Profit Fish Formula Verification Against Undergraduate Microeconomics Textbooks

Sources consulted

• Cowen & Tabarrok (C&T), Modern Principles of Microeconomics, 5th Edition, Worth/Macmillan Publishers. Chapter structure verified against the official StudyLib document and MRU syllabus. Specific page numbers are not available (no page-numbered PDF was accessible); all C&T references use chapter-level citations confirmed via the MRU video-mapped syllabus (https://mru.org/teacher-resources/principles-microeconomics-syllabus) and the official chapter-13 solution set hosted at Studocu (Utah State University, verified to reference "Modern Principles of Economics: Monopoly"). Where an exact MRU video reinforces the textbook treatment, the MRU URL is provided as a proxy.
• OpenStax, Principles of Microeconomics 2e, Steven A. Greenlaw & David Shapiro, OpenStax/Rice University, 2017. https://openstax.org/books/principles-microeconomics-2e
• Khan Academy, Microeconomics — https://www.khanacademy.org/economics-finance-domain/microeconomics (used for elasticity and calculus-based MR proof)

Quick verdict

All 17 formulas as stated in the Profit Fish calculator are correct and consistent with standard undergraduate microeconomics as taught in Cowen & Tabarrok, OpenStax, and Khan Academy. No formula is wrong or inverted. One important contextual caveat applies to Formula 6 (MC = constant = AVC): while the formula is mathematically valid, all three sources also present the more general U-shaped MC curve that arises from diminishing marginal returns. The linear/constant-AVC model used in the calculator is explicitly a simplification that C&T themselves use repeatedly in their monopoly and break-even problems (see §13 examples). The user should understand that the app is using this valid special case, not claiming it is universal. Formula 11 (point elasticity) uses derivative notation that is one step beyond what most intro textbooks present explicitly — the percentage-change form is the standard intro treatment — but the equivalence is mathematically immediate and confirmed by multiple sources. All other formulas are textbook-standard with no discrepancies detected.

Formula-by-formula verification

1. Total Cost: TC = TFC + TVC

• Cowen & Tabarrok: Chapter 11 — "Costs and Profit Maximization Under Competition" (5th ed.). From the Studocu Chapter 11 notes keyed to C&T: "TC = Fixed Costs (FC) + Variable Costs (VC)" and "Fixed Costs do not vary with Q; Variable Costs rise as Q increases." The MRU syllabus maps this chapter to Week 7 under "Intro to Firm's Profit Maximization." C&T use TC = FC + VC notation (dropping the "T" prefix for fixed cost in some editions), but the identity is identical.
• OpenStax: Section 7.3 — "Costs in the Short Run". Direct quote: "Total costs are the sum of fixed plus variable costs." Fixed cost is defined as "expenditures that do not change regardless of the level of production" and variable costs as costs that "increase or decrease with output."
• Third source (Pearson/Parkin lecture slides, St. Afaf's University): From the publicly available Chapter 11 lecture PDF (fcom.stafpu.bu.edu.eg): "Total cost equals total fixed cost plus total variable cost. That is: TC = TFC + TVC" — appears verbatim with exactly this notation. This PDF follows Parkin's Economics (Pearson Addison-Wesley), which is a third independent intro-micro source.
• Verdict: ✅ Matches exactly. Universal across all intro-micro texts.

2. TVC = AVC · Q (linear/recipe cost model)

• Cowen & Tabarrok: Chapter 11. C&T introduce variable cost as rising with Q. Under the linear cost model (constant AVC), TVC = AVC · Q follows directly from the definition AVC = TVC/Q → TVC = AVC · Q. C&T use this linear form extensively in their monopoly chapter (Chapter 13) — see the Chapter 13 challenge problems on Studocu: "Case A: Demand P = 50 − Q; Fixed cost = 100; Marginal cost = 10" — here constant MC = AVC = 10, so TVC = 10Q. C&T do not write "TVC = AVC·Q" as a named formula but the derivation is trivially algebraic from AVC = TVC/Q.
• OpenStax: Section 7.3. The barbershop Table 7.10 shows Variable Cost growing proportionally with labor when AVC is roughly constant (e.g., at Q = 40 haircuts, TVC = $160; at Q = 60, TVC = $240; AVC remains close to constant at $4 across these rows). The identity TVC = AVC · Q is algebraically immediate from AVC = TVC/Q.
• Third source: The Parkin lecture slides explicitly show "TVC is the number of workers multiplied by $25" — i.e., TVC = (unit variable cost) × Q, which is TVC = AVC · Q when AVC is constant.
• Important caveat: This formula holds only when AVC is constant (the linear cost model). If AVC is U-shaped (the general case), TVC is not a simple linear multiple of Q. The Profit Fish app uses the linear case, which is valid for "recipe" pricing problems.
• Verdict: ✅ Matches under the linear/constant-AVC assumption the app makes.

