# Elasticity

### Elasticity

Including:
The Price Elasticity of Demand
The Midpoint Formula
(Price) Elasticity of Supply
Income Elasticity of Demand
Cross (Price) Elasticity of Demand
Tax Incidence and Elasticity

Elasticity is a term used in economics to denote responsiveness. It is a measurement of the responsiveness, or sensitivity, of one variable to a change in another variable.

Elasticity has a variety of uses in economics. For example, when a company is deciding on a pricing strategy, it needs to know which proposed price will generate the most total revenue. The elasticity measurement will answer that question. Another example would be a producers' tax proposed on the sale of a particular product. How much of the tax will the producers pay, and how much of it will the consumers end up paying through higher prices? How much tax will be collected? How much will this change the quantity of the product that consumers purchase? The answers depend on the elasticity measurement. In the extreme case of price controls imposed by the government: price controls distort market equilibrium, but how much will output fall with price controls? Elasticity again determines the answer.

Many students have trouble understanding elasticity because the concept is often taught with the use of formulas that look cumbersome, and are difficult to memorize. The way I explain it here is different. I teach the concepts behind the formulas. All it takes is an understanding of the concepts, and the formulas fall into place. There is no need for memorization. Students should have no difficulty with the numbers or formulas involved if they understand the concepts of elasticity.

So here is the first, and most important concept: the discussions relating to specific types of elasticities, with links in the list to the left, follow from an understanding of this:

All types of elasticities are numbers, which represent measurements based on two variables. These measurements are formed using division, and are based on percentage changes in each variable used.

These measurements are formed by dividing the percentage change in one (dependent) variable by the percentage change in another (independent) variable. This should be easier to understand by realizing that it is a formula that will yield a number; by putting the dependent variable on top the result will be a measure of the dependent variable’s responsiveness to changes in the independent variable.

The results of such a formula will center around the number 1 (that is the case, mathematically, when division is used to compare 2 numbers). Higher than 1 means, mathematically, that the top number is larger than the bottom number; lower than 1, the bottom number is larger than the top number; and equal to 1, the two numbers are identical.

Higher than 1 (representing a change in the top number that is greater than the change in the bottom number) is relatively responsive, and is called elastic; lower than 1 (representing a change in the bottom number that is greater than the change in the top number) is relatively unresponsive and is called inelastic; equal to 1 means that the changes are neutral, and is called unit elastic.

This covers all situations, but the extreme cases where one variable does not change at all with any change in the other variable (either the top or the bottom of the formula equals zero) have special names: Perfectly elastic (zero for the bottom number, or infinite elasticity) and perfectly inelastic (zero for the top number, or zero elasticity).

The price elasticity of demand is probably the first, and most common, type of elasticity encountered in an economics class.

The price elasticity of demand is a measure of responsiveness, or sensitivity, of demand to a change in price. If a business decides to raise the price of a product, it will probably sell a lower quantity of that product, since for a normal good the demand curve is downward-sloping. Or if it decides to lower the price, it will probably sell a higher quantity of that product.

The relative change in the quantity sold, compared to the change in price, is the elasticity of demand. The price elasticity of demand is measured by taking the percentage change in quantity and dividing by the percentage change in price. It is shown mathematically by the formula:

p.e.d. = %change in quantity / %change in price

This formula will always yield a negative number for a normal good. This is due to the law of demand. For a normal good, the price and quantity demanded will always move in opposite directions, due to a downward-sloping demand curve. If it does not yield a negative number, then it is not a normal good but an inferior good. Often, a normal good is assumed for purposes of economic analysis, and it will be shown as the absolute value (the number without the negative sign). This assumption is based on the fact that the negative sign is irrelevant to the analysis. The relationship is still negative, but the negative sign is assumed to be there and is not shown. There are exceptions, for example with a demand equation, where the negative sign is important to the analysis.

For the remainder of this section, in order to keep from repeating this relationship over and over, assume that p.e.d. (or the price elasticity of demand), refers to the absolute value; forget about the negative sign involved, unless it becomes relevant to the discussion.

