Suppose the average revenue of a short run perfectly competitive firm is 2 and its Marginal cost and fixed cost is given as: MC=3Q2 -8Q+6 and TFC=10 then,
A. Derive the function of TC, AVC and TR
B. Calculate equilibrium price and quantity
A. Find the profit at the equilibrium point and identify whether the firm makes positive profit, normal profits or incurs loss.
B. What price is needed for the firm to stay in the market?
To derive the functions of TC, AVC, and TR, we need to find the total cost (TC) and average variable cost (AVC) functions using the given marginal cost (MC) and fixed cost (TFC).
A. To find the total cost (TC), we need to integrate the marginal cost (MC) function with respect to quantity (Q).
MC = 3Q^2 - 8Q + 6
Integrating MC with respect to Q, we get:
TC = ∫(3Q^2 - 8Q + 6) dQ
= Q^3 - 4Q^2 + 6Q + C
Since TFC = 10, we can find the constant C:
10 = 0^3 - 4(0)^2 + 6(0) + C
C = 10
Therefore, the total cost (TC) function is:
TC = Q^3 - 4Q^2 + 6Q + 10
To find the average variable cost (AVC), we need to divide the total variable cost (TVC) by quantity (Q). Since TFC is the fixed cost and doesn't change with quantity, TVC equals TC - TFC.
TVC = TC - TFC
TVC = Q^3 - 4Q^2 + 6Q + 10 - 10
TVC = Q^3 - 4Q^2 + 6Q
AVC = TVC / Q
AVC = (Q^3 - 4Q^2 + 6Q) / Q
AVC = Q^2 - 4Q + 6
The total revenue (TR) function is given as an average revenue (AR) of 2. Since AR equals total revenue (TR) divided by quantity (Q), we can find TR by multiplying AR by Q.
AR = TR / Q
2 = TR / Q
TR = 2Q
B. To calculate the equilibrium price and quantity, we set the average revenue (AR) equal to the marginal cost (MC) and solve for Q.
AR = MC
2 = 3Q^2 - 8Q + 6
Simplifying the above equation, we get:
3Q^2 - 8Q + 4 = 0
Using the quadratic formula, we find two possible solutions for Q:
Q = (8 ± √(8^2 - 4(3)(4))) / (2(3))
Q = (8 ± √(64 - 48)) / 6
Q = (8 ± √16) / 6
Q = (8 ± 4) / 6
The two possible values for Q are:
Q1 = 2
Q2 = 2/3
To find the equilibrium price, we substitute the equilibrium quantity (Q1 or Q2) into the average revenue (AR) function.
P = AR / Q
P = 2 / 2
P = 1
Therefore, the equilibrium price is 1 and the equilibrium quantity is 2.
A. To find the profit at the equilibrium point, we first need to find the total revenue (TR) and total cost (TC) at the equilibrium quantity (Q1 = 2).
TR = 2Q1
TR = 2(2)
TR = 4
TC = Q1^3 - 4Q1^2 + 6Q1 + 10
TC = 2^3 - 4(2)^2 + 6(2) + 10
TC = 8 - 16 + 12 + 10
TC = 14
Profit = TR - TC
Profit = 4 - 14
Profit = -10
The firm incurs a loss of 10 at the equilibrium point.
B. To determine the price needed for the firm to stay in the market, we need to consider the average variable cost (AVC). If the price is below the AVC, the firm incurs a loss and may exit the market.
At Q1 = 2, the AVC is:
AVC = Q1^2 - 4Q1 + 6
AVC = 2^2 - 4(2) + 6
AVC = 4 - 8 + 6
AVC = 2
Therefore, the firm needs a price equal to or higher than 2 to stay in the market.