Percentage Change Approximation

Suppose we know that Z = A x B. If A changes by D A and B changes by D B, what will be the change in Z? What about percentage changes?

We can start by noting that Z + D Z = (A + D A) x (B + D B). Multiplying out the terms on the right, we have Z + D Z = AB + BD A + AD B + D AD B. But since Z = AB, this simplifies to D Z = BD A + AD B + D AD B. For small changes, D AD B will be an order of magnitude smaller and so we have as an approximation, D Z » BD A + AD B. To convert this to percentage changes, we can divide by Z on the left side and the equal term AB on the right to obtain » + . In words, the percentage change in the product of two variables is approximately equal to the sum of their component percentage changes.

What if Z = A/C? Does a similar relationship hold? As before, let A change by D A and C change by D C to obtain Z + D Z = . To obtain percentage changes, divide the left side by Z and the right side by the equal amount A/C (equivalent to multiplying by C/A). Then = 1 + = x . Next we will subtract 1 from each side, but on the right side this 1 will take the form to get a common denominator. This leaves us with = . The first and third terms cancel, and we can then factor out the common term to obtain =

For small values of D C, is approximately equal to 1, which gives us our final result:

if Z = A/C, then » - .

In the text example, real income (Z) is equal to nominal income (A) divided by the price level index (C). Consequently, the approximation applies: the percentage change in real income » the percentage change in nominal income minus the percentage change in the price level.

There is a shortcut to these relationships as well: Note that for any Z, the differential of the natural logarithm of Z is d(lnZ) = dZ/Z. At the limit as dZ approaches zero, the term on the right is simply the point-estimate of the percentage change in Z-an approximation of . Suppose as before that Z = AB. Taking the natural log of both sides, we have lnZ = lnA + lnB. Differentiating, we obtain dZ/Z = dA/A + dB/B as before. Likewise, if Z = A/C, then lnZ = lnA - lnC so that d(lnZ) = dZ/Z = dA/A - dC/C.