The problem states that an unregulated electric company is a monopolist and faces a demand curve of $Q = 100 - 50P$. The marginal cost is constant and equal to 1. We need to find the profit-maximizing price.

1. Problem Description

The problem states that an unregulated electric company is a monopolist and faces a demand curve of $Q = 100 - 50P$. The marginal cost is constant and equal to

1. We need to find the profit-maximizing price.

2. Solution Steps

First, we need to express the inverse demand function, which expresses price as a function of quantity. From $Q = 100 - 50P$, we can solve for $P$:

$50P = 100 - Q$

$P = \frac{100 - Q}{50}$

$P = 2 - \frac{1}{50}Q$

The total revenue (TR) is the product of price and quantity:

$TR = P \times Q = (2 - \frac{1}{50}Q)Q = 2Q - \frac{1}{50}Q^2$

The marginal revenue (MR) is the derivative of total revenue with respect to quantity:

$MR = \frac{d(TR)}{dQ} = 2 - \frac{2}{50}Q = 2 - \frac{1}{25}Q$

To maximize profit, the monopolist will produce where marginal revenue equals marginal cost (MC). In this case, $MC = 1$.

$MR = MC$

$2 - \frac{1}{25}Q = 1$

$\frac{1}{25}Q = 1$

$Q = 25$

Now, we substitute this quantity back into the inverse demand function to find the profit-maximizing price:

$P = 2 - \frac{1}{50}Q = 2 - \frac{1}{50}(25) = 2 - \frac{25}{50} = 2 - \frac{1}{2} = 2 - 0.5 = 1.5$

3. Final Answer

The profit-maximizing price is 1.

5.