A monopolist faces the demand function P = 140 − 2Q, with P denoting the market price and Q denoting the quantity demanded.

There is a fixed cost of 200, the total variable cost function is T V C(Q) = 1 3Q3 − 10Q2 + 188Q.

(a) Obtain an expression for the profit function f(Q), defined for Q > 0.

(b) Determine the derivative of the profit function.

(c) For what value(s) of Q the derivative of the profit function equals to zero? Explain your answer carefully

Expert Answer

The Profit function is defined as the difference between Revenue function and Cost function, where the Revenue function is defined as the price times the number of products.

So starting with the revenue we have:

[tex]R(Q)=PQ[/tex]

where P is the market price and Q the quantity demanded, using the demand function we can write:

[tex]R(Q)=(140-2Q)Q[/tex]

a) Thus the Profit function is

[tex]P(Q)=R(Q)-C(Q)\\\\P(Q)=140Q-2Q^2 -(13Q^3-10Q^2 +188Q)\\\\P(Q)=140Q-2Q^2 -13Q^3+10Q^2-188Q\\\\P(Q)= -13Q^3 +8Q^2-44Q[/tex]

b) We can determine the derivative of the profit function using the result on part a).

[tex]P(Q)= -13Q^3 +8Q^2-44Q\\P \,'(Q)=-39Q^2 +16Q-44[/tex]

c) We can set the profit function equal to 0

[tex]P \, '(Q)= 0[/tex]

[tex]-39 Q^2 + 16 Q - 44 =0[/tex]

Since we cannot factor we will have to use quadratic formula.

[tex]Q = \cfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

where the coefficients are

[tex]a=-39\\b = 16\\ c=-44[/tex]

So we get

[tex]Q = \cfrac{-16\pm \sqrt{16^2-4(-39)(-44)}}{2(-39)}[/tex]

Notice that we get a negative number inside the square root if we evaluate it.

[tex]Q = \cfrac{-16\pm \sqrt{-6608}}{2(-39)}[/tex]

Since we get a negative number inside the root, then there are no real values of Q where the derivative of the profit function is equal to 0.