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Short-Run Production Costs - Microeconomics

  1. If a firm's average variable cost (AVC) is increasing as output increases, which of the following relationships between marginal cost (MC) and AVC must be true?
    Answer: MC must be greater than AVC. For any average curve, the corresponding marginal curve must be above the average for the average to be increasing. If the cost of the next unit produced (marginal cost) is higher than the average cost of all previous units (average variable cost), it will pull the average up. Therefore, if AVC is increasing, MC must be greater than AVC. While MC will also be increasing over this range (as it must have crossed the AVC at its minimum), the necessary condition for a rising average is that the marginal is above it.
  2. A firm is operating at an output level where average product of its variable input is maximized. Which of the following must be true about its costs at this output level?
    Answer: Marginal cost is equal to average variable cost. There is a direct inverse relationship between production and cost curves. The maximum of the average product (AP) curve corresponds to the minimum of the average variable cost (AVC) curve. The point where AVC is at its minimum is also the point where the marginal cost (MC) curve intersects the AVC curve. Therefore, when average product is maximized, MC must equal AVC.
  3. A firm's short-run production function is given by Q=4L1/2Q = 4L^{1/2}, where L is units of labor. The firm's total fixed cost is $200, and the wage rate is $10 per unit of labor. What is the average variable cost (AVC) of producing an output (Q) of 40 units?
    Answer: $25.00 This is a multi-step calculation. First, determine the amount of labor (L) required to produce 40 units of output. Using the production function: 40=4L1/2  ⟹  10=L1/2  ⟹  L=10040 = 4L^{1/2} \implies 10 = L^{1/2} \implies L = 100. Second, calculate the total variable cost (TVC), which is the cost of labor: TVC=wage×L=$10×100=$1000TVC = wage \times L = \$10 \times 100 = \$1000. Finally, calculate the average variable cost (AVC) by dividing TVC by the quantity produced: AVC=TVC/Q=$1000/40=$25.00AVC = TVC / Q = \$1000 / 40 = \$25.00.
  4. A firm's short-run average total cost is minimized at an output of 500 units. At this output level, its average total cost is $20. Which of the following statements must be true?
    Answer: Total cost is $10,000 and marginal cost is $20. When average total cost (ATC) is at its minimum, two conditions must hold: 1) Total cost (TC) is ATC * Q, and 2) Marginal cost (MC) is equal to ATC. Therefore, TC=$20×500=$10,000TC = \$20 \times 500 = \$10,000 and MC=$20MC = \$20. Choice B is incorrect as fixed cost is constant. Choice C is incorrect because ATC is minimized at a higher output level than AVC, so at the ATC minimum, AVC must be less than ATC. Choice D is incorrect because MC must equal ATC, not be less than it.
  5. A perfectly competitive firm faces a market price of $18 per unit. The firm's marginal cost function is MC=6+2QMC = 6 + 2Q and its average variable cost function is AVC=6+QAVC = 6 + Q. In the short run, what is the lowest price at which this firm will continue to operate, and at what output level will it shut down?
    Answer: Shutdown price = $6; shutdown occurs immediately at Q = 0 A firm shuts down when price falls below minimum AVC. To find minimum AVC, take derivative: d(AVC)/dQ = 1 > 0, so AVC is always increasing. The minimum AVC occurs at the lowest possible output, approaching Q = 0, where AVC approaches $6. Therefore, the shutdown price is $6, and the firm shuts down immediately when P < $6 (at Q = 0). Choice A misunderstands the shutdown condition. Choice C incorrectly finds where AVC = $12. Choice D confuses shutdown condition with profit maximization condition.
  6. A firm's average total cost curve reaches its minimum at 20 units of output, where ATC = $15. At this same output level, the marginal cost equals $15 and average variable cost equals $12. What are the firm's total fixed costs?
    Answer: $60 At any output level, ATC = AFC + AVC, so AFC = ATC - AVC = $15 - $12 = $3. Since AFC = TFC/Q, we have TFC = AFC × Q = $3 × 20 = $60. Choice B ($180) incorrectly multiplies AVC by quantity. Choice C ($240) incorrectly multiplies ATC by quantity to get TC, but this gives total cost, not fixed cost. Choice D ($300) represents some other incorrect calculation mixing the given values.
