CHAPTER 8: PROFIT MAXIMIZATION AND COMPETITIVE SUPPLY

IPMX Managerial Economics — Complete Solution Manual

CHAPTER OVERVIEW

Topic: How competitive firms decide how much to produce, when to shut down, and what determines long-run industry equilibrium.

Core Logic Chain:

• Short run: firm chooses output where P = MC (if P ≥ AVC). Below AVC → shut down.
• Long run: firms enter/exit until P = min AC → zero economic profit.
• Industry supply = horizontal sum of firm supply curves.
• LR equilibrium depends on whether it is a constant-cost or increasing-cost industry.

PART I: QUESTIONS FOR REVIEW — FULLY SOLVED

Question 1

Why would a firm that incurs losses choose to produce rather than shut down?

Concept Being Used: Short-run shutdown rule vs. operating rule. The key distinction is between fixed costs (sunk in the short run) and variable costs (avoidable).

Why This Concept Applies: In the short run, fixed costs are already paid regardless of production decision. The only relevant comparison is whether revenue covers variable costs.

Formula / Theory:

• Produce if: TR ≥ TVC → P ≥ AVC
• Shut down if: P < AVC (minimum average variable cost)
• Loss from producing = TFC − (P − AVC) × q
• Loss from shutting down = TFC

Economic Intuition: Think of fixed costs as a hotel booking you've already paid for — you can't get a refund. Whether you stay home or travel, you lose the room cost. You only avoid the travel expense (variable cost). Similarly, a firm avoids variable costs by shutting down, but cannot recover fixed costs.

Step-by-Step Answer:

1. When a firm produces at a loss, it means TR < TC.
2. But TC = TVC + TFC. If TR > TVC, then the revenue more than covers variable costs.
3. The excess (TR − TVC) contributes toward paying fixed costs.
4. Shutting down saves TVC but does NOT save TFC.
5. Therefore: if P > AVC → loss from producing < loss from shutting down.
6. Only shut down when TR < TVC (i.e., P < AVC), meaning production cannot even cover its own direct costs.

Interpretation: A firm producing at a loss in the short run is making a rational decision. It is minimizing its loss, not maximizing profit. The firm will exit in the long run if the situation doesn't improve.

Final Answer: A loss-making firm produces if and only if P ≥ AVC. At that point, operating revenue covers variable costs and contributes something toward unavoidable fixed costs. Shutting down would only worsen the loss by the contribution margin (P − AVC) × q.

Question 2

Explain why the industry supply curve is not the long-run industry marginal cost curve.

Concept Being Used: Industry-level vs. firm-level marginal cost; input price effects from industry expansion.

Step-by-Step Answer:

1. The long-run industry supply curve traces equilibrium prices as the number of firms adjusts.
2. As industry output expands (via firm entry), input demand rises.
3. If input prices rise (increasing-cost industry), each firm's MC curve shifts upward.
4. The LR supply curve therefore traces out a locus of new equilibrium points — it incorporates the effect of input price changes on each firm's cost.
5. The industry MC would simply sum up individual firm MCs at a point in time, ignoring these dynamic input price effects.

Final Answer: The LR industry supply curve ≠ LR industry MC because, as industry output expands, input prices may rise, shifting each firm's cost curves. The supply curve captures the new equilibrium price after full adjustment, not just the marginal cost at a static point.

Question 3

In long-run equilibrium, all firms earn zero economic profit. Why?

Concept Being Used: Free entry and exit mechanism in perfect competition.

Step-by-Step Answer:

1. If profit > 0 → new firms enter → industry supply increases → market price falls → profit falls.
2. This continues until π = 0.
3. If profit < 0 → firms exit → supply falls → price rises → profit rises.
4. This continues until π = 0.
5. The only stable equilibrium is zero economic profit, because any deviation triggers entry or exit.

Important Nuance: "Zero economic profit" means the firm earns exactly its opportunity cost — the normal rate of return. Accounting profit is positive; only excess profit disappears.

Final Answer: Free entry and exit is the self-correcting mechanism. Positive profit attracts entrants; negative profit triggers exit. The only stable long-run state is π = 0 — where firms earn exactly their opportunity cost and no one has an incentive to enter or leave.

Question 4

What is the difference between economic profit and producer surplus?

Concept Being Used: Two distinct measures of firm performance.

Definitions and Formulas:

• Economic Profit (π): π = TR − TC = TR − TVC − TFC
• Producer Surplus (PS): PS = TR − TVC = area above MC, below price

Relationship: PS = π + TFC

Intuition: Producer surplus measures the gain from selling output relative to the variable costs of producing it. It ignores fixed costs. Economic profit subtracts all costs including fixed.

