I Have the Power

In econometrics generally and energy modeling specifically, variables in equations often include an exponent. In particular, this is often seen in price variables. That is, the price is raised to an exponent to impose a magnitude and direction for the responsiveness (of energy or sales) to changes in its price. In the case of electricity, the price is typically negative and fractional (e.g., -0.2. -0.1), which suggests that increases in price result in decreased consumption. If the absolute value of the exponent is greater than one (1.0), the variable is said to be elastic. If the absolute value of the exponent is less than one (1.0), the variable is said to be inelastic.

The meaning of the exponent is clear when the number is an integer greater than 1.

The following figure depicts x raised to the powers of 1, 2, 3, and 4. As the magnitude of the exponent increases, the steepness of the curve increases. That is, x⁴ is much steeper than x¹. In fact, x¹ is barely visible below, as the magnitude of x⁴ swamps it.

That seems simple enough. Now let’s extend the idea. What is the mathematical interpretation of a decimal exponent? The best way to think about this is in terms of fractions, rather than decimals. Since any terminating decimal can be written as the quotient of two integers, this is easily represented by the following example:

Once the decimal is converted into a fraction, we can express the exponent more generally as follows:

The denominator (b) of the fraction is the root of the base (x) and the numerator (a) is the exponent to which the root is raised, which can also be written more intuitively as follows:

The following is a numerical example:

The following figure depicts x raised to the powers of 1/4, 2/4, 3/4, and 4/4 (respectively equal to 0.25, 0.5, 0.75, and 1). As in the case of the integer exponents above, the steepness of the curve increases as the magnitude of the exponent increases. Viewed alternatively, the curve flattens (i.e., gets closer to 0) as the decimal gets smaller. Thus, x^1/4 is closer to 0 on the y-axis than x^4/4 (which is equivalent to x¹).

As alluded to above, we often use negative exponents in energy forecasting.

The rule for evaluating this expression is to take the multiplicative inverse of the number and raise it to the positive power. Thus, the above expression becomes:

We can extend this idea to negative fractional exponents:

This can also be expressed as:

Let’s return to our original numerical example, except this time with a negative exponent:

After all of the intermediate steps, you should note that the result (1/125) is simply the inverse of the original result (125).

With this knowledge, you can now interpret exponents with a deeper level of understanding for your modeling with MetrixND. More importantly, you are now well-equipped to speak intelligently to the next 10^th grader you meet.

There’s more great information on a variety of load forecasting topics available in the forecasting section of the Itron website and we invite you to also register for our regular free webinars. Let us help you improve your forecasts! Contact us at forecasting@itron.com.