Profit Maximization - Microeconomics

A perfectly competitive firm has been producing 15 units at a market price of $30. Due to a change in input costs, the firm's marginal cost at 15 units increases from $30 to $35, while marginal cost at 14 units becomes $32 and at 13 units becomes $29. Assuming the market price remains $30, how should the firm adjust its production?
Answer: Reduce production to 13 units where marginal cost is closest to but below the market price When a perfectly competitive firm faces changing costs, you need to apply the fundamental profit-maximization rule: produce where marginal cost equals marginal revenue (which equals market price in perfect competition). Initially, the firm was optimally producing 15 units where MC = MR = $30. After the cost increase, you must find the new quantity where MC equals the $30 market price. Looking at the new marginal costs: at 15 units, MC = $35; at 14 units, MC = $32; at 13 units, MC = $29. Since the firm should produce where MC equals price ($30), and the closest match is 13 units where MC = $29 (just below the market price), the firm should reduce to 13 units. Producing beyond this point would mean MC > MR, reducing profit on additional units. Answer A incorrectly suggests sunk cost thinking - there's no "commitment" to previous output levels in competitive markets. Answer B misunderstands the optimization rule by focusing on the loss per unit rather than finding where MC = MR. The "$2 loss per unit" language also confuses marginal analysis with average cost thinking. Answer D reflects a common misconception about spreading costs - this is irrelevant for short-run profit maximization and would actually worsen the situation since MC > MR for units beyond 13. Remember: in perfect competition, always produce where MC = MR (market price). When costs change, immediately recalculate this optimal point rather than sticking with previous decisions or trying to "spread" costs.
A perfectly competitive firm has a total cost function of $TC = 100 + 10q + q^2$ . What is the equation of the firm's short-run supply curve?
Answer: $P = 10 + 2q$ for $P \ge 10$ A competitive firm's short-run supply curve is its marginal cost (MC) curve above its average variable cost (AVC) curve. First, find MC: $MC = dTC/dq = 10 + 2q$ . Next, find AVC: The variable cost is $VC = 10q + q^2$ , so $AVC = VC/q = 10 + q$ . The firm will only produce if price is greater than or equal to the minimum AVC. The minimum of $AVC = 10 + q$ occurs at q=0, where AVC=$10. This is the shutdown price. Therefore, the supply curve is given by P = MC for all prices at or above the minimum AVC. The supply curve is $P = 10 + 2q$ for $P \ge 10$ .
A perfectly competitive firm is producing a positive output level, q*, where market price equals marginal cost. For q* to be the profit-maximizing output, which of the following conditions is also required?
Answer: The marginal cost curve must be upward sloping at q*. The condition P = MC is the first-order condition for profit maximization; it identifies a point where profit is at a local maximum or minimum. To ensure it is a maximum, the second-order condition must be satisfied. This condition states that the marginal cost curve must be rising (i.e., its slope must be positive) at the point of intersection with the price line. If MC were falling, producing more would lead to MC falling further below the price, increasing profit and implying the current point is not a maximum.
A perfectly competitive firm faces a market price of $12 per unit. Its total cost function is $TC = 50 + 4Q + 0.5Q^2$ . If the firm currently produces 8 units and is considering whether to increase production to 10 units, what should the firm conclude about this change?
Answer: The firm should not increase production because marginal cost exceeds marginal revenue at 10 units For a perfectly competitive firm, MR = P = $12. The marginal cost function is MC = dTC/dQ = 4 + Q. At Q = 10, MC = 4 + 10 = $14. Since MC ($14) > MR ($12) at 10 units, the firm should not increase production to 10 units. Choice A is incorrect because MC > MR at 10 units. Choice C is incorrect because the change in profit from 8 to 10 units is negative (each additional unit beyond optimal adds negative marginal profit). Choice D focuses on average cost rather than the marginal analysis needed for profit maximization.
A perfectly competitive firm's total cost function is $TC = 100 + 5Q + Q^2$ . If the market price falls from $25 to $15, how does this affect the firm's profit-maximizing output and total profit?