• In the short run: PS > π (by the amount of fixed costs)
• In the long run (with π = 0): PS = TFC (the return on sunk capital)
• When TFC = 0 (constant-cost LR): PS = π = 0

Final Answer: Economic profit = TR − TC (includes fixed costs). Producer surplus = TR − TVC (excludes fixed costs). The gap between them equals fixed cost: PS = π + FC.

Question 5

Why do firms enter an industry when they know that in the long run economic profit will be zero?

Concept Being Used: Opportunity cost, short-run vs. long-run dynamics, normal vs. abnormal profit.

Step-by-Step Answer:

1. Firms enter to capture short-run positive profits before the market adjusts.
2. In the long run, zero economic profit means firms earn exactly their opportunity cost (the normal rate of return on invested capital).
3. This is not "nothing" — it means the firm does as well as it would in its next best alternative.
4. Entrepreneurs who enter at the right time earn above-normal returns before competition erodes them.
5. Economic zero profit is consistent with accounting profit that covers the cost of capital.

Final Answer: Firms enter because short-run profits exist. Zero long-run economic profit means they earn their required rate of return — exactly what capital could earn elsewhere. It is not a loss; it is the normal competitive return. Early entrants earn more while the window lasts.

Question 6

Explain the decrease in American automobile manufacturers from many (early 1900s) to three large ones (end of century). Not due to antitrust enforcement failure.

Concept Being Used: Economies of scale, minimum efficient scale, natural industry consolidation.

Step-by-Step Answer:

1. Auto manufacturing exhibits very large economies of scale — the minimum efficient scale (MES) requires massive production volumes.
2. As technology advanced (assembly lines, robotics, global sourcing), the cost advantages of scale increased.
3. Firms that achieved sufficient scale produced at far lower unit cost than small competitors.
4. Small manufacturers could not achieve MES → their AC was higher → they could not survive at the market price set by large firms.
5. Eventually, only firms large enough to reach MES survived.
6. This is competitive consolidation via cost structure, not antitrust failure.

Final Answer: The auto industry has a cost structure characterized by very large economies of scale. As the industry matured, firms that could not achieve minimum efficient scale faced persistently higher average costs. They exited, not due to predatory pricing or antitrust failure, but because the inherent cost structure favored very large producers.

Question 7

Industry X is in long-run equilibrium with zero profit. If the product price falls, no firm can survive. Agree or disagree?

Concept Being Used: Long-run adjustment process; shutdown vs. exit distinction.

Answer: Disagree (with important nuance)

Step-by-Step Reasoning:

1. In LR equilibrium, P = min AC → zero profit.
2. If price falls below min AC, all firms make losses in the short run.
3. Firms begin to exit (not shut down — they leave the industry entirely in the long run).
4. As firms exit, industry supply falls → price rises back toward min AC.
5. The process continues until enough firms have exited that the remaining firms can again earn zero profit at the new LR equilibrium.
6. Some firms will survive — just fewer of them.
7. The statement would be true only if the price permanently stays below the minimum possible AC for any firm, which is unlikely unless demand permanently collapses to zero.

Caveat: If all firms are truly identical and price permanently stays below even the minimum input cost (impossible in a functioning economy), then no firm survives. But in reality, some lower-cost firms or firms in different cost situations will survive as others exit.

Final Answer: Disagree. Exit by some firms raises the market price back toward the new equilibrium. Fewer firms, not zero firms, will survive.

Question 8

Increase in demand for movies also increases actor salaries. Is the LR supply curve for films likely to be horizontal or upward sloping?

Concept Being Used: Increasing-cost industry vs. constant-cost industry in the long run.

Step-by-Step Answer:

1. When demand for films increases, more films are made → demand for inputs (actors, directors, crew) increases.
2. Actor salaries rise because they are scarce specialized inputs.
3. Rising input prices mean each film's production cost increases as industry output expands.
4. This is an increasing-cost industry — input prices are not fixed but rise with industry scale.
5. In increasing-cost industries, the LR supply curve is upward sloping.
6. Each expansion in film production comes at progressively higher cost per unit.

Final Answer: Upward sloping. Film production is an increasing-cost industry. As more films are made, competition for scarce specialized talent drives up salaries, increasing average costs. The LR supply curve reflects these rising input costs.

Question 9

True or false: A firm should always produce at an output at which long-run average cost is minimized.

Concept Being Used: Profit maximization vs. cost minimization; when they coincide.

