Answer to Question #274855 in Microeconomics for Stephany

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

$u=5xy, P_x=4, P_y=2, I=1200$

(a) For homothetic Preferences, $MRS_{xy}=\frac{MUx}{MUy}=\frac{y}{x}$

$then \ MRS(\lambda x,\lambda y)=\frac{\lambda y}{\lambda x}=\lambda^0 MRS$

$\forall \lambda \gt0,$

L₁ thus MRS, homogenous function of degree zero in x,y, hence preference are homothetic

(b) Now at equation

$MRS=\frac{px}{py}$

$\frac{y}{x}=\frac{px}{py}$

$ypy=xpx$

from Bc$\to ypy+xpx=I$

so$\ 2xpx=I$

$x^*=\frac{I}{2px}, \ y^*=\frac{I}{2py},$

$so (x,y)=(\frac{1200}{8},\frac{1200}{4}) \\=(150,300)$

(C)IOC - locus of all combinations of two goods when only icome varies and prices are unchanged

so IOC: $\frac{y}{x}=2, \ y=2x$

Along which optimal combinations lie

(d) As $\frac{dx'}{dI}=\frac{1}{2px}\gt 0, \ \frac{dy'}{dI}=\frac{1}{2py}\gt 0$

so as income rises optimal combination rises so It is Normal goods

(e) as Engel graph between income and Q of the good

thus it is upward sloping with slope= 2px

$=2\times4=8\\$