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Unformatted text preview: ucsc supplementary notes ams/econ 11b Percentage Change and Elasticity c 2008, Yonatan Katznelson 1. Relative and percentage rates of change The derivative of a differentiable function y = f ( x ) describes how the function changes. The value of the derivative at a point, f ( x ), gives the instantaneous rate of change of the function at that point, which we can understand in more practical terms via the approxi mation formula Δ y = f ( x + Δ x ) f ( x ) ≈ f ( x ) · Δ x, which is accurate when Δ x is sufficiently small, (see SN5 for more details). This formula describes the change Δ y in terms of the change Δ x , which is very useful in a variety of contexts. In other contexts however, it is sometimes more appropriate to describe the change in y in relative terms. In other words, to compare the change in y to its previous value. Suppose, for example that you learn that a firm’s revenue increased by $2.5 million over the past year. An increase in revenue is generally a good thing, but how good is a $2.5 million increase? The answer depends on the size of the firm, and more specifically, on the amount of last year’s revenue. If in the previous year the firm’s revenue was $2.1 million, then an increase of $2.5 million means that revenue more than doubled, which means that the firm’s business is growing very robustly. On the other hand, if last year’s revenue was $350 million, then $2.5 million is an increase of less than 1%, which is far less impressive. The relative change in the value of a variable or function is simply the ratio of the change in value to the starting value. I.e., the relative change in y = f ( x ) is given by Δ y y = f ( x + Δ x ) f ( x ) f ( x ) . Likewise, the relative rate of change (rroc) of a function y = f ( x ) is obtained by dividing the derivative of the function by the function itself, (1.1) rroc = f ( x ) f ( x ) = 1 y · dy dx . Example 1. Suppose that y = x 2 + 1, then dy dx = 2 x , and the relative rate of change of y with respect to x is 1 y · dy dx = 2 x x 2 + 1 . When x = 4, the relative rate of change is 2 · 4 / (4 2 + 1) = 8 / 17 ≈ . 47, and when x = 10, the relative rate of change is 2 · 10 / (10 2 + 1) = 20 / 101 ≈ . 198. The relative change, and the relative rate of change are often expressed in the percentage terms, in which case they are usually called the percentage change and the percentage rate 1 2 of change , respectively. Thus, the percentage change in y is given by %Δ y = Δ y y · 100% = f ( x + Δ x ) f ( x ) f ( x ) · 100% , and the percentage rate of change (%roc) of the function y = f ( x ) is simply the relative rate of change multiplied by 100%, %roc = rroc · 100% = f ( x ) f ( x ) · 100% = 1 y · dy dx · 100% ....
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 Spring '08
 QIAN
 Derivative, Supply And Demand

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