Answer: FALSE

Step-by-Step Reasoning:

1. A firm maximizes profit where P = LR MC, not where LR AC is minimized.
2. Min LR AC only coincides with profit-maximizing output when P = min LR AC — which is the zero-profit long-run competitive equilibrium.
3. If P > min LR AC, the firm should produce MORE (at P = LR MC > min LR AC).
4. The firm would sacrifice profit by stopping at the AC-minimizing output when price is high.
5. Producing at min AC when price is above min AC would leave profitable units unproduced.

Final Answer: False. The firm maximizes profit at P = LR MC. Production at min LR AC is correct only in long-run competitive equilibrium where P = min AC. At any higher price, the firm should produce beyond the minimum AC output.

Question 10

Can there be constant returns to scale in an industry with an upward-sloping supply curve?

Concept Being Used: Distinction between production technology (returns to scale) and input market effects.

Answer: Yes

Step-by-Step Reasoning:

1. Returns to scale describes the production function's technical properties — doubling inputs doubles output.
2. An upward-sloping supply curve can arise from rising input prices as the industry expands, even if production technology exhibits CRS.
3. If more firms enter and compete for specialized land, labor, or materials, input prices rise → AC rises with industry output → upward-sloping LR supply.
4. CRS in technology ≠ constant costs for the industry, because input prices are not fixed.

Final Answer: Yes. CRS means the production function is linearly homogeneous. But if input prices rise as the industry expands (increasing-cost industry due to input market effects), the LR supply curve is upward sloping — even with CRS technology.

Question 11

What assumptions are necessary for a market to be perfectly competitive? Why is each important?

Final Answer: Each assumption is important because it prevents a firm from gaining the power to set price above MC. Collectively, these assumptions ensure the perfectly competitive outcome: P = MC, efficient resource allocation, and zero long-run economic profit.

Question 12

How does a competitive market ensure increased output in response to a demand increase? Does your answer change if the government imposes a price ceiling?

Without a price ceiling (free market adjustment):

1. Demand shifts right → price rises above LR equilibrium → firms earn positive profit.
2. In the short run: existing firms expand (move up their MC curves) → output increases partially.
3. Positive profit attracts new entrants in the long run.
4. New entry increases market supply → price falls back toward equilibrium.
5. New LR equilibrium: more firms, more output, same price (constant-cost) or slightly higher price (increasing-cost).

With a price ceiling below equilibrium:

1. The ceiling prevents price from rising to signal increased demand.
2. No profit signal → no entry incentive → no long-run supply expansion.
3. Result: shortage (QD > QS at ceiling price).
4. The market cannot self-correct through entry.

Final Answer: Without a ceiling, price rises signal profit opportunities, triggering entry and output expansion until a new equilibrium. With a ceiling below the equilibrium price, the price signal is suppressed, no entry occurs, and the shortage persists.

Question 13

Government subsidizes every acre of land used to grow tobacco. How does this affect the LR supply curve?

Step-by-Step:

1. The subsidy reduces the effective cost of a key input (land) for tobacco farmers.
2. This shifts each firm's AC curve downward.
3. In the long run, entry occurs because existing firms now earn positive profit (P > new min AC).
4. More entry → supply increases → price falls → LR supply curve shifts rightward and/or downward.
5. For a constant-cost industry: LR supply is horizontal at a lower price level.
6. For an increasing-cost industry: LR supply is upward-sloping but at lower cost levels.

Final Answer: The subsidy reduces the cost of a key input, shifting AC downward. In the long run, more firms enter, and the LR supply curve shifts downward (rightward), leading to lower equilibrium prices and higher output.

Question 14

Vacuum cleaners available from local stores and catalogues.

Part (a): Will all sellers earn zero economic profit if they charge the same price? Yes, assuming all sellers have similar cost structures. In a competitive market with identical costs, zero economic profit is the long-run equilibrium outcome.

Part (b): If one local seller owns his building (pays no rent), is he earning positive economic profit? No. Economic profit accounts for opportunity cost. The seller's implicit cost includes the market rent he could earn by renting the building to someone else. Even though he pays no accounting rent, his economic cost includes this opportunity cost. Zero economic profit means he earns exactly what the building could earn in its next best use.

Part ©: Does the seller who pays no rent have incentive to lower prices? No. The no-rent seller is already earning zero economic profit (once opportunity cost is accounted for). Lowering price would create economic losses. Raising price would attract competition. There is no reason to deviate from the market price.

Final Answer (a): Yes, zero economic profit in the long run. Final Answer (b): No — the opportunity cost of the building (market rent) is an implicit economic cost. Economic profit remains zero. Final Answer ©: No — the owner is at the competitive equilibrium. Lowering price creates losses.

PART II: EXERCISES — FULLY SOLVED

Exercise 1: Fill in the Table and Analyze Output Choice

Given Data: P = $60, C(q) values from q=0 to q=11.

