Hello ive been struggling with this problem for about 2 days now could someone walk me through it?
Suppose a firm faces a downward sloping demand curve given by the equation Q = 100 - (1/3)P. The firm's cost function is given by the equation C = 30 + (1/4)Q^2. Find the Profit Maximizing level of output.
thank you
always always always, MC=MR.
First rearrange the demand function to be P=f(Q). That is P=33.33 - Q/3
Now then Total revenue is P*Q. So TR=33.33Q -(Q^2)/3
MR is the first derivitive of TR. So MR=33.33 - (2/3)Q
MC is the first derivitive of TC. So MC=(1/2)Q
MC=MR - use algebra and solve for Q. Take it from here
oops, my bad algebra. I divided by 3 instead of multiplying by 3. So, P should be P=300 - 3Q.
But follow the same methodology as before starting from here.
To find the profit-maximizing level of output, we need to determine the quantity at which the firm's total revenue (TR) minus its total cost (TC) is maximized.
1. Find the total revenue (TR):
Total revenue is equal to the quantity sold (Q) multiplied by the price (P). In this case, the price is determined by the demand curve equation: P = 100 - (1/3)Q.
Therefore, TR = Q * P = Q * (100 - (1/3)Q).
2. Find the total cost (TC):
The cost function is given by C = 30 + (1/4)Q^2.
Therefore, TC = 30 + (1/4)Q^2.
3. Find the profit function (𝜋):
The profit function is equal to the total revenue minus the total cost.
Therefore, 𝜋 = TR - TC = Q * (100 - (1/3)Q) - (30 + (1/4)Q^2).
4. Maximize the profit:
To find the quantity (Q) at which the profit is maximized, we need to take the derivative of the profit function with respect to Q, set it equal to zero, and solve for Q.
𝜕𝜋/𝜕Q = 100 - (2/3)Q - (1/4)Q^2 = 0.
Rearrange the equation:
(1/4)Q^2 - (2/3)Q + 100 = 0.
Solve the quadratic equation using any method (factoring, quadratic formula, etc.) to find the values of Q.
Once you have the values of Q, substitute them back into the profit function and find the corresponding profit for each value of Q.
Select the quantity (Q) that gives the highest profit. This will be the profit-maximizing level of output.
Note: It is important to check the second derivative of the profit function to ensure that the solution obtained is indeed a maximum. If the second derivative is negative, then the solution is a maximum.