Question

How does a firm maximize the profits? Give an algebraic explanation.

Short answer

In summary, to maximize profits algebraically, a firm must follow these steps: 1. Define total revenue (TR), total cost (TC), and profit (π). 2. Use the profit function, π(Q) = TR(Q) - TC(Q), which represents the firm's profit as a function of output quantity (Q). 3. Find the marginal revenue (MR) and marginal cost (MC) functions by taking derivatives of TR and TC, respectively, with respect to Q. 4. Set MR(Q) equal to MC(Q) to find the profit maximization condition. 5. Solve the resulting equation for Q to find the optimal level of output (Q*), which maximizes the firm's profit. For example, given TR(Q) = 100Q - 0.5Q^2 and TC(Q) = 20 + 5Q, we determined that the firm maximizes its profit by producing 95 units of output.

Step-by-step solution

Define total revenue, total cost, and profit

Total revenue (TR) is the amount of money a firm receives from selling its goods or services. It is calculated as the price (P) multiplied by the quantity (Q) of goods sold. Mathematically, TR = P*Q. Total cost (TC) is the amount of money a firm spends to produce its goods or services. It includes both fixed costs (expenses that don't change with output) and variable costs (expenses that change with output). Profit (π) is the difference between total revenue and total cost. Mathematically, π = TR - TC.

Introduce the profit function

The profit function represents the firm's profit as a function of the quantity of output it produces. To maximize profit, the firm must find the level of output (Q) that maximizes the profit function. We can write the profit function as: π(Q) = TR(Q) - TC(Q)

Explain marginal revenue and marginal cost

Marginal revenue (MR) is the additional revenue a firm receives from selling one more unit of its product. It is the derivative of total revenue with respect to quantity: MR = dTR/dQ. Marginal cost (MC) is the additional cost a firm incurs from producing one more unit of its product. It is the derivative of total cost with respect to quantity: MC = dTC/dQ.

Profit maximization condition

A firm maximizes its profit when its marginal revenue equals its marginal cost. Mathematically, this can be expressed as: MR(Q) = MC(Q). To find the optimal level of output (Q*), we need to solve this equation for Q.

Illustrate an example

Let's take a hypothetical firm with given total revenue and total cost functions: TR(Q) = 100Q - 0.5Q^2 TC(Q) = 20 + 5Q First, calculate the profit function: π(Q) = (100Q - 0.5Q^2) - (20 + 5Q) Now, find the marginal revenue and marginal cost functions: MR(Q) = d(TR)/dQ = 100 - Q MC(Q) = d(TC)/dQ = 5 Set MR equal to MC and solve for Q: 100 - Q = 5 Q = 95 Thus, in this example, the firm maximizes its profit by producing 95 units of output.

Key concepts

Total Revenue

Consider a lemonade stand as a simple example. If a glass of lemonade sells for \(2, and 50 glasses are sold, the total revenue would be \)100. This can be expressed algebraically as Total Revenue (TR) = Price per Unit (P) * Quantity Sold (Q). It represents the starting point for a business's financial analysis.

Total revenue is critical for assessing a company's performance over time. It not only reflects the firm's ability to sell its product but also indicates market demand and pricing power. To increase TR, the firm could either improve its sales volume or increase prices, assuming demand remains stable.

Total Cost

Continuing with our lemonade example, if it costs \(0.50 in materials to make each glass of lemonade and there's a daily stand rental fee of \)10, the total cost (TC) for selling 50 glasses would be the sum of the cost of materials (\(25) and the fixed rental fee. Mathematically, TC = Total Fixed Costs (TFC) + Total Variable Costs (TVC), where the variable costs change with the level of production.

Analyzing TC allows a business to track efficiency, identify cost-saving opportunities, and inform pricing strategies. Understanding and managing TC is as crucial as generating revenue since it directly affects profitability.

Marginal Revenue

To optimize sales, it's vital to understand the benefits of selling 'just one more.' This incremental gain is known as marginal revenue (MR). Put into an equation, MR equals the change in total revenue (ΔTR) divided by the change in quantity sold (ΔQ), simplified as MR = ΔTR/ΔQ. If selling an additional glass of lemonade increases the stand's revenue from \)100 to \(102, the MR of selling that glass is \)2.

In a more competitive market, or if market saturation occurs, MR could decrease as additional units are sold. Businesses must monitor MR to ensure that producing more will indeed be beneficial to profits rather than detrimental.

Marginal Cost

Producing more isn't always straightforward; it often comes with additional costs. Here we look at marginal cost (MC), the cost of producing 'one more unit.' Mathematically, MC = ΔTC/ΔQ. If our lemonade stand, now making 51 glasses instead of 50, incurs an additional cost of \(0.50 for that extra glass (from materials), the MC would be that \)0.50.

Understanding MC is crucial for profit maximization strategies. Increased production might reach a point where MC exceeds MR, signaling that further production would not be profitable and could lead to losses.

Profit Function

Ultimately, a business exists to make a profit. The profit function is a formula that encapsulates the economic viability of a business's operations. It is calculated by subtracting total costs from total revenues: Profit (π) = TR - TC. By graphing this function against different levels of output (Q), we can visualize the point at which profit is maximized – it's the peak of the profit curve.

In our textbook scenario, when Profit (π) = (100Q - 0.5Q^2) - (20 + 5Q), calculating dπ/dQ gives us the marginal profit. Setting MR = MC and solving for Q provides us with the precise quantity (Q*) that maximizes the firm's profit.