Concept Being Used: Short-run profit maximization: produce where MR = MC = P. For a competitive firm, MR = P always.

Formula: MC = ΔC/Δq; MR = ΔR/Δq = P; Profit = TR − TC; TR = P × q.

Part (a): Complete Table at P = $60

Optimal output at P = $60: q* = 10 units (profit = $190).

Reasoning: MR = MC logic. The 10th unit: MR = 60 > MC = 55 → produce it. The 11th unit: MR = 60 < MC = 65 → don't produce it.

Part (b): What happens when P falls from $60 to $50?

At P = $50, compute TR = 50q and new profits:

Verification: 9th unit: MR=50 > MC=45 → produce. 10th unit: MR=50 < MC=55 → don't.

Final Answers:

• At P = $60: q* = 10, profit = $190
• At P = $50: q* = 9, profit = $95
• Price fall reduces both optimal output (10 → 9) and profit (190 → 95).

Exercise 2: Effect of Changing Fixed Costs

Concept Being Used: Fixed costs do not affect the profit-maximizing output choice. They shift the profit level but not the optimal quantity.

Why: In the short run, the firm's output rule is P = MC. Fixed costs are sunk and do not appear in MC. The firm produces the same quantity regardless of fixed costs.

Analysis at P = $60:

From Exercise 1, optimal q = 10 in all cases (unchanged by FC).

Note: C(10) at FC=$150 = (410−100+150) = 460. At FC=$200: 510.

General Conclusion:

Fixed costs have NO effect on the firm's short-run output choice. They reduce profit dollar-for-dollar. The profit-maximizing output is determined solely by the intersection of P and MC, which is unaffected by fixed costs.

This is a critical managerial insight: sunk costs should not drive production decisions.

Exercise 3: Short-Run Supply Curve and Industry Supply

Concept Being Used: Firm supply curve = MC curve above AVC. Industry supply = horizontal sum of firm supply curves.

Part (a): Derive the firm's short-run supply curve

From the data, variable cost (VC) = C − FC = C − 100:

Minimum AVC occurs at approximately q = 7 (AVC ≈ $24.57). The MC at q=7 is $22.

Shutdown price: The firm shuts down if P < min AVC ≈ $24.57.

Supply curve: For P ≥ $24.57, produce where P = MC. The supply schedule is:

Part (b): Industry supply with 100 identical firms

Each firm supplies q at a given price. Industry supply Q = 100 × q_firm.

Final Answers:

• Firm supply: produce at P = MC for P ≥ min AVC ≈ $24.57.
• Industry supply: Q = 100 × q_firm (horizontal sum of all firm supply curves).

Exercise 4: Watchmaking Firm — Maximize Profit

Given: C = 200 + 2q², MC = 4q, Fixed Cost = $200, P = $100.

Concept Being Used: Profit maximization in a competitive firm: set P = MC.

Part (a): Optimal output

Set P = MC: $100=4q\Rightarrow q^{*}=25$

Part (b): Profit level

$\pi =TR-TC=P·q-C$ $=100(25)-[200+2(25{)}^2]$ $=2500-200-1250=\$1,050$

Part ©: Minimum price for positive output

The firm produces as long as P ≥ min AVC. $AVC=\frac{VC}{q}=\frac{2q^2}{q}=2q$

AVC increases with q. Minimum AVC → 0 as q → 0. Therefore, the firm will produce a positive output at any price P > 0.

Intuition: Since MC = 4q passes through the origin, and AVC = 2q is always below MC, the firm always covers variable costs when P > 0. The shutdown price is effectively $0.

Final Answers:

• (a) q* = 25 watches
• (b) Profit = $1,050
• © Any P > $0 induces positive output (shutdown price = $0)

Exercise 5: Producer Surplus and Profit

Given: MC(q) = 3 + 2q, P = $9, AVC(q) = 3 + q, FC = $3.

Concept Being Used: P = MC to find optimal output; PS = area above MC below P; Profit = PS − FC.

Part (a): Optimal output

$P=MC:9=3+2q\Rightarrow 2q=6\Rightarrow q^{*}=3$

Part (b): Producer surplus

PS = area of triangle above MC(q) below P, from 0 to q* = 3. $PS={\int }_0^3[P-MC(q)]dq={\int }_0^3[9-(3+2q)]dq={\int }_0^3(6-2q)dq$ $={[6q-q^2]}_0^3=18-9=\$9$

Part ©: Profit (positive, negative, or zero?)

Variable Cost: VC = AVC × q = (3 + q) × q = 3q + q²

At q = 3: VC = 9 + 9 = $18. TC = FC + VC = 3 + 18 = $21. TR = 9 × 3 = $27.

