From Total Output to Marginal Effects
A revenue or production model describes a total: total revenue for a given quantity sold, or total output for a given amount of an input. A first derivative describes how that total changes as the input changes. A second derivative describes how the first derivative changes.
This distinction matters in economic and production settings. A firm’s revenue can still be rising even as each additional unit sold contributes less additional revenue than the previous one. Likewise, total production can continue to rise even while each additional worker adds less output than the previous worker. In each case, the second derivative helps describe the trend in the marginal effect, not by itself the trend in the total.
Marginal Revenue and Marginal Product
Suppose \(R(q)\) is a differentiable revenue function, where \(q\) is the quantity sold. Its derivative \(R'(q)\) is called marginal revenue: it describes the rate at which revenue changes as quantity changes. If \(P(L)\) is a differentiable production function, where \(L\) is an input such as labor, \(P'(L)\) is called marginal product: it describes the rate at which output changes as the input changes.
The sign of the second derivative gives a directional interpretation. A positive second derivative means the first derivative is increasing; a negative second derivative means it is decreasing. In a revenue model, these are increasing and diminishing marginal revenue. In a production model, they are increasing and diminishing marginal product.
| Second-derivative sign | Marginal revenue | Marginal product |
|---|---|---|
| \(R''(q)>0\) or \(P''(L)>0\) | Increasing with quantity | Increasing with the input |
| \(R''(q)<0\) or \(P''(L)<0\) | Diminishing with quantity | Diminishing with the input |
| Second derivative is zero at one point | That value alone does not establish an increasing or diminishing trend nearby | That value alone does not establish an increasing or diminishing trend nearby |
Why the Second Derivative Gives This Information
The interpretation follows from the definition of a derivative. The second derivative \(R''\) is the derivative of \(R'\), so it gives the rate of change of marginal revenue. More precisely, if \(R''\) is negative throughout an interval, marginal revenue decreases from any smaller quantity to any larger quantity in that interval.
To prove the negative case, choose any \(x_1,x_2\in I\) with \(x_1<x_2\). The function \(f'\) is continuous on \([x_1,x_2]\) and differentiable on \((x_1,x_2)\), since \(f\) is twice differentiable on \(I\). The Mean Value Theorem applied to \(f'\) gives some \(c\in(x_1,x_2)\) such that
Here \(f''(c)<0\), while \(x_2-x_1>0\). Their product is negative, so \(f'(x_2)-f'(x_1)<0\), or \(f'(x_2)<f'(x_1)\). This proves that \(f'\) is strictly decreasing. If instead \(f''>0\) throughout the interval, the product is positive, so \(f'(x_2)>f'(x_1)\); this proves the positive case.
Worked Example: Diminishing Marginal Revenue
Worked Example: Diminishing Marginal Revenue
Suppose revenue, in monetary units, is modeled by \(R(q)=120q-3q^2\) for \(0\le q\le20\). Find and interpret the marginal revenue and its trend. Compare marginal revenue at \(q=5\) and \(q=10\).
Differentiate the revenue function:
Thus marginal revenue at quantity \(q\) is \(120-6q\). Differentiate again:
Since \(R''(q)<0\) throughout the stated interval, marginal revenue is strictly decreasing there. At \(q=5\),
At \(q=10\),
The marginal-revenue value has fallen from \(90\) to \(60\). This does not mean total revenue has fallen over that range. In fact,
Total revenue is higher at \(q=10\) than at \(q=5\), even though marginal revenue is lower. As a check, differentiating \(120q-3q^2\) gives \(120-6q\), and differentiating \(120-6q\) gives \(-6\), as used above.
Worked Example: Diminishing Marginal Product While Output Rises
Worked Example: Diminishing Marginal Product While Output Rises
Consider the production model \(P(L)=30L-L^2\) on \(0\le L\le10\), where \(L\) represents labor input. Find the marginal product and determine whether it diminishes. Then compare total output at \(L=2\) and \(L=8\).
Differentiate to obtain marginal product:
Differentiate once more:
The second derivative is negative throughout the model’s interval, so marginal product is strictly decreasing. Its values at the specified inputs are
The marginal product decreases from \(26\) to \(14\). But the total output values are
Output rises from \(56\) to \(176\), even as marginal product diminishes. Indeed, \(P'(L)=30-2L>0\) for \(0\le L\le10\), so the total production function is increasing on this interval. To check the calculations, differentiating \(30L-L^2\) gives \(30-2L\), and differentiating \(30-2L\) gives \(-2\).
Worked Example: Marginal Product That First Increases, Then Diminishes
Worked Example: A Change in the Marginal-Product Trend
Let \(P(L)=2L^2-\frac{1}{3}L^3\) for \(0\le L\le3\). Determine where marginal product increases and where it diminishes. Check the marginal-product values at \(L=1,2,3\).
The first derivative is
The second derivative is
For \(0\le L<2\), \(4-2L>0\), so marginal product is increasing. For \(2<L\le3\), \(4-2L<0\), so marginal product is decreasing. At \(L=2\), the second derivative is zero; the sign on either side establishes the change in trend.
Evaluate the marginal product at the three inputs:
The values rise from \(3\) to \(4\), then fall to \(3\), in agreement with the second-derivative signs. As a check, differentiating \(2L^2-\frac{1}{3}L^3\) gives \(4L-L^2\), and differentiating that expression gives \(4-2L\).
What Diminishing Marginal Returns Does—and Does Not—Say
In a production model, diminishing marginal returns means that marginal product decreases as the input increases over the interval where \(P''(L)<0\). It does not, by itself, say that total production decreases. Whether total production rises or falls is determined by the sign of \(P'(L)\): a positive marginal product means output is increasing, while a negative marginal product means output is decreasing.
The same distinction applies to revenue. The sign of \(R''(q)\) describes the trend in marginal revenue. The sign of \(R'(q)\), not the sign of \(R''(q)\), describes whether revenue is increasing or decreasing. Also, a decreasing marginal revenue need not be negative: it may remain positive throughout the interval under consideration.
These interpretations require attention to the interval and the model. A second derivative that is negative at one point gives a local sign at that point; to conclude that marginal effect decreases throughout an interval, the second derivative must be negative throughout that interval. If the model is intended only for a restricted range of quantities or inputs, its conclusions should be limited to that range.
A Common Interpretation Error
A frequent mistake is to treat a negative second derivative as proof that the total is decreasing. For the revenue model in the first example, one might reason incorrectly:
That conclusion does not follow. The second derivative shows that marginal revenue is decreasing. For this model, at \(q=5\),
Thus revenue is increasing at \(q=5\), even though its rate of increase is itself decreasing. The correction is to match each derivative with the question it answers: \(R'\) describes the direction of change in revenue, while \(R''\) describes the direction of change in marginal revenue.
Check Your Understanding
Use derivatives to distinguish what happens to a total from what happens to its marginal effect.
- If \(R''(q)<0\) throughout an interval, what happens to marginal revenue as \(q\) increases?
- Does diminishing marginal revenue necessarily mean that total revenue is decreasing? Explain which derivative determines the direction of change in total revenue.
- For \(P(L)=40L-2L^2\), find \(P'(L)\) and \(P''(L)\). On what interval is marginal product diminishing?
- A production model has \(P'(L)>0\) and \(P''(L)<0\) on an interval. Describe what happens to output and to marginal product there.
- Why is a second-derivative value of zero at a single input not enough, by itself, to conclude that marginal product changes from increasing to decreasing?
