Cross Elasticity of Demand

The cross elasticity of demand (or cross-price elasticity of demand)  ϵ_AB refers to the sensitivity of the demand for item A q_A to changes in the price of item B p_B:

\begin{equation*}\epsilon_{AB} = \frac{\Delta\mathit{q}_{A}}{\Delta\mathit{p}_{B}} = \frac{\frac{{dq}_{A}}{\mathit{q}_{A}}}{\frac{{dp}_{B}}{p_B}}\end{equation*}

In microeconomics, individuals' utility (or material well-being) is assumed to depend on their access to, or consumption of, bundles of items. It is also contingent on individuals seeking to maximize their utility. 

The precise relationship between the consumption of particular items and an individual’s utility is determined by the person’s preferences. Individuals are constrained in their consumption by their budgets and by the prices of the items.  

If the price of an item increases, then the individual’s ability to obtain utility through consumption of that item is reduced. It may be possible to obtain utility more efficiently through consumption of a different item instead. The extent to which this is possible depends on the substitutability of other items with the item whose price has increased.

For example, an individual may like to consume hazelnuts and peanuts. Ideally the individual likes to consume both types of nuts (due to diminishing marginal utility).  

However, if the price of one nut type – say peanuts – increases, then the individual can afford fewer peanuts. This would lead them to reduce peanut consumption and replace some of this product with hazelnuts. Hazelnuts and peanuts are thus substitutes in consumption. 

Note that the reverse is also true if the price of peanuts falls and becomes more affordable. The individual will then obtain utility more efficiently by reducing hazelnut consumption and increasing peanut consumption. This means that in the case of substitutes, the cross elasticity of demand is positive  ϵ_{HazelnutsPeanuts} > 0.

The interrelationship between items is key in the cross elasticity of demand. Some items are consumed together as complements rather than as replacements or substitutes. 

For example, fuel is required to power vehicles. If the price of either one of these items changes, demand for both goods will move in the same direction. 

If the price of vehicles increases then demand for the vehicles will fall. Because of this the demand for the fuel that powers the vehicles will also fall. Likewise, if the price of fuel rises then demand for both fuel and vehicles will fall. So, in the case of complements, the cross elasticity of demand is negative  ϵ_FuelVehicles < 0.

Other items have conceivably very little to do with each other. For example, if the price of yogurt changes then this would not be expected to affect the demand for smartphones. The cross elasticity of demand between these items should be close to zero  ϵ_{SmartphonesYogurt} ≈ 0. 

However, note that if an item that makes up a large or necessary part of individuals’ budgets becomes less affordable, people might buy fewer goods in general because they are relatively poorer. This will then lead indirectly to change in the demand for other seemingly unrelated items.

For example, if smartphones triple in price, people might decide they need smartphones but end up buying much less yogurt because they have less leftover money to spend. But, this still doesn’t mean that smartphones and yogurt are truly substitutes; there’s a different effect at play. See income elasticity of demand for details about this type of effect.

As with the standard price elasticity of demand for a single item, a cross elasticity of demand can be elastic (Iϵ_ABI ≥ 1), unit elastic (Iϵ_ABI = 1), inelastic (Iϵ_ABI < 1), perfectly inelastic (Iϵ_ABI = 0) or (in theory) perfectly elastic (Iϵ_ABI → ∞). This reflects the extent of the change in quantity demanded relative to a change in price.

Further reading

To better understand the difference between substitutes and complements and its importance for policy makers, read “Marijuana and Alcohol Evidence Using Border Analysis and Retail Sales Data” (2018). This paper from researchers Baggio, Chong and Kwon from the University of Connecticut and Georgia State University reflects on the effect of legalizing cannabis on the consumption of alcohol.

More generally, it has recently been reported that current cross-price elasticities are lower than previously estimated (Auer and Papies, Journal of the Academy of Marketing Science, 2019). The median estimate of ϵ_AB is about 0.1, implying typically only a small degree of substitutability between items. The authors mention firms’ focus on the development of brand loyalty as one reason why consumers may switch less between items than traditional economic theory would predict.

Good to know

The United States Department of Agriculture Economic Research Service provides a data tool to access estimates of price elasticities. This includes cross elasticities of demand, both for major commodities and countries (particularly the USA and China). The estimates are derived from research published by US academic and government sources.