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- Demand theory: A. From the indifference curve/budget constraint diagram, derive the individual's demand curve for x1.

- Demand theory: A. From the indifference curve/budget constraint diagram, derive the individual's demand curve for x1.

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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please answer question a b c

### Demand Theory #### A. Deriving the Individual’s Demand Curve for \(x_1\) To derive the demand curve for \(x_1\), we start with an indifference curve and budget constraint diagram. 1. **Indifference Curve**: Represents various combinations of two goods that provide the individual with the same level of satisfaction. 2. **Budget Constraint**: Shows all the combinations of two goods that the individual can afford given their income and the prices of the goods. By analyzing the point where the highest indifference curve is tangent to the budget constraint, we can determine the optimal consumption bundle. As the price of \(x_1\) changes, the budget constraint pivots, and a new tangency point is found. Plotting these new bundles as the price changes helps to derive the demand curve for \(x_1\). There is no graph or diagram provided in this image, but typically, an indifference curve is a downward-sloping convex curve, and a budget constraint is a straight line. The point of tangency between the two illustrates the equilibrium point where utility is maximized given the budget constraint. This point shifts as the price of \(x_1\) varies, giving us data points for the demand curve.
Transcribed Image Text:### Demand Theory #### A. Deriving the Individual’s Demand Curve for \(x_1\) To derive the demand curve for \(x_1\), we start with an indifference curve and budget constraint diagram. 1. **Indifference Curve**: Represents various combinations of two goods that provide the individual with the same level of satisfaction. 2. **Budget Constraint**: Shows all the combinations of two goods that the individual can afford given their income and the prices of the goods. By analyzing the point where the highest indifference curve is tangent to the budget constraint, we can determine the optimal consumption bundle. As the price of \(x_1\) changes, the budget constraint pivots, and a new tangency point is found. Plotting these new bundles as the price changes helps to derive the demand curve for \(x_1\). There is no graph or diagram provided in this image, but typically, an indifference curve is a downward-sloping convex curve, and a budget constraint is a straight line. The point of tangency between the two illustrates the equilibrium point where utility is maximized given the budget constraint. This point shifts as the price of \(x_1\) varies, giving us data points for the demand curve.
### Econometrics Problem Set **Instructions:** Please answer the following questions based on the provided economic model. 1. Graphically derive the market demand curve for \( x_1 \) in a universe with only two economic agents (individuals). 2. Let \( U(x_1, x_2) = x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} \), the price of \( x_1 \) equal \( p_1 \), and the price of \( x_2 \) equal \( p_2 \). Income equals \( M \). Derive the individual demand curves for \( x_1 \) and \( x_2 \) in terms of income and prices. 3. What are the optimal quantities if \( p_1 = 4 \), \( p_2 = 2 \), and \( M \) equals $1,000? **Solutions:** ### 1. Market Demand Curve for \( x_1 \) - Construct a graphical representation considering two individuals in the market. - Each individual's demand curve should be plotted, and the market demand curve will be the horizontal summation of both individual demand curves. ### 2. Deriving Individual Demand Curves Given the utility function \( U(x_1, x_2) = x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} \), budget constraint \( p_1 x_1 + p_2 x_2 = M \): - Use the Lagrangian method to find the optimal consumption bundles for each individual. - Set up the Lagrangian: \[ \mathcal{L} = x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} + \lambda (M - p_1 x_1 - p_2 x_2) \] - Solve for \( x_1 \) and \( x_2 \) by setting partial derivatives equal to zero. ### 3. Optimal Quantities Calculation Using the derived demand functions, substitute the given values \( p_1 = 4 \), \( p_2 = 2 \), and \( M = 1,000 \) into the equations to find the optimal quantities of \( x_1 \) and \( x_2 \). **Note:** Provide all mathematical steps and derivations in
Transcribed Image Text:### Econometrics Problem Set **Instructions:** Please answer the following questions based on the provided economic model. 1. Graphically derive the market demand curve for \( x_1 \) in a universe with only two economic agents (individuals). 2. Let \( U(x_1, x_2) = x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} \), the price of \( x_1 \) equal \( p_1 \), and the price of \( x_2 \) equal \( p_2 \). Income equals \( M \). Derive the individual demand curves for \( x_1 \) and \( x_2 \) in terms of income and prices. 3. What are the optimal quantities if \( p_1 = 4 \), \( p_2 = 2 \), and \( M \) equals $1,000? **Solutions:** ### 1. Market Demand Curve for \( x_1 \) - Construct a graphical representation considering two individuals in the market. - Each individual's demand curve should be plotted, and the market demand curve will be the horizontal summation of both individual demand curves. ### 2. Deriving Individual Demand Curves Given the utility function \( U(x_1, x_2) = x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} \), budget constraint \( p_1 x_1 + p_2 x_2 = M \): - Use the Lagrangian method to find the optimal consumption bundles for each individual. - Set up the Lagrangian: \[ \mathcal{L} = x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} + \lambda (M - p_1 x_1 - p_2 x_2) \] - Solve for \( x_1 \) and \( x_2 \) by setting partial derivatives equal to zero. ### 3. Optimal Quantities Calculation Using the derived demand functions, substitute the given values \( p_1 = 4 \), \( p_2 = 2 \), and \( M = 1,000 \) into the equations to find the optimal quantities of \( x_1 \) and \( x_2 \). **Note:** Provide all mathematical steps and derivations in
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