Question

Given the constant elasticity demand function as :

𝑃=𝑎𝑃𝑏 𝑤ℎ𝑒𝑟𝑒 𝑏 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑙𝑠𝑡𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑚𝑎𝑛𝑑

a. Show that Marginal Revenue of this function is proportional to the price

(let 𝐾=(1𝑎)1𝑏⁄ ). To simplify the equation.

b. calculate the Marginal Revenue when 𝑒𝑞𝑢𝑖𝑙𝑃=𝑏=−2 and when b=-10.

c. what does your answer mean in terms of revenue facing the firm?

d. If 𝑏=−𝛼 what would this imply for Marginal Revenue and Total Revenue of the firm.

e. Explain how Marginal Revenue and profit maximization would be affected if demand was inelastic.

Answer 1

constant utility demand function is given as :

$p=ap^b$

b- elasticity of demand

a.

$p=ap^b\\$

$\frac{1}{a}=\frac{p^b}{p}\\$

$\frac{1}{a}=pb^{-1}$

$p=(\frac{1}{a})^{\frac{1}{b-1}}$

$Revenue=p.Q\\=(\frac{1}{a})^{\frac{1}{b-1}}$

$MR=p$

Therefore, marginal revenue of the function is proportional to the price.

b

when equil$p=b=-2$

$p=ap^b\\(-2)=a(-2)^{-2}$

$a=-8$

$\implies MR=(\frac{1}{-8})^{\frac{1}{-2-1}}\\MR=-2$

when equil $p=b=-10$

$p=ap^b\\(-10)=a(-10)^{-10}\\a=-10^{11}$

$\implies MR =-10$

c.

The marginal revenue, MR is negative in both the cases. this means that the toital revenue facing the firm is decreasing.

d.

$\frac{g}{b} =-\alpha,$ The marginal revenue of the firm would be $-\alpha$ and the total revenue of the firm would be decreasing.