Cross-price elasticity of demand
Contents
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 ![]() ![]() ![]() |
This depends on the (own)-price elasticity of demand of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
substitution effect | ![]() ![]() ![]() ![]() |
A change in the price of ![]() ![]() ![]() |
This tends to make the cross-price elasticity of demand more positive, because increasing price of ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
complementary effects | ![]() ![]() ![]() ![]() |
A change in the price of ![]() ![]() ![]() |
This tends to make the cross-price elasticity of demand more negative, because increasing price of ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 ![]() ![]() |
Do ![]() ![]() |
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 |