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

Set up the profit function

The firm's profit (\(\Pi\)) is defined as total revenue (\(TR\)) minus total cost (\(TC\)). Since price (\(P\)) is given as 54 and the total revenue is the price times quantity (\(Q\)), profit function can be expressed as: \[ \Pi = TR - TC = P \cdot Q - (10Q + 2Q^2) \]Substituting the given price:\[ \Pi = 54Q - 10Q - 2Q^2 \]Simplifying, we have:\[ \Pi = 44Q - 2Q^2 \]

Differentiate the profit function

To find the profit-maximizing quantity, find the derivative of the profit function with respect to \(Q\), and then set this derivative equal to zero.The derivative of the profit function \( \Pi = 44Q - 2Q^2 \) with respect to \(Q\) is:\[ \frac{d\Pi}{dQ} = 44 - 4Q \]

Solve for the profit-maximizing quantity

Set the derivative equal to zero to find the critical points:\[ 44 - 4Q = 0 \]Solve for \(Q\):\[ 4Q = 44 \]\[ Q = \frac{44}{4} = 11 \]Thus, the profit-maximizing quantity is 11 units.

Verify the second derivative test

To ensure that this is a maximum, apply the second derivative test. Differentiate the first derivative \(44 - 4Q\) with respect to \(Q\) again:\[ \frac{d^2\Pi}{dQ^2} = -4 \]Since \(\frac{d^2\Pi}{dQ^2} = -4\) is negative, the function is concave, confirming that the critical point is indeed a maximum.

Key concepts

Profit Maximization

In a perfectly competitive market, the goal of a firm is to maximize its profit. Profit maximization refers to finding the quantity of output that allows the firm to achieve the highest possible profit.
To calculate profit, we subtract the total cost (TC) from the total revenue (TR). In mathematical terms, it looks like this:

• Profit (\( \Pi \)) = Total Revenue (\( TR \)) - Total Cost (\( TC \))

Here, the price (\( P \)) remains constant due to the market's competitive nature, meaning that firms are price-takers, setting the price equal to demand. Thus, total revenue is defined as the market price times the quantity produced:

• Total Revenue = \( P \times Q \)

With this setup, the firm aims to find the optimal Q (quantity) where the derivative of its profit (\( \frac{d\Pi}{dQ} \)) equals zero, marking a critical point for profit maximization.

Marginal Analysis

Marginal analysis is a crucial tool in determining the firm's optimal production level. It involves examining the additional benefits of increasing production by one more unit, which boils down to finding where the marginal revenue (MR) equals marginal cost (MC).
In perfectly competitive markets, since each unit can be sold at the market price, marginal revenue equals the price (\( MR = P \)). Meanwhile, the marginal cost is depicted by the derivative of the total cost with respect to quantity, \( MC = \frac{dTC}{dQ} \). Therefore, the equation for profit maximization in terms of marginal analysis becomes:

• Set \( MR = MC \), or since \( MR = P \), then \( P = MC \)

The solution happens when additional revenue from selling one more unit equals the additional cost of producing that unit. In our exercise, by using the profit derivative, finding \( \frac{d\Pi}{dQ} = 0 \), we directly see where the MR equals MC.

Total Cost Function

The total cost function is an expression representing the total cost incurred by a firm to produce a particular level of output. In mathematical form, it's provided as:

• \( TC = 10Q + 2Q^2 \)

The total cost function includes fixed costs, which do not change with the level of output, and variable costs, which do vary as production levels rise or fall. However, in our case, such simplistic quadratic expression illustrates that fixed costs are negligible or zero (as indicated by no constant term).
To find marginal cost, differentiate the total cost function with respect to quantity. Here, \( \frac{dTC}{dQ} = 10 + 4Q \), meaning MC rises as production increases due to the second term, \( 4Q \). This reflects increasing costs with greater output, a typical scenario due to factors such as diminishing returns or resource constraints. Understanding total cost helps businesses decide how much to produce efficiently.