profit is to find the slope of the profit function at a given level of output. The
graph in Figure 2A.1 shows how this is done. We start by specifying a particu-
lar level of output, say, Q 5. Next, we draw a tangent line that just touches the
profit graph at this output level. Finally, we find the slope of the tangent line.
By careful measurement (taking the ratio of the “rise over the run” along the
line), we find the slope to be exactly 1 (that is, the tangent happens to be a 45
line). Thus, the marginal profit at Q 5 is measured as $1,000 per additional
1,000 units or, equivalently, $1 per unit.
The upward-sloping tangent shows that profit rises as output increases.
Marginal profit measures the steepness of this slope, that is, how quickly profit
rises with additional output. In the graph in Figure 2A.1, tangents also are
drawn at output levels Q 10 and Q 15. At Q 15, profit falls with increases
in output; marginal profit (the slope of the tangent) is negative. At Q 10, the
tangent line is horizontal; marginal profit (again its slope) is exactly zero.
Marginal analysis can identify the optimal output level directly, dispensing
with tedious enumeration of candidates. The principle is this:The manager’s objective is maximized when the marginal value with respect to that
objective becomes zero (turning from positive to negative).To maximize profit, the marginal principle instructs us to find the output for
which marginal profit is zero. To see why this is so, suppose we are considering
an output level at which marginal profit is positive. Clearly, this output cannot
be optimal because a small increase would raise profit. Conversely, if marginal
profit is negative at a given output, output should be decreased to raise profit.
In Figure 2A.1, profit can be increased (we can move toward the revenue peak)
if current output is in either the upward- or downward-sloping region.
Consequently, the point of maximum profit occurs when marginal profit is
neither positive nor negative; that is, it must be zero. This occurs at output
Q 10 thousand, where the tangent’s slope is flat, that is, exactly zero.DIFFERENTIAL CALCULUS To apply the marginal principle, we need a simple
method to compute marginal values. (It would be tedious to have to compute
rates of change by measuring tangent slopes by hand.) Fortunately, the rules of
differential calculus can be applied directly to any functional equation to derive
marginal values. The process of finding the tangent slope commonly is referred
to as taking the derivative of(or differentiating) the functional equation.^1 To illus-
trate the basic calculus rules, let y denote the dependent variable and x the(^1) The following are all equivalent statements:
- The slope of the profit function at Q 5 is $1 per unit of output.
- The derivative of the profit function at Q 5 is $1 per unit of output.
- The marginal profit at Q 5 is $1 per unit of output.
- At Q 5, profit is rising at a rate of $1 per unit of output.
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