,
the partial derivative of the quantity with respect to its price.Ceteris Paribus
In mathematics, the "other-things-equal
assumption" is embodied in the concept of the partial derivative. Suppose,
for example, that the amount of Pepsi consumed (QP) depends
on its price (PP), the price of Coca-Cola (PC),
consumer incomes (I), and preferences (Z) according to the relationship
QP =
f(PP, PC, I, Z). If
we wish to investigate the effect of a change in the amount of Pepsi brought
about by a change in its price, other things equal, we would compute ,
the partial derivative of the quantity with respect to its price.
Consider the following example from
Rao and Miller. They estimate the demand for Ceylonese Tea to be QC
= 1.85 - 2.10PC + 0.20PB + 1.56PI,
where QC is the quantity of Ceylonese tea demanded, PC
is its price, and PB and PI are the prices
of Brazilian coffee and Indian tea, respectively (all variables are expressed
in logarithms.) If we wish to know what happens to the quantity of Ceylonese
tea demanded when its price changes, with no change in the prices of Brazilian
coffee or Indian tea, we can simply calculate 
=
- 2.10. This tells us that each unit increase in the (logarithm of the) price
of Ceylonese tea reduces its (logarithm of) consumption by 2.10. Likewise, each
increase in the price of Brazilian coffee is found to increase the quantity
of Ceylonese tea, all else constant, by 
= 0.20.