Question

1A.) The utility function and the prices are the following:
U = 40 x1 + 20 x2
P1=4,   P2=3 and I =1,200

What is the optimal amount of x₁?

1B.) The utility function and the prices are the following:
U = 3 x₁ + 33 x₂
P₁=37,   P₂=12 and I =5,004
What is the optimal amount of x₂?

Answer 1

Part 1) We have the following information

U(X₁,X₂) = 40X₁ + 20X₂

We have a case of perfect substitutes

The marginal rate of substitution between X₁ and X₂ is 2/1, which is a constant, independent of the quantities consumed of the goods. The indifference curves between the two goods are straight lines.

Marginal utility of X₁ = ∂U/∂X₁ = 40

Marginal utility of X₂ = ∂U/∂X₂ = 20

Using lagrangian multiplier

µ = 40X₁ + 20X₂ + λ(1200 – P₁X₁ – P₂X₂)

Mathematically, the first-order conditions

∂µ/∂X₁ = 40 – λP₁

40 = λP₁

∂µ/∂X₂ = 20 – λP₂

20 = λP₂

could both hold only if 2/1 = P₁/P₂, which would happen by coincidence. Usually, the consumer will choose to be at a corner solution, spending all her money on the good for which a_i/P_i is highest (a_i is coefficient of the ith good in utility function and Pi is the price of ith good).

So, if 40/P₁>20/P₂, then the consumer will chose

X₁ = 1200/P₁

On the other hand if 40/P₁<20/P₂, then the consumer will chose

X₂ = 1200/P₂

In the present case we have P₁ = 4, and P₂ = 3

40/P₁ = 40/4 = 10

20/P₂ = 20/3 = 6.67

Since, 40/P₁>20/P_2­ so consumer will only consume X₁, and the amount of X₁ consumed by the consumer is

X₁ = 1200/P₁

X₁ = 1200/4

X₁ = 300

Part 2) We have the following information

U(X₁,X₂) = 3X₁ + 33X₂

We have a case of perfect substitutes

The marginal rate of substitution between X₁ and X₂ is 3/33, which is a constant, independent of the quantities consumed of the goods. The indifference curves between the two goods are straight lines.

Marginal utility of X₁ = ∂U/∂X₁ = 3

Marginal utility of X₂ = ∂U/∂X₂ = 33

Using lagrangian multiplier

µ = 3X₁ + 33X₂ + λ(5004 – P₁X₁ – P₂X₂)

Mathematically, the first-order conditions

∂µ/∂X₁ = 3 – λP₁

3 = λP₁

∂µ/∂X₂ = 33 – λP₂

33 = λP₂

could both hold only if 3/33 = P₁/P₂, which would happen by coincidence. Usually, the consumer will choose to be at a corner solution, spending all her money on the good for which a_i/P_i is highest (a_i is coefficient of the ith good in utility function and Pi is the price of ith good).

So, if 3/P_X1>33/P_X2, then the consumer will chose

X₁ = 5004/P₁

On the other hand if 3/P₁<33/P₂, then the consumer will chose

X₂ = 5004/P₂

In the present case we have P₁ = 37, and P₂ = 12

3/P₁ = 3/37 = 0.081

33/P₂ = 33/12 = 2.75

Since, 3/P₁<33/P_2­ so consumer will only consume X₂, and the amount of X₂ consumed by the consumer is

X₂ = 5004/P₂

X₂ = 5004/12

X₂ = 417