Question
Suppose that Bridget and Erin spend their incomes on two goods, food (F) and clothing (C).
Bridgetโs preferences are represented by the utility function ๐(๐น, ๐ถ) = 10๐น๐ถ, while Erinโs
preferences are represented by the utility function ๐(๐น, ๐ถ) = 15๐น
2๐ถ
2
.
a) With food on the horizontal axis and clothing on the vertical axis, identify on a graph the
set of points that give Bridget the same level of utility as the bundle (10, 5). Do the same
for Erin on a separate graph.
b) On the same two graphs, identify the set of bundles that give Bridget and Erin the same
level of utility as the bundle (15, 8).
c) Do you think Bridget and Erin have the same preferences or different preferences?
Explain
Bridgetโs preferences are represented by the utility function ๐(๐น, ๐ถ) = 10๐น๐ถ, while Erinโs
preferences are represented by the utility function ๐(๐น, ๐ถ) = 15๐น
2๐ถ
2
.
a) With food on the horizontal axis and clothing on the vertical axis, identify on a graph the
set of points that give Bridget the same level of utility as the bundle (10, 5). Do the same
for Erin on a separate graph.
b) On the same two graphs, identify the set of bundles that give Bridget and Erin the same
level of utility as the bundle (15, 8).
c) Do you think Bridget and Erin have the same preferences or different preferences?
Explain
Answers
GPT-4o mini
To analyze the preferences of Bridget and Erin based on their utility functions, we will first calculate the utility levels at the given bundles, and then draw the corresponding indifference curves.
### a) Indifference Curves for Given Bundles
**Bridget's Utility Function:**
\[ U_B(F, C) = 10FC \]
Calculating utility for Bridget at the bundle (10, 5):
\[ U_B(10, 5) = 10(10)(5) = 500 \]
**Indifference Curve for Bridget:**
The indifference curve for Bridget where \(U_B = 500\) can be expressed as:
\[ 10FC = 500 \]
\[ FC = 50 \]
To find the indifference curve, we can rearrange to express \(C\) in terms of \(F\):
\[ C = \frac{50}{F} \]
**Graph for Bridget:**
Draw the curve \(C = \frac{50}{F}\) in the first quadrant with food on the x-axis and clothing on the y-axis.
---
**Erin's Utility Function:**
\[ U_E(F, C) = 15F^2C^2 \]
Calculating utility for Erin at the bundle (10, 5):
\[ U_E(10, 5) = 15(10^2)(5^2) = 15(100)(25) = 37500 \]
**Indifference Curve for Erin:**
The indifference curve for Erin where \(U_E = 37500\) can be expressed as:
\[ 15F^2C^2 = 37500 \]
\[ F^2C^2 = 2500 \]
Taking the square root:
\[ FC = 50 \]
Again, rearranging gives:
\[ C = \frac{50}{F} \]
**Graph for Erin:**
Draw the curve \(C = \frac{50}{F}\) in the first quadrant with food on the x-axis and clothing on the y-axis.
### b) Indifference Curves for Different Bundles
Now we will calculate the utility for both individuals at the second bundle (15, 8).
**Bridget's Utility at (15, 8):**
\[ U_B(15, 8) = 10(15)(8) = 1200 \]
**Indifference Curve for Bridget:**
For \(U_B = 1200\):
\[ 10FC = 1200 \]
\[ FC = 120 \]
Rearranging gives:
\[ C = \frac{120}{F} \]
**Graph for Bridget:**
Draw the second curve \(C = \frac{120}{F}\) on the same graph.
---
**Erin's Utility at (15, 8):**
\[ U_E(15, 8) = 15(15^2)(8^2) = 15(225)(64) = 21600 \]
**Indifference Curve for Erin:**
For \(U_E = 21600\):
\[ 15F^2C^2 = 21600 \]
\[ F^2C^2 = 1440 \]
Taking the square root gives:
\[ FC = 60 \]
Rearranging gives:
\[ C = \frac{60}{F} \]
**Graph for Erin:**
Draw the second curve \(C = \frac{60}{F}\) on the same graph as the first.
### c) Preferences Analysis
Bridget and Erin have different preferences despite both representing their preferences with indifference curves.
1. **Utility Functions:** The functional forms of their utility functions are different. Bridget has a linear product form capturing her preference for the combinations of food and clothing, whereas Erin has a squared term indicating increasing marginal utility from her consumption of both goods.
