Question

Imagine you've started a new pizza restaurant. It costs you about $5 to produce a pizza.

Last week you sold 500 pizzas for $10 each.

This week you raised your price and sold 150 pizzas for $16 each.

What price should you be charging?

Answer 1

Solution:

Derive functions:

$500=(a\times 10) + b$

$150=(a\times 16) + b$

$350=-6a$

$a = \frac{350}{-6}$

$a = -58.3$

Substitute from the function to get b:

$500 = (-58.3\times 10)+b$

$500 = -583+b$

$500+583=b$

$b=1083$

Therefore:

$Q =-58.3P+1083$

Substitute to derive P:

$-58.3P+1083=Q$

$-58.3P=Q-1083$

$P=\frac{-Q}{-58.3}-\frac{1083}{-58.3}$

$P=-0.02Q+18.6$

Find TR:

$TR = (P\times Q)$

$TR=(-0.02Q\times Q) +(18.6\times Q)$

$TR=-0.02Q^{2} +18.6Q$

Find MR:

The MR is the derivative of the total revenue with respect to demand:

Therefore:

$MR =-0.04Q+18.6$

Substitute to derive the Price:

$5=-0.04Q+18.6$

$0.04Q=18.6-5$

$Q=\frac{13.6}{0.04}$

$Q=340$

Substitute for Price:

$P=-0.02Q+18.6$

$P=(-0.02\times 340)+18.6$

$P=-6.8+18.6$

$P=11.8$

The price that should be charged is = 11.8