a The Greek letter n (pi) is frequently used in economics and business to denote profits. b The symbol A (delta) denotes difference or change. Thus, marginal profit is expressed as An = TCq - TCq_ 1. c Average profit (n) equals total profit (n) divided by total output (Q): n = n/Q.
Measure of the steepness of a line tangent
A straight line that touches a curve at only one point
Graphing Total, Marginal, and Average Relations
Knowledge of the geometric relations among totals, marginals, and averages can prove useful in managerial decision making. Figure 2.2(a) presents a graph of the profit-to-output relation given in Table 2.2. Each point on the curve represents a combination of output and total profit, as do columns 1 and 2 of Table 2.2. The marginal and average profit figures from Table 2.2 have been plotted in Figure 2.2(b).
Just as there is an arithmetic relation among totals, marginals, and averages in the table, so too there is a corresponding geometric relation. To see this relation, consider the average profit per unit of output at any point along the total profit curve. The average profit figure is equal to total profit divided by the corresponding number of units of output. Geometrically, this relation is represented by the slope of a line from the origin to any point on the total profit curve. For example, consider the slope of the line from the origin to point B in Figure 2.2(a). Slope is a measure of the steepness of a line and is defined as the increase (or decrease) in height per unit of movement along the horizontal axis. The slope of a straight line passing through the origin is determined by dividing the Y coordinate at any point on the line by the corresponding X coordinate. Using A (read delta) to designate change, slope = AY/AX = (Y2 - Y1)/(X2 - Xa). Because X1 and Y1 are zero for any line going through the origin, slope = Y2/X2 or, more generally, slope = Y/X. Thus, the slope of the line 0B can be calculated by dividing $93, the Y coordinate at point B, by 3, the X coordinate at point B. This process involves dividing total profit by the corresponding units of output. At any point along a total curve, the corresponding average figure is given by the slope of a straight line from the origin to that point. Average figures can also be graphed directly, as in Figure 2.2(b), where each point on the average profit curve is the corresponding total profit divided by quantity.
The marginal relation has a similar geometric association with the total curve. In Table 2.2, each marginal figure is the change in total profit associated with a one-unit increase in output. The rise (or fall) in total profit associated with a one-unit increase in output is the slope of the total profit curve at that point.
Slopes of nonlinear curves are typically found geometrically by drawing a line tangent to the curve at the point of interest and determining the slope of the tangent. A tangent is a line that touches but does not intersect a given curve. In Figure 2.2(a), the marginal profit at point A is
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