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

The quantity that the firm should produce to maximize profit can be obtained by solving the quadratic equation formed by equalizing marginal cost and marginal revenue. The Q value obtained will be the quantity to produce for maximum profits.

Step-by-step solution

Understanding Profits and Quantity

The profit maximization condition for a firm is when Marginal Revenue equals Marginal Cost (MR = MC). This is where the profit is the highest as the cost of producing an additional unit equals the revenue gained from selling that unit.

Equalize Marginal Revenue and Marginal Cost

We set the Marginal Revenue equal to Marginal Cost: \(MR = MC\)\nHence, \(20 - 8Q = Q^2\)

Solve the Equation for Q

By moving terms, the equation becomes a quadratic equation in the form of \(0 = Q^2 + 8Q - 20\). Solve this quadratic equation to get the values of Q

Applying Quadratic Formula to Solve for Q

We can solve for Q by using the quadratic formula \(Q = [-b ± sqrt(b^2 - 4ac)] / 2a\), where \(a = 1\), \(b = 8\) and \(c = -20\). By substituting these values in the formula, solve for Q

Key concepts

Marginal Revenue

Marginal revenue (MR) is a critical component in understanding a monopoly's decision-making process. It represents the additional income that a firm receives for selling one more unit of a good or service. In a perfectly competitive market, marginal revenue equals the price of the good, as firms are price takers. However, in a monopoly, the firm is a price setter and faces a downward-sloping demand curve.

This means that to sell more units, a monopolist must lower the price, which affects the revenue of previously sold units. Thus, the marginal revenue is not simply the price of the good, but the price minus the loss in revenue from having to lower the price on all units sold. The formula for marginal revenue in a monopoly context often includes the demand curve's price function, as in the exercise where it is expressed as \(MR = P - 8Q\). To maximize profits, a monopolist will produce units up to the point where MR equals marginal cost (MC).

Marginal Cost

Marginal cost (MC) is the cost of producing one additional unit of a product. It is crucial for a firm to understand its marginal cost in order to make effective production and pricing decisions. In profit maximization, a firm continues to produce additional units as long as the revenue from selling a unit exceeds the marginal cost of producing it.

In the context of a monopoly, and as seen in our textbook problem, the marginal cost is not constant. It can change with the level of output because of economies of scale, the use of different technologies, or resource constraints. For instance, the MC formula \(MC=Q^2\) in the exercise indicates that the cost increases with the square of the quantity, suggesting the presence of increasing marginal costs. To find the profit-maximizing level of output, the monopolist will equate this marginal cost with its marginal revenue.

Demand Curve

The demand curve is a graphical representation of the relationship between the price of a good and the quantity demanded by consumers at those prices. For a monopoly, the demand curve is also the firm's average revenue curve and is typically downward sloping. This slope reflects the law of diminishing marginal utility; as consumers buy more of a good, the less they are willing to pay for additional units.

In a monopolistic setting, the shape of the demand curve plays a pivotal role in determining prices and output. A steeper curve indicates less price sensitivity, while a flatter curve suggests that consumers' purchase quantity responds significantly to price changes. In the given exercise, the demand curve is depicted by the equation \(P=20-4Q\), which indicates that the price falls as the quantity demanded increases. The monopolist uses this curve alongside marginal revenue and marginal cost calculations to identify the profit-maximizing quantity to produce.