y = 2x

Budget constraint: I = PxX + PyY =

1,200 = 4X + 2Y

1,200 = 4X + 2(2X) = 4X + 4X = 8X

1,200 = 8X

X = 150

Y = 2X = 2 x 150 = 300

Consumers will purchase 150 quantities of X and 300 quantities of Y to maximize utility.

 

c.). The income offer curve is as below:

 

d.). The two goods are normal goods.

 

e.). The graph is as below: