{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

11Asn6

11Asn6 - ucsc supplementary notes ams/econ 11b Percentage...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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% ....
View Full Document

{[ snackBarMessage ]}

Page1 / 8

11Asn6 - ucsc supplementary notes ams/econ 11b Percentage...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online