What is cross price elasticity of demand, what does it mean when it's negative or positive?

In economics we are very interested in the concept of elasticities, and one of the more difficult elasticities to understand is the cross price elasticity.  I believe this is because the cross price elasticity of demand is the only elasticity that considers two different goods simultaneously in the same equation.  The price elasticity of demand considers a percentage change in price and quantity, but both are for the same good.  The price elasticity of supply again considers the percentage change in price and quantity, and again both are for the same good.  Finally we have the income elasticity of demand which considers the percentage change in quantity of a good, and the percentage change of income, which seems to be easier to understand because of its name.

When working with cross price elasticities the magnitude of the number is not what we are most
interested in.  For most applications, economists only want to know if the cross price elasticity of demand is positive, negative or zero.  Let's now define the math for the cross price elasticity of demand, and explore the different scenarios that would result in a positive, negative, or zero cross price elasticity of demand.

Cross price elasticity of demand equation:
(Percent change in Quantity of one good)/(Percent change in Price of another good)
or

Remember that the cross price elasticity of demand equation requires two different goods.  Now, let's imagine Coke as one good, and Pepsi as the other.  If the price of Coke goes up by 20%, then 10% more people will buy Pepsi.  This means that the percent change in price of Coke is positive, and the percent change in quantity demanded of Pepsi is positive.  Since both changes are positive, the cross price elasticity of demand will be positive.  The cross price elasticity of demand measure in this example is 10/20 or 0.5.  Since Coke and Pepsi are substitutes, we can confirm that if the cross price elasticity of demand is positive, then we are dealing with substitutes.

Note that this relationship holds true even if the price of Coke were to drop.  If the price of coke went down by 20%, then less people would buy Pepsi because Coke is cheaper.  We can assume demand for Pepsi drops 10%.  This gives us -10/-20 or 0.5 which is what we had before.  It is important to realize that the end result is still positive, because the goods are still substitutes.

Now we will consider computers and monitors.  If the price of computers goes up by 10%, we will probably see a decline in consumer demand for monitors by 10%.  This results in the equation -10/10 or -1.  Since the change in price is positive, and the change in quantity is negative, the cross price elasticity of demand measure will be negative.  This means that the goods are complements, which computers and monitors are.

Remember that it doesn't matter if the price goes up or down, the final sign of the cross price elasticity of demand will remain the same depending on whether the goods are substitutes or complements.

Finally it is possible for the cross price elasticity of demand to be zero.  While getting a perfect zero is rare, any number near zero indicates that the goods simply are not related.  For example, if the price of exercise DVDs went up, we would expect almost no change in the quantity of fish food demanded by people.

SUMMARY:
 If CPED > 0 the goods are substitutes
If CPED < 0 the goods are complements
If CPED = 0 (or very close) the goods are not related