Question

Consider a perfectly competitive firm that has a total cost of producing output given by: \(T C=10 Q+2 Q^{2}\). The market price is \(P=54\). Find the profitmaximizing quantity produced by the firm.

Short answer

The profit-maximizing quantity is 11 units.

Step-by-step solution

Identify the Total Cost (TC) function

The total cost (TC) function provided in the problem is given as: \(TC = 10Q + 2Q^2\). This function represents the total cost incurred by the firm to produce a quantity \(Q\) of the product.

Define the Profit Function

In a perfectly competitive market, the firm maximizes profit where marginal cost (MC) equals the market price (\(P\)). Profit (\(\pi\)) is defined as total revenue (\(TR\)) minus total cost (\(TC\)), where \(TR = PQ\). Therefore, the profit function is: \(\pi = PQ - (10Q + 2Q^2)\).

Find the Marginal Cost (MC) Function

The marginal cost (MC) is the derivative of the total cost (TC) function with respect to quantity (Q). Thus, the MC function is: \(MC = \frac{d(TC)}{dQ} = \frac{d(10Q + 2Q^2)}{dQ} = 10 + 4Q\).

Set MC equal to Market Price

To find the profit-maximizing quantity, set the marginal cost equal to the market price. Thus, we solve the equation: \(10 + 4Q = 54\).

Solve for Quantity (Q)

Subtract 10 from both sides: \(4Q = 44\). Divide by 4: \(Q = 11\). Therefore, the profit-maximizing quantity produced by the firm is 11 units.

Key concepts

Profit Maximization

Profit maximization is a crucial concept for firms in a perfectly competitive market. The goal is to produce a quantity of output that allows a firm to achieve the highest possible profit. In such markets, firms are price takers, meaning they cannot influence the market price of the product they sell. Instead, they must sell their goods at the prevailing market price.

To maximize profit, a firm equates its marginal cost (MC), the cost of producing one additional unit of output, with the market price (P) of the product. This is because, in a perfectly competitive market, profit is maximized where marginal cost equals marginal revenue, and marginal revenue in perfect competition is the market price of the good.

By ensuring that each additional unit's cost is precisely the revenue generated from selling it, firms ensure that they are not underselling or overselling their product relative to their costs. This allows them to efficiently manage production and align closely with the market forces.

Marginal Cost

Marginal Cost (MC) is a key factor in the decision-making process of firms, especially in a perfectly competitive environment. It represents the additional cost a firm incurs when producing one more unit of output. Understanding and calculating marginal cost is vital for determining the optimal level of production.

In the given example, the marginal cost function is derived from the total cost (TC) function, which is mathematically expressed as the derivative of the TC function with respect to quantity (Q). For instance, with a TC function given by \(TC = 10Q + 2Q^2\), the marginal cost would be derived as \(MC = \frac{d(TC)}{dQ} = 10 + 4Q\). This equation shows that marginal cost increases as the quantity produced rises, highlighting the principle of increasing costs in production.

A firm uses the marginal cost to determine the profit-maximizing output level, as it helps in comparing production costs with market prices. When marginal cost equals market price, it indicates that producing an additional unit will not increase profitability, reinforcing the importance of marginal cost in profit maximization.

Total Cost Function

The total cost function is a mathematical expression that provides the total cost incurred by a firm for producing a specific quantity of output. It combines fixed and variable costs into a single function that can be used for analysis and decision-making.

For example, the total cost function \(TC = 10Q + 2Q^2\) illustrates how total production costs depend on the quantity of goods produced. Here, the term \(10Q\) represents linear, or predictable, costs related directly to output levels (often variable costs), while \(2Q^2\) captures quadratic components reflecting costs that increase with the square of production, typically associated with increasing complexity or inefficiencies at higher production levels.

Understanding the total cost function helps firms in planning and forecasting, allowing them to determine the cost implications of different output levels. It becomes especially critical when calculating marginal costs, necessary for setting the right production quantity and achieving profit maximization in a perfectly competitive market.