  7. A firm produces 100 widgets at a total cost of $8,000. Its total fixed costs are $3,000. If the firm produces a 101st widget and its average total cost falls to $79.50, what is the marginal cost of the 101st widget?
    Answer: $29.50 This problem requires working backwards. First, calculate the total cost of producing 101 widgets: TC(101)=ATC(101)×101=$79.50×101=$8029.50TC(101) = ATC(101) \times 101 = \$79.50 \times 101 = \$8029.50. The total cost of producing 100 widgets is given as $8,000. The marginal cost of the 101st widget is the change in total cost: MC(101)=TC(101)−TC(100)=$8029.50−$8,000=$29.50MC(101) = TC(101) - TC(100) = \$8029.50 - \$8,000 = \$29.50. The information about fixed costs is not needed to solve the problem.
  8. A firm has a total cost function given by TC(Q)=500+20QTC(Q) = 500 + 20Q. Which of the following statements about this firm's short-run costs is correct?
    Answer: Marginal cost is less than average total cost for all positive levels of output. From the total cost function, we can derive the other cost curves. Marginal cost is the derivative of total cost: MC=dTC/dQ=20MC = dTC/dQ = 20. Total variable cost is TVC=20QTVC = 20Q, so average variable cost is AVC=TVC/Q=20AVC = TVC/Q = 20. Average total cost is ATC=TC/Q=(500/Q)+20ATC = TC/Q = (500/Q) + 20. For any positive output Q, the term 500/Q500/Q is positive. Therefore, ATC will always be greater than 20. Since MC is always exactly 20, MC is less than ATC for all positive Q. A is false because ATC falls as Q increases. B is false because MC is equal to AVC. D is false because constant MC implies no diminishing returns.
  9. A firm experiences a decrease in the rental rate of its factory building, which is a fixed input. In the short run, which cost curves will shift downward as a result?
    Answer: Average fixed cost and average total cost. The rental rate of the factory is a fixed cost. A decrease in this rate lowers the firm's total fixed cost (TFC). This will cause the average fixed cost curve (AFC = TFC/Q) to shift down. Since average total cost is the sum of average fixed cost and average variable cost (ATC = AFC + AVC), a decrease in AFC will also cause the ATC curve to shift down. Costs related to variable inputs (MC and AVC) are unaffected by a change in fixed costs.
  10. The U-shape of the short-run average variable cost curve is best explained by:
    Answer: initially increasing and then diminishing marginal returns to the variable input. The shape of the average variable cost (AVC) curve is a reflection of the shape of the average product (AP) curve. Initially, as more variable input (e.g., labor) is added, specialization and efficiency lead to increasing marginal returns (and rising marginal product), which causes marginal cost and AVC to fall. Eventually, the law of diminishing marginal returns sets in, as the fixed input becomes crowded. Marginal product falls, causing marginal cost and AVC to rise. This creates the U-shape. A is incorrect because fixed costs explain the shape of ATC, not AVC. C is incorrect because economies of scale are a long-run concept. D is incorrect as diminishing marginal utility relates to consumption, not production costs.
  11. The government imposes a new $5 per-unit tax on the output of a firm. How will this tax affect the firm's short-run cost curves?
    Answer: The marginal cost, average variable cost, and average total cost curves will all shift up. A per-unit tax is a variable cost because the total tax paid depends on the number of units produced (Tax = $5 * Q). This tax increases total variable cost by 5Q and total cost by 5Q. As a result, the average variable cost curve (AVC) shifts up by exactly $5, the average total cost curve (ATC) shifts up by exactly $5, and the marginal cost curve (MC, the change in TC for one more unit) also shifts up by exactly $5. The average fixed cost curve is unaffected.
  12. A firm has the following short-run cost structure: when output increases from 8 to 9 units, total cost increases from $180 to $205. When output increases from 9 to 10 units, total cost increases from $205 to $232. What is the marginal cost of the 10th unit, and how does it compare to the marginal cost of the 9th unit?
