### Understanding the Utility Function and Marginal Utilities Given the utility function: \[ u(x_1, x_2) = \alpha \cdot \ln(x_1) + \beta \cdot \ln(x_2) \] - **Marginal Utility of \(x_1\):** This is calculated as the partial derivative of \(u\) with respect to \(x_1\). - **Marginal Utility of \(x_2\):** This is calculated as the partial derivative of \(u\) with respect to \(x_2\). Determine if these marginal utilities are increasing, diminishing, or constant based on \(\alpha\), \(\beta\), and the function's characteristics. ### Indifference Curve Details An indifference curve represents a set of bundles of goods between which a consumer is indifferent. - The graph shows an indifference curve for the specified utility function with \( u \approx 1.565 \). - **Bundle \(G(5, 4)\):** A point on the indifference curve with coordinates \(x_1 = 5\) and \(x_2 = 4\). - **Bundle \(H(10, 0.25)\):** Another point on the curve with coordinates \(x_1 = 10\) and \(x_2 = 0.25\). ### Calculating the Marginal Rate of Substitution (MRS) The \(MRS_{12}\) is calculated at different bundles and helps understand consumer preference. - **At Bundle \(G(5, 4)\):** Calculate \(MRS_{12}\) using \(\alpha = 0.8\) and \(\beta = 0.2\). - **At Bundle \(H(10, 0.25)\):** Calculate \(MRS_{12}\) with the same values for \(\alpha\) and \(\beta\). Discuss what these values imply about the convexity of preferences regardless of the graph’s depiction. ### Graph Explanation The graph is a standard utility representation: - **Axes:** \(x_1\) on the horizontal axis and \(x_2\) on the vertical axis. - **Curve:** Shows constant utility \( u \approx 1.565 \). - **Dotted Lines:** Highlight the values at Bundles \(G\) and \(H\). Understanding how these concepts and calculations
### Understanding the Utility Function and Marginal Utilities Given the utility function: \[ u(x_1, x_2) = \alpha \cdot \ln(x_1) + \beta \cdot \ln(x_2) \] - **Marginal Utility of \(x_1\):** This is calculated as the partial derivative of \(u\) with respect to \(x_1\). - **Marginal Utility of \(x_2\):** This is calculated as the partial derivative of \(u\) with respect to \(x_2\). Determine if these marginal utilities are increasing, diminishing, or constant based on \(\alpha\), \(\beta\), and the function's characteristics. ### Indifference Curve Details An indifference curve represents a set of bundles of goods between which a consumer is indifferent. - The graph shows an indifference curve for the specified utility function with \( u \approx 1.565 \). - **Bundle \(G(5, 4)\):** A point on the indifference curve with coordinates \(x_1 = 5\) and \(x_2 = 4\). - **Bundle \(H(10, 0.25)\):** Another point on the curve with coordinates \(x_1 = 10\) and \(x_2 = 0.25\). ### Calculating the Marginal Rate of Substitution (MRS) The \(MRS_{12}\) is calculated at different bundles and helps understand consumer preference. - **At Bundle \(G(5, 4)\):** Calculate \(MRS_{12}\) using \(\alpha = 0.8\) and \(\beta = 0.2\). - **At Bundle \(H(10, 0.25)\):** Calculate \(MRS_{12}\) with the same values for \(\alpha\) and \(\beta\). Discuss what these values imply about the convexity of preferences regardless of the graph’s depiction. ### Graph Explanation The graph is a standard utility representation: - **Axes:** \(x_1\) on the horizontal axis and \(x_2\) on the vertical axis. - **Curve:** Shows constant utility \( u \approx 1.565 \). - **Dotted Lines:** Highlight the values at Bundles \(G\) and \(H\). Understanding how these concepts and calculations
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![### Understanding the Utility Function and Marginal Utilities
Given the utility function:
\[ u(x_1, x_2) = \alpha \cdot \ln(x_1) + \beta \cdot \ln(x_2) \]
- **Marginal Utility of \(x_1\):** This is calculated as the partial derivative of \(u\) with respect to \(x_1\).
- **Marginal Utility of \(x_2\):** This is calculated as the partial derivative of \(u\) with respect to \(x_2\).
Determine if these marginal utilities are increasing, diminishing, or constant based on \(\alpha\), \(\beta\), and the function's characteristics.
### Indifference Curve Details
An indifference curve represents a set of bundles of goods between which a consumer is indifferent.
- The graph shows an indifference curve for the specified utility function with \( u \approx 1.565 \).
- **Bundle \(G(5, 4)\):** A point on the indifference curve with coordinates \(x_1 = 5\) and \(x_2 = 4\).
- **Bundle \(H(10, 0.25)\):** Another point on the curve with coordinates \(x_1 = 10\) and \(x_2 = 0.25\).
### Calculating the Marginal Rate of Substitution (MRS)
The \(MRS_{12}\) is calculated at different bundles and helps understand consumer preference.
- **At Bundle \(G(5, 4)\):** Calculate \(MRS_{12}\) using \(\alpha = 0.8\) and \(\beta = 0.2\).
- **At Bundle \(H(10, 0.25)\):** Calculate \(MRS_{12}\) with the same values for \(\alpha\) and \(\beta\).
Discuss what these values imply about the convexity of preferences regardless of the graph’s depiction.
### Graph Explanation
The graph is a standard utility representation:
- **Axes:** \(x_1\) on the horizontal axis and \(x_2\) on the vertical axis.
- **Curve:** Shows constant utility \( u \approx 1.565 \).
- **Dotted Lines:** Highlight the values at Bundles \(G\) and \(H\).
Understanding how these concepts and calculations](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec3b779b-d0d6-4768-9dcd-21f6961c75e0%2F34360603-149e-410d-bedc-c3b805bcff1a%2Foeaqtds_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Utility Function and Marginal Utilities
Given the utility function:
\[ u(x_1, x_2) = \alpha \cdot \ln(x_1) + \beta \cdot \ln(x_2) \]
- **Marginal Utility of \(x_1\):** This is calculated as the partial derivative of \(u\) with respect to \(x_1\).
- **Marginal Utility of \(x_2\):** This is calculated as the partial derivative of \(u\) with respect to \(x_2\).
Determine if these marginal utilities are increasing, diminishing, or constant based on \(\alpha\), \(\beta\), and the function's characteristics.
### Indifference Curve Details
An indifference curve represents a set of bundles of goods between which a consumer is indifferent.
- The graph shows an indifference curve for the specified utility function with \( u \approx 1.565 \).
- **Bundle \(G(5, 4)\):** A point on the indifference curve with coordinates \(x_1 = 5\) and \(x_2 = 4\).
- **Bundle \(H(10, 0.25)\):** Another point on the curve with coordinates \(x_1 = 10\) and \(x_2 = 0.25\).
### Calculating the Marginal Rate of Substitution (MRS)
The \(MRS_{12}\) is calculated at different bundles and helps understand consumer preference.
- **At Bundle \(G(5, 4)\):** Calculate \(MRS_{12}\) using \(\alpha = 0.8\) and \(\beta = 0.2\).
- **At Bundle \(H(10, 0.25)\):** Calculate \(MRS_{12}\) with the same values for \(\alpha\) and \(\beta\).
Discuss what these values imply about the convexity of preferences regardless of the graph’s depiction.
### Graph Explanation
The graph is a standard utility representation:
- **Axes:** \(x_1\) on the horizontal axis and \(x_2\) on the vertical axis.
- **Curve:** Shows constant utility \( u \approx 1.565 \).
- **Dotted Lines:** Highlight the values at Bundles \(G\) and \(H\).
Understanding how these concepts and calculations
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