Answer: Output decreases by 5 units, and profit decreases by $75 MC = dTC/dQ = 5 + 2Q. At P = $25: 25 = 5 + 2Q, so Q = 10. Profit = 25(10) - (100 + 5(10) + 10²) = 250 - 250 = $0. At P = $15: 15 = 5 + 2Q, so Q = 5. Profit = 15(5) - (100 + 5(5) + 5²) = 75 - 150 = -$75. Output decreases by 5 units, profit decreases by $75. Choice A uses incorrect profit calculation. Choice C incorrectly calculates the output change. Choice D underestimates the profit decrease.
A perfectly competitive firm is initially in a long-run equilibrium where it is earning zero economic profit. The government then imposes an annual lump-sum license fee on every firm in the industry. In the short run, how will this tax affect the firm's optimal output and profit?
Answer: Output will remain the same and profit will become negative. A lump-sum license fee is a fixed cost. An increase in fixed costs does not affect the firm's marginal cost (MC) or its average variable cost (AVC). Since the profit-maximizing rule in the short run is to produce where Price = MC, and neither P nor MC has changed, the firm's optimal output level will remain the same. However, the increase in fixed costs will raise the firm's total costs, and thus its average total cost. Since the firm was previously earning zero economic profit (P = ATC), the increase in ATC means that now P < ATC, and the firm will earn negative economic profit (a loss).
The market demand in a perfectly competitive industry is $Q_D = 1,200 - 10P$ and market supply is $Q_S = 20P - 300$ . A typical firm has marginal cost $MC(q) = 10 + 5q$ . What is the profit-maximizing output (q) for a typical firm?
Answer: 8 units This is a two-step problem. First, find the market equilibrium price (P) by setting market demand equal to market supply. $1,200 - 10P = 20P - 300$ . Solving for P: $1,500 = 30P$ , so $P = 50$ . Second, a competitive firm maximizes profit by producing the quantity (q) where price equals its marginal cost. Set the equilibrium price equal to the firm's MC: $50 = 10 + 5q$ . Solving for q: $40 = 5q$ , so $q = 8$ . The profit-maximizing output for a typical firm is 8 units.
A competitive firm's total cost function is $TC = 64 + 4q + q^2$ . The firm will earn zero economic profit if the market price is:
Answer: $20 A firm earns zero economic profit when price is equal to the minimum of its average total cost (ATC). First, find the ATC function: $ATC = TC/q = 64/q + 4 + q$ . The minimum of the ATC curve occurs where marginal cost (MC) equals ATC. First, find MC: $MC = dTC/dq = 4 + 2q$ . Now, set MC = ATC: $4 + 2q = 64/q + 4 + q$ . This simplifies to $q = 64/q$ , which means $q^2 = 64$ , so $q = 8$ . This is the quantity at which ATC is minimized. To find the price that results in zero profit, substitute this quantity back into either the MC or ATC function. Using MC: $P = MC(8) = 4 + 2(8) = 4 + 16 = \$20$ . Using ATC: $P = ATC(8) = 64/8 + 4 + 8 = 8 + 4 + 8 = \$20$ .
A competitive firm has a marginal cost function $MC = 6 + 0.5Q$ and faces a market price of $18. The firm also has the option to invest in new technology that would change its marginal cost function to $MC = 8 + 0.4Q$ . Should the firm adopt the new technology, and why?
Answer: Yes, because the new technology allows the firm to produce more units profitably When analyzing technology adoption decisions, you need to compare how the change affects profit at the optimal output level, not just look at marginal cost in isolation. First, find the profit-maximizing output under each technology. A competitive firm maximizes profit where price equals marginal cost. With the original technology: $18 = 6 + 0.5Q$ , so $Q = 24$ units. With the new technology: $18 = 8 + 0.4Q$ , so $Q = 25$ units. The key insight is that "profitably" means producing units where price exceeds marginal cost. Under the original technology, the 25th unit would cost $MC = 6 + 0.5(25) = 18.5$ , which exceeds the $18 price. Under the new technology, the 25th unit costs exactly $18, making it profitable to produce. The new technology allows production of one additional profitable unit. Option A incorrectly focuses only on the intercept increase from 6 to 8, ignoring that the slope decreased from 0.5 to 0.4. Option B is wrong because at the original profit-maximizing output of 24 units, the new technology actually increases marginal cost from $18 to $17.6 - but this misses the point that optimal output changes. Option C incorrectly states that output decreases when it actually increases from 24 to 25 units. Remember: technology adoption decisions depend on how changes affect profit at the new optimum, not just marginal cost comparisons at arbitrary output levels. Always recalculate the profit-maximizing quantity under each scenario.
