Each firm in a perfectly competitive market has a short-run total cost of TC=75+500Q-5Q²+0.5Q², where MC=500-10Q+1.5Q²
a) Calculate the output that minimizes the firm's AVC
b) What is the firm's shutdown price?
To find the output that minimizes the firm's Average Variable Cost (AVC), we need to differentiate the Average Variable Cost function (AVC) with respect to output (Q) and solve for the output where the derivative equals zero.
a) Calculate the output that minimizes the firm's AVC:
Step 1: Calculate the firm's Average Variable Cost (AVC):
The Average Variable Cost (AVC) is calculated by dividing the Total Variable Cost (TVC) by the output (Q). We already have the Total Cost (TC) equation, so we need to subtract the fixed cost (75) from the Total Cost (TC) equation to obtain the Total Variable Cost (TVC) equation:
TVC = TC - Fixed Cost = 500Q - 5Q² + 0.5Q³
Next, divide the Total Variable Cost (TVC) by the output (Q) to get the Average Variable Cost (AVC):
AVC = TVC / Q = (500Q - 5Q² + 0.5Q³) / Q
AVC = 500 - 5Q + 0.5Q²
Step 2: Differentiate the Average Variable Cost (AVC) equation with respect to output (Q):
dAVC/dQ = -5 + Q
Step 3: Set the derivative equal to zero and solve for Q:
-5 + Q = 0
Q = 5
Therefore, the output that minimizes the firm's Average Variable Cost (AVC) is Q = 5.
b) To find the firm's shutdown price, we need to determine the price at which the firm covers its variable costs but fails to cover its fixed costs. In other words, the shutdown price occurs at the point where the Average Variable Cost (AVC) equals the Average Revenue (AR), assuming that the firm continues production as long as the price exceeds the AVC.
Step 1: Calculate the Average Revenue (AR):
Average Revenue (AR) is the same as the market price in a perfectly competitive market. However, the question does not provide information about the market price. Therefore, we cannot determine an exact shutdown price without knowing the market price.
Nevertheless, we can explain the concept of the shutdown price. When the Average Variable Cost (AVC) exceeds the price (AR), the firm is unable to cover its variable costs. However, as long as the price (AR) exceeds the Average Variable Cost (AVC), the firm will continue production to minimize its losses and try to cover at least its variable costs.
Therefore, the shutdown price occurs when the price (AR) falls below the Average Variable Cost (AVC), causing the firm to shut down or cease production in the short run.