### Utility Function and Expenditure Analysis#### Problem 1: Finding the Expenditure FunctionAn individual has a utility function given by $ U(X, Y) = \min\{4X^2, Y^2\} $. The task is to find the expenditure function as a function of the prices $ p_x, p_y $ and utility $ U $.The options for the expenditure function $ E(p_x, p_y, U) $ are:- a. $ E(p_x, p_y, U) = U(p_x + p_y) $- b. $ E(p_x, p_y, U) = U^2(p_x + 2p_y) $- c. $ E(p_x, p_y, U) = \sqrt{U \left(\frac{p_x}{2} + p_y\right)} $- d. $ E(p_x, p_y, U) = \sqrt{U(p_x p_y)} $The correct answer is:- c. $ E(p_x, p_y, U) = \sqrt{U \left(\frac{p_x}{2} + p_y\right)} $#### Problem 2: Utility Maximization with a Given Expenditure FunctionEugene has the following expenditure function:\[ E(p_x, p_y, U) = p_x^{1/2} p_y^{1/2} U. \]Given that Eugene has an income of 200 and the prices are $ p_x = 1 $, $ p_y = 4 $, we need to determine the maximum utility he can achieve.The options for the maximum utility are:- a. 25- b. 50- c. 100- d. 200The correct answer is:- c. 100#### Explanation of Solution to Problem 2To solve this, we use the given expenditure function:\[ E(p_x, p_y, U) = p_x^{1/2} p_y^{1/2} U. \]Substitute the given values:\[ 200 = (1)^{1/2} (4)^{1/2} U \]\[ 200 = \sqrt{1} \cdot 2 \cdot U \]\[ 200 = 2U \]\[ U = \frac{200

Note: Since you've asked multiple questions, we will solve the first question for you. If you want…

An individual has utility function U(X, Y) = min{4X2, Y2). Find the expenditure function as a function of prices Px. Py, U. %3D O a. E(p. Py, U = U(px + Py) O b. E(px. Py. U) = U²(p, + 2p,) O c. E(px, Py, U) = VUG + P,) %3D %3D O d. E(px. Py, U) = VU(PxPy) %3D Clear my choice

An individual has utility function U(X, Y) = min{4X2, Y2). Find the expenditure function as a function of prices Px. Py, U. %3D O a. E(p. Py, U = U(px + Py) O b. E(px. Py. U) = U²(p, + 2p,) O c. E(px, Py, U) = VUG + P,) %3D %3D O d. E(px. Py, U) = VU(PxPy) %3D Clear my choice

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Concept explainers

Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.

Question

### Utility Function and Expenditure Analysis

#### Problem 1: Finding the Expenditure Function
An individual has a utility function given by $ U(X, Y) = \min\{4X^2, Y^2\} $. The task is to find the expenditure function as a function of the prices $ p_x, p_y $ and utility $ U $.

The options for the expenditure function $ E(p_x, p_y, U) $ are:

- a. $ E(p_x, p_y, U) = U(p_x + p_y) $
- b. $ E(p_x, p_y, U) = U^2(p_x + 2p_y) $
- c. $ E(p_x, p_y, U) = \sqrt{U \left(\frac{p_x}{2} + p_y\right)} $
- d. $ E(p_x, p_y, U) = \sqrt{U(p_x p_y)} $

The correct answer is:
- c. $ E(p_x, p_y, U) = \sqrt{U \left(\frac{p_x}{2} + p_y\right)} $

#### Problem 2: Utility Maximization with a Given Expenditure Function
Eugene has the following expenditure function:
\[ E(p_x, p_y, U) = p_x^{1/2} p_y^{1/2} U. \]

Given that Eugene has an income of 200 and the prices are $ p_x = 1 $, $ p_y = 4 $, we need to determine the maximum utility he can achieve.

The options for the maximum utility are:

- a. 25
- b. 50
- c. 100
- d. 200

The correct answer is:
- c. 100

#### Explanation of Solution to Problem 2
To solve this, we use the given expenditure function:
\[ E(p_x, p_y, U) = p_x^{1/2} p_y^{1/2} U. \]

Substitute the given values:

\[ 200 = (1)^{1/2} (4)^{1/2} U \]
\[ 200 = \sqrt{1} \cdot 2 \cdot U \]
\[ 200 = 2U \]
\[ U = \frac{200

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