Question

Do the following utility functions represent the same preferences? Please provide your reasoning

(a) u(X1; XY) = X1 X2, V(X; X2) = 3(x] X2)2 +6 (b) u(x]; X») = X1X2, V(X]; X2) =-3(x1x2)2 +6 (c) u(x1,x2)=X]X2,v(x1,x2)=lnx1 +lnx2 (d) u(x]; X) = X1 X2, V(X1; X2) = x1 + x2

Answer 1

Two utility functions have same preference if their MRS is identical.

MRS = MU1/MU2, where

MU1 = $\partial$ U/$\partial$x1 and

MU2 = $\partial$ U/$\partial$x2

(a)

(i) U = x1.x2

MU1 = x2

MU2 = x1

MRS = x2/x1

(ii) v = 3(x1x2)² + 6

MU1 = 6x1x2.(x2)

MU2 = 6x1x2.(x1)

MRS = x2/x1

Therefore, U and V represent same preferences.

(b)

(i) U = x1.x2

MU1 = x2

MU2 = x1

MRS = x2/x1

(ii) v = - 3(x1x2)² + 6

MU1 = - 6x1x2.(x2)

MU2 = - 6x1x2.(x1)

MRS = x2/x1

Therefore, U and V represent same preferences.

(c)

(i) U = x1.x2

MU1 = x2

MU2 = x1

MRS = x2/x1

(ii) v = ln x1 + ln x2

MU1 = 1 / x1

MU2 = 1 / x2

MRS = (1 / x1) / (1 / x2) = x2/x1

Therefore, U and V represent same preferences.

(d)

(i) U = x1.x2

MU1 = x2

MU2 = x1

MRS = x2/x1

(ii) v = x1 + x2

MU1 = 1

MU2 = 1

MRS = 1/1 = 1

Therefore, U and V do not represent same preferences.