You don’t really have to worry about memorizing this formula, but it helps to be able to recognize it when you see it. You do need to remember that the price elasticity of demand is a measurement of the relative change in quantity to a given change in price, and that by using division to measure this comparison, a higher number for a result will mean more responsiveness, or more sensitivity, of quantity to a price change.

The elasticity is higher if it is more responsive or more sensitive.

Using division as a method of measurement highlights another important concept. If the quantity changes by the same percentage as the price change, then the answer will have a value of 1. If the quantity changes more than the price, the answer will have a value greater than 1. If the quantity changes less than the price change, the answer will have a value less than one. This distinction is important because for most types of analysis, the p.e.d. - relative to 1 - is more important than the actual number itself. In other words, the difference between a price elasticity of 3.5 and a price elasticity of 1.5 is not as important as the fact that each number is greater than 1. This importance will be illustrated in the section below about the effect of price elasticity of demand on total revenue.

The price elasticity of demand measurement (absolute value), as compared to 1, calls for some terminology that you need to know:

If price elasticity of demand is equal to 1, it is called unit elastic

If price elasticity of demand is greater than 1, it is called elastic

If price elasticity of demand is less than 1, it is called inelastic

In other words, a unit elastic demand is when the quantity changes by the same percentage as the price change. An elastic demand is when the quantity changes by a larger percentage than the price change. An inelastic demand is when the quantity changes by a smaller percentage than the price change.

You may encounter the extreme cases where either the price change or the quantity change is equal to zero. If there is no price change for any given range of quantities (the formula yields an answer that divides the quantity change by zero, which is defined mathematically as infinity), then it is called perfectly elastic. If there is no quantity change for any given range of prices (the formula yields an answer that divides zero by some number, which is defined mathematically as zero), then it is called perfectly inelastic.

At this time, an example should be helpful.

Suppose that a company sells a product for \$2, and at this price the quantity sold is 100 units. If the company raised the price to \$3 it could sell 75 units. What is the price elasticity of demand?

To arrive at the answer, start with the change in quantity. The quantity sold will go from 100 units to 75 units: a decrease of 25 units (100 - 75 = 25). This would be a change, from the beginning units sold of 100, of 25/100, or 25%.

Now, figure the change in price, using the same method. The price increased from \$2 to \$3, an increase of \$1. This would be a change, from the beginning price of \$2, of \$1/\$2, or 50%.

Now that you have the percentages, simply divide the quantity percentage by the price percentage, you get:

25/50 = 0.5. Always show this number as a fraction. If rounding is necessary, the answers could vary, and the course instructor may set the rules for rounding.

In this example, the absolute value of the result, 0.5, is less than 1. So using the terminology defined above, this would be an inelastic demand. The quantity changed by a lower percentage than the price changed.

Perhaps another example, showing a result of an elastic, as opposed to an inelastic, answer would be helpful. Suppose you had a situation similar to the example above, where the original (beginning) price is \$2, and the original (beginning) quantity sold is 100 units. But now, suppose that in increase in the price from \$2 to \$3 means that the quantity sold changes from 100 to 25. Using the exact same method to calculate the price elasticity of demand, you get:

The quantity sold will go from 100 units to 25 units: a decrease of 75 units (100 - 25 = 75). This would be a change, from the beginning units sold of 100, of 75/100, or 75%. The price change is exactly the same as in the example above, so that percentage change is still 50%. To get the price elasticity of demand, you divide the percentage change in quantity by the percentage change in price, or in this case 75/50, or 1.5. The price elasticity of demand in this case is greater than 1, since 1.5 > 1. So using the terminology defined above, this would be an elastic demand. The quantity changed by a higher percentage than the price changed.

Determinants of the price elasticity of demand:

Availability of substitutes
Degree of necessity
Share of budget
Time frame

The relationship between the price elasticity of demand and total revenue:

The importance of the price elasticity of demand for a business can be shown by the effect that it has on total revenue. The business will want to know whether a proposed price change will increase or decrease total revenue.

