Question

The monopolistic competitor faces a demand curve given by Q(p)=50-5p. It's cost function is C(y)=4y. What is optimal level of output and price? (5 marks)

Answer 1

Given the demand function, the inverse demand can be written as:

$Q(p)=50−5p$

$Y= 50-5p$

$5p=50-Y$

$P=10-\frac{1}{5}Y$

The cost function is given as:

$c(Y) = 4Y$

Thus, given the demand and cost, the profit function can be written as:

Profit(π) = Total Revenue−Total Cost

$Profit [π]= (10−\frac{1}{5}Y)Y−4Y$

Differentiating the profit function to solve for profit-maximizing output:

$=> \frac{∂π}{∂Y}=10−\frac{2}{5}Y−4=0$

$=>6−\frac{2}{5}Y−4=0$

$=>\frac {2}{5}Y=6$

$Y=\frac{6×5}{2}=15$

Hence, the profit-maximizing output is 15 units. 

Given the output, it can be substituted in the inverse demand function to find the price:

$p=10−[\frac{1}{5}×15]$

$p =10−3$

$p=7$

The optimal price for the monopolist competitor to charge is 7.