2. **Different Shapes of Indifference Curves:**
- Bridget's indifference curves have a rectangular hyperbola shape, indicating a constant marginal rate of substitution.
- Erin's curves would also be hyperbolic, but the rates of substitution would change more dramatically due to the squares in her utility function, indicating a preference for balance in her consumption that grows more rapidly.
3. **Different Levels of Utility:** At every level of consumption, the same utility level may not represent the same bundles of goods for both individuals. Therefore, the nature of their preferences and how they trade-off between F and C differ significantly.
As a result, while both individuals derive utility from the same types of goods, they express their preferences in different forms, leading to different indifference curve shapes and indicating that they indeed have different preferences.
### a) Indifference Curves for Given Bundles
**Bridget's Utility Function:**
\[ U_B(F, C) = 10FC \]
Calculating utility for Bridget at the bundle (10, 5):
\[ U_B(10, 5) = 10(10)(5) = 500 \]
**Indifference Curve for Bridget:**
The indifference curve for Bridget where \(U_B = 500\) can be expressed as:
\[ 10FC = 500 \]
\[ FC = 50 \]
To find the indifference curve, we can rearrange to express \(C\) in terms of \(F\):
\[ C = \frac{50}{F} \]
**Graph for Bridget:**
Draw the curve \(C = \frac{50}{F}\) in the first quadrant with food on the x-axis and clothing on the y-axis.
---
**Erin's Utility Function:**
\[ U_E(F, C) = 15F^2C^2 \]
Calculating utility for Erin at the bundle (10, 5):
\[ U_E(10, 5) = 15(10^2)(5^2) = 15(100)(25) = 37500 \]
**Indifference Curve for Erin:**
The indifference curve for Erin where \(U_E = 37500\) can be expressed as:
\[ 15F^2C^2 = 37500 \]
\[ F^2C^2 = 2500 \]
Taking the square root:
\[ FC = 50 \]
Again, rearranging gives:
\[ C = \frac{50}{F} \]
**Graph for Erin:**
Draw the curve \(C = \frac{50}{F}\) in the first quadrant with food on the x-axis and clothing on the y-axis.
### b) Indifference Curves for Different Bundles
Now we will calculate the utility for both individuals at the second bundle (15, 8).
**Bridget's Utility at (15, 8):**
\[ U_B(15, 8) = 10(15)(8) = 1200 \]
**Indifference Curve for Bridget:**
For \(U_B = 1200\):
\[ 10FC = 1200 \]
\[ FC = 120 \]
Rearranging gives:
\[ C = \frac{120}{F} \]
**Graph for Bridget:**
Draw the second curve \(C = \frac{120}{F}\) on the same graph.
---
**Erin's Utility at (15, 8):**
\[ U_E(15, 8) = 15(15^2)(8^2) = 15(225)(64) = 21600 \]
**Indifference Curve for Erin:**
For \(U_E = 21600\):
\[ 15F^2C^2 = 21600 \]
\[ F^2C^2 = 1440 \]
Taking the square root gives:
\[ FC = 60 \]
Rearranging gives:
\[ C = \frac{60}{F} \]
**Graph for Erin:**
Draw the second curve \(C = \frac{60}{F}\) on the same graph as the first.
### c) Preferences Analysis
Bridget and Erin have different preferences despite both representing their preferences with indifference curves.
1. **Utility Functions:** The functional forms of their utility functions are different. Bridget has a linear product form capturing her preference for the combinations of food and clothing, whereas Erin has a squared term indicating increasing marginal utility from her consumption of both goods.
2. **Different Shapes of Indifference Curves:**
- Bridget's indifference curves have a rectangular hyperbola shape, indicating a constant marginal rate of substitution.
- Erin's curves would also be hyperbolic, but the rates of substitution would change more dramatically due to the squares in her utility function, indicating a preference for balance in her consumption that grows more rapidly.
3. **Different Levels of Utility:** At every level of consumption, the same utility level may not represent the same bundles of goods for both individuals. Therefore, the nature of their preferences and how they trade-off between F and C differ significantly.
As a result, while both individuals derive utility from the same types of goods, they express their preferences in different forms, leading to different indifference curve shapes and indicating that they indeed have different preferences.