Question

Imagine two goods that, when consumed individually, give increasing utility with increasing amounts
consumed (they are individually monotonic) but that, when consumed together, detract from the utility
that the other one gives. (One could think of milk and orange juice, which are fine individually but
which, when consumed together, yield considerable disutility.)
a. Propose a functional form for the utility function for the two goods just described.
b. Find the MRS between the two goods with your functional form.
c. Which (if any) of the general assumptions that we make regarding preferences and utility
functions does your functional form violate?

Answer 1

a. One possible functional form that utility function exhibit for two goods case (x and y) is:

U(x,y) = (x+y) - 3xy

The given utility function shows:

When y = 0 ; U(x,y) = x -> Shows increasing utility as x increases individually

When x= 0; U(x,y) = y -> Shows increasing utility as y increases individually

When x > 0 and y > 0 ; U(x,y) <0

b. The MRS (Marginal substitution of x for y) is given by MRSxy = MUx/MUy

For the given utility function U(x,y) = (x+y) - 3xy

MUx = 1-3y

MUy = 1 -3x

Hence,

MRSxy = MUx/MUy = (1-3y)/(1-3x)

c. The given utility function U(x,y) = (x+y) - 3xy does not exhibit weakly monotone preference as it exhibit convex preference implying that the given utility function is concave & hence, quasi-concave.