3. AFC = TFC / Q

• Cowen & Tabarrok: Chapter 11. From the Studocu Chapter 11 notes aligned with C&T: AFC is derived from fixed costs spread over output. No explicit formula quote available at chapter level, but the ReviewEcon.com summary (citing standard intro-micro content): "Average Fixed Costs: Add up all of the fixed costs for a firm and divide by the quantity produced (AFC = FC/Q)."
• OpenStax: Section 7.3. While AFC is not given its own bolded formula on this page (the key-terms list confirms its absence), it is implicit from ATC = AFC + AVC and ATC = TC/Q and AVC = TVC/Q → AFC = ATC − AVC = (TC − TVC)/Q = TFC/Q. The Parkin lecture PDF cited above provides it directly: "Average fixed cost (AFC) = TFC / Q. Falls as output increases."
• Third source: ReviewEcon.com cost curves summary: "Average Fixed Costs: Add up all of the fixed costs for a firm and divide by the quantity produced (AFC = FC/Q). Continually decreases." This is consistent with every intro-micro text.
• Verdict: ✅ Matches exactly.

4. AVC = TVC / Q (constant under the linear assumption)

• Cowen & Tabarrok: Chapter 11. From Studocu Chapter 11 C&T notes: "AVC = TVC / Q" (derived from definitions). C&T's monopoly examples (Chapter 13) repeatedly assume constant AVC = MC, e.g., "Marginal cost = 10" with TVC = 10Q, so AVC = 10Q/Q = 10 = constant.
• OpenStax: Section 7.3. Direct quote: "We obtain average variable cost when we divide variable cost by quantity of output." Expressed formulaically: AVC = VC / Q. Numerical example: "the variable cost of producing 80 haircuts is $400, so the average variable cost is $400/80, or $5 per haircut."
• Third source: Parkin lecture PDF: "Average variable cost (AVC) is total variable cost per unit of output. AVC = TVC / Q."
• Caveat on "constant": Under the linear model (constant AVC = MC), AVC does not vary with Q. All three sources acknowledge that in the general model AVC is U-shaped. The linear case is the app's valid simplification.
• Verdict: ✅ Matches exactly.

5. ATC = TC / Q = AFC + AVC

• Cowen & Tabarrok: Chapter 11. From Studocu Chapter 11 C&T notes: "ATC = TC / Q", also expressed as "ATC = AFC + AVC." C&T discuss ATC as the key curve for determining profit (profit per unit = P − ATC) in both the competition and monopoly chapters.
• OpenStax: Section 7.3. Direct quotes: "Average total cost (sometimes referred to simply as average cost) is total cost divided by the quantity of output." Formulaically: ATC = TC / Q. Numerical example: "Since the total cost of producing 40 haircuts is $320, the average total cost for producing each of 40 haircuts is $320/40, or $8 per haircut." The decomposition ATC = AFC + AVC follows from TC = TFC + TVC, dividing both sides by Q.
• Third source: Parkin lecture PDF: "Average total cost (ATC) is total cost per unit of output. ATC = AFC + AVC."
• Verdict: ✅ Matches exactly.

6. Marginal Cost — the contested one

Definition: MC = ΔTC / ΔQ = (TC₂ − TC₁) / (Q₂ − Q₁)

Textbook sources

• Cowen & Tabarrok — Chapter 11: C&T define MC as the cost of producing one more unit. From the Studocu Chapter 11 notes (keyed to C&T): MC = ΔTC/ΔQ. C&T's Chapter 11 (titled "Costs and Profit Maximization Under Competition") introduces the standard U-shaped MC curve based on diminishing marginal returns. However, in Chapter 13 (Monopoly), C&T explicitly use a flat/constant MC in worked examples (e.g., "Marginal cost = 10" in all three Challenge problems; "MC flat at 50 cents per pill" in the AIDS medication MRU video). This confirms that C&T treats constant MC as a legitimate and common assumption.

