Donald derives utility from only two goods, carrots (Q.) and donuts (Qa). His utility function is as follows: U(Q..Qa) = Q.*Q« Donald has an income (1) of $120 and the price of carrots (P.) and donuts (Pa) are both $1. a. What is Donald's budget constraint? b. What is Donald's utility-maximizing condition? c. What quantities of carrots and donuts will maximize Donald's utility?
Donald derives utility from only two goods, carrots (Q.) and donuts (Qa). His utility function is as follows: U(Q..Qa) = Q.*Q« Donald has an income (1) of $120 and the price of carrots (P.) and donuts (Pa) are both $1. a. What is Donald's budget constraint? b. What is Donald's utility-maximizing condition? c. What quantities of carrots and donuts will maximize Donald's utility?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Explain why two indifference
![**Donald Derives Utility from Carrots and Donuts**
Donald derives utility from two goods, carrots (\(Q_c\)) and donuts (\(Q_d\)). His utility function is defined as:
\[ U(Q_c, Q_d) = Q_c \times Q_d \]
Donald has an income (\(I\)) of $120, and the price of carrots (\(P_c\)) and donuts (\(P_d\)) are both $1.
**a. What is Donald's budget constraint?**
Donald's budget constraint can be expressed as:
\[ P_c \times Q_c + P_d \times Q_d = I \]
Since \(P_c\) and \(P_d\) are both $1:
\[ Q_c + Q_d = 120 \]
**b. What is Donald's utility-maximizing condition?**
Donald's utility-maximizing condition involves equalizing the marginal utility per dollar spent on each good. Therefore:
\[ \frac{MU_c}{P_c} = \frac{MU_d}{P_d} \]
Given the utility function \(U(Q_c, Q_d) = Q_c \times Q_d\), the marginal utilities are:
\[ MU_c = Q_d \quad \text{and} \quad MU_d = Q_c \]
So, the condition becomes:
\[ \frac{Q_d}{1} = \frac{Q_c}{1} \]
Therefore, \(Q_d = Q_c\).
**c. What quantities of carrots and donuts will maximize Donald's utility?**
Using the budget constraint and utility-maximizing condition:
\[ Q_c + Q_d = 120 \quad \text{and} \quad Q_c = Q_d \]
Substitute \(Q_c = Q_d\) into the budget constraint:
\[ 2Q_c = 120 \]
\[ Q_c = 60 \quad \text{and} \quad Q_d = 60 \]
**d. Holding Donald's income and \(P_d\) constant at $120 and $1 respectively, what is Donald's demand function for carrots?**
Donald's demand function for carrots, holding income and price of donuts constant, depends solely on how the price of carrots affects quantity. Since \(\frac{P_c}{P_d} = 1\), demand for carrots remains:
\[ Q_c = \frac{I}{2P_c} \]
**e. Suppose that a tax](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbe4d004-b0c7-4fb9-bd84-121b248ea820%2F4347ad4a-bb80-4268-8f89-1d95f52316bf%2Fllubuz_processed.png&w=3840&q=75)
Transcribed Image Text:**Donald Derives Utility from Carrots and Donuts**
Donald derives utility from two goods, carrots (\(Q_c\)) and donuts (\(Q_d\)). His utility function is defined as:
\[ U(Q_c, Q_d) = Q_c \times Q_d \]
Donald has an income (\(I\)) of $120, and the price of carrots (\(P_c\)) and donuts (\(P_d\)) are both $1.
**a. What is Donald's budget constraint?**
Donald's budget constraint can be expressed as:
\[ P_c \times Q_c + P_d \times Q_d = I \]
Since \(P_c\) and \(P_d\) are both $1:
\[ Q_c + Q_d = 120 \]
**b. What is Donald's utility-maximizing condition?**
Donald's utility-maximizing condition involves equalizing the marginal utility per dollar spent on each good. Therefore:
\[ \frac{MU_c}{P_c} = \frac{MU_d}{P_d} \]
Given the utility function \(U(Q_c, Q_d) = Q_c \times Q_d\), the marginal utilities are:
\[ MU_c = Q_d \quad \text{and} \quad MU_d = Q_c \]
So, the condition becomes:
\[ \frac{Q_d}{1} = \frac{Q_c}{1} \]
Therefore, \(Q_d = Q_c\).
**c. What quantities of carrots and donuts will maximize Donald's utility?**
Using the budget constraint and utility-maximizing condition:
\[ Q_c + Q_d = 120 \quad \text{and} \quad Q_c = Q_d \]
Substitute \(Q_c = Q_d\) into the budget constraint:
\[ 2Q_c = 120 \]
\[ Q_c = 60 \quad \text{and} \quad Q_d = 60 \]
**d. Holding Donald's income and \(P_d\) constant at $120 and $1 respectively, what is Donald's demand function for carrots?**
Donald's demand function for carrots, holding income and price of donuts constant, depends solely on how the price of carrots affects quantity. Since \(\frac{P_c}{P_d} = 1\), demand for carrots remains:
\[ Q_c = \frac{I}{2P_c} \]
**e. Suppose that a tax
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