If a representative firm with long-run total cost given by TC = 2,000 + 20q + 5q² operates in a competitive industry where the market demand is given by Q_D = 10,000 - 40P,the long-run equilibrium output of the individual firm's will be:

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In the long-run equilibrium in a competitive market, firms produce at the minimum possible average total cost (ATC) point, which occurs where marginal cost (MC) intersects ATC. To find the long-run equilibrium output of the representative firm, we can start by finding its long-run marginal cost (LMC) and ATC. LMC = dTC/dq = 20 + 5² ATC = TC/q = 2,000/q + 20 + 5²/q Note that ² is just a constant and does not change with output or price. Next, we can set LMC equal to the market price (P) to find the output level that minimizes ATC: 20 + 5² + 40²/q = 40 Simplifying: q = (2/3)(²) Plugging in the values, we get: q = (2/3)(5) = 3.33 However, firms can only produce integer units of output, so the closest possible output level is 3 units. We can check that this is indeed the output level that minimizes ATC: ATC = 2,000/3 + 20 + 5(5)/3 = 43.33 Finally, we verify that this output level is indeed profitable for the firm: Total revenue = Pq = 40(3) = 120 Total cost = 2,000 + 20(3) + 5(5)(3) = 215 Profit = 120 - 215 = -95 Since the firm is making a negative profit with q = 3, it will not choose this output level. Instead, it will choose the next highest integer output level that still minimizes ATC: ATC(q=4) = 2,000/4 + 20 + 5(5)/4 = 38.75 Total revenue = Pq = 40(4) = 160 Total cost = 2,000 + 20(4) + 5(5)(4) = 240 Profit = 160 - 240 = -80 Again, the firm is making a negative profit, so it would not choose q = 4. The next highest integer output level is q = 5, which gives: ATC(q=5) = 2,000/5 + 20 + 5(5)/5 = 44 Total revenue = Pq = 40(5) = 200 Total cost = 2,000 + 20(5) + 5(5)(5) = 275 Profit = 200 - 275 = -75 This is still a negative profit, so we try q = 6: ATC(q=6) = 2,000/6 + 20 + 5(5)/6 = 47.5 Total revenue = Pq = 40(6) = 240 Total cost = 2,000 + 20(6) + 5(5)(6) = 310 Profit = 240 - 310 = -70 Finally, with q = 7, we get: ATC(q=7) = 2,000/7 + 20 + 5(5)/7 = 50 Total revenue = Pq = 40(7) = 280 Total cost = 2,000 + 20(7) + 5(5)(7) = 345 Profit = 280 - 345 = -65 Since q = 7 gives the highest possible output level that still minimizes ATC, this is the long-run equilibrium output level for the representative firm.

If a Representative Firm with Long-Run Total Cost Given by TC