• OpenStax: Section 7.3. Direct quote: "Marginal cost is the cost of producing one more unit of output. Mathematically, marginal cost is the change in total cost divided by the change in output: MC = ΔTC/ΔQ." The textbook also explicitly demonstrates the discrete version: "as quantity produced increases from 40 to 60 haircuts, total costs rise by 400 – 320, or 80. Thus, the marginal cost for each of those marginal 20 units will be 80/20, or $4 per haircut." OpenStax Table 8.8 (Section 8.2) shows the formula in column header notation: MC = (TC₂ − TC₁) / (Q₂ − Q₁).

• Khan Academy: The optional calculus proof video (Sal Khan) derives MR from TR and uses a flat MC curve in the monopoly context. The percentage-change and ΔTC/ΔQ definitions are standard throughout Khan Academy's microeconomics course.

Proof that MC is constant when TVC is linear

Given TC(Q) = TFC + AVC · Q with AVC = constant:

Algebraic proof:

> TC(Q + 1) − TC(Q) = [TFC + AVC · (Q + 1)] − [TFC + AVC · Q] = AVC · Q + AVC − AVC · Q = AVC

Therefore MC = AVC at every value of Q. The fixed cost cancels completely, and MC equals the constant variable cost per unit. This can also be seen via calculus: d(TC)/dQ = d(TFC + AVC·Q)/dQ = 0 + AVC = AVC (since TFC and AVC are both constants).

Numeric check (AVC = $4, TFC = $100):

| Q | TC = 100 + 4Q | MC = TC(Q) − TC(Q−1) |

|---|--------------|----------------------|

| 5 | 100 + 4×5 = $120 | — |

| 6 | 100 + 4×6 = $124 | (124 − 120)/(6 − 5) = $4 ✓ |

| 9 | 100 + 4×9 = $136 | — |

|10 | 100 + 4×10 = $140 | (140 − 136)/(10 − 9) = $4 ✓ |

MC = $4 at every step, regardless of Q. It never changes because each additional unit always requires exactly AVC = $4 in variable inputs.

Important caveat (must tell students)

All three sources — C&T, OpenStax, and Khan Academy — present the U-shaped MC curve as the general case. The U-shape arises from diminishing marginal returns to variable inputs: as a firm uses more of a variable input (e.g., labor) against a fixed input (e.g., capital), eventually each additional unit of labor adds less output, so more labor is needed per unit of output, raising MC. OpenStax Section 7.3 states this explicitly: "As the firm increases the quantity produced, it will eventually experience diminishing returns."

The linear/constant-MC model used in the Profit Fish calculator is a valid and widely-used simplification for:

• Monopoly pricing problems (C&T Chapter 13 uses it throughout)
• "Recipe" or unit-cost problems where each unit requires the same ingredients
• Break-even and shutdown analysis at the intro level

Verdict: ✅ Definition MC = ΔTC/ΔQ matches all sources. Proof that MC = AVC when TVC is linear is mathematically airtight. The app's use of constant MC is a valid simplification consistent with C&T's own practice.

7. P(Q) = a − bQ (linear inverse demand)

• Cowen & Tabarrok: Chapter 13 (Monopoly) and Chapter 5 (Elasticity). The MRU video "Maximizing Profit Under Monopoly" (Alex Tabarrok) states: "If we were to write the demand curve in inverse form, as P is equal to A minus B times Q, then the marginal revenue curve is equal to A minus 2B times Q." This is C&T's exact notation for linear inverse demand. The Studocu Chapter 13 solution set confirms this with worked problems: "P = 50 − Q", "P = 100 − 2Q", "P = 120 − 12Q" — all in the a − bQ form.
• OpenStax: Section 9.2. OpenStax's Table 9.5 shows a linear demand schedule (P = 1,300 − 100Q implicitly, from the data). The tip box states: "For a straight-line demand curve, MR and demand have the same vertical intercept. As output increases, marginal revenue decreases twice as fast as demand." OpenStax does not write out the P = a − bQ equation explicitly in 2e but the concept is identical.
• Third source (Khan Academy calculus proof): The Khan Academy optional proof writes: "P = mQ + b" (their slope-intercept form, where m < 0), equivalent to P = a − bQ with a = b (intercept) and b = −m (slope magnitude). This is the same formula under different variable names.
• Notation note: C&T use A (capital) and B for the intercept and slope coefficient; the app uses lowercase a and b. This is a notational convention only — mathematically identical.
• Verdict: ✅ Matches all sources. Flag: notation varies (A/B vs. a/b) but the functional form is universal in intro micro.

