firms demand function,p=24-0.54q and the average cost function,AC=Q2-8Q+36+3/Q,calculate the level of output Q, which maximize total revenue,maximizes profits
But what do you need here?
Sra
A) Total revenue is the product of the quantity sold and the price at which they were sold.
(total revenue) = Q*P
= Q*(24 - .5Q)
= -.5Q^2 + 24Q
This quadratic function will have a maximum at Q = -24/(2*(-.5)) = 24
Total revenue will be maximized when Q = 24.
B) The profit on each unit is the difference between the selling price and the cost. The total profit will be the product of the unit profit and the number of units sold.
(total profit) = Q*(P - AC)
= Q*((24 - .5Q) - (Q^2 - 8Q + 36 + 3/Q))
= 24Q - .5Q^2 - Q^3 + 8Q^2 - 36Q - 3
= -Q^3 + 7.5Q^2 - 12Q - 3
This will be maximized when the derivative is zero.
-3Q^2 + 15Q - 12 = 0
Q^2 - 5Q + 4 = 0 (divide by -3)
(Q - 1)(Q - 4) = 0
This suggests that quantities of 1 or 4 will maximize profit. Using our equation for total profit, we find that profit for Q=1 is negative. Profit will be maximized when Q = 4.
Here is a plot of profit versus quantity sold.
To find the level of output that maximizes total revenue and profits, we need to first calculate the total revenue and profit functions.
Total revenue (TR) is defined as the product of price (p) and quantity (q):
TR = p * q
Profit (π) is defined as total revenue minus total cost (TC):
π = TR - TC
We will calculate these functions step by step.
1. Total Revenue Function (TR):
TR = p * q
= (24 - 0.54q) * q
= 24q - 0.54q^2
2. Total Cost Function (TC):
To calculate the total cost, we need to find the average cost (AC) and then multiply it by the quantity (Q).
Average Cost (AC):
AC = Q^2 - 8Q + 36 + 3/Q
Total Cost (TC):
TC = AC * Q
= (Q^2 - 8Q + 36 + 3/Q) * Q
= Q^3 - 8Q^2 + 36Q + 3
Now that we have the total revenue function (TR) and the total cost function (TC), we can move on to finding the level of output that maximizes total revenue, which will also maximize profits.
To maximize total revenue, we need to find the value of q that maximizes the TR function. This can be done by finding the derivative of TR with respect to q, and setting it equal to zero, then solving for q.
d(TR)/dq = 24 - 1.08q
Setting this derivative equal to zero:
24 - 1.08q = 0
Solving for q:
1.08q = 24
q = 24 / 1.08
q ≈ 22.22
Therefore, the level of output (Q) that maximizes total revenue is approximately 22.22 units.