For a monopolist’s product, the demand function is 𝑝 = 50/ 𝑥 and the
average cost function is AC=0.5 + (1000/x). find the profit-maximizing
output and price. At this level, show marginal revenue is equal to marginal
cost

### Problem Setup:
We are given:
1. **Demand function**: \( p = \frac{50}{x} \), where \( p \) is the price and \( x \) is the quantity demanded.
2. **Average cost function**: \( AC = 0.5 + \frac{1000}{x} \).

### Step-by-Step Solution:

#### 1. **Revenue Function**:
Revenue \( R(x) \) is given by the product of price \( p \) and quantity \( x \):
\[
R(x) = p \times x = \frac{50}{x} \times x = 50
\]
So, the revenue is constant at 50, which is unusual, but we proceed with it.

#### 2. **Cost Function**:
The total cost \( C(x) \) can be obtained by multiplying the average cost by the quantity:
\[
C(x) = AC \times x = \left( 0.5 + \frac{1000}{x} \right) \times x = 0.5x + 1000
\]

#### 3. **Profit Function**:
Profit \( \Pi(x) \) is the difference between revenue and cost:
\[
\Pi(x) = R(x) - C(x) = 50 - (0.5x + 1000)
\]
\[
\Pi(x) = 50 - 0.5x - 1000 = -0.5x - 950
\]

#### 4. **Maximizing Profit**:
To find the profit-maximizing output, we differentiate the profit function with respect to \( x \) and set it equal to zero:
\[
\frac{d\Pi(x)}{dx} = \frac{d}{dx} (-0.5x - 950) = -0.5
\]
Setting \( \frac{d\Pi(x)}{dx} = 0 \) yields no solution since the derivative is a constant.

There seems to be an issue with the setup where the profit is decreasing with increasing \( x \). Let's re-examine the problem and focus on the conditions for marginal revenue and marginal cost, which might be more useful.

#### 5. **Marginal Revenue and Marginal Cost**:
From the revenue function \( R(x) = 50 \), the marginal revenue \( MR \) is 0, as it does not depend on \( x \).

The marginal cost \( MC \) is the derivative of the total cost:
\[
C(x) = 0.5x + 1000
\]
\[
MC = \frac{dC(x)}{dx} = 0.5
\]

At this level, the marginal revenue is already 0, which is less than the marginal cost of 0.5. Thus, profit does not increase with output, and the firm should not produce.

### Conclusion:
The profit-maximizing output is **zero**, since marginal revenue equals zero and marginal cost exceeds it at all levels of production. There is no profitable level of production with the given functions.

---

Would you like more details or further clarification on any part? Here are some related questions to deepen understanding:

1. How does a monopolist determine profit-maximizing output in general?
2. What is the relationship between marginal revenue and price elasticity of demand?
3. How does average cost influence the monopolist's decision-making?
4. Can a monopolist operate if marginal revenue is less than marginal cost?
5. How does total revenue behave in cases of perfectly inelastic demand?

**Tip**: In monopoly pricing, always check the price elasticity of demand to understand how sensitive revenue is to changes in output levels.

For a monopolist’s product, the demand function is 𝑝 = 50/𝑥 and the average cost function is AC=0.5 + (1000/x). Find the profit-maximizing output and price. At this level, show marginal revenue is equal to marginal cost.

Learn how to find the profit-maximizing output and price for a monopolist's product with a demand function p = 50/x and an average cost function AC = 0.5 + 1000/x. This problem explores revenue, cost, and profit functions, and explains the conditions where marginal revenue equals marginal cost.

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