8. TR = P · Q

• Cowen & Tabarrok: Chapter 11 and Chapter 13. From Studocu Chapter 11 C&T notes: "Profit = TR − TC" with TR implicitly = P × Q. The MRU "Calculating Monopoly Profit" video states: "to find a monopolist's profit, we need to find how much money this firm is spending and subtract it from how much money this firm is making... that's just another way of saying: Find total cost and subtract it from total revenue." Total revenue = P × Q is universal and never defined otherwise.
• OpenStax: Section 7.1. Quote: "Total revenue is the income the firm generates from selling its products. We calculate it by multiplying the price of the product times the quantity of output sold." Also Section 9.2: "this is 5 × $800 = $4000" (TR = P × Q numeric confirmation).
• Third source: OpenStax Key Terms Ch.7: "revenue: income from selling a firm's product; defined as price times quantity sold."
• Verdict: ✅ Matches universally. No possible discrepancy.

9. MR = a − 2bQ (linear demand, same intercept, twice the slope)

Derivation:

TR = P · Q = (a − bQ) · Q = aQ − bQ²

MR = dTR/dQ = a − 2bQ

• Cowen & Tabarrok: Chapter 13 (Monopoly). The MRU video "Maximizing Profit Under Monopoly" (Tabarrok, https://mru.org/courses/principles-economics-microeconomics/monopoly-profit-maximization-price-aids-medication): "Take a linear demand curve, then the marginal revenue curve begins at the same point on the vertical axis as the demand curve, and it has twice the slope." And directly: "If we were to write the demand curve in inverse form, as P = A − B × Q, then the marginal revenue curve is equal to A − 2B × Q." This is the exact formula. The MRU "Office Hours: Calculating Monopoly Profit" video also states: "The shortcut to finding the marginal revenue curve is to simply double the slope of our demand curve." The Studocu Chapter 13 solutions show this applied: "P = 50 − Q → MR = 50 − 2Q", "P = 100 − 2Q → MR = 100 − 4Q", "P = 120 − 12Q → MR = 120 − 24Q".

• OpenStax: Section 9.2. Exact quote from Tip box: "For a straight-line demand curve, MR and demand have the same vertical intercept. As output increases, marginal revenue decreases twice as fast as demand, so that the horizontal intercept of MR is halfway to the horizontal intercept of demand." The numerical tables confirm this: with demand P = 1,300 − 100Q (implicit from data), MR steps show exactly twice the decline rate. Note: OpenStax 2e does not write the algebraic formula a − 2bQ in the main text, but the "twice as fast" rule is stated explicitly.

• Third source (Khan Academy): Optional calculus proof — Sal Khan writes P = mQ + b, TR = mQ² + bQ, dTR/dQ = 2mQ + b. Quote: "it has the same y-intercept as our demand curve... but it has twice the slope." The MRU "Office Hours" video carries out the exact algebra: given P = 60 − 2Q, MR = 60 − 4Q, set MR = MC = 20, solve Q = 20, P = $60 − 2(20) = $20 (or $60 checked against demand).

• Verdict: ✅ Matches exactly across C&T, OpenStax, and Khan Academy. The derivation is standard and the formula a − 2bQ is universally correct for linear inverse demand.

10. AR = TR/Q = P (average revenue equals price for a single-price firm)

• Cowen & Tabarrok: Chapter 13. From Studocu Chapter 13 C&T content: the demand curve for a monopolist is described as the AR curve. MRU video transcript confirms: "The demand curve tells us the maximum willingness to pay" — for a non-discriminating firm, P = AR at every quantity.
• OpenStax: Section 8.2. Quote: "marginal revenue (MR) is constant and equal to price (P)" for competitive firms; for monopoly, the demand curve = AR curve is standard. AR = TR/Q = (P·Q)/Q = P is algebraically immediate. The IB Econ video cited earlier explicitly writes: "Average revenue (AR) = TR / quantity of output = P × Q / Q = P (price)."
• Third source: Khan Academy Microeconomics. The relationship AR = P = D is a universal result in intro micro for any single-price firm (price taker or price maker that charges one price).
• Verdict: ✅ Matches exactly. AR = P is universally standard.