$\pi =TR-TC=27-21=+\$6\text{(POSITIVE PROFIT)}$

Cross-check: π = PS − FC = 9 − 3 = $6. ✓

Final Answers:

• (a) q* = 3 units
• (b) Producer Surplus = $9
• © Positive profit = $6 (the firm earns above its opportunity cost in the short run)

Exercise 6: Is the Firm Maximizing Profit?

Given: C = 50 + 4q + 2q², MC = 4 + 4q, P = $20, Current output = 5 units.

Concept Being Used: At profit maximum: P = MC. Check if this holds at q = 5; find the true optimum.

Step 1: Check if q = 5 is optimal. At q = 5: MC = 4 + 4(5) = 24. But P = 20. Since MC = 24 > P = 20, the firm is overproducing. It should produce less.

Step 2: Find profit-maximizing q. $P=MC:20=4+4q\Rightarrow 4q=16\Rightarrow q^{*}=4$

Step 3: Compare profits. At q = 4: π = 20(4) − [50 + 16 + 32] = 80 − 98 = −$18 (loss). At q = 5: π = 20(5) − [50 + 20 + 50] = 100 − 120 = −$20 (larger loss).

So the firm is losing less at q = 4 than at q = 5.

Step 4: Long-run decision. In the long run, find min AC: $AC=\frac{50}{q}+4+2q$ $\frac{d(AC)}{dq}=-\frac{50}{q^2}+2=0\Rightarrow q^2=25\Rightarrow q_{min\_AC}=5$ $\text{Min AC}=\frac{50}{5}+4+10=10+4+10=\$24$

Since P = $20 < min AC = $24, the firm cannot cover average costs. In the long run, this firm will exit.

If the price rises to $24 (LR competitive equilibrium), the firm would produce q = 5 (where P = min AC = $24).

Final Answers:

• The firm at q = 5 is NOT profit-maximizing (MC = $24 > P = $20). It should produce q* = 4.
• In the long run, with P = $20 < min AC = $24, the firm exits. LR equilibrium only if P = $24.

Exercise 7: Cost Curves and Price Ranges

Given: C(q) = 4q² + 16, MC = 8q.

Concept Being Used: Deriving cost components; finding shutdown price, break-even price, and supply conditions.

Part (a): Cost components

Part (b): Graph description

• MC = 8q → straight line through origin, steeper slope.
• AVC = 4q → straight line through origin, half the slope of MC.
• AC = 4q + 16/q → U-shaped curve, reaching minimum where MC = AC.
• MC intersects AVC at origin; MC intersects AC at minimum AC.

Key property: Since AVC = 4q is always increasing from zero, and MC = 2 × AVC always (MC = 8q = 2 × 4q), the MC lies above AVC at every positive q.

Part ©: Output that minimizes average cost

$\frac{d(AC)}{dq}=4-\frac{16}{q^2}=0\Rightarrow q^2=4\Rightarrow q^{*}=2$ $\text{Min AC}=4(2)+\frac{16}{2}=8+8=\$16$

Verify: MC at q = 2: 8(2) = $16 = min AC. ✓ (MC = AC at minimum AC — the standard condition.)

Part (d): Price range for positive output

The firm produces for P ≥ min AVC. Since AVC = 4q → min AVC as q → 0 is $0. Therefore the firm produces for any P > 0.

Supply function: P = MC = 8q → q = P/8.

Part (e): Price range for negative profit

At supply quantity q = P/8: $\pi =TR-TC=P·\frac{P}{8}-4{\left(\frac{P}{8}\right)}^2-16=\frac{P^2}{8}-\frac{P^2}{16}-16=\frac{P^2}{16}-16$

Negative profit: π < 0 → P²/16 < 16 → P² < 256 → P < $16.

Part (f): Positive profit requires P > $16.

Summary:

• Min AC at q = 2, price = $16
• Positive output for any P > 0
• Negative profit for P < $16
• Positive profit for P > $16
• Break-even (zero profit) at P = $16

Exercise 8: Competitive Firm Cost and Supply Curve

Given: C(q) = q³ − 8q² + 30q + 5.

Concept Being Used: Derive MC, AC, AVC; find shutdown price; identify supply curve.

Part (a): Cost curves

• MC = 3q² − 16q + 30
• AVC = VC/q = (q³ − 8q² + 30q)/q = q² − 8q + 30
• AC = q² − 8q + 30 + 5/q

Minimum AVC: dAVC/dq = 2q − 8 = 0 → q = 4 Min AVC = 16 − 32 + 30 = $14

Verify with MC: MC at q = 4 = 3(16) − 64 + 30 = 48 − 64 + 30 = $14 ✓ (MC = min AVC at shutdown point)

Part (b): Price range for zero output

Firm supplies zero output if P < min AVC = $14.

