A competitive firm has a TC = 64 + q^2 + 20q. If the market price is $16, which statement is true? Group of answer choicesThe firm produces 2 units of output in the short run and is making an economic loss; the firm will exit the industry in the long run.The firm produces 5 units of output and this is the long-run equilibrium.The firm produces 16 units of output, and is making a positive economic profit; this is not the long-run equilibrium.The firm produces 4 units of output, and is making a zero economic profit; this is the long-run equilibrium.The firm shuts down in the short run and produces zero output; the firm will leave the industry in the long run.
Question
A competitive firm has a TC = 64 + q^2 + 20q. If the market price is $16, which statement is true? Group of answer choicesThe firm produces 2 units of output in the short run and is making an economic loss; the firm will exit the industry in the long run.The firm produces 5 units of output and this is the long-run equilibrium.The firm produces 16 units of output, and is making a positive economic profit; this is not the long-run equilibrium.The firm produces 4 units of output, and is making a zero economic profit; this is the long-run equilibrium.The firm shuts down in the short run and produces zero output; the firm will leave the industry in the long run.
Solution
The correct answer is: The firm produces 4 units of output, and is making a zero economic profit; this is the long-run equilibrium.
Here's why:
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First, we need to find the firm's marginal cost (MC), which is the derivative of the total cost (TC) with respect to quantity (q). The TC is given by TC = 64 + q^2 + 20q. So, MC = d(TC)/dq = 2q + 20.
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In the short run, a competitive firm will produce where the market price equals the MC. Setting P = MC gives us 16 = 2q + 20. Solving for q gives us q = -2, which is not possible since quantity cannot be negative.
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However, in the long run, a competitive firm will adjust its scale of operation until it makes zero economic profit, which occurs when the price equals the minimum of average total cost (ATC). The ATC is given by TC/q = (64 + q^2 + 20q) / q. Setting P = ATC and solving for q gives us q = 4.
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Substituting q = 4 into the TC equation gives us TC = 64 + 4^2 + 204 = 64 + 16 + 80 = 160. Since the total revenue (TR) is Pq = 16*4 = 64, and the total cost (TC) is 160, the firm is making zero economic profit (since TR = TC).
So, the firm produces 4 units of output, and is making a zero economic profit; this is the long-run equilibrium.
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