Answer to Question #209387 in Microeconomics for james

Question

An individual has the utility function 𝑼 = 𝟒𝑿 𝟎.𝟓𝒀 𝟎.𝟓 and can buy good X at £2 a unit and good Y at £8 a unit. If their budget is £100. Find the combination of X and Y that they should purchase to maximize utility

Answer 1

Individual utility function given by

$U=4X^{0.5}Y^{0.5}........(1)$

Price of good X. Px=2

Price of good Y. PY=8

Equition of the budget line is

$PX(X)+PY(Y)=M\\2X+8Y=100........(2)$

Consumer maximizes his utility at a point where $\frac{MUX}{MUY}=\frac{PX}{PY}$

$\frac{\frac{∆U}{∆X}}{\frac{∆U}{∆Y}}=\frac{2}{8}$

$\frac{4(0.5)X^{-0.5}Y^{0.5}}{4X^{0.5}(0.5Y^{-0.5})}=\frac{1}{4}$

$\frac{Y^{0.5+0.5}}{X^{0.5+0.5}}=\frac{1}{4}$

$\frac{Y}{X}=\frac{1}{4}$

$Y=\frac{X}{4}$

$2X+8(\frac{X}{4})=100\\2X+2X=100\\4X=100\\X=25\\Y=\frac{X}{4}={25}{4}=6.25$

Combination (X,Y)=(25,6) should be purchased to maximize utility.