# Cross-price elasticity of demand

## Definition

Suppose  and  are two commodities. The cross-price elasticity of demand of  with respect to  measures the fractional change in the demand of  in response to a fractional change in the unit price of . Note that the price of  is not changed in the process.

Formally, if  and  denote the unit prices of  and  and  and  denote the quantities demanded for  and , the cross-price elasticity is given by any of the following four equivalent formulations:



Note that although only  and  appear in the expression for cross-price elasticity, the value  could also affect the value of cross-price elasticity. More specifically, the formula makes sense against a backdrop of the value of  and all the other determinants of demand.

### Interpretation in terms of demand curve

The cross-price elasticity of demand cannot be computed by looking at any single instance of the usual demand curve or logarithmic demand curve for either  or . Rather, it measures the speed of expansion/contraction of the demand curve for  with respect to a price change in .

The price of  is possibly one of the determinants of demand for the quantity demanded of . The price of  is thus one of the exogenous parameters to the demand curve of . A change in this parameter leads to a shift (expansion or contraction) in the demand curve for  and the magnitude of this shift at the price level  is what the cross-price elasticity measures.

## Properties

For the entire discussion, we assume that both  and  are ordinary goods (quantity demanded decreases with increase in price) and normal goods (quantity demanded increases with increase in disposable income). In the absence of these assumptions, more interesting and complicated variations of the possibilities below can occur.

### Factors affecting cross-price elasticity of demand

There are three kinds of factors affecting cross-price elasticity of demand. As above,  is the good whose price is changed and we measure the effect of this change on the quantity of  demanded.

Effect Condition on relationship between goods Explanation of effect How it affects cross-price elasticity of demand
income effect none A finite income imposes a budget constraint. An increase in the price of  changes the quantity consumed and hence the total money spent on . This in turn changes the effective available income for other purposes, including the consumption of . This depends on the (own)-price elasticity of demand of . If its magnitude is greater than , the total money spent on  declines with an increase in price, leaving more money free for consuming  -- thus it tends to make the cross-price elasticity of demand more positive. If the magnitude of own-price elasticity of demand of  is less than , the total money spent on  increases as price increases, leaving less money for , thus it tends to make the cross-price elasticity of demand more negative.
substitution effect  and  are substitute goods: they substitute (partially or wholly) for each other -- i.e., having more of  reduces the marginal utility of obtaining , and vice versa A change in the price of  affects the quantity of  demanded. By the law of demand, the quantity demanded decreases with an increase in price. This in turn affects the utility function of , and hence, the quantity demanded at a given price. This tends to make the cross-price elasticity of demand more positive, because increasing price of   lower quantity of   increase in utility of consuming   increase in quantity of  demanded at a given price
complementary effects  and  are complementary goods: having more of  increases the marginal utility of  and vice versa. A change in the price of  affects the quantity of  demanded. By the law of demand, the quantity demanded decreases with an increase in price. This in turn affects the utility function of , and hence, the quantity demanded at a given price. This tends to make the cross-price elasticity of demand more negative, because increasing price of   lower quantity of   decrease in utility of consuming   decrease in quantity of  demanded at a given price

### Sign

The sign of the cross-price elasticity of demand depends on the relative extent of operation of the various factors that influence the cross-price elasticity of demand noted above. In some cases, the sign is unambiguous, while in others, it is ambiguous.

Is the magnitude of own-price elasticity of demand of  greater than or less than ? Do  and  complement or substitute for each other? Conclusion about sign of cross-price elasticity
less than  complement negative
less than  substitute ambiguous but typically positive; positive if substitution effect dominates, which would be the case for close substitutes.
greater than  complement ambiguous; sign depends on whether the complementary effect or income effect is stronger.
greater than  substitute positive