**Compensation for a Rise in Price**Charlie’s utility over goods X and Y is given by the function U(X,Y) = XY, where \( P_x = P_y = 1 \) and m = 100. Due to the pandemic, the price of X (\( P_x \)) doubled. The government wants to compensate people like Charlie for their suffering due to the rise in price and offer them a payment.You are hired to determine how much to offer. Keep in mind that the government does not want to overpay but aims to offer only the necessary amount and no more.a. Use the Cobb-Douglas “shortcut” to find the demand for X and Y.b. Charlie’s indirect utility function is given by V(\( P_x, P_y, m \)) = ____________________.c. Fill in the table below:\[\begin{array}{|c|c|c|c|c|c|}\hlineP_x & P_y & m & X^* & Y^* & U^* \\\hline\text{Before the price increase} & & & & & \\\hline\text{Price of X doubled} & & & & & \\\hline\text{Price of X doubled. The government pays Charlie to allow him to afford the old bundle} & & & & & \\\hline\text{Price of X doubled. The government pays Charlie to allow him to reach the old utility level} & & & & & \\\hline\end{array}\]d. What is the right compensation?

Answered: Compensation for a rise in price…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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**Compensation for a Rise in Price**

Charlie’s utility over goods X and Y is given by the function U(X,Y) = XY, where \( P_x = P_y = 1 \) and m = 100. Due to the pandemic, the price of X (\( P_x \)) doubled. The government wants to compensate people like Charlie for their suffering due to the rise in price and offer them a payment.

You are hired to determine how much to offer. Keep in mind that the government does not want to overpay but aims to offer only the necessary amount and no more.

a. Use the Cobb-Douglas “shortcut” to find the demand for X and Y.

b. Charlie’s indirect utility function is given by V(\( P_x, P_y, m \)) = ____________________.

c. Fill in the table below:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
P_x & P_y & m & X^* & Y^* & U^* \\
\hline
\text{Before the price increase} & & & & & \\
\hline
\text{Price of X doubled} & & & & & \\
\hline
\text{Price of X doubled. The government pays Charlie to allow him to afford the old bundle} & & & & & \\
\hline
\text{Price of X doubled. The government pays Charlie to allow him to reach the old utility level} & & & & & \\
\hline
\end{array}
\]

d. What is the right compensation?

Expert Solution

Introduction

Here we are given the well behaved utility function of the consumer and also the budget of the consumer. And hence the consumer will choose his consumption bundle such that his utility gets maximized within given budget.

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d. What is the right compensation?
e. The graph below shows Charlie’s indifference curves and the original budget line. Illustrate your answer on the graph and show the optimal points that you found above.

**Graph Description:**

- The graph is a grid with three downward-sloping indifference curves labeled U1, U2, and U3. These curves represent different levels of utility.
- The indifference curves are convex to the origin, illustrating that as one moves along the curve, the rate at which one good is substituted for the other changes.
- The original budget line intersects these indifference curves, indicating combinations of two goods that Charlie can afford.
- The points where the budget line is tangent to the indifference curves represent the optimal points of consumption that maximize Charlie's utility given the budget constraint.

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