A price-taking firm is maximizing profit by producing 500 units. The market price is $80 per unit, and the firm is earning $10,000 in economic profit. What is the firm's average total cost (ATC) at this level of production?
Answer: $60 Economic profit is calculated as (Price - Average Total Cost) × Quantity, or $\pi = (P - ATC) \times q$ . We are given $\pi = \$10,000$ , P = $80, and q = 500. We can plug these values into the formula and solve for ATC: $10,000 = (80 - ATC) \times 500$ . First, divide both sides by 500: $10,000 / 500 = 80 - ATC$ , which gives $20 = 80 - ATC$ . Rearranging the equation to solve for ATC yields $ATC = 80 - 20 = \$60$ .
A competitive firm's marginal cost curve is given by $MC = 2Q + 6$ . The firm faces a market price of $18. However, the firm also pays a monthly licensing fee of $100 regardless of output level. What is the firm's profit-maximizing output level?
Answer: 6 units, because this is where MR equals MC regardless of the fixed licensing fee In perfect competition, profit maximization occurs where MR = MC, and MR = P = $18. Setting 18 = 2Q + 6 gives 2Q = 12, so Q = 6. Fixed costs (like the licensing fee) do not affect the profit-maximizing output level because they don't affect marginal cost or marginal revenue. Choice B incorrectly assumes fixed costs affect optimal output. Choice C incorrectly suggests fixed costs reduce optimal output. Choice D incorrectly focuses on revenue maximization rather than profit maximization.
A competitive firm has the total cost function $TC = 27 + 3q + (1/3)q^2$ . In the long run, this firm will exit the market if the price falls below:
Answer: $9 In the long run, a firm will exit the market if the price falls below the minimum of its average total cost (ATC). The exit point is P = min(ATC). First, find ATC: $ATC = TC/q = 27/q + 3 + (1/3)q$ . To find the minimum of ATC, set MC equal to ATC. First, find MC: $MC = dTC/dq = 3 + (2/3)q$ . Now, set MC = ATC: $3 + (2/3)q = 27/q + 3 + (1/3)q$ . This simplifies to $(1/3)q = 27/q$ , which gives $q^2 = 81$ , so $q = 9$ . This is the output level where ATC is minimized. To find the exit price, plug q=9 into the ATC or MC function: $P = MC(9) = 3 + (2/3)(9) = 3 + 6 = \$9$ .
A firm's marginal cost function is given by $MC(q) = q^2 - 18q + 90$ . If the market price is $18, what is the firm's profit-maximizing level of output?
Answer: 12 units To find the profit-maximizing output, set price equal to marginal cost: $18 = q^2 - 18q + 90$ . Rearranging gives the quadratic equation $q^2 - 18q + 72 = 0$ . Factoring the quadratic yields $(q - 6)(q - 12) = 0$ , so the possible output levels are q=6 and q=12. Profit is maximized only when the marginal cost curve is upward-sloping. The MC curve is a U-shaped parabola with its minimum at $q = -b/(2a) = -(-18)/(2*1) = 9$ . Therefore, at q=6, MC is downward-sloping (profit is minimized at this point relative to its surroundings), and at q=12, MC is upward-sloping. The profit-maximizing output is 12 units.
A perfectly competitive firm faces a market price of $28. Its total cost function is given by $TC(q) = 50 + 8q + q^2$ . What is the firm's profit-maximizing quantity of output?
Answer: 10 units A perfectly competitive firm maximizes profit by producing the quantity (q) where price (P) equals marginal cost (MC), provided that P is greater than or equal to average variable cost (AVC). First, find the marginal cost by taking the derivative of the total cost function: $MC = dTC/dq = 8 + 2q$ . Next, set P = MC: $28 = 8 + 2q$ . Solving for q gives $20 = 2q$ , so $q = 10$ . We must check if the firm should operate. AVC = $(8q + q^2)/q = 8 + q$ . At q=10, AVC = 18. Since P=$28 > AVC=$18, the firm should produce. Thus, the profit-maximizing quantity is 10 units.