11. Price Elasticity of Demand: E_d = (% ΔQ) / (% ΔP), equivalently E_d = (dQ/dP) · (P/Q) = −(1/b) · (P/Q)

• Cowen & Tabarrok: Chapter 5 (Elasticity). The MRU syllabus maps Chapter 5 to Week 3: "Calculating the Elasticity of Demand." From the StudyLib 5th edition overview: "Here is the formula: Elasticity of demand = E_d = % Δ Q Demanded / % Δ Price". C&T present the percentage-change formula as the primary intro-level definition. The point-elasticity form (dQ/dP) · (P/Q) is equivalent but not always written out explicitly at intro level; it appears in the MRU elasticity and monopoly markup videos.

• OpenStax: Section 5.1. Direct quote: "The price elasticity of demand is the percentage change in the quantity demanded of a good or service divided by the percentage change in the price." Formula shown: Price Elasticity = % change in quantity / % change in price. OpenStax 2e uses the midpoint method as its primary calculation approach for intro students; it does not present the derivative form (dQ/dP)·(P/Q) in the main text. It does explicitly warn: "It is a common mistake to confuse the slope of either the supply or demand curve with its elasticity."

• Third source: Khan Academy elasticity article: "price elasticity of demand is the percentage change in the quantity demanded of a good or service divided by the percentage change in the price." The Khan Academy calculus proof video and the monopoly/elasticity connection video both use the point-elasticity form E = (dQ/dP) · (P/Q) in that context.

• Equivalence derivation for the linear demand case Q = (a − P)/b:

dQ/dP = −1/b (slope of direct demand)

Therefore: E_d = (dQ/dP) · (P/Q) = (−1/b) · (P/Q)

|E_d| = (1/b) · (P/Q)

This is exactly what the app states. The sign is negative (price ↑ → quantity ↓), but by intro-micro convention, elasticity is often reported as the absolute value.

• ⚠️ Partial flag: The point-elasticity formula E = (dQ/dP)·(P/Q) is mathematically equivalent to the percentage-change formula but requires calculus. OpenStax 2e and Mankiw intro texts do not show the derivative form explicitly in the main text — they use midpoint-method percentage changes. C&T's main text presents the percentage-change form, though their companion videos use the derivative form in the monopoly markup context. The app's presentation is correct but is slightly more advanced than what purely intro students see first. Recommendation: add a note that the percentage-change definition is the intro-level version, and the formula E = (P/Q)·(1/b) is the point-elasticity form that follows immediately from it when demand is linear.
• Verdict: ✅ Formula is correct. ⚠️ Note that intro textbooks lead with % ΔQ / % ΔP; the point-elasticity derivative form is equivalent but one level of abstraction up.

12. Accounting Profit = TR − TC (explicit costs only)

• Cowen & Tabarrok: Chapter 11. From Studocu Chapter 11 C&T notes: "Profit = TR − TC. This is Economic Profit." C&T treat the default "profit" as economic profit in their framework (opportunity costs included) and explicitly distinguish accounting profit. C&T note that accounting profit = TR minus only explicit costs. The distinction appears in their discussion of implicit costs and opportunity costs in Chapter 11.
• OpenStax: Section 7.1. Direct quote: "Accounting profit is a cash concept. It means total revenue minus explicit costs—the difference between dollars brought in and dollars paid out." Includes depreciation in explicit costs. From OpenStax Key Terms: "accounting profit: total revenues minus explicit costs, including depreciation."
• Third source: OpenStax Section 7.1 worked example: Revenue $200,000 − Explicit costs ($50,000 rent + $35,000 clerk) = $115,000 accounting profit. This exactly matches TR − TC(explicit).
• Verdict: ✅ Matches. Accounting profit = TR − explicit costs is universally standard.

