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Derive the Marshallian demand function.

Derive the Marshallian demand function.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Suppose that a consumer is given by the utility function \( U(q_x, q_y) = 2 \ln q_x + \ln q_y \). This expression represents a consumer's utility function, where: - \( q_x \) and \( q_y \) are quantities of goods \( x \) and \( y \), respectively. - The utility function \( U \) is a logarithmic function, showing that utility depends logarithmically on the quantities of the two goods. - The coefficient 2 in front of \( \ln q_x \) indicates that the utility derived from each unit of good \( x \) is twice as impactful as that from good \( y \). In this type of function, the consumer gains satisfaction or utility from consuming goods \( x \) and \( y \), and the goal is typically to maximize this utility given budget constraints.
Transcribed Image Text:Suppose that a consumer is given by the utility function \( U(q_x, q_y) = 2 \ln q_x + \ln q_y \). This expression represents a consumer's utility function, where: - \( q_x \) and \( q_y \) are quantities of goods \( x \) and \( y \), respectively. - The utility function \( U \) is a logarithmic function, showing that utility depends logarithmically on the quantities of the two goods. - The coefficient 2 in front of \( \ln q_x \) indicates that the utility derived from each unit of good \( x \) is twice as impactful as that from good \( y \). In this type of function, the consumer gains satisfaction or utility from consuming goods \( x \) and \( y \), and the goal is typically to maximize this utility given budget constraints.
**Derive the Marshallian demand function.** To derive the Marshallian demand function, follow these steps: 1. **Understand the Consumer's Objective:** - Consumers aim to maximize their utility subject to a budget constraint. 2. **Set Up the Utility Maximization Problem:** - Let \( U(x_1, x_2) \) be the utility function where \( x_1 \) and \( x_2 \) are the quantities of two goods. - The budget constraint is \( p_1x_1 + p_2x_2 = M \), where \( p_1 \) and \( p_2 \) are prices of the goods and \( M \) is the consumer's income. 3. **Formulate the Lagrangian:** - \(\mathcal{L}(x_1, x_2, \lambda) = U(x_1, x_2) + \lambda (M - p_1x_1 - p_2x_2)\) 4. **Find the First-Order Conditions:** - Differentiate the Lagrangian with respect to \( x_1 \), \( x_2 \), and \(\lambda\). - \(\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial U}{\partial x_1} - \lambda p_1 = 0\) - \(\frac{\partial \mathcal{L}}{\partial x_2} = \frac{\partial U}{\partial x_2} - \lambda p_2 = 0\) - \(\frac{\partial \mathcal{L}}{\partial \lambda} = M - p_1x_1 - p_2x_2 = 0\) 5. **Solve the System of Equations:** - Use these equations to solve for \( x_1 \) and \( x_2 \) in terms of prices \( p_1, p_2 \) and income \( M \). 6. **Interpret the Marshallian Demand:** - The solutions \( x_1^*(p_1, p_2, M) \) and \( x_2^*(p_1, p_2, M) \) represent the Marshallian demand functions for goods 1 and 2. This approach derives the
Transcribed Image Text:**Derive the Marshallian demand function.** To derive the Marshallian demand function, follow these steps: 1. **Understand the Consumer's Objective:** - Consumers aim to maximize their utility subject to a budget constraint. 2. **Set Up the Utility Maximization Problem:** - Let \( U(x_1, x_2) \) be the utility function where \( x_1 \) and \( x_2 \) are the quantities of two goods. - The budget constraint is \( p_1x_1 + p_2x_2 = M \), where \( p_1 \) and \( p_2 \) are prices of the goods and \( M \) is the consumer's income. 3. **Formulate the Lagrangian:** - \(\mathcal{L}(x_1, x_2, \lambda) = U(x_1, x_2) + \lambda (M - p_1x_1 - p_2x_2)\) 4. **Find the First-Order Conditions:** - Differentiate the Lagrangian with respect to \( x_1 \), \( x_2 \), and \(\lambda\). - \(\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial U}{\partial x_1} - \lambda p_1 = 0\) - \(\frac{\partial \mathcal{L}}{\partial x_2} = \frac{\partial U}{\partial x_2} - \lambda p_2 = 0\) - \(\frac{\partial \mathcal{L}}{\partial \lambda} = M - p_1x_1 - p_2x_2 = 0\) 5. **Solve the System of Equations:** - Use these equations to solve for \( x_1 \) and \( x_2 \) in terms of prices \( p_1, p_2 \) and income \( M \). 6. **Interpret the Marshallian Demand:** - The solutions \( x_1^*(p_1, p_2, M) \) and \( x_2^*(p_1, p_2, M) \) represent the Marshallian demand functions for goods 1 and 2. This approach derives the
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