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Chapter 2
Optimal Decisions Using Marginal Analysis
Marginal Revenue Marginal revenue is the amount of additional revenue that comes with a unit increase in output and sales. The marginal revenue (MR) of an increase in unit sales from Q0 to Q1 is Marginal revenue [Change in Revenue]/[Change in Output] ¢R/¢Q [R1 R0]/[Q1 Q0] For instance, the MR earned by increasing sales from 2.0 to 2.1 lots is [268.8 260.0]/[2.1 2.0] $88 thousand per lot. where 268.8 is the revenue from selling 2.1 lots and 260.0 is the revenue from selling 2.0 lots. The graphic depiction of the MR between two quantities is given by the slope of the line segment joining the two points on the revenue graph. In turn, marginal revenue at a given sales quantity has as its graphic counterpart the slope of the tangent line touching the revenue graph. To calculate the marginal revenue at a given sales output, we start with the revenue expression (Equation 2.3), R 170Q 20Q2, and take the derivative with respect to quantity: MR 170 40Q.
[2.8]
We can use this formula to compute MR at any particular sales quantity. For example, marginal revenue at Q 3 is MR 170 (40)(3) $50 thousand; that is, at this sales quantity, a small increase in sales increases revenue at the rate of $50,000 per additional lot sold. A SIMPLIFYING FACT
Recall that the firm’s market-clearing price is given by
Equation 2.2: P 170 20Q. Note the close similarity between the MR expression in Equation 2.8 and the firm’s selling price in Equation 2.2. This similarity is no coincidence. The following result holds: For any linear (i.e., straight-line) demand curve with an inverse demand equation of the form P a bQ, the resulting marginal revenue is MR a 2bQ. In short, the MR equation has the same intercept and twice the slope as the firm’s price equation.5 5 If P a bQ, it follows that R PQ aQ bQ2. Taking the derivative with respect to Q, we find that MR dR/dQ a 2bQ. This confirms the result described.
