Part 2 beginning with "now assume" There are two commodities: z and y. Let the consumer's consumption set be Rị and his preferencerelation on his consumption set be represented by u(z, y) = VE + 3y. When his income is equal to1 and the prices of good z, Pz = 1 and of good y, Py = 1,1. Solve the utility maximization problem.2. Does the solution calculated here obey the "equal bang for bnek' condition? Explain.Now assume that the preference relation on his consumption set be represented by u(z, y) = 3V7+y.When his income is equal to 1 and the prices of good z, Pz = 1 and of good y, Py = 1,1. Solve the utility maximization problem.2. Does the solution calculated here obey the "equal bang for buck' condition? Explain.

Answer: Part 2: Given, Utility function: ux, y=3x+y Income=1 px=1py=1 (1). To solve the maximization…

Answered: Now assume that the preference relation…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Part 2 beginning with "now assume"

There are two commodities: z and y. Let the consumer's consumption set be Rị and his preference
relation on his consumption set be represented by u(z, y) = VE + 3y. When his income is equal to
1 and the prices of good z, Pz = 1 and of good y, Py = 1,
1. Solve the utility maximization problem.
2. Does the solution calculated here obey the "equal bang for bnek' condition? Explain.
Now assume that the preference relation on his consumption set be represented by u(z, y) = 3V7+y.
When his income is equal to 1 and the prices of good z, Pz = 1 and of good y, Py = 1,
1. Solve the utility maximization problem.
2. Does the solution calculated here obey the "equal bang for buck' condition? Explain.

Expert Solution

Step 1

Answer:

Part 2:

Given,

Utility function:

$u (x, y) = 3 \sqrt{x} + y$

Income=1

$p_{x} = 1 p_{y = 1}$

(1).

To solve the maximization problem let us set the Lagrange function :

$L = u (x, y) + λ (I - p_{x} x - p_{y} y) w h e r e, I = i n c o m e p_{x} = p r i c e o f g o o d x p_{y} = p r i c e o f g o o d y x = u n i t s o f g o o d x y = u n i t s o f g o o d y L = 3 \sqrt{x} + y + λ (1 - 1 x - 1 y) N o w, s o l v e f o r p a r t i a l d e r i v a t i o n o f t h e L a g r a n g e f u n c t i o n a n d s e t t h e m e q u a l t o z e r o \frac{\partial L}{\partial x} = 3 \times \frac{1}{2} x^{- \frac{1}{2}} - λ \frac{\partial L}{\partial x} = \frac{3}{2} x^{- \frac{1}{2}} - λ \frac{\partial L}{\partial x} = 0 \frac{3}{2} x^{- \frac{1}{2}} - λ = 0 \frac{3}{2} x^{- \frac{1}{2}} = λ . . . . . . e q u a t i o n 1 \frac{\partial L}{\partial y} = 1 - λ \frac{\partial L}{\partial y} = 0 1 = λ . . . . . . . . . . . e q u a t i o n 2 D i v i d e e q u a t i o n 1 b y 2 \frac{\frac{3}{2} x^{- \frac{1}{2}}}{1} = \frac{λ}{λ} \frac{3}{2} x^{- \frac{1}{2}} = 1 \frac{3}{2 x^{\frac{1}{2}}} = 1 \frac{3}{2} = x^{\frac{1}{2}} x = {(\frac{3}{2})}^{2} = \frac{9}{4} = 2.25$

x=2.25 will maximize the utility of the consumer given the income and prices.

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