Question

Suppose a consumer utility function is given by U(X,Y)=XY+8X the price of good X and Y are 2 birr and 6 birr respectively the consumer has total income of 160 birr to be spent on two goods

A) find the utility maximizing quantities of good X and Y

B) find MRXSY at equilibrium

Answer 1

a)

According to the given utility function, Marginal utility from X would be:

$U(X,Y)=XY+8X\\\frac{dU(X,Y)}{dX}=Y+8$

And, Marginal utility from Y would be:

$U(X,Y)=XY+8X\\\frac{dU(X,Y)}{dY}=X$

At profit-maximizing level, 

$\frac{MUX}{PX}=\frac{MUY}{PY}.....(2)\\\frac{Y+8}{2}=\frac{X}{6}\\6Y+48=2X\\2X−6Y=48\\X−3Y=24$

Using, budget contrant and second equation (2), puting $X=24+3Y$ in budget constraint equation:

$2X+6Y=160\\2(24+3Y)+6Y=160\\48+6Y=160\\6Y=160−48\\Y=\frac{112}{6}\\Y=18.667$

Using the value of Y, value of X would be:

$X=24+3Y\\X=24+3(18.667)\\X=80$

b)

At the profict maximizing level, MRS_XYwould be:

$=\frac{MUX}{MUY}\\=\frac{Y+8}{X}\\=\frac{18.667+8}{80}\\=0.34$