Question

A consumer’s utility function is U(x₁,x₂)=3x₁+x₂^1/3. If the consumer weakly prefers the bundle (x₁’,x₂’) to the bundle (x₁’’,x₂’’), will he necessarily also weakly prefer the bundle (x₁’+1,x₂’) to the bundle (x₁’’+1,x₂’’)?

Answer 1

We have utility function U(x₁,x₂)=3x₁+x₂^1/3

Let us assume U(x₁,x₂) = 100

100 = 3x₁+x₂^1/3

x₂^1/3 = 100 - 3x₁

x₂ = (100 – 3x₁)³

The consumer weakly prefers the bundle (x1’,x2’) to the bundle (x1’’,x2’’) means that the consumer is indifferent between the two bundles and these bundles correspond to points on the same indifference curve with same utility value.

Put x₁^’= 1, then x₂^’= (100 – 3 x 1)³ = 912673

Put x₁^”= 2, then x₂^” = (100 – 3 x 2)³ = 830584

Now the bundle (x₁’+1,x₂’) = ( 1+ 1, 912673)

= (2, 912673)

The corresponding utility from the utility function is

U(2 , 912673) = 3x2 + 912673^1/3

= 103

U = 103 corresponds to a different indifference curve from that of U = 100.

Similarly the bundle (x₁”+1,x₂”) = ( 2+ 1, 830584)

= (3, 830584)

The corresponding utility from the utility function is

U(3, 830584) = 3x3 + 830584^1/3

= 103

Hence, the bundles (x₁’+1,x₂’) and (x₁”+1,x₂”) are on the same indifference curve with U = 103.

This proves that if the consumer weakly prefers the bundle (x₁’,x₂’) to the bundle (x₁’’,x₂’’), he will necessarily also weakly prefer the bundle (x₁’+1,x₂’) to the bundle (x₁’’+1,x₂’’).