What is Cost Curves, LRAC & Exercises in Class 12 Economics NCERT?

Interactive NCERT Class 12 Economics lesson on Cost Curves, LRAC & Exercises

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Cost Curves, LRAC & Exercises

🎓 Class 12 Economics CBSE Theory Chapter 3 — Production and Costs ⏱ ~28 min

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Class 12 · Introductory Microeconomics · Chapter 3 · Part 3

Costs of Production — Short-Run, Long-Run Curves & Exercises

A factory must pay rent on its land even when no shirts are stitched. It must also pay for thread and electricity that rise as production rises. Out of this very ordinary fact comes the entire short-run cost theory of microeconomics: fixed and variable costs, their averages, the marginal cost, and the famous U-shape that shows up in every textbook in the world. This final part of NCERT Chapter 3 turns the production story of Parts 1–2 into a cost story, derives every cost curve, and solves all 30 NCERT exercises with full model answers.

3.8 From Production to Costs

To produce output, the firm must employ inputs. But a given level of output can typically be produced in many ways: in Table 3.1 of Part 1, an output of 50 units can be made by (L = 6, K = 3), (L = 4, K = 4) or (L = 3, K = 6). With input prices given, the firm picks the least expensive combination. So for every level of output, the firm chooses the least-cost input combination — the resulting relationship between output and minimum cost is the firm's cost function^?.

📖 Cost Function — NCERT

The cost function describes the least cost of producing each level of output, given the prices of the factors of production and the technology.

3.9 Short-Run Costs

In the short run, some factors are fixed and some are variable. The cost of employing the fixed inputs is the Total Fixed Cost (TFC)^?: it does not change with the level of output. Whatever the firm produces — even if it produces zero — TFC is the same. The cost of employing the variable inputs is the Total Variable Cost (TVC)^?: it increases as output increases. Adding the two gives Total Cost.

TC = TVC + TFC …(3.6)

To produce more output, the firm must hire more variable inputs, so TVC rises with q and therefore so does TC. TFC stays put.

A Numerical Example — NCERT Table 3.3

Below is NCERT's Table 3.3 — the short-run cost schedule of a typical firm with TFC = ₹20.

Table 3.3 — Total, Average and Marginal Costs (TFC = ₹20)
Output q	TFC (₹)	TVC (₹)	TC (₹)	AFC (₹)	AVC (₹)	SAC / ATC (₹)	SMC / MC (₹)
0	20	0	20	—	—	—	—
1	20	10	30	20.00	10.00	30.00	10
2	20	18	38	10.00	9.00	19.00	8
3	20	24	44	6.67	8.00	14.67	6
4	20	29	49	5.00	7.25	12.25	5
5	20	33	53	4.00	6.60	10.60	4
6	20	39	59	3.33	6.50	9.83	6
7	20	47	67	2.86	6.71	9.57	8
8	20	60	80	2.50	7.50	10.00	13
9	20	75	95	2.22	8.33	10.55	15
10	20	95	115	2.00	9.50	11.50	20

Per-Unit (Average) Costs and Marginal Cost — Definitions

The Short-run Average Cost (also called Average Total Cost, ATC) is the total cost per unit of output:

SAC = ATC = TC ÷ q …(3.7)

The Average Variable Cost is the total variable cost per unit of output:

AVC = TVC ÷ q …(3.8)

The Average Fixed Cost is the total fixed cost per unit:

AFC = TFC ÷ q …(3.9)

By construction:

SAC = AVC + AFC …(3.10)
Equivalently: ATC = AFC + AVC

The Short-run Marginal Cost is the change in total cost per unit change in output:

SMC = MC = ΔTC ÷ Δq …(3.11)
Where Δ ("delta") means "change in".

NCERT's worked check at q = 5: change in TC = TC(5) − TC(4) = 53 − 49 = 4. Change in q = 1. SMC = 4 ÷ 1 = 4. (Note: in the table version the values vary slightly; the principle stands.)

⭐ A Useful Identity in the Short Run

In the short run, fixed cost cannot be changed. So when output rises, the entire change in TC comes from the change in TVC. Therefore SMC = ΔTC/Δq = ΔTVC/Δq. The sum of all marginal costs up to any q gives the total variable cost at that q. The area under the SMC curve up to any output level equals the TVC at that level.

3.10 Shapes of the Short-Run Cost Curves

TFC, TVC and TC — Total Cost Curves

Place output on the X-axis and cost on the Y-axis. The TFC curve is a horizontal straight line at the constant fixed-cost value c₁ — TFC does not change with q. TVC starts at the origin (zero output → zero variable cost) and rises through the points dictated by Table 3.3. TC is the vertical sum of TFC and TVC, so the TC curve has the same shape as TVC, lifted upward by the constant TFC.

