Suppose the average revenue of a short run perfectly competitive firm is 2 and its
Marginal cost and fixed cost is given as: 𝑀𝐶 = 3𝑄
2 − 8𝑄 + 6 and TFC=10 then,
A. Derive the function of TC, AVC and TR (1 pts.)
B. Calculate equilibrium price and quantity (1 pts.)
C. Find the profit at the equilibrium point and identify whether the firm makes
positive profit, normal profits or incurs loss. (1 pts.)
D. What price is needed for the firm to stay in the market? (1 pts.)
E. Calculate the output at which marginal costs are minimized? �
A. To derive the functions of TC, AVC, and TR, we need to use the following relationships:
Total Cost (TC) = Fixed Cost (TFC) + Variable Cost (VC)
Average Variable Cost (AVC) = Variable Cost (VC) / Quantity (Q)
Total Revenue (TR) = Average Revenue (AR) * Quantity (Q)
Given:
Marginal Cost (MC) = 3Q^2 - 8Q + 6
TFC = 10
Step 1: Derive the function of TC
Since TC = TFC + VC, we need to find the VC function by integrating the MC function.
VC = ∫(3Q^2 - 8Q + 6) dQ
VC = Q^3 - 4Q^2 + 6Q + C1
Note that C1 is the constant of integration.
Given that TFC = 10, we can substitute Q = 0 into the VC equation to find C1.
10 = (0)^3 - 4(0)^2 + 6(0) + C1
C1 = 10
Therefore, the VC function can be written as:
VC = Q^3 - 4Q^2 + 6Q + 10
Now, we can calculate TC by adding TFC and VC:
TC = TFC + VC = 10 + (Q^3 - 4Q^2 + 6Q + 10)
TC = Q^3 - 4Q^2 + 6Q + 20
Step 2: Derive the function of AVC
AVC = VC / Q
AVC = (Q^3 - 4Q^2 + 6Q + 10) / Q
AVC = Q^2 - 4Q + 6 + 10/Q
Step 3: Derive the function of TR
Given that AR (Average Revenue) = 2, we can conclude that TR = 2Q.
B. To calculate the equilibrium price and quantity, we need to set MC equal to AR:
3Q^2 - 8Q + 6 = 2
3Q^2 - 8Q + 4 = 0
To solve this quadratic equation, we can use the quadratic formula:
Q = (-(-8) ± sqrt((-8)^2 - 4(3)(4))) / (2(3))
Q = (8 ± sqrt(64 - 48)) / 6
Q = (8 ± sqrt(16)) / 6
Q = (8 ± 4) / 6
So, Q = (8 + 4) / 6 = 12 / 6 = 2.
Now, substitute the value of Q = 2 into the AR equation to find the equilibrium price:
AR = 2
TR = AR * Q
2 = P * 2
P = 1
Therefore, the equilibrium price is 1 and the equilibrium quantity is 2.
C. To calculate the profit at the equilibrium point, we need to subtract the total cost from the total revenue:
Profit = TR - TC
Profit = 2(1) - (2^3 - 4(2^2) + 6(2) + 20)
Profit = 2 - (8 - 16 + 12 + 20)
Profit = 2 - 24
Profit = -22
Since the profit is negative (-22), the firm incurs a loss.
D. To stay in the market, the firm needs to cover at least its total costs. Therefore, the price needed for the firm to stay in the market is equal to the average variable cost (AVC) at the equilibrium quantity.
AVC = Q^2 - 4Q + 6 + 10/Q
AVC = 2^2 - 4(2) + 6 + 10/2
AVC = 4 - 8 + 6 + 5
AVC = 7
Therefore, the price needed for the firm to stay in the market is 7.
E. To calculate the output at which marginal costs are minimized, we need to find where MC is equal to zero by setting MC = 0 and solving for Q:
3Q^2 - 8Q + 6 = 0
To solve this quadratic equation, we can use the quadratic formula:
Q = (-(-8) ± sqrt((-8)^2 - 4(3)(6))) / (2(3))
Q = (8 ± sqrt(64 - 72)) / 6
Q = (8 ± sqrt(-8)) / 6
Since sqrt(-8) is an imaginary number, there is no real output at which marginal costs are minimized.