Question

Consider the following utility function: U(x1,x2) X₁^1/3 X₂

Suppose a consumer with the above utility function faces prices p_{1 =} 2 and
p_{2 =} 3 and he has an income m = 12. What’s his optimal bundle to consume?

Answer 1

U(x1,x2) = X₁^1/3 X₂

P1 = 2 and P2 = 3. Income m =12

Consumers optimise where their Marginal Rate of Substitution between x1 and x2 equals the price ratio of x1 and x2

MRS = MUx1/MUx2

Differentiating the utility function with respect to x1 and x2 we get,

MUx1 =

MUx2 =

MRS = (x2)/(3x1)

Price ratio of x1 and x2 equals = 2/3

Hence at equilibrium,

(x2)/(3x1) = 2/3

x2 = 2*x1

The consumer's budget constraint can be written as:

P(x1)*x1 + P(x2)*x2 = m

2*x1+3*x2 = 12

Put x2 = 2*x1, we get

2*x1+6*x1 = 12

x1 = 1.5

x2 = 2*x1 = 1.5*2 = 3

Hence optimal consumption bundle, x1=1.5 and x2 = 3