13. Economic Profit = TR − TC − Opportunity Cost (= TR − explicit costs − implicit costs)

• Cowen & Tabarrok: Chapter 11. From Studocu Chapter 11 C&T notes: "TC = explicit costs + implicit costs; Economic Profit = TR − TC (where TC includes opportunity costs)." C&T use Sandy's bakery example: "Under scenario iii, Sandy would be wise to close up shop and take the job. She is making a paper profit but an economic loss, i.e., it becomes a loss once opportunity costs are correctly taken into account."
• OpenStax: Section 7.1. Direct quote: "Economic profit is total revenue minus total cost, including both explicit and implicit costs." Worked example: Revenue $200,000 − Explicit costs $85,000 − Implicit cost (forgone salary) $125,000 = −$10,000 economic profit. OpenStax Key Terms: "economic profit: total revenues minus total costs (explicit plus implicit costs)."
• Third source: The formula as stated in the app — Economic Profit = TR − TC − Opportunity Cost — is a correct alternative way to write the same thing, where "TC" refers to explicit (accounting) costs and "Opportunity Cost" refers to implicit costs. This decomposition is standard.
• Verdict: ✅ Matches. The app's formulation is fully consistent.

14. Profit-Maximizing Rule: MR = MC

• Cowen & Tabarrok: Chapter 11 (competition) and Chapter 13 (monopoly). From Studocu Chapter 13 C&T content: "The rule of rational life: MR = MC." The text states: "Both [competitive firms and monopolies] follow this rule. For competitive firms, MR = MC. Of course, for competitive firms, price equals marginal revenue, so this fact is somewhat hidden: We usually emphasize P = MC under competition. With monopoly firms, this fact is much more blatant: MR = MC is stated many times in this chapter." MRU video: "So how does a monopolist maximize profit? By producing at the level of output where marginal revenue is equal to marginal cost."
• OpenStax: Section 8.2. Direct quote: "The firm's profit-maximizing choice of output will occur where MR = MC (or at a choice close to that point)." Section 9.2: "a profit-maximizing monopoly should follow the rule of producing up to the quantity where marginal revenue is equal to marginal cost—that is, MR = MC."
• Third source: Khan Academy microeconomics; MRU "Maximizing Profit Under Monopoly" — both state MR = MC as the universal profit-maximization condition.
• Verdict: ✅ Matches universally and without any discrepancy.

15. Monopolist's closed-form: Q = (a − AVC) / (2b) and P = (a + AVC) / 2

Derivation:

Given P = a − bQ and MC = AVC = constant.

MR = a − 2bQ

Set MR = MC: a − 2bQ = AVC

→ 2bQ = a − AVC

→ **Q* = (a − AVC) / (2b)**

Then P = a − b·Q = a − b·(a − AVC)/(2b) = a − (a − AVC)/2 = (2a − a + AVC)/2 = (a + AVC)/2

• Cowen & Tabarrok: Chapter 13. C&T do not present this closed-form formula as a named equation in the textbook, but it is derived in their worked examples. From the Studocu Chapter 13 challenge problems: "Case A: P = 50 − Q, MC = 10 → MR = 50 − 2Q → set 50 − 2Q = 10 → Q = 20 → P = 50 − 20 = $30." Verification: (50 + 10)/2 = 30 ✓ and (50 − 10)/(2·1) = 20 ✓. The MRU "Office Hours: Calculating Monopoly Profit" video carries out the exact same algebraic steps, stating: "Set that [MR] equal to the marginal revenue and solve... you'll arrive at a profit-maximizing quantity of 20. Step three... plug Q into the demand curve and solve for the maximum price."
• OpenStax: Section 9.2. OpenStax uses tables rather than closed-form algebra to find Q. The numerical tables show MR = MC at Q = 5, P = $800 for a specific example. The Tip box confirms the geometry: "MR and demand have the same vertical intercept. As output increases, marginal revenue decreases twice as fast as demand." This is sufficient to derive Q and P* algebraically, even if OpenStax does not write the closed form.
• Third source: Khan Academy's calculus proof video demonstrates the algebra directly. The general formula follows immediately from MR = a − 2bQ and MC = constant.
• Notation note: C&T use MC (not AVC) in their monopoly setting. The equivalence MC = AVC under the linear/constant cost model is established in Formula 6. When they write "MC = 10," they mean the variable cost per unit is $10, which equals AVC = $10 (and TVC = 10Q). The app's use of AVC in place of MC is valid and correct under the linear model.
• Verdict: ✅ Derivation is correct. C&T confirm numerically (not algebraically in the general formula, but the worked examples match exactly). OpenStax confirms the underlying geometry. The closed form Q = (a − AVC)/(2b) and P = (a + AVC)/2 is the standard result.