Part ©: Supply curve

For P ≥ $14, the supply curve is: P = MC = 3q² − 16q + 30 (above min AVC = $14). For P < $14: q = 0.

Part (d): Price to supply exactly 6 units

$P=MC\text{ at }q=6:P=3(36)-16(6)+30=108-96+30=\$42$

Final Answers:

• (a) MC = 3q²−16q+30; AVC = q²−8q+30; AC = q²−8q+30+5/q
• (b) Zero output if P < $14 (min AVC)
• © Supply: P = MC for P ≥ $14
• (d) To supply 6 units: P = $42

Exercise 9: Production Function to Supply Curve

Given: q = 9x^(1/2) (production function), Fixed costs = $1,000, Variable input cost = $4,000/unit.

Concept Being Used: Deriving total cost function from production function, then applying P = MC for supply.

Part (a): Total cost function C(q)

From production function: q = 9x^(1/2) → x = q²/81.

Variable Cost = input price × x = $4,000 × (q²/81) = 4000q²/81.

$C(q)=1000+\frac{4000q^2}{81}$

Part (b): Supply curve

$MC=\frac{dC}{dq}=\frac{8000q}{81}$

Set P = MC: $P=\frac{8000q}{81}\Rightarrow q=\frac{81P}{8000}$

This is the firm's supply function.

Part ©: At P = $1,000

$q^{*}=\frac{81\times 1000}{8000}=\frac{81}{8}=10.125\approx 10.125\text{ units}$

Profit calculation: $TR=1000\times \frac{81}{8}=\$10,125$ $VC=\frac{4000\times (81/8{)}^2}{81}=\frac{4000\times 81}{64}=\$5,062.50$ $TC=1000+5062.50=\$6,062.50$ $\pi =10125-6062.50=\$4,062.50$

Final Answers:

• (a) C(q) = 1,000 + 4,000q²/81
• (b) Supply: q = 81P/8,000 (or P = 8,000q/81)
• © At P=$1,000: q = 10.125 units, profit = $4,062.50

Exercise 10: Perfect Competition — Market Equilibrium

Given: QD = 6,500 − 100P; QS = 1,200P; C(q) = 722 + q²/200; MC(q) = q/100.

Concept Being Used: Market equilibrium; firm vs. market supply; long-run zero-profit condition.

Part (a): Short-Run Equilibrium

Market clearing: QD = QS $6500-100P=1200P\Rightarrow 6500=1300P\Rightarrow P^{*}=\$5$ $Q^{*}=1200(5)=6000\text{ units}$

Firm's output: From MC = P: $\frac{q}{100}=5\Rightarrow q^{*}=500\text{ units per firm}$

Number of firms: N = Q/q = 6000/500 = 12 firms

Profit per firm: $\pi =5(500)-[722+\frac{{500}^2}{200}]=2500-722-1250=\$528$

Part (b): Long-run adjustment

π = $528 > 0 → Firms will enter in the long run. Entry increases supply, lowers price, until π = 0.

Part ©: Long-run equilibrium price (P = min AC)

$AC=\frac{722}{q}+\frac{q}{200}$ $\frac{d(AC)}{dq}=-\frac{722}{q^2}+\frac{1}{200}=0\Rightarrow q^2=722\times 200=144,400\Rightarrow q=380$ $\text{Min AC}=\frac{722}{380}+\frac{380}{200}=1.90+1.90=\$3.80$

In the long run: P = $3.80, each firm produces q = 380. Profit = 0.

Part (d): Shutdown price in the short run

$AVC=\frac{q/200·q}{q}=\frac{q}{200}$

Min AVC → $0 as q → 0. Any positive price induces positive output. The firm will not shut down for any P > 0.

Final Answers:

• (a) P* = $5, Q* = 6,000, q_firm = 500, π = $528
• (b) Firms enter (positive profit)
• © Long-run price = $3.80 (zero profit at min AC)
• (d) Shutdown price = $0 (firm produces at any P > 0 in SR)

Exercise 11: Output, Profit, and Producer Surplus

Given: C(q) = 450 + 15q + 2q², MC = 15 + 4q, P = $115.

