Question

Prove, through the use of derivatives, that if a firm is trying to maximize its profits it should produce where marginal revenue equals marginal cost.

Short answer

To maximize profits, a firm should produce where its Marginal Revenue (MR) equals its Marginal Cost (MC). This can be proved using calculus by calculating the derivatives of the Total Revenue (TR) and Total Cost (TC) functions with respect to quantity (Q) and setting them equal. The derivative of the profit function must be 0, meaning $ \frac{{d(Profit)}}{{dQ}} = 0 $. By the rules of derivatives, we can rewrite this as $ MR - MC = 0 $, which simplifies to the fundamental profit maximization condition: $ MR = MC $.

Step-by-step solution

Understand Basic Concepts

Before applying calculus, it's crucial to understand the fundamental economic concepts at play. In a commercial scenario, a firm's Total Revenue (TR) is the total income from selling a particular quantity of goods or services, while Total Cost (TC) is the full expense incurred in producing that quantity. Profit, accordingly, is expressed as the difference between these two figures: \[ Profit = TR - TC. \] The term 'marginal' is used in economics to signify the effect of producing one additional unit. Marginal Revenue, thus, represents the additional revenue acquired from selling one more unit of output, and Marginal Cost is the additional cost incurred from producing that added unit.

Calculate Marginal Revenue and Marginal Cost

Marginal Revenue and Marginal Cost can be calculated using the derivatives of the Total Revenue and Total Cost functions respectively. If \( Q \) represents the quantity of goods, then: \[ MR = \frac{{d(TR)}}{{dQ}} \] and \[ MC = \frac{{d(TC)}}{{dQ}} \]

Define Profit Maximization Condition

For a firm to reach its profit-maximizing production level, the slope of the profit function (the derivative of the profit function with respect to quantity) should be 0. This is because at the maximum point of a function, the slope of the tangent is 0. Hence: \[ \frac{{d(Profit)}}{{dQ}} = \frac{{d(TR - TC)}}{{dQ}} = 0 \]

Establish Equality of Marginal Revenue and Marginal Cost

To clarify above step, this can be rewritten using the rules of derivatives as follows: \[ \frac{{d(TR)}}{{dQ}} - \frac{{d(TC)}}{{dQ}} = 0 \] At this point, we can see that the derivative of Total Revenue with respect to Quantity is Marginal Revenue (MR), and the derivative of Total Cost with respect to Quantity is Marginal Cost (MC). Therefore: \[ MR - MC = 0 \] Ultimately, this equation simplifies to: \[ MR = MC \] So, for the profit maximization scenario, particularly in the realm of perfect competition, the producing firm should thus equate Marginal Revenue and Marginal Cost.

Key concepts

Marginal Revenue

Marginal Revenue (MR) is a fundamental concept in understanding how businesses make decisions about production. It refers to the additional income a company earns from selling one more unit of its product.
To calculate MR, we look at the total revenue change that results from selling that additional unit. In mathematical terms, MR is the derivative of the Total Revenue (TR) function with respect to quantity. This is expressed as:

• (MR) = \( \frac{{d(TR)}}{{dQ}} \)

This formula shows the rate at which revenue changes as quantity changes, making it a critical consideration for pricing and output decisions.
When a firm thinks about increasing production, it needs to determine if the extra revenue generated by one more unit exceeds the extra cost of making it.
In competitive markets, prices are often taken as given, meaning MR can just be the market price. However, in imperfect markets, understanding how MR changes with quantity is crucial for setting optimal production levels.

Marginal Cost

Marginal Cost (MC) indicates the additional cost a firm incurs in producing one more unit of a product. Understanding MC is crucial for firms aiming to maximize profitability, as it helps in deciding how much to produce.
From a mathematical perspective, MC is the derivative of the total cost (TC) function with respect to the quantity of output produced, expressed as:

• (MC) = \( \frac{{d(TC)}}{{dQ}} \)

This formula allows firms to estimate the cost impact of producing one more unit, which is central to profit maximization strategies.
At the production level where marginal revenue equals marginal cost (MR = MC), firms achieve optimal output. Here, producing additional units would not increase profit since the cost of each added unit equals the revenue it generates.
Understanding the behavior of MC is vital, especially because economies or diseconomies of scale can influence costs as production increases or decreases, impacting the slope of the MC curve.

Total Revenue

Total Revenue (TR) refers to the total income a firm receives from selling its goods or services. It is a straightforward concept defined by the product of the quantity sold and the selling price per unit:

• TR = Quantity Sold × Price per Unit

This principle serves as the foundation for understanding revenue dynamics and is a baseline for further calculations like Marginal Revenue.
In profit maximization, total revenue is analyzed alongside total cost to determine profitability.
The relationship between TR and quantity is critical; changes in quantity affect total revenue, and this is where derivatives become useful.
For instance, knowing how TR changes with each additional unit sold provides insight into price elasticity and market demand.
In settings like perfect competition, the TR curve tends to be linear, whereas in monopoly or oligopoly settings, it's nonlinear, reflecting more complex dynamics of demand and price changes as output changes.