Marginal rate of substitution

Question

Problem:

Luke likes to consumer CDs (good1) and pizzas (good 2). His preference over both goods is given by the utility function

U(x1; x2) = x²₁ x⁴₂.

If Luke allocates $200 to spend on both goods and if a case of CDs costs $20 and a pizza costs $10, how many cases of CDs and pizzas would he consume in order to maximize his utility subject to his income. Show your work and illustrate your answer graphically.

Summary:

The problem in economics in price theory deals with deriving maximum marginal utility and marginal rate of substitution.

Answer:

U(x₁, x₂) = x₁²x₂⁴

P₁ = 20, P₂ = 10

MUx₁ = 2x₁x₂⁴

MUx₂ = 4x₁²x₂³

Therefore, MRS = ½(x₂/x₁)

Now, for optimization, MRS = P₁/P₂ = 20/10 = 2

Now, ½(x₂/x₁) = 2

x₂ = 4x₁

Putting this value into the budget equation:

20x₁ + 10x₂ = 200

20x₁ + 40x₁ = 200

x₁ = 10/3

and x₂ = 40/3