Figure 3.G — TFC, TVC and TC curves (NCERT Fig 3.3 style). TFC is a horizontal straight line at c₁. TVC starts at the origin and rises (initially flatter, then steeper as MC increases). TC = TFC + TVC is the vertical sum.

The AFC Curve — A Rectangular Hyperbola

AFC = TFC ÷ q. Since TFC is a constant, as q rises, AFC falls. As q approaches zero, AFC becomes very large; as q grows toward infinity, AFC approaches zero. Multiplying any q with its corresponding AFC always gives back the constant TFC. Geometrically, the AFC curve is a rectangular hyperbola^? — and the area under any rectangle drawn from a point on AFC down to the axes equals TFC.

The SMC Curve — U-Shaped from the Law of Variable Proportions

By the Law of Variable Proportions (Part 1), the marginal product of a factor first rises and then falls. With the factor's price given, when MP rises, the variable input requirement per extra unit of output shrinks — and so the additional cost falls. When MP falls, the requirement per extra unit grows — and so the additional cost rises. The SMC curve therefore first falls, reaches a minimum, and then rises: a classic U-shape.

The AVC Curve — Also U-Shaped

For the first unit of output, SMC and AVC coincide. As output rises, SMC falls; AVC, being the average of marginal costs, also falls but less steeply than SMC. After SMC's minimum, SMC begins to rise. Yet AVC keeps falling as long as SMC remains below it. Once SMC has risen high enough to exceed AVC, AVC starts rising too. The AVC curve is therefore U-shaped — and the SMC curve cuts the AVC curve from below at the minimum point of AVC.

The SAC (ATC) Curve — Also U-Shaped, Lying Above AVC

SAC = AVC + AFC. Initially both AVC and AFC fall, so SAC falls. After AVC has bottomed and begun rising, AFC continues to fall — and so long as AFC's fall outweighs AVC's rise, SAC keeps falling. Eventually AVC's rise overtakes AFC's fall, and SAC turns upward. SAC is therefore U-shaped; it lies above AVC by exactly the value of AFC; and its minimum occurs to the right of AVC's minimum.

🟣 The Two "MC Cuts at Minimum" Results

As long as SAC is falling, SMC is less than SAC. When SAC is rising, SMC is greater than SAC. So SMC cuts SAC from below at the minimum point of SAC.

Similarly, SMC cuts AVC from below at the minimum point of AVC.

These two intersection rules are direct consequences of the same averaging logic that gave us "MP cuts AP at AP-max" in Part 2.

Figure 3.H — Family of short-run cost curves (NCERT Fig 3.8 style). SMC cuts AVC at its minimum P (output q₁), and cuts SAC at its minimum S (output q₂). Note q₂ > q₁: SAC's minimum lies to the right of AVC's minimum because AFC continues to fall after AVC has begun rising.

Figure 3.I — Cost curves plotted from NCERT Table 3.3. AFC falls steadily (rectangular hyperbola). AVC, SAC (ATC) and SMC (MC) are U-shaped. SMC reaches its minimum first; AVC's minimum follows; SAC's minimum is the last and the highest output of the three turning points.

3.11 Long-Run Costs

In the long run, all inputs are variable — there are no fixed factors and therefore no fixed costs. Total cost and total variable cost coincide. NCERT defines:

LRAC = TC ÷ q …(3.13)
LRMC = (TC at q₁ units) − (TC at q₁ − 1 units) …(3.14)
Equivalently: LRMC = ΔTC ÷ Δq

Just as in the short run, the sum of all marginal costs up to any output level equals the total cost at that level (because there is no fixed component to add).

Shapes of the Long-Run Cost Curves — Driven by Returns to Scale

Returns to scale (Part 2) tells us how output reacts to a proportional scaling of inputs. Now we ask: how does average cost behave as output rises in the long run?

⬇️

IRS → LRAC falls

If we want to double output, inputs need to be increased less than double. So cost (= input price × inputs used) rises less than double — average cost falls.

➡️

CRS → LRAC flat

A proportional increase in inputs gives a proportional increase in output. Cost and output rise in the same proportion — average cost stays constant.

⬆️

DRS → LRAC rises

To double output, inputs need to rise more than double. Cost rises more than double — average cost rises with output.

It is empirically argued that a typical firm's lifecycle moves through IRS → CRS → DRS as it grows. So the LRAC curve typically falls (downward-sloping part = IRS), bottoms out (CRS region) and then rises (upward-sloping part = DRS). The shape is a U.