Part 1: Optimal output $P=MC:115=15+4q\Rightarrow 4q=100\Rightarrow q^{*}=25$

Part 2: Profit $TC=450+15(25)+2(625)=450+375+1250=\$2,075$ $TR=115\times 25=\$2,875$ $\pi =2875-2075=\$800$

Part 3: Producer surplus

Method 1 — Direct calculation: $PS=TR-TVC=2875-(15\times 25+2\times 625)=2875-375-1250=\$1,250$

Method 2 — Area calculation: $PS={\int }_0^{25}[115-(15+4q)]dq={\int }_0^{25}(100-4q)dq={[100q-2q^2]}_0^{25}=2500-1250=\$1,250$

Cross-check: PS = π + FC = 800 + 450 = $1,250 ✓

Final Answers:

• q* = 25 units
• Profit = $800
• Producer Surplus = $1,250

Exercise 12: Film Developing — Long-Run Equilibrium and Technology Value

Given: C(q) = 50 + 0.5q + 0.08q², MC = 0.5 + 0.16q.

Part (a): Is industry in long-run equilibrium at P = $8.50?

Firm's optimal output: P = MC → 8.50 = 0.5 + 0.16q → q = 50.

Profit: $\pi =8.50(50)-[50+0.5(50)+0.08(2500)]=425-50-25-200=\$150>0$

Since π > 0, industry is NOT in LR equilibrium. Firms will enter.

Long-run equilibrium price (P = min AC): $AC=\frac{50}{q}+0.5+0.08q$ $\frac{d(AC)}{dq}=-\frac{50}{q^2}+0.08=0\Rightarrow q^2=625\Rightarrow q=25$ $\text{Min AC}=\frac{50}{25}+0.5+0.08(25)=2+0.5+2=\$4.50$

LR equilibrium price = $4.50.

Part (b): Maximum willingness to pay for new technology (25% cost reduction)

New cost function: C_new = 0.75 × [50 + 0.5q + 0.08q²] = 37.5 + 0.375q + 0.06q² New MC = 0.375 + 0.12q

If one store adopts the technology first (others still at old equilibrium, P = $4.50): Optimal output: 4.50 = 0.375 + 0.12q → q = 34.375 $TR=4.50\times 34.375=\$154.69$ $T{C}_{new}=37.5+0.375(34.375)+0.06(34.375{)}^2=37.5+12.89+70.90=\$121.29$ ${\pi }_{with\_tech}=154.69-121.29\approx \$33.40$

This profit is what the technology generates above the normal competitive return.

Maximum willingness to pay = ~$33.40 per period for the technology.

Final Answers:

• (a) Industry is NOT in LR equilibrium. LR price = $4.50 (at min AC with q = 25).
• (b) Maximum payment for technology = ~$33.40 per period.

Exercise 13: Hot Dog Vendors — Competitive Market and Permits

Given: MC = $1.50/hot dog, max capacity = 100/vendor/day, QD = 4,400 − 1,200P.

Part (a): If P = $2, how many hot dogs does each vendor want to sell?

P = $2 > MC = $1.50. Firm maximizes by producing up to capacity. Each vendor wants to sell 100 hot dogs (at capacity).

Part (b): Will price stay at $2?

In perfect competition: P = MC = $1.50. Entry drives price to MC. No, price will fall to $1.50 (long-run competitive equilibrium where P = MC = $1.50).

Part ©: Number of vendors if Q = 100 each and QD must equal supply

At P = $1.50: QD = 4,400 − 1,200(1.50) = 4,400 − 1,800 = 2,600. Each vendor sells 100: N = 2,600/100 = 26 vendors.

Part (d): With 20 permits, price with 100 hot dogs each

Total supply: 20 × 100 = 2,000 hot dogs. Find P from demand: 2,000 = 4,400 − 1,200P → 1,200P = 2,400 → P = $2.00.

Part (e): Maximum price for a permit

Profit per vendor per day = (P − MC) × q = (2.00 − 1.50) × 100 = $50/day.

A vendor would pay up to $50/day for a permit (this is the economic rent earned by the scarce permit).

Final Answers:

• (a) 100 hot dogs (at capacity)
• (b) Price falls to $1.50 (competitive equilibrium)
• © 26 vendors
• (d) P = $2.00
• (e) Maximum permit price = $50/day

Exercise 14: Effect of a $1 Unit Sales Tax on a Competitive Firm

Given: Tax = $1/unit; original market price = $5; many firms in competitive industry.

Part (a): How does the tax affect cost curves?

The tax is equivalent to an increase in marginal and average variable cost by $1.

• MC shifts up by $1: MC_new = MC + 1.
• AVC shifts up by $1: AVC_new = AVC + 1.
• AC shifts up by $1: AC_new = AC + 1.
• Fixed costs are unaffected.

Part (b): Short-run effects on price, output, and profit

• The taxed firm's supply curve shifts upward by $1.
• All firms in the industry are affected → industry supply shifts left.
• In the short run: price rises (by less than $1, depending on demand/supply elasticities).
• Individual firm output decreases (moving down the MC curve with new tax).
• Profit decreases — potentially becoming negative.

