Assume that B Corporation has the following total profit function equation: Total Profit \(=\pi=-\$ 3,000-\$ 2,400 \mathrm{Q}+\$ 350 \mathrm{Q}^{2}-\$ 8.333 \mathrm{Q}^{3}\) a) Compute the marginal profit b) Determine its output quantities

Understanding the total profit function is foundational in business economics. It represents the overall earnings a company makes from selling its goods after all costs have been subtracted. In mathematical terms, if we denote total profit by \( \pi \), and it is expressed as a function of output quantities \( Q \), the equation might include terms for fixed costs, variable costs per unit, revenues per unit, and even more complex relationships.

For instance, B Corporation's total profit function is given by the equation \( \pi(Q) = -\$ 3,000 - \$ 2,400Q + \$ 350Q^2 - \$ 8.333Q^3 \). This equation includes a fixed cost of \$3,000, a variable cost of \$2,400 per unit of output, and revenue represented by the terms \( \$350Q^2 \) and \( -\$8.333Q^3 \). The last two terms suggest that the revenue per unit changes as the quantity produced changes, which is a common situation for products with economies or diseconomies of scale.

To make informed decisions, business managers need to understand how changing output levels affect total profits. Any output level that maximizes the total profit function is of particular interest since that is where the company earns the most money, all other factors remaining constant.

The derivative of the profit function is a critical tool for analyzing business performance. It provides us with the marginal profit, which indicates how much the total profit changes when the output quantity \( Q \), is increased by one unit. Essentially, it tells us the additional profit from selling one more unit.

To calculate the marginal profit for B Corporation, we differentiate the previously mentioned total profit function with respect to \( Q \), yielding \( d\pi(Q)/dQ = -\$2,400 + \$700Q - \$25Q^2 \). This expression simplifies to the marginal profit function, which is crucial in determining the most profitable level of production. When the derivative of the profit function equals zero, it often reflects maximum or minimum points for profit, which can guide businesses on where to set their production levels.

Understanding Marginal Profit

In B Corporation's case, the marginal profit function decreases as \( Q \) increases, suggesting diminishing returns at higher quantities. If a company keeps increasing its production beyond the point where marginal profit turns negative, it will begin to incur losses on additional units produced, hence the need to find the optimal production quantity.

Output quantities relate to the number of units a company produces and sells. Finding the optimal output quantity is a key objective in maximizing profit. It involves understanding at which production levels the costs are covered and profit starts to increase, and recognizing when profits peak before they start to decrease again.

For B Corporation, solving \( -\$2,400 + \$700Q - \$25Q^2 = 0 \) yields the output quantities where the marginal profit is zero – these are the critical points that could represent maximum or minimum profit scenarios. Using the quadratic formula, we find two such output quantities, \( Q_1 = 24 \) and \( Q_2 = 4 \).

Decision Points for Production

Output quantities at which the marginal profit is zero are important decision points. These points guide business strategy; for instance, a company would not usually produce beyond the quantity where marginal profit is zero unless it has strategic reasons, such as attempting to gain market share. It's essential to not only calculate these points but also understand their economic implications within the greater context of a company’s operations and competitive environment.