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

a) For homothetic preference;

$MRS_{xy}=\frac{MU_{X}}{MU_{y}}=\frac{y}{x}$

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

Thus $MRS$ ,homogenous function of degree zero in x, y, P_ref are homothetic

b)At equilibrium, $MRS=\frac{P_{x}}{P_{y}}$

$\frac{y}{x}=\frac{P_{x}}{P_{y}}$

$yP_{y}=xP_{x}$

$xP_{x}+yP_{y}=I$ so $2xP_{x}=I$

$\therefore x=\frac{I}{2Px}$ and $y=\frac{I}{2Py}$

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

c) Locus of all combinations of two goods, when only income varies and prices remain unchanged.

So IOC,$\frac{y}{x}=2$

$y=2x$ along which optimal combinations lie

d)As $\frac{\delta x}{\delta I}=\frac{I}{2Px}>0, \frac{\delta y}{\delta I}=\frac{I}{2Py}>0$

So as income rises, optimal combinations rises. So the goods are normal good

e)

As $x=\frac{I}{2Px}, I=(2Px)X$