The LRMC Curve and Its Relation to LRAC

For the first unit of output, LRMC and LRAC are the same. Then as output rises, LRAC falls (in the IRS region), bottoms out, and rises (in the DRS region). As long as LRAC is falling, LRMC must be less than LRAC. When LRAC is rising, LRMC must be greater than LRAC. So LRMC is also U-shaped, and LRMC cuts LRAC from below at LRAC's minimum point.

Figure 3.J — Long-run marginal cost (LRMC) and long-run average cost (LRAC) curves. Both are U-shaped. LRMC cuts LRAC from below at the minimum point M of LRAC (output q₁). The downward-sloping segment of LRAC corresponds to IRS; the bottom corresponds to CRS; the upward-sloping segment corresponds to DRS.

🟪 LRAC as the Envelope of SAC Curves

Although NCERT does not state this explicitly, a useful intuition for the U-shape of LRAC is the envelope idea: at each output q, the LRAC equals the lowest point on the relevant short-run SAC curve (because in the long run the firm can choose the optimal plant size). As output rises, the firm "switches" to ever-larger plants. The LRAC curve, drawn through the lowest reachable cost for each q, traces out the bottom of all short-run SAC curves — and is therefore U-shaped.

LET'S EXPLORE — Build the Cost Schedule from Scratch

Take a simple TC schedule: q = 0,1,2,3,4,5,6 with TC = ₹15, 25, 33, 39, 47, 60, 80.
Identify TFC by reading TC at q = 0.
Compute TVC for each q as TC − TFC.
Compute AFC, AVC, SAC and SMC for each positive q.
Identify the q at which AVC reaches its minimum and the q at which SAC reaches its minimum. Verify SAC-min lies to the right of AVC-min.

Sample finding. TFC = ₹15 (TC at q=0). TVC: 0, 10, 18, 24, 32, 45, 65. AFC: 15, 7.5, 5, 3.75, 3, 2.5. AVC: 10, 9, 8, 8, 9, 10.83. SAC: 25, 16.5, 13, 11.75, 12, 13.33. SMC: 10, 8, 6, 8, 13, 20. AVC bottoms between q = 3 and q = 4 (value 8). SAC bottoms at q = 4 (value 11.75). SAC-min at q = 4 indeed lies to the right of AVC-min at q ≈ 3.5 — the AFC fall (3.75 → 3 → 2.5) keeps pulling SAC down even after AVC has begun to rise.

THINK ABOUT IT — The Roadside Tea Stall

A tea stall pays ₹50/day rent for the cart and ₹100/day for licensing — TFC = ₹150.
Each cup of tea costs ₹3 in milk, sugar and tea leaves; cooking gas adds ₹0.50; cup costs ₹0.50 — TVC per cup = ₹4.
Suppose the stall sells 50, 100, 150 or 200 cups in a day. Compute TVC, TC, AFC, AVC, SAC for each.
What happens to AFC as output rises? What happens to AVC?
Identify the lowest-SAC output level. Why is SAC monotonically falling here?

Sample finding. TFC = 150. TVC = 4q. So at q = 50: TVC = 200, TC = 350, AFC = 3.00, AVC = 4.00, SAC = 7.00. At q = 100: TVC = 400, TC = 550, AFC = 1.50, AVC = 4.00, SAC = 5.50. At q = 150: TVC = 600, TC = 750, AFC = 1.00, AVC = 4.00, SAC = 5.00. At q = 200: TVC = 800, TC = 950, AFC = 0.75, AVC = 4.00, SAC = 4.75. AFC falls steadily; AVC stays constant (linear TVC). SAC keeps falling — because AFC keeps falling and AVC is flat. In real life AVC will eventually rise (overtime wages, slower service per cup at extreme volume), turning SAC U-shaped. The flat-AVC simplification of this stall is why a small business is so eager to spread fixed costs across more units.

📝 Competency-Based Questions — Apply, Analyse, Evaluate, Create

Scenario. A bakery has TFC = ₹40 and the following TVC schedule:
q = 0,1,2,3,4,5,6,7. TVC = ₹0, 10, 18, 28, 42, 60, 84, 120.

Q1. Compute the TC, AFC, AVC, SAC and SMC schedules. Identify the q at which AVC and SAC reach their respective minima.

L3 Apply

Answer. TC = TFC + TVC: 40, 50, 58, 68, 82, 100, 124, 160. AFC = 40/q: —, 40, 20, 13.33, 10, 8, 6.67, 5.71. AVC = TVC/q: —, 10, 9, 9.33, 10.5, 12, 14, 17.14. SAC = TC/q: —, 50, 29, 22.67, 20.5, 20, 20.67, 22.86. SMC = ΔTC/Δq: —, 10, 8, 10, 14, 18, 24, 36. AVC minimum at q = 2 (₹9). SAC minimum at q = 5 (₹20). SAC's minimum is to the right of AVC's, as predicted by theory.

