To maximize profit in a perfectly competitive market, the firm should produce at the output level where marginal cost (MC) equals the market price (P).
The total cost (TC) function is given as:
TC = 1/3Q^3 - 5Q^2 + 20Q + 50
To find the marginal cost, we need to take the derivative of the total cost function with respect to Q:
MC = dTC/dQ = Q^2 - 10Q + 20
A. To maximize profit, set MC equal to the market price (P) and solve for Q:
Q^2 - 10Q + 20 = 6
Q^2 - 10Q + 14 = 0
Solving this quadratic equation will give us the level of output (Q) that maximizes profit. Using the quadratic formula, we find:
Q ≈ 2.34 or Q ≈ 7.66
Since Q represents the level of output, it cannot be a fraction or negative value in this context. Therefore, the firm should produce approximately 7 units of output to maximize profit.
B. To determine the level of profit at equilibrium, we need to calculate total revenue (TR) and subtract total cost (TC) at the output level where MC = P.
TR = P * Q
TR = 6 * 7 = 42
Profit (π) = TR - TC
Profit (π) = 42 - TC
To calculate the specific level of profit, we need the total cost equation. Substituting the value of Q into the TC equation, we can find the total cost at equilibrium.
C. To determine the minimum price required by the firm to stay in the market, we need to consider the level of average variable cost (AVC). If the price falls below AVC, the firm would be better off shutting down in the short run.
AVC = TVC / Q
AVC = (TC - TFC) / Q
However, the total fixed cost (TFC) is not provided in the given information. Without the TFC value, we cannot determine the minimum price required to stay in the market.
Therefore, we can answer parts A and B, but we cannot determine the minimum price required to stay in the market without the total fixed cost (TFC) information.