Total revenue, by definition, is equal to the price times the quantity sold (TR=PxQ). [Sometimes, when dealing with elasticity, the term “total expenditures” may be used instead of “total revenue”, but it has the same meaning].

Note what happens to the results of this formula (TR=PxQ) if a price change is involved. Due to the law of demand, the price will move in one direction, and the quantity sold will move in the other direction. Unless the price change and quantity change are both for the same percentage (unit elastic), then total revenue will also change whenever a price change is involved. The question is, does total revenue increase, or decrease? The answer depends on the direction of the price change along with the price elasticity of demand.

If the price elasticity of demand is elastic (greater than 1), then that means that the quantity change is more than the price change. So total revenue (price times quantity) will decrease for a price increase, and increase for a price decrease.

If the price elasticity of demand is inelastic (less than 1), then the quantity change is less than the price change. So total revenue (price times quantity), increases for a price increase, and decreases for a price decrease.

In summary:

Elastic demand, price increase: total revenue decreases
Elastic demand, price decrease: total revenue increases
Inelastic demand, price increase: total revenue increases
Inelastic demand, price decrease: total revenue decreases
Unit elastic demand: total revenue does not change

So, you can see from this, that the price elasticity is an important element in the pricing decisions of businesses.

One more thing about the relationship between the price elasticity of demand and total revenue: As long as demand is either elastic or inelastic, total revenue can always be increased with a price change in the proper direction. The only point where total revenue is maximized is the point where the price elasticity of demand is unit elastic.

The Midpoint Formula

The standard method for computing the price elasticity of demand has one major drawback. It is based on the assumption that the company is starting from a position of a specific price and quantity sold, and is considering a change in price. But what about the case where the company is not really "starting" from one particular point, but instead only wants to compare the results of two different possible prices? In other words, it does not care whether one price is considered to be the beginning price and the other is considered to be the ending price; it just wants to compare the difference between two possible prices.

The drawback is that, using the standard method for calculating the price elasticity of demand, you often get two different answers depending on which price you call the beginning, and which price you call the ending. Recall that in the first example used for that analysis, the prices being considered were: \$2, with sales of 100 units, and \$3, with sales of 75 units. The analysis used \$2 as the beginning price, and arrived at a price elasticity of demand of 0.5.

But suppose that \$3, with the exact same sales of 75 units, was used as the beginning price, and \$2, with the exact same sales of 100 units, was used as the ending price: the same situation, using the same numbers, but looking at them from the opposite direction. In this case, the price elasticity of demand would be calculated as follows:

The percentage change in quantity would be 25/75, or 33%. The percentage change in price would be 1/3, or 33%. The price elasticity of demand would be 33/33, or 1.

Going from \$2 to \$3 gives an answer of 0.5, but going from \$3 to \$2 gives an answer of 1. Two different answers calculated from the same numbers. Neither method is better than the other if specifying a beginning price is not relevant to the situation.

To correct for this discrepancy, the midpoint formula is considered to be a superior method.

In economics class, the textbook or the instructor might define the midpoint formula as something like this:

Midpoint Formula:

p.e.d. = [(Q2 - Q1) / ((Q1 + Q2) / 2)] / [(P2 - P1) / ((P1 + P2) / 2)]

Pretty intimidating, right? But luckily, you do not have to actually memorize this formula in order to know how to do the calculations. You only need to understand the concepts behind it, and the calculations will be easy.
This formula does not change the concept of the price elasticity of demand. Refer back to the beginning of this section on elasticity, and the first and most important concept of elasticity listed: elasticity (in this case, the price elasticity of demand) is always going to be the percentage change in quantity divided by the percentage change in price. The midpoint formula does not change that. The midpoint formula only changes the method of arriving at these percentages; all you have to know is the concept behind the different method of arriving at the percentages, and you know how to use the midpoint formula.

