Question

if a consumer has a utility function u(X,Y) = XY4 what fraction of her income will she spend on Y

Answer 1

Given that ,

The consumer has a utility function $U(X,Y)=XY^4$

This utility function shows as Cobb Douglas utility function .

Generally utility function would be written as $U=X^aY^b$

$U=XY^4$

Good X $X=\frac{a}{a+b}.\frac{M}{PX}$

Good Y $X=\frac{b}{a+b}.\frac{M}{PY}$

Where,

PX-price of good X

PY-price of good Y

M- money income

Then

Demand for good Y

$X=\frac{b}{a+b}.\frac{M}{PY}=X=\frac{4}{1+4}.\frac{M}{PY}\\X=\frac{4}{5}.\frac{M}{PY}\\=\frac{4M}{5PY}$

Then, Total expenditure on good Y=price of good Y × demand for good Y

$PY×\frac{4M}{5PY}\\\frac{4M}{5}$

Fraction of income spent on good Y=total expenditure of good Y÷total income

$=\frac{\frac{4M}{5}}{M}=\frac{4M}{5}×\frac{1}{M}\\=\frac{4}{5}$