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.

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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.