Given demand curve P=400-.05Q, at what price is total revenue maximized?
There are a number of ways to do this problem. Let me use simple calculus.
Total Revenue is P*Q=400Q-.05Q^2. Take the first derivitive wrt Q. MR=400-.1Q To maximize, find Q when MR=zero. Q=4000 Plug this into the original demand equation..... (take it from here)
To maximize total revenue, we can use the concept of marginal revenue (MR). Marginal revenue is the additional revenue generated from selling one additional unit of a product.
In this case, the demand curve is given by P = 400 - 0.05Q, where P represents the price and Q represents the quantity demanded. The total revenue (TR) is given by the product of price and quantity, which is TR = P * Q.
To find the price at which total revenue is maximized, we need to find the quantity that maximizes marginal revenue (MR) and then substitute this quantity into the demand equation to calculate the corresponding price.
First, we differentiate the total revenue equation with respect to quantity (Q) to find the marginal revenue function:
MR = dTR/dQ
Taking the derivative of TR = (400 - 0.05Q) * Q with respect to Q, we get:
MR = 400 - 0.1Q
To maximize total revenue, we set MR equal to zero and solve for Q:
0 = 400 - 0.1Q
0.1Q = 400
Q = 4000
So, the quantity of the product that maximizes total revenue is Q = 4000.
Next, substitute this value of Q back into the demand equation to find the corresponding price (P):
P = 400 - 0.05Q
P = 400 - 0.05(4000)
P = 400 - 200
P = 200
Therefore, the price at which total revenue is maximized is $200.