given the demand function Q=9-.5P how can you figure out the optimal price if the objective of the firm is to mazimize revenues?? PLEASE HELP
more a calclus question with an economics application. First, set the equation for total revenue TR. TR is simply price times quantity -- P*Q. Multiply your equation by P.
P*Q = 9P-.5P^2 = TR
Next, find the first derivitive wrt to P. TR' = 9-P. Find the P such that TR'=0. Obviously P=9. This is the revenue (not profit) maximizing price.
To figure out the optimal price that maximizes revenue, you can follow these steps:
1. Start with the demand function Q = 9 - 0.5P, where Q represents quantity and P represents price.
2. Calculate the total revenue (TR) by multiplying the price (P) by the quantity (Q). This can be written as P * Q.
TR = P * Q
Replace Q in the equation with the demand function:
TR = P * (9 - 0.5P)
Simplify the equation to get: TR = 9P - 0.5P^2
3. Differentiate the total revenue equation with respect to price (P) to find the derivative TR':
TR' = d(TR)/dP = 9 - P
This represents the rate of change of total revenue with respect to price.
4. Set the derivative TR' equal to zero to find the critical points:
9 - P = 0
Solve for P:
P = 9
This is the price that maximizes revenue (not profit).
Therefore, the optimal price for maximizing revenue in this case is P = 9.