Question

Suppose a single-price monopoly's demand curve is given by \(P=20-4 Q,\) where \(P\) is price and \(Q\) is quantity demanded. Marginal revenue is \(M R=20-\) \(8 Q .\) Marginal cost is \(M C=Q^{2} .\) How much should this firm produce in order to maximize profit?

Short answer

This firm should produce 2 units in order to maximize profit.

Step-by-step solution

Set Marginal Revenue Equal to Marginal Cost

To find the quantity that maximizes profit for this firm, we start by setting the marginal revenue (MR) equal to the marginal cost (MC). We have the two equations as \(MR = 20-8Q\) and \(MC = Q^2\). So, we can write it as \(20-8Q = Q^2\).

Rearrange the Equation

The equation is quadratic so we want to rearrange it into a standard form. That means we need to have the terms ordered in descending power, with the constant term on the other side. So, we rearrange our equation to result in \(Q^2 + 8Q - 20 = 0\).

Solve for Q using Quadratic Formula

We can use the quadratic formula to find the solutions to this equation. The quadratic formula is \(Q = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where a, b, and c are the coefficients from the quadratic equation. In this case, a is 1, b is 8, and c is -20. Plugging these values into the quadratic formula gives two solutions: \(Q = -10\) and \(Q = 2\).

Discard Non-sensible Solution

Quantity can't be negative in economic context. Hence discard Q = -10. Thus, the quantity that maximizes the firm's profit is \(Q = 2\).

Key concepts

Understanding Marginal Revenue

Marginal Revenue (MR) is a crucial concept in economics, especially when dealing with monopolies. Simply put, marginal revenue is the additional income that a firm earns by selling one more unit of its product. In mathematical terms, it's the derivative of the total revenue with respect to quantity.

For a monopoly, which is characterized by having complete control over its market, the marginal revenue curve typically lies below the demand curve. This is because the firm must lower the price to sell an additional unit, thus reducing the revenue gained from previous units.

In our exercise, the marginal revenue is expressed as the function:\[MR = 20 - 8Q\].
This equation shows that as the quantity (Q) increases, the marginal revenue decreases. This pattern is typical for monopolistic markets, where the firm has to constantly adjust prices to maintain sales increases.

The Role of Marginal Cost

Marginal Cost (MC) represents the cost of producing one additional unit of a product. It is pivotal for firms to understand and calculate marginal cost to make informed production decisions.

For a monopoly, the marginal cost can guide price setting and production levels to ensure profitability. Remember, a lower marginal cost compared to marginal revenue signals that producing more could be profitable.

In the exercise scenario, the marginal cost is given by:\[MC = Q^2\].
This equation implies that with every additional unit produced, the cost increases at a quadratic rate. Thus, as production grows, the costs rise significantly, requiring careful consideration of how many units should be produced.

Maximizing Profit

Profit maximization is the ultimate goal for a monopoly, like any other firm. The core strategy for achieving this is by producing the quantity of goods where Marginal Revenue (MR) equals Marginal Cost (MC).

At this point, any additional production would cost more than the revenue it generates, and thus not be profitable. Economically, this is summarized by the equation:\[MR = MC\].

In the provided exercise, we’ve set the equations for MR and MC equal to each other to find the optimal production level. That equation is: \[20 - 8Q = Q^2\].
Solving this gives us the quantity that results in maximum profit. The solutions indicate that producing 2 units maximizes the monopoly's profit, while the "negative" quantity solution is economically irrelevant.

• This method empowers firms to balance their production levels and financial gains effectively.
• It shows the importance of understanding both cost and revenue dynamics in monopolistic markets.