Math Problem Statement
What is the demand function for x and y? U=u(x,y) = xy^5. Edgar's income is $300, the price of x is $10, and the price of y is $5.
Solution
To derive the demand functions for and based on Edgar's utility function and the given constraints, we'll use the utility maximization approach. Edgar maximizes his utility subject to his budget constraint. Here's how we proceed:
Step 1: Utility Function
The utility function is given as: where and represent quantities of two goods, and represents total utility.
Step 2: Budget Constraint
Edgar's income is $300, the price of is $10, and the price of is $5. So, the budget constraint is: This represents the total expenditure on the two goods.
Step 3: Setting Up the Lagrangian
To maximize utility subject to the budget constraint, we use the Lagrangian method. The Lagrangian is given by: where is the Lagrange multiplier.
Step 4: First-Order Conditions
We take the partial derivatives of with respect to , , and , and set them equal to zero.
-
With respect to :
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With respect to :
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With respect to :
Step 5: Solving the System of Equations
From the first condition, we have: From the second condition, we have: Substituting into , we get:
Now substitute into the budget constraint:
Thus, . Now, using , we get:
Step 6: Demand Functions
The demand functions are: Thus, the demand for and is fixed based on the given income and prices.
Summary:
- Demand for :
- Demand for :
Would you like further details on the steps or the concepts used here?
Here are 5 related questions you might find useful:
- How do you derive a demand function using a Cobb-Douglas utility function?
- What is the relationship between income and demand in this case?
- How do changes in the prices of or affect the demand functions?
- What is the meaning of the Lagrange multiplier in this optimization problem?
- How do you solve for demand when the utility function is not Cobb-Douglas?
Tip: In utility maximization problems, remember to use the budget constraint to eliminate one of the variables, simplifying the system of equations for easier solving.
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Math Problem Analysis
Mathematical Concepts
Utility Maximization
Demand Function
Lagrangian Optimization
Partial Derivatives
Budget Constraint
Formulas
Utility function: U(x, y) = xy^5
Budget constraint: 10x + 5y = 300
Lagrangian: L = xy^5 + λ(300 - 10x - 5y)
Theorems
Lagrange Multiplier Method
Suitable Grade Level
Undergraduate (Economics or Intermediate Calculus level)
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