Chapter 18: Problem 662

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

Expert verified

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.

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