For the midpoint formula, instead of dividing the change in quantity by the beginning quantity, and the change in price by the beginning price, simply divide the change in quantity by the average of the two quantities, and the change in price by the average of the two prices. Using the average avoids having to designate a beginning and an ending.

In the example above, the change in quantity is 25; the average of the two quantities is 87.5 ((100 + 75) / 2). The percentage change in quantity, then - using the midpoint formula - is 25 / 87.5, or 28.57%.

The change in price is \$1; the average of the two prices is \$2.50 ((\$2 + \$3) / 2). The percentage change in price, then, using the midpoint formula, is 1 / 2.5, or 40%.

The price elasticity of demand, using the midpoint formula, is 28.57 / 40, or 0.71 (your instructor may have you use a different method for rounding).

Recall that the standard method yielded an answer of 0.5 for a price increase and 1.0 for a price decrease, two different answers using the same numbers. The midpoint formula, which is considered superior, yields only one answer, 0.71.

(Price) Elasticity of Supply

The elasticity of supply, also called the price elasticity of supply, is a measurement of the responsiveness of the quantity supplied to a change in price. The formula for the price elasticity of supply is:

Price elasticity of supply = the percentage change in quantity supplied divided by the percentage change in price.

This formula will yield a positive number, since the quantity supplied changes in the same direction as a change in price (due to the law of supply, and an upward sloping supply curve).

The numerical value of the price elasticity of supply depends on the ability of suppliers to readily change production inputs.

In the short run, some input levels will be fixed. In the long run, all inputs are variable. Time is the predominant factor in determining the price elasticity of supply. The longer the time frame, the more elastic the supply is.

Income Elasticity of Demand

Income elasticity of demand measures the responsiveness of the quantity demanded to a change in income. The formula for the income elasticity of demand is:

Income elasticity of demand = the percentage change in the quantity demanded divided by the percentage change in income

A normal good would have an income elasticity of demand that is greater than zero (a positive number).

An inferior good would have an income elasticity of demand that is less than zero (a negative number).

For normal goods, a necessity would have a relatively low income elasticity of demand while a luxury would have a relatively high income elasticity of demand.

Cross (Price) Elasticity of Demand

The cross elasticity of demand, also called the cross-price elasticity of demand, measures the degree to which different goods are related. It measures the responsiveness of quantity demanded of one good to a price change of another good.

The formula for the cross elasticity of demand is:

cross elasticity of demand = the percentage change in the quantity demanded of one good divided by the percentage change in the price of another good

If this formula yields a positive number, it means that as the price of one good increases, the quantity demanded of the other good increases: the goods are substitutes. The higher the cross elasticity, the closer the goods serve as substitutes.

If this formula yields a negative number, it means that as the price of one good increases, the quantity demanded of the other good decreases: the goods are complements. Consumers tend to buy the two goods together, as if they were a "package deal".

Tax Incidence and Elasticity

The combination of the price elasticity of demand and the price elasticity of supply will determine whether the consumer or the firm pay for a given tax increase.

A tax on the sale of goods (sales tax, excise tax) will ultimately be paid by either the consumer or the firm based on elasticities, regardless of who the government actually levies the tax on.

If the consumer ultimately pays the tax, it means that the tax incidence falls on the consumer. If the firm ultimately pays the tax, it means that the tax incidence ultimately falls on the firm.

The less elastic the demand and more elastic the supply, the more the tax incidence falls on the consumer.

The more elastic the demand and the less elastic the supply the more the tax incidence falls on the firm.

This is because with an inelastic demand, consumers will tend to spend more money with a tax increase. The decrease in the quantity purchased will be less than the increase in the tax. But with an elastic demand, consumers will purchase a smaller quantity with a tax increase, leaving firms to pay for the tax increase with lower revenue.

To the extent that consumers pay the tax, business costs and revenues will not be affected. To the extent that businesses pay the tax, the tax represents a cost of production, and supply will be decreased.