a U xy b U xy13 c U minxy2 d U 2x 3y e U x2 y2 xy 2 A

a) U = xy

b) U = (xy)^1/3

c) U = min(x,y/2)

d) U = 2x + 3y

e) U = x^2 y^2 + xy

2. All homogeneous utility functions are homothetic. Are any of the above

functions homothetic but not homogeneous? Show your work.

Solution

Utility function is homogeneous if for some degree k , U(tx,ty) = t^k U(x,y) .

And it is homothetic , if it is a monotonic transformation of homogeeneous functions.

(a) U(x,y) = xy

U(tx,ty) = (tx) (ty)

= t² xy

It is homogeneous of degree 2.

(b) U(x,y) = (xy)^1/3

U(tx,ty) = (tx)^1/3(ty)^1/3

= t^2/3 (xy)^1/3

It is homogeneous of degree 2/3.

(c) U(x,y) = min(x,y/2)

It is homogeneous of degree 1.

(d) U(x,y) = 2x + 3y

U(tx,ty) = 2(tx) + 3(ty)

= t(2x + 3y)

It is homogeneous of degree 1.

(e) U(x,y) = x²y² + xy

Now, multiply t with x and y :

U(tx, ty) = (tx)² (ty)² + (tx)(ty)

= t⁴ (x² y²) + t² xy

= t² (t²x² y² + xy)

It is not homogeneous but it is homothetic.

Hence, (e) is homothetic but not homogeneous.