(a) Explanation The given information: The utility functions are given below: Utility ( X, Y ) = X + Y (1) The budget constrain for two goods X and Y can be written as follows. P X and P Y indicate the price of goods X and the price of good Y, respectively. ( P X × X ) + ( P Y × Y ) − Income = 0 (2) Utility is maximized at the point where the budget constraint is tangent with the indifference curves. This can be written as follows: Φ = Utility − λ ( Budget constraint function ) (3) Substitute Equation (1) and Equation (2) in Equation (3) to derive the utility maximizing function. I indicates the income. Φ = X + Y − λ ( ( P X × X ) + ( P Y × Y ) − I ) (5) Differentiate Equation (5) to derive the value of X as follows: ∂ ( Φ ) ∂ ( X ) = ∂ ( X + Y − λ ( ( P X × X ) + ( P Y × Y ) − I ) ) ∂ ( X ) 0 = 0.5 X − 0.5 − λ P X λ P X = 1 2 X − 0.5 λ = 1 2 P X X 0.5 (6) Differentiate Equation (5) to derive the value of Y as follows: ∂ ( Φ ) ∂ ( Y ) = ∂ ( X + Y − λ ( ( P X × X ) + ( P Y × Y ) − I ) ) ∂ ( Y ) 0 = 0.5 Y − 0.5 − λ P Y λ P Y = 1 2 Y − 0.5 λ = 1 2 P Y Y 0.5 (7) Differentiate Equation (5) to derive the value of λ as follows: ∂ ( Φ ) ∂ ( λ ) = ∂ ( X + Y − λ ( ( P X × X ) + ( P Y × Y ) − I ) ) ∂ ( λ ) 0 = I − ( P X × X ) − ( P Y × Y ) (8) Divide Equation (6) by Equation(7) as follows: 1 2 P X X 0 (b) To determine The quantities of X and Y. (c) To determine The marginal utility of income.

Identify the shape of the indifference curve.

Pearson eText Microeconomics -- Instant Access (Pearson+)

9th Edition

ISBN: 9780136879572

Author: Robert Pindyck, Daniel Rubinfeld

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Question

Chapter 4.A, Problem 4E

(a)

To determine