A company is working on the market of perfect competition. Its cost function TC=Q^2-4Q+64 and demand function on the product that company produces is Q=400-20P. Calculate: a. optimal quantity of production, at which company maximizes profits b. market proce at which company maximizes profits c. break-even price of this company d. total revenue at profit maximazation
To find the optimal quantity of production at which the company maximizes profits, we need to find the quantity that corresponds to the highest point on the profit curve.
First, we need to find the total revenue (TR) function from the demand function. Total revenue is the price (P) multiplied by the quantity (Q):
TR = P * Q
We can substitute Q from the demand function into the total revenue equation:
TR = P * (400 - 20P)
TR = 400P - 20P^2
Next, we need to find the total cost (TC) function from the given cost function:
TC = Q^2 - 4Q + 64
To find the profit (π), we subtract the total cost from the total revenue:
π = TR - TC
π = 400P - 20P^2 - (Q^2 - 4Q + 64)
π = 400P - 20P^2 - Q^2 + 4Q - 64
To maximize profits, we need to find the derivative of the profit function with respect to quantity (Q) and set it equal to zero:
dπ/dQ = -2Q + 4 = 0
2Q = 4
Q = 2
a. The optimal quantity of production at which the company maximizes profits is Q = 2.
To find the market price at which the company maximizes profits, we can substitute the quantity we found (Q = 2) into the demand function:
Q = 400 - 20P
2 = 400 - 20P
20P = 398
P ≈ 19.9
b. The market price at which the company maximizes profits is approximately $19.90.
To find the break-even price, we need to set the total cost equal to zero and solve for price (P):
TC = Q^2 - 4Q + 64
0 = Q^2 - 4Q + 64
Using the quadratic formula, we can solve for Q:
Q = (4 ± √(4^2 - 4*1*64)) / (2*1)
Q = (4 ± √(16 - 256)) / 2
Q = (4 ± √(-240)) / 2
Since the square root of a negative number is not real, there are no real solutions for Q, meaning the company does not achieve a break-even price.
c. The company does not have a break-even price.
To find the total revenue at profit maximization, we can substitute the quantity (Q = 2) into the total revenue function:
TR = P * Q
TR = 19.9 * 2
TR ≈ 39.8
d. The total revenue at profit maximization is approximately $39.80.
To find the optimal quantity of production at which the company maximizes profits, we need to find the quantity level that equates marginal cost (MC) to marginal revenue (MR).
First, let's find the marginal cost function (MC) from the total cost function (TC). The marginal cost is the derivative of the total cost with respect to quantity (Q):
MC = d(TC)/dQ
Given the total cost function TC = Q^2 - 4Q + 64, we can find the derivative:
MC = d/dQ (Q^2 - 4Q + 64)
= 2Q - 4
Next, let's find the marginal revenue function (MR) from the demand function. The marginal revenue is the derivative of the total revenue with respect to quantity:
MR = d(TR)/dQ
Since total revenue (TR) is the product of quantity (Q) and price (P):
TR = Q * P
And the demand function is Q = 400 - 20P, we can express price as P = (400 - Q) / 20:
MR = d/dQ (Q * (400 - Q) / 20)
= (400 - 2Q) / 20
= 20 - 0.1Q
Now, equate marginal cost (MC) to marginal revenue (MR) to find the optimal quantity:
MC = MR
2Q - 4 = 20 - 0.1Q
Simplifying the equation:
2Q + 0.1Q = 20 + 4
2.1Q = 24
Q = 24 / 2.1
Q ≈ 11.43
The optimal quantity of production, at which the company maximizes profits, is approximately 11.43.
Now, let's calculate the market price at which the company maximizes profits. We can substitute the optimal quantity back into the demand function:
Q = 400 - 20P (original demand function)
11.43 = 400 - 20P
Rearranging the equation:
20P = 400 - 11.43
20P ≈ 388.57
P ≈ 19.43
The market price at which the company maximizes profits is approximately $19.43.
Next, let's calculate the break-even price of this company. The break-even price occurs when the total cost (TC) equals the total revenue (TR):
TR = TC
Since TR is the product of quantity (Q) and price (P), and TC is given as Q^2 - 4Q + 64:
Q * P = Q^2 - 4Q + 64
Substituting the demand function into the equation:
(400 - 20P) * P = (400 - 20P)^2 - 4(400 - 20P) + 64
Simplifying the equation, we find the break-even price P:
P ≈ 21.76
The break-even price of this company is approximately $21.76.
Finally, to calculate the total revenue at profit maximization, we substitute the optimal quantity into the demand function:
Q = 400 - 20P
Q ≈ 400 - 20(19.43)
Q ≈ 10.14
Now, multiply the optimal quantity by the market price to get the total revenue:
Total Revenue = Q * P
≈ 10.14 * 19.43
≈ $197.31
Therefore, the total revenue at profit maximization is approximately $197.31.