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It should be clear by now that any variable on the right-hand side of the demand function can serve as the basis for its own elasticity. Recall that the price of another good might very well have an impact on the demand for a good or service, so we should be able to define an elasticity with respect to the other price, as well. That elasticity is called the cross-price elasticity of demand and takes on the same structure as own-price elasticity and income elasticity of demand, as represented in Equation 26.
Note how similar in structure this equation is to own-price elasticity. The only difference is that the subscript on P is now y, indicating the price of some other good, Y, instead of the own-price, X. This cross-price elasticity of demand measures how sensitive the demand for good X is to changes in the price of some other good, Y, holding all other things constant. For some pairs of goods, X and Y, when the price of Y rises, more of good X is demanded. That is, the cross-price elasticity of demand is positive. Those goods are defined to be substitutes.Substitutes are defined empirically. If the cross-price elasticity of two goods is positive, they are substitutes, irrespective of whether someone would consider them “similar.”
This concept is intuitive if you think about two goods that are seen to be close substitutes, perhaps like two brands of beer. When the price of one of your favorite brands of beer rises, what would you do? You would probably buy less of that brand and more of a cheaper brand, so the cross-price elasticity of demand would be positive.
Alternatively, two goods whose cross-price elasticity of demand is negative are defined to be complements. Typically, these goods would tend to be consumed together as a pair, such as gasoline and automobiles or houses and furniture. When automobile prices fall, we might expect the quantity of autos demanded to rise, and thus we might expect to see a rise in the demand for gasoline. Ultimately, though, whether two goods are substitutes or complements is an empirical question answered solely by observation and statistical analysis. If, when the price of one good rises the demand for the other good also rises, they are substitutes. If the demand for that other good falls, they are complements. And the result might not immediately resonate with our intuition. For example, grocery stores often put something like coffee on sale in the hope that customers will come in for coffee and end up doing their weekly shopping there as well. In that case, coffee and, say, cabbage could very well empirically turn out to be complements even though we do not normally think of consuming coffee and cabbage together as a pair (i.e., that the price of coffee has a relation to the sales of cabbage).
For substitute goods, an increase in the price of one good would shift the demand curve for the other good upward and to the right. For complements, however, the impact is in the other direction: When the price of one good rises, the quantity demanded of the other good shifts downward and to the left.