Part ©: Entry/exit — long-run effects

If the price rise is less than $1 (which it is with any non-zero demand elasticity), firms face losses → firms exit.

Exit continues until the surviving firms can earn zero profit at the new equilibrium. For a constant-cost industry: in the long run, price rises by exactly $1 (the full amount of the tax), output per firm returns to pre-tax level, but there are fewer firms.

Final Answers:

• (a) Tax shifts MC, AVC, and AC up by $1 each.
• (b) In SR: price rises slightly (< $1), output falls, profit falls (possibly negative).
• © Firms exit in the LR until zero profit is restored. Long-run price rises by full $1 (constant-cost industry).

Exercise 15: Tax on Polluters, Subsidy to Non-Polluters

Given: 10% sales tax on polluters; 10% revenue subsidy to non-polluters. Identical LR AC before policy.

Part (a): Short-run and long-run effects on price, output, industry output

Short Run:

• Polluters: 10% tax raises their effective cost → MC rises → output falls → their supply curve shifts left.
• Non-polluters: 10% subsidy reduces effective cost → MC falls → output rises → their supply curve shifts right.
• Net effect on industry price depends on relative sizes of these two groups.
• If equal numbers: price may be roughly stable, but polluters shrink and non-polluters expand.
• If polluters' contraction dominates: price rises. If non-polluters' expansion dominates: price falls.

Long Run:

• The tax creates a permanent cost disadvantage for polluters: they face higher effective prices.
• Polluters exit the industry as they cannot earn zero economic profit.
• Non-polluters earn positive profit (effective subsidy lowers costs) → non-polluters enter.
• Long-run equilibrium: industry is dominated by non-polluters; polluters are largely or entirely replaced.
• Price adjusts to the new LR equilibrium AC of non-polluters (lower than polluters' AC).

Part (b): Can the budget always be balanced?

Tax revenue = 10% × (polluters' revenue) = 0.10 × P × Q_polluters. Subsidy cost = 10% × (non-polluters' revenue) = 0.10 × P × Q_nonpolluters.

Budget balances only if: Q_polluters = Q_nonpolluters.

In general, as polluters exit and non-polluters expand, this equality need not hold.

• In the LR, as polluters exit, tax revenue falls while subsidy to remaining non-polluters rises.
• Budget becomes unbalanced (deficit) as non-polluter output exceeds polluter output.

Therefore: This policy cannot always be achieved with a balanced budget. The asymmetric responses of polluters and non-polluters to cost changes mean revenues and subsidy payments will diverge over time.

Final Answers:

• (a) SR: Polluters contract; non-polluters expand; price is ambiguous. LR: Polluters exit; non-polluters dominate; price falls toward non-polluter min AC.
• (b) No. Budget balance requires equal output from both groups — this holds in the initial period but breaks down as the market adjusts. The policy is not self-financing in the long run.

PART III: LEARNING ENHANCEMENTS

Key Formula Sheet — Chapter 8

Important Equilibrium Conditions

Common IPMX Exam Traps

1. "Zero profit means loss" — Wrong. Zero economic profit includes opportunity cost. Accounting profit is positive.
2. "Fixed costs affect optimal output" — Wrong. Fixed costs are sunk. Only variable/marginal costs matter for quantity decisions.
3. "Competitive firms have no producer surplus" — Wrong. PS = TR − TVC, which is positive whenever P > AVC.
4. "If price falls, the firm always shuts down" — Wrong. Firm produces if P ≥ min AVC.
5. "LR supply is industry MC" — Wrong. LR supply incorporates input price changes as industry expands.
6. "Minimum AC and profit-maximizing output always coincide" — Only in LR competitive equilibrium.

Concept Comparison Tables

Constant-Cost vs. Increasing-Cost Industry (Long Run)

Short Run vs. Long Run Decision Rules

Quick Revision Summary

Perfect competition requires: many firms, identical product, free entry, price-taking behavior.

Short-run output rule: P = MC (above min AVC). Below min AVC → shut down.

Long-run zero profit: Entry/exit drives π → 0. All firms earn exactly their opportunity cost.

Producer surplus is the area above MC and below price. It equals profit + fixed cost.

Long-run supply reflects not just firm MC but also input price changes from industry expansion:

• Constant cost → horizontal LR supply.
• Increasing cost → upward-sloping LR supply.

Policy impacts: Taxes shift cost curves up → firms exit in LR → price rises by full tax (constant-cost). Subsidies on inputs → cost curves down → entry → price falls.

End of Chapter 8 Solution Manual