Question

We have noted that u(x) is invariant to positive monotonic transforms. One common transformation is the logarithmic transform, ln(u(x)). Take the logarithmic transform of the utility function; p1 x1 + p2 x2 = y and assume an interior solution, so assume x1 > 0 and x2 > 0, then, using that as the utility function, derive the Marshallian demand functions, and verify that they are identical to those derived.

Answer 1

Taking the example of Cobb Douglas utility function to explain this question

U(x,y) =x1x2

Initial demand functions

p1x1+p2x2=y ------- (1)

MU (x1)/ MU (x2) = p1/ p2 ------- (*)

where MU = Marginal utility

x2/ x1 = p1/ p2 ------- (**)

x2= p1 x1/p2

Substituing x2 in budget constraint (1)

p1x1 + p2 (p1x1/p2) = y

2p1x1 = y

x1=y/2p1 (Marshallian demand of x1)

x2 = p1/p2 (y / 2p1)

x2 = y/2p2 (Marshallian demand of x2)

Now, with positive monotonic transformation

ln U (x1, x2) = ln (x1x2)

ln U(x1, x2) = ln x1 + ln x2 ------(2)

Again using consumer's utility maximising condition

MUx1/MUx2= 1/x1 / 1/x2 (Calculated from (2))

MUx1/MUx2= x2/x1 =p1/p2

The above condition is same as (**) implying the same procedure and the same Marshallian demand functions. Hence, monotonic transformation does not have any impact.