A competitive firm's marginal cost is $MC = 15 + 2q$ , and the market price is $55. The government introduces a per-unit subsidy of $10 paid to the firm for each unit it produces. After the subsidy is introduced, the firm's output will:
Answer: increase by 5 units. Without the subsidy, the firm sets P=MC: $55 = 15 + 2q$ , which means $40 = 2q$ , so $q_1 = 20$ . A per-unit subsidy can be viewed in two ways: it either reduces the firm's marginal cost or increases its marginal revenue. Viewing it as a reduction in MC, the effective marginal cost becomes $MC' = MC - subsidy = (15 + 2q) - 10 = 5 + 2q$ . The firm sets P = MC': $55 = 5 + 2q$ , which means $50 = 2q$ , so $q_2 = 25$ . The output increases from 20 to 25, which is an increase of 5 units.
A competitive firm is producing at an output level where the market price is $40, its marginal cost is $32, and its average total cost is $35. To maximize its profit, this firm should:
Answer: increase its output. The rule for profit maximization is to produce up to the point where marginal revenue (MR) equals marginal cost (MC). For a perfectly competitive firm, price equals marginal revenue, so MR = $40. Currently, the firm's MR ($40) is greater than its MC ($32). This means that producing an additional unit of output would add more to revenue than it would to cost, thereby increasing profit. Therefore, the firm should increase its output until its MC rises to equal the market price of $40.
In a perfectly competitive market, a firm's total revenue is $1,000, its total cost is $800, and its marginal cost is $25. The market price is $20. To maximize profit, this firm should:
Answer: decrease output. For a firm in a perfectly competitive market, the market price is equal to its marginal revenue (MR). Here, the price is $20, so MR = $20. The firm's marginal cost (MC) is given as $25. Since MC ($25) > MR ($20), the last unit produced added more to cost than it did to revenue, which means profit decreased. To maximize profit, the firm should reduce its output until its marginal cost falls to the level of the market price ($20). The information about total revenue and total cost indicates the firm is currently profitable, but not maximizing its profit.
A competitive firm discovers that at its current output of 12 units, marginal revenue is $40, marginal cost is $35, and average total cost is $45. The firm is considering three options: maintain current output, increase to 13 units (where MC = $42), or decrease to 11 units (where MC = $33). Which option maximizes profit?
Answer: Increase to 13 units, because moving toward the point where MC = MR increases total profit The firm should produce where MC = MR = $40. Currently at 12 units, MC ($35) < MR ($40), so the firm should increase output. At 13 units, MC ($42) is closer to MR ($40) than at 12 units, so moving to 13 units increases profit even though MC slightly exceeds MR. The 12th unit adds $5 to profit (MR - MC = $40 - $35). Choice A uses imprecise reasoning. Choice B stops short of the optimum. Choice C incorrectly focuses on ATC rather than marginal analysis.
A perfectly competitive firm has a total cost function $TC = 20 + 15q + q^2$ . The market price falls to $25. What is the firm's profit (or loss) in the short run if it produces the optimal quantity?
Answer: A loss of $5 First, find the optimal quantity by setting P = MC. The marginal cost is $MC = dTC/dq = 15 + 2q$ . So, $25 = 15 + 2q$ , which gives $10 = 2q$ , or $q=5$ . Next, check the shutdown condition. AVC = $(15q+q^2)/q = 15+q$ . At q=5, AVC = 20. Since P=$25 > AVC=$20, the firm should produce. Finally, calculate profit at q=5: Profit = TR - TC = $(P \times q) - (20 + 15q + q^2)$ = $(25 \times 5) - (20 + 15(5) + 5^2)$ = $125 - (20 + 75 + 25)$ = $125 - 130 = -5$ . The firm incurs a loss of $5.
A competitive firm faces a market price that is exactly equal to the minimum point on its average variable cost (AVC) curve. At this price, the firm's profit-maximizing strategy is to:
Answer: be indifferent between producing the output where MC = AVC and shutting down. The price equal to the minimum AVC is the shutdown price. If the firm produces the quantity where P = MC (which is also where MC=AVC), its total revenue (P × q) will exactly equal its total variable cost (AVC × q). In this case, the firm's loss will be equal to its total fixed costs. If the firm shuts down (produces q=0), its loss will also be equal to its total fixed costs. Since both actions result in the exact same loss, the firm is indifferent between producing and shutting down.

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