Find the level of output which total revenue will be max

[Guest]

2. The demand curve function of a good is given by

Q = 12.5 – 0.125P

where

Q denotes units of output
P denotes price

and the total cost function is

[tex]TC = Q^3-10Q^2 + 85Q + 10[/tex]

a) Find the level of output which total revenue (TR) will be maximized.

b) Derive total profit (π) function and then find the level of output which total profit (π).

[Guest]

We have to find the maximum of the function:
[tex]TC = Q^3-10Q^2 + 85Q + 10 = (12.5 - 0.125P)^3 - 10(12.5-0.125P)^2 + 85(12.5 - 0.125P) + 10[/tex]
the derivative is [tex]3(12.5 - 0.125P)^2*(-0.125) - 20(12.5 - 0.125P) + 85[/tex]
Solve it and write the roots:
[tex]3(12.5 - 0.125P)^2*(-0.125) - 20(12.5-0.125P) + 85 = 0[/tex]

There is a nice tool here for loving quadratic equations:
http://www.math10.com/en/algebra/quadra ... ation.html

[Guest]

When you find the roots one of them is the maximum and the other one is the minimum.
You can check which one is min. and max. by simply replacing P with the root and see which value ot TC is bigger.

[Guest]

You are a leading consultant with a firm producing milk within Nakuru County. One of your main roles is to advice the firm on price strategies that would lead to maximize profits. The firm is a monopolist which sells in two distinct markets, one of which is completely sealed off from the other.
In line of your assignment, you establish that the total demand for the firms output is given by the following equation:
Q = 50 – 0.5P
The demand for the firms output in the two markets is:
Q1 = 32 – 0.4P1
Q2 = 18 – 0.1 P2
Where Q= total output
P= Price
Q1= Output sold in market 1
Q2= Output sold in market 2
P1= Price charged in market 1
P2= Price charged in market 2
The total cost of production is given by C= 50 + 40Q, where C= total cost of producing a unit of milk.





Required:
a) The total output that the firm must produce in order to maximize profits
b) What price must be charged in each market in order to maximize profits
c) How much profit would the firm earn if it sold the output as a single price, and if the firm discriminates
d)The price elasticity of demand for the two markets at the equilibrium price and quantity.