Economics, 3rd Edition

Calculus

The demand curve is often depic-

ted as a linear curve but there is 

no reason why it should be linear 

and can be curvilinear. To measure 

elasticity accurately in this case 

economists use calculus. The rules 

of calculus applied to a demand 

curve give a far more accurate 

measurement of the price elasticity 

of demand at a particular point.

For a linear demand function, the 

approximation to the point elasticity 

at the initial price and quantity is 

given by:

(Q

2

− Q

1

)

(P

2

− P

1

)

·

P

1

Q

1

which gives exactly the same res-

ult as the point elasticity which is 

defined in terms of calculus and is 

given by:

dQ

dP

·

P

Q

Point elasticity defined in terms of 

calculus gives a precise answer; all 

the other formulae are approxima-

tions of some sort. The formula looks 

similar but it must be remembered 

that what we are talking about in 

this instance is an infinitesimally 

small change in quantity follow-

ing an infinitesimally small change 

in price. In the formula, dQ/dP  is 

the derivative of a linear demand 

function. Given our basic linear 

equation of the form Q

= a − bP,  

the power function rule gives dQ/dP  

as the coefficient of P

− (−b).

Take the following demand 

equation:

Q

= 60 − 3P

To find the price elasticity of 

demand when price 

= 15 first of all 

we need to find Q:

Q

= 60 − 3P

Q

= 60 − 3(15)

Q

= 15

We calculate dQ/dP  as 

−3.

Substitute this into the formula 

to get:

Price elasticity of demand

= −3a

15

15

b

Price elasticity of demand

= −3