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Marginal rate of substitution

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

678_marginal rate of substitution.png

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