16. Break-Even: TR = TC (quadratic roots give the break-even range)

For linear demand P = a − bQ and linear cost TC = TFC + AVC·Q:

TR = TC

(a − bQ)·Q = TFC + AVC·Q

aQ − bQ² = TFC + AVC·Q

−bQ² + (a − AVC)·Q − TFC = 0

or equivalently: bQ² − (a − AVC)·Q + TFC = 0

This is a quadratic in Q with two roots (when the discriminant is positive), giving the lower and upper break-even quantities.

• Cowen & Tabarrok: Chapter 11. C&T define break-even in the competition chapter: a firm breaks even when P = ATC (i.e., TR/Q = TC/Q → TR = TC). The quadratic algebra for the linear-demand monopoly case is not spelled out as a named formula in C&T, but it follows directly from their treatment of TR, TC, and profit. The Studocu Chapter 13 worked problems demonstrate: "What entrepreneurs really care about—total profit: Case A: TR = $30 × 20 = $600, TC = 100 + 10×20 = $300, Profit = $300." Break-even is where this equals zero.
• OpenStax: Section 8.2. Quote: "At this price and output level, where the marginal cost curve is crossing the average cost curve, the price the firm receives is exactly equal to its average cost of production. We call this the break even point." Equivalently P = ATC → TR = TC. OpenStax does not present the quadratic version (which is specific to the monopoly/linear demand case), but the TR = TC definition is confirmed.
• Third source: The Parkin lecture slides note: "If price falls exactly on the break even point where the MC and AC curves cross, the firm earns zero profits." Zero profit ↔ TR = TC.
• Caveat: The quadratic TR = TC formulation is the calculator-specific application of the break-even concept to the linear demand + linear cost model. It is mathematically correct and flows naturally from the textbook definition, but intro textbooks do not present it as a named formula because they typically work with given prices (not demand curves) in the competition chapter. For the monopoly/calculator context, the derivation is entirely correct.
• Verdict: ✅ TR = TC definition matches all sources. The quadratic formulation is a correct mathematical consequence of the linear demand + linear cost model.

17. Shutdown Condition: P < AVC (equivalently TR < TVC)

• Cowen & Tabarrok: Chapter 11. From Studocu Chapter 11 C&T notes: "If price falls below AC, firms shut down and exit an industry." More precisely for the short-run shutdown decision, C&T distinguish the short-run condition (shut down if P < AVC) from the long-run exit condition (exit if P < ATC). The MRU "Entry, Exit, and Supply Curves" videos confirm this treatment.
• OpenStax: Section 8.2. Multiple direct quotes:
• "if price < minimum average variable cost, then firm shuts down"
• "If P < AVC, then the firm stops producing and only incurs its fixed costs."
• "The intersection of the average variable cost curve and the marginal cost curve, which shows the price below which the firm would lack enough revenue to cover its variable costs, is called the shutdown point."

Equivalence P < AVC ↔ TR < TVC: since P < AVC ↔ P·Q < AVC·Q ↔ TR < TVC (because TVC = AVC·Q when AVC is the average — this holds exactly when AVC is constant, and approximately otherwise). OpenStax confirms: "If the farmer cannot pay workers (the variable costs), then it has to shut down."

• Third source: Lumen Learning "The Shutdown Point": "If price is below the minimum average variable cost, the firm must shut down." Also: "profit = total revenue – (fixed costs + variable cost)" with the explicit demonstration that when TR < TVC, losses from operating exceed fixed costs alone, so shutdown minimizes losses.
• Verdict: ✅ Matches all sources exactly. P < AVC (shut down) is the universal short-run shutdown condition.

How the model relates to real cost curves

Intro-micro textbooks, including C&T and OpenStax, present two distinct cost models:

The general (U-shaped) model: MC is upward-sloping because of diminishing marginal returns. As a firm adds variable inputs (labor) against fixed capital, each additional worker eventually adds less output than the previous one. More labor per unit → higher cost per unit → rising MC. This produces the familiar U-shaped AVC and ATC curves. OpenStax Section 7.2 states: "As we add workers, the marginal product increases at first, but sooner or later additional workers will have decreasing marginal product." This is the model used in most diagrams of perfect competition and monopoly in textbooks.

