Question

Derive an expression for total revenue.

Find its critical value. Is it a min or max? Explain how you know.

4. Suppose you have the following demand curve: Pa = a - bQ Derive an expression for total revenue. Find its critical value. Is this a minimum or maximum? How do you know? Explain/Show work.

Answer 1

Answer:

Given, demand function P_d=a-bQ

We know that Total Revenue (TR)=Price*Quantity=aQ-bQ²

Critical value:

Critical value is the value when a function will have minimum or maximum value and it is possible when the first derivative is zero. Thus first derivative of TR w.r.t. Q we get MR

Hence, $MR=\frac{\partial TR}{\partial Q}=0=>\frac{\partial}{\partial Q}(aQ-bQ^2)=0=>a-2bQ=0=>Q=\frac{a}{2b}$

Now if the second derivative is positive it is a minimum and if it is negative then it is maximum.

Thus, the second derivative of the above function, $\frac{\partial}{\partial Q}(a-2bQ)=-2b$

Thus, the second derivative is negative (i.e. -2b is negative, but b should be greater than 0).

Therefore, we have a maximum value.