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 xx and yy 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: U(x,y)=xy5U(x, y) = x y^5 where xx and yy represent quantities of two goods, and UU represents total utility.

Step 2: Budget Constraint

Edgar's income is $300, the price of xx is $10, and the price of yy is $5. So, the budget constraint is: 10x+5y=30010x + 5y = 300 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: L=xy5+λ(30010x5y)\mathcal{L} = x y^5 + \lambda (300 - 10x - 5y) where λ\lambda is the Lagrange multiplier.

Step 4: First-Order Conditions

We take the partial derivatives of L\mathcal{L} with respect to xx, yy, and λ\lambda, and set them equal to zero.

  1. With respect to xx: Lx=y510λ=0y5=10λ\frac{\partial \mathcal{L}}{\partial x} = y^5 - 10\lambda = 0 \quad \Rightarrow \quad y^5 = 10\lambda

  2. With respect to yy: Ly=5xy45λ=0xy4=λ\frac{\partial \mathcal{L}}{\partial y} = 5x y^4 - 5\lambda = 0 \quad \Rightarrow \quad x y^4 = \lambda

  3. With respect to λ\lambda: Lλ=30010x5y=0\frac{\partial \mathcal{L}}{\partial \lambda} = 300 - 10x - 5y = 0

Step 5: Solving the System of Equations

From the first condition, we have: y5=10λy^5 = 10\lambda From the second condition, we have: xy4=λx y^4 = \lambda Substituting λ=xy4\lambda = x y^4 into y5=10λy^5 = 10\lambda, we get: y5=10xy4y=10xy^5 = 10x y^4 \quad \Rightarrow \quad y = 10x

Now substitute y=10xy = 10x into the budget constraint: 10x+5(10x)=30010x+50x=30060x=300x=510x + 5(10x) = 300 \quad \Rightarrow \quad 10x + 50x = 300 \quad \Rightarrow \quad 60x = 300 \quad \Rightarrow \quad x = 5

Thus, x=5x = 5. Now, using y=10xy = 10x, we get: y=10(5)=50y = 10(5) = 50

Step 6: Demand Functions

The demand functions are: x=5andy=50x = 5 \quad \text{and} \quad y = 50 Thus, the demand for xx and yy is fixed based on the given income and prices.

Summary:

  • Demand for xx: x=5x = 5
  • Demand for yy: y=50y = 50

Would you like further details on the steps or the concepts used here?

Here are 5 related questions you might find useful:

  1. How do you derive a demand function using a Cobb-Douglas utility function?
  2. What is the relationship between income and demand in this case?
  3. How do changes in the prices of xx or yy affect the demand functions?
  4. What is the meaning of the Lagrange multiplier in this optimization problem?
  5. 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)