The linear (constant-MC) model: When each unit of output requires the same fixed "recipe" of variable inputs — e.g., each pizza requires exactly $3 of ingredients regardless of volume — AVC is constant and MC = AVC at all output levels. This is the model the Profit Fish calculator uses. It is not a simplification that C&T apologize for; they use it repeatedly:

• C&T Chapter 13 (Monopoly): All three Challenge problems use constant MC ("Marginal cost = 10"). The MRU AIDS medication video uses MC = $0.50/pill, flat. The MRU "Office Hours" video uses MC = 20, constant.
• OpenStax Ch. 9.2: The textbook's monopoly tables and the Tip box about "twice as fast" implicitly use constant or slowly rising MC. The constant-MC case is the standard for intro monopoly problems.
• C&T and OpenStax both teach break-even using TC = TFC + AVC·Q (constant-AVC form) in intro worked problems.

The linear model is not wrong — it is a deliberate, pedagogically common abstraction that captures the essential economics of pricing decisions without requiring students to locate the minimum of a U-shaped curve. The key caveat for students: in the real world (and in the U-shaped model), MC will eventually rise, and the profit-maximizing Q is determined by MR = MC at that rising point, not by a flat intersection.

Textbook passages that use the constant-MC case explicitly:

• C&T MRU video: "We're going to make [MC] flat at 50 cents per pill." (Maximizing Profit Under Monopoly)
• OpenStax Section 9.2: "the profit-maximizing quantity will occur where MR = MC" (with a constant-MC example in many courses that use OpenStax).
• Studocu Chapter 13 C&T solutions: All problems in the Challenges section use constant MC.

Where Cara can read more

Cowen & Tabarrok (Modern Principles of Microeconomics, 5th edition)

| Topic | Chapter | MRU Video Equivalent |

|-------|---------|---------------------|

| TC, TVC, TFC, AFC, AVC, ATC, MC | Ch. 11 — Costs and Profit Maximization Under Competition | https://mru.org/courses/principles-economics-microeconomics/maximizing-profit-competition |

| Profit = TR − TC, Shutdown, Break-even | Ch. 11 | https://mru.org/courses/principles-economics-microeconomics/entry-exit-supply-curves-increasing-costs |

| Elasticity of Demand (% formula) | Ch. 5 — Elasticity and Its Applications | https://mru.org/courses/principles-economics-microeconomics/elasticity-demand-definition |

| Monopoly MR = MC, Q, P | Ch. 13 — Monopoly | https://mru.org/courses/principles-economics-microeconomics/monopoly-profit-maximization-price-aids-medication |

| MR formula, twice slope of demand | Ch. 13 | https://mru.org/courses/principles-economics-microeconomics/monopoly-profit-maximization-price-aids-medication |

| Economic vs. accounting profit | Ch. 11 | https://mru.org/courses/principles-economics-microeconomics/introduction-competitive-firm |

Note: Specific page numbers in C&T 5th edition are not confirmed (no page-numbered PDF was accessible). All chapter-level citations are confirmed via the official MRU syllabus and Studocu solution sets keyed to C&T.

OpenStax Principles of Microeconomics 2e

| Topic | Section | URL |

|-------|---------|-----|

| TC, TVC, TFC, AFC, AVC, ATC, MC definitions | 7.3 — Costs in the Short Run | https://openstax.org/books/principles-microeconomics-2e/pages/7-3-costs-in-the-short-run |

| Accounting vs. Economic Profit | 7.1 — Explicit and Implicit Costs | https://openstax.org/books/principles-microeconomics-2e/pages/7-1-explicit-and-implicit-costs-and-accounting-and-economic-profit |

| MR = MC, Shutdown (P < AVC), Break-even | 8.2 — How Perfectly Competitive Firms Make Output Decisions | https://openstax.org/books/principles-microeconomics-2e/pages/8-2-how-perfectly-competitive-firms-make-output-decisions |

| Monopoly MR = MC, Q, P | 9.2 — How a Profit-Maximizing Monopoly Chooses Output and Price | https://openstax.org/books/principles-microeconomics-2e/pages/9-2-how-a-profit-maximizing-monopoly-chooses-output-and-price |

| Price Elasticity of Demand | 5.1 — Price Elasticity of Demand | https://openstax.org/books/principles-microeconomics-2e/pages/5-1-price-elasticity-of-demand-and-price-elasticity-of-supply |

Document prepared: verification against Cowen & Tabarrok Modern Principles of Microeconomics 5th ed. (chapter-level, MRU-confirmed), OpenStax Principles of Microeconomics 2e (section-level, direct URL), and Khan Academy Microeconomics (supplementary). No page numbers for C&T are confirmed; MRU video URLs serve as the accessible proxy for C&T content.