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.

To derive the demand functions for $ x $ and $ y $ 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:
\[
U(x, y) = x y^5
\]
where $ x $ and $ y $ represent quantities of two goods, and $ U $ represents total utility.

### Step 2: Budget Constraint
Edgar's income is $300, the price of $ x $ is $10, and the price of $ y $ is $5. So, the budget constraint is:
\[
10x + 5y = 300
\]
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:
\[
\mathcal{L} = x y^5 + \lambda (300 - 10x - 5y)
\]
where $ \lambda $ is the Lagrange multiplier.

### Step 4: First-Order Conditions
We take the partial derivatives of $ \mathcal{L} $ with respect to $ x $, $ y $, and $ \lambda $, and set them equal to zero.

1. **With respect to $ x $:**
   \[
   \frac{\partial \mathcal{L}}{\partial x} = y^5 - 10\lambda = 0 \quad \Rightarrow \quad y^5 = 10\lambda
   \]

2. **With respect to $ y $:**
   \[
   \frac{\partial \mathcal{L}}{\partial y} = 5x y^4 - 5\lambda = 0 \quad \Rightarrow \quad x y^4 = \lambda
   \]

3. **With respect to $ \lambda $:**
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = 300 - 10x - 5y = 0
   \]

### Step 5: Solving the System of Equations
From the first condition, we have:
\[
y^5 = 10\lambda
\]
From the second condition, we have:
\[
x y^4 = \lambda
\]
Substituting $ \lambda = x y^4 $ into $ y^5 = 10\lambda $, we get:
\[
y^5 = 10x y^4 \quad \Rightarrow \quad y = 10x
\]

Now substitute $ y = 10x $ into the budget constraint:
\[
10x + 5(10x) = 300 \quad \Rightarrow \quad 10x + 50x = 300 \quad \Rightarrow \quad 60x = 300 \quad \Rightarrow \quad x = 5
\]

Thus, $ x = 5 $. Now, using $ y = 10x $, we get:
\[
y = 10(5) = 50
\]

### Step 6: Demand Functions
The demand functions are:
\[
x = 5 \quad \text{and} \quad y = 50
\]
Thus, the demand for $ x $ and $ y $ is fixed based on the given income and prices.

### Summary:
- **Demand for $ x $**: $ x = 5 $
- **Demand for $ y $**: $ y = 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 $ x $ or $ y $ 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.

Learn how to derive the demand functions for goods x and y based on the utility function U(x, y) = xy^5. With Edgar's $300 income, prices at $10 and $5 respectively, we maximize utility using the Lagrangian method and partial derivatives. This guide is ideal for students studying economics and calculus.

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