Suppose that a demand function is linear, that is, q=m-np, where m and n are positive constants. Show that E=1 at the midpoint of the demand curve on the interval 0 greater than or equal to P greater than or equal to m/n, that is, at p=m/(2n)
First, let's calculate the elasticity of demand at the midpoint of the demand curve, which is given by E=(dq/dp)*(p/q).
Given the demand function q=m-np, the derivative of q with respect to p is -n.
Therefore, the elasticity of demand at the midpoint p=m/(2n) is E=(-n)*(m/(2n))/(m-n*(m/(2n)))=(-m/(2(m-n(m/(2n))))=-m/(2m-m/2)=-(2/3).
Hence, we can see that the elasticity of demand is equal to 1 at the midpoint of the demand curve when p=m/(2n).