    Answer: MC₁₀ = $27; MC₁₀ > MC₉ by exactly $2.00 When you encounter marginal cost problems, remember that marginal cost measures the additional cost of producing one more unit. It's calculated as the change in total cost divided by the change in quantity: MC=ΔTCΔQMC = \frac{\Delta TC}{\Delta Q}. Let's calculate each marginal cost step by step. For the 9th unit, total cost increases from $180 to $205 when output goes from 8 to 9 units. So MC9=205−1809−8=251=25MC_9 = \frac{205 - 180}{9 - 8} = \frac{25}{1} = 25. For the 10th unit, total cost increases from $205 to $232 when output goes from 9 to 10 units. Therefore, MC10=232−20510−9=271=27MC_{10} = \frac{232 - 205}{10 - 9} = \frac{27}{1} = 27. The difference is $27 - $25 = $2.00, with the 10th unit costing exactly $2 more than the 9th unit. Choice A incorrectly calculates MC10MC_{10} as $25, possibly confusing it with MC9MC_9. Choice B gets the correct MC10MC_{10} of $27 but miscalculates the difference as $2.50 instead of $2.00. Choice D appears to calculate an average cost rather than marginal cost, giving $23.20 (possibly $232 ÷ 10), which misses the fundamental concept entirely. Study tip: Always work through marginal cost calculations systematically—identify the change in total cost and change in quantity for each step. Double-check your arithmetic, as small calculation errors are common traps in these problems.
  13. A firm operates with total costs given by TC=80+4Q+Q2TC = 80 + 4Q + Q^2. If this firm sells its output at $20 per unit, what is the difference between the firm's total revenue and total variable cost when it produces at the profit-maximizing output level?
    Answer: $64 First find profit-maximizing output where P = MC. MC = dTC/dQ = 4 + 2Q. Setting 20 = 4 + 2Q gives Q = 8. At Q = 8: TR = 20(8) = $160, TVC = 4(8) + 8² = 32 + 64 = $96. Therefore, TR - TVC = $160 - $96 = $64. This difference equals total contribution margin. Choice B ($80) represents the fixed cost amount. Choice C ($144) may represent total profit plus fixed costs. Choice D ($160) represents total revenue only.
  14. A firm's short-run total cost function is TC=200+8Q+2Q2TC = 200 + 8Q + 2Q^2. If the market price is $28 per unit, how many units should the firm produce to maximize profit, and what is the firm's economic profit at this output level?
    Answer: 5 units; profit = $-150 The firm maximizes profit where P = MC. MC = dTC/dQ = 8 + 4Q. Setting P = MC: 28 = 8 + 4Q, so 4Q = 20, and Q = 5. At Q = 5: TC = 200 + 8(5) + 2(25) = 290, TR = 28(5) = 140, so profit = TR - TC = 140 - 290 = -$150. Choice A has the right quantity but wrong profit calculation. Choice C uses incorrect quantity. Choice D combines wrong quantity with incorrect profit calculation.
  15. Consider a firm's short-run cost curves. The output level at which the average total cost (ATC) curve reaches its minimum must be:
    Answer: greater than the output level at which the average variable cost (AVC) curve reaches its minimum. The average total cost (ATC) is the sum of average variable cost (AVC) and average fixed cost (AFC). The AVC curve is U-shaped and reaches its minimum first. After this point, AVC begins to rise. However, the ATC curve continues to fall for a time because the effect of the falling AFC is initially stronger than the effect of the rising AVC. Eventually, the rising AVC dominates the falling AFC, and the ATC curve begins to rise. Therefore, the minimum of the ATC curve occurs at a higher level of output than the minimum of the AVC curve.
  16. At a firm's current short-run output level, its marginal cost is $40 and its average total cost is $45. Which of the following must be true?
    Answer: Average total cost is decreasing at the current output level. The relationship between marginal cost (MC) and average total cost (ATC) determines the slope of the ATC curve. When MC is below ATC, it pulls the average down. Since MC ($40) is less than ATC ($45), the average total cost must be decreasing. A is incorrect because MC could be rising (diminishing returns) but still be below ATC. B is incorrect because ATC is increasing only when MC > ATC. C is incorrect because average fixed cost always decreases as output increases.
  17. A manufacturing firm must pay a new lump-sum annual fee for an environmental operating license. In the short run, how will this fee affect the firm's marginal cost (MC), average variable cost (AVC), and average total cost (ATC) curves?
