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
C. Find the profit at the equilibrium point and identify whether the firm makes positive profit, normal profits or incurs loss.
D. What price is needed for the firm to stay in the market?
E. Calculate the output at wich the marginal costs are minimizes
A. To derive the function of total cost (TC), average variable cost (AVC), and total revenue (TR), we can use the formulas:
TC = TFC + TVC
AVC = TVC / Q
TR = P * Q
Given that TFC = 10, we can plug it into the formula for TC:
TC = 10 + TVC
B. To calculate the equilibrium price and quantity, we need to determine where the marginal cost (MC) equals the average revenue (AR). Since the firm is in a perfectly competitive market, AR is equal to the price (P).
MC = AR = P
To find the equilibrium quantity, we set MC equal to the derivative of the total cost function (TC):
3Q^2 - 8Q + 6 = d(TVC) / dQ
Solving this quadratic equation, we find the equilibrium quantity (Q):
Q = 1
Now, substitute Q = 1 into the marginal cost equation to find the equilibrium price (P):
MC = 3(1)^2 - 8(1) + 6 = 1
Therefore, the equilibrium price is 1. The equilibrium quantity is 1.
C. To find the profit at the equilibrium point, we need to calculate the total revenue (TR) and subtract the total cost (TC):
TR = P * Q = 1 * 1 = 1
TC = TFC + TVC = 10 + TVC
Profit = TR - TC = 1 - (10 + TVC)
To determine whether the firm makes positive profit, normal profits, or incurs a loss, we need more information about the total variable cost (TVC). Without this information, we cannot determine the firm's profit.
D. To determine the price needed for the firm to stay in the market, we need to consider the average variable cost (AVC). Normally, in the short run, a firm should stay in the market if the price (P) is greater than or equal to the AVC.
Therefore, we need the function for TVC to calculate AVC and determine the price needed for the firm to stay in the market.
E. To calculate the output at which marginal costs are minimized, we need to find the quantity (Q) that corresponds to the minimum point on the marginal cost curve.
To find the minimum, we can take the derivative of the marginal cost equation (MC) with respect to Q and set it equal to zero:
d(MC) / dQ = 6Q - 8 = 0
Solving this equation, we find:
Q = 4/3
Therefore, the output at which marginal costs are minimized is 4/3.