Question

Consumer spends $360 per week on two goods X and Y. PX=$3 and PY=$2.Hisutility

functionisU=2X2Y.

What quantities of X and Y does he buy each week in equilibrium?.

Answer 1

we have: I=$360, U=2X²Y, P_X=$3, P_Y=$2

thus $U_{xx}={ d\over dx}({ dU\over dx})=4Y, U_{yy}={ d\over dy}({ dU\over dy})=0$

$U_{xy}={ d\over dx}({ dU\over dy})=4X={ d\over dy}({ dU\over dx})=U_{yx}$

$U_{xx}={ d\over dx}({ dU\over dx})=4Y, U_{yy}={ d\over dy}({ dU\over dy})=0$

At equilibrium

${ MU_x\over MU_y}={ 4XY\over 2X^2}={ P_x\over P_y}\implies2YP_y=XP_x$

$YP_y+XP_x=I$

$3YP_y=I$

$Y={ I\over 3P_x}=60$

$YP_y={ 1\over 2}XP_x\implies{ 1\over 2}XP_x+XP_x=I$

$\implies X={ 2I\over 3P_x}=80$