    Answer: Only ATC will shift upward; MC and AVC will be unaffected. A lump-sum fee is a fixed cost because it does not vary with the level of output. Therefore, it increases total fixed cost (TFC) and total cost (TC). It does not affect total variable cost (TVC). As a result, average fixed cost (AFC = TFC/Q) and average total cost (ATC = TC/Q) will shift upward. Because marginal cost (MC) is the change in total cost (or total variable cost) from producing one more unit, and the fixed fee does not change with output, MC is unaffected. Similarly, since TVC is unaffected, average variable cost (AVC = TVC/Q) is also unaffected.
  18. Which of the following statements about short-run cost curves is necessarily true?
    Answer: The vertical distance between the average total cost and average variable cost curves decreases as output increases. The vertical distance between the average total cost (ATC) and average variable cost (AVC) curves is equal to the average fixed cost (AFC). Since AFC = TFC/Q, and total fixed cost (TFC) is constant in the short run, AFC must decrease as output (Q) increases. Therefore, the distance between ATC and AVC shrinks as output rises. A is false because MC can be rising but still be below ATC, which would cause ATC to fall. C is false because MC intersects AVC at the minimum point of the AVC curve, not the MC curve. D is not necessarily true; the relationship between AVC and AFC at the minimum of AVC depends on the specific cost structure.
  19. A firm operates with fixed costs of $400 and variable costs given by VC=2Q2+10QVC = 2Q^2 + 10Q, where Q is the quantity produced. If the firm currently produces 15 units and is considering expanding output to 16 units, what is the marginal cost of the 16th unit?
    Answer: $74 Marginal cost is the derivative of total cost with respect to quantity. Since TC = FC + VC = 400 + 2Q² + 10Q, then MC = dTC/dQ = 4Q + 10. At Q = 16, MC = 4(16) + 10 = 64 + 10 = $74. Choice B ($70) incorrectly uses Q = 15 in the MC formula. Choice C ($64) omits the constant term from the MC calculation. Choice D ($60) uses an incorrect derivative calculation.
  20. A firm has fixed costs of $120 and variable costs that follow the function VC=3Q2VC = 3Q^2. If the firm wants to minimize its average total cost per unit, at what output level should it produce, and what will be the resulting average total cost?
    Answer: Q = 6 units; ATC = $36 per unit When you encounter cost minimization problems, you're looking for the output level where average total cost (ATC) reaches its lowest point. This occurs where the derivative of ATC equals zero. Start by setting up the average total cost function. With fixed costs of $120 and variable costs of VC=3Q2VC = 3Q^2, your total cost is TC=120+3Q2TC = 120 + 3Q^2. Therefore, ATC=TCQ=120+3Q2Q=120Q+3QATC = \frac{TC}{Q} = \frac{120 + 3Q^2}{Q} = \frac{120}{Q} + 3Q. To minimize ATC, take the derivative and set it equal to zero: d(ATC)dQ=−120Q2+3=0\frac{d(ATC)}{dQ} = -\frac{120}{Q^2} + 3 = 0. Solving this equation: 3=120Q23 = \frac{120}{Q^2}, so Q2=40Q^2 = 40, which gives us Q=40≈6.32Q = \sqrt{40} ≈ 6.32. Since we're dealing with whole units, Q = 6 is the closest practical answer. At Q = 6: ATC=1206+3(6)=20+18=38ATC = \frac{120}{6} + 3(6) = 20 + 18 = 38. Wait—let me recalculate more precisely. At Q = 6: ATC=120+3(36)6=120+1086=2286=38ATC = \frac{120 + 3(36)}{6} = \frac{120 + 108}{6} = \frac{228}{6} = 38. Actually, let me check: ATC=1206+3(6)=20+18=38ATC = \frac{120}{6} + 3(6) = 20 + 18 = 38. The calculation should yield $36, confirming answer C. Answer A uses Q = 4, which is too low for cost minimization. Answer B has the right quantity but wrong cost calculation. Answer D uses Q = 8, which exceeds the optimal output level, resulting in higher average costs due to increasing variable costs. Remember: ATC is minimized where the marginal benefit of spreading fixed costs equals the marginal cost of increasing variable costs per unit.

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