Q2. Verify two properties of the cost curves from this schedule: (i) at q = 5, SMC is approximately equal to SAC; (ii) at q = 2, SMC is approximately equal to AVC.

L4 Analyse

Answer. At q = 5, SMC = ₹18 and SAC = ₹20. Close but not exactly equal (the table is in discrete steps). The principle holds: SAC reaches its minimum at q = 5 and SMC is on the way up through the SAC curve at this output. At q = 2, SMC = ₹8 and AVC = ₹9. Again close: AVC's minimum is at q = 2 and SMC has crossed AVC near this output. Both properties — SMC cutting AVC at AVC-min and SMC cutting SAC at SAC-min — are visible in the data, with small discrete-table approximation errors.

Q3. The bakery's owner says, "I will keep increasing output indefinitely because my AFC keeps falling." Diagnose the error in this reasoning.

L5 Evaluate

Answer. AFC indeed falls continuously, but the relevant per-unit cost is SAC = AFC + AVC. Beyond a certain q, AVC starts rising fast enough to more than offset the AFC fall. From q = 5 onwards SAC rises (20 → 20.67 → 22.86) — even though AFC is still falling. The owner is also ignoring the demand side: the firm's profit depends on price minus SAC × q, not on AFC alone. Profit-maximisation requires looking at SMC and price together (Chapter 4), not at AFC alone.

Q4. Suppose rent rises so TFC jumps from ₹40 to ₹60. Without recomputing the entire table, predict precisely how each of the following curves shifts: AFC, AVC, SAC, SMC.

L4 Analyse

Answer. A rise in TFC affects only fixed-cost components. AFC = TFC/q rises at every q (the entire AFC curve shifts upward). SAC = AFC + AVC rises by exactly the same amount AFC has risen (the entire SAC curve shifts upward by the increase in AFC). AVC is unchanged because it depends only on variable costs. SMC = ΔTC/Δq is also unchanged because the change in TC across two adjacent output levels is the change in TVC alone — fixed cost cancels out in the difference. So a rise in TFC shifts only AFC and SAC up; AVC and SMC are not affected.

HOT Q5. Indian Railways has very high fixed costs (track, signalling, stations) and very low marginal cost per additional passenger. Use the cost-curve framework to argue why pricing rail seats at marginal cost would be socially desirable but financially ruinous, and propose two pricing rules that resolve the trade-off.

L6 Create

Answer. Pricing at SMC equates the price an extra passenger pays with the extra resource used to carry her — socially efficient (no passenger willing to pay above the marginal social cost is excluded). However, with very high TFC, SMC lies far below SAC for most relevant output ranges, so price = SMC × q does not cover TFC + TVC. The Railways would run perpetual losses. Two resolution rules: (i) Two-part tariff: charge a fixed access fee (covers TFC) plus a per-trip fee equal to SMC; this combines efficiency with cost recovery. (ii) Ramsey pricing: charge a mark-up over SMC that is inversely proportional to demand elasticity — price-insensitive segments (business class, peak hours) pay a bigger mark-up, price-sensitive segments (sleeper class, off-peak) closer to SMC. Both rules close the gap between SMC and SAC without sacrificing all efficiency.

🎯 Assertion–Reason Questions

Assertion (A): The Average Fixed Cost (AFC) curve is a rectangular hyperbola.
Reason (R): AFC equals TFC divided by output, and TFC is constant — so AFC × q always equals the same constant TFC.

Options: (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true, R is false. (d) A is false, R is true.

Correct answer: (a) — A is true (NCERT explicitly states AFC is a rectangular hyperbola). R is the algebraic reason: a curve y = c/x has the property y · x = c (a constant) — that is the defining property of a rectangular hyperbola. R is the correct explanation of A.

Assertion (A): The Short-run Marginal Cost curve cuts the AVC curve from below at the minimum point of AVC.
Reason (R): When SMC is less than AVC, AVC is falling; when SMC is greater than AVC, AVC is rising — so SMC must equal AVC at AVC's minimum.

Correct answer: (a) — A is the standard NCERT result. R is the precise mechanism: as long as SMC < AVC, AVC keeps falling (the marginal pulls the average down); once SMC > AVC, AVC rises. At AVC-min, AVC transitions from falling to rising, which can happen only if SMC = AVC at that point. R is the correct explanation of A.

Assertion (A): In the long run, there are no fixed costs.
Reason (R): The U-shape of the LRAC curve arises because of increasing returns to scale (IRS) at low output, constant returns at the bottom, and decreasing returns to scale (DRS) at high output.

Correct answer: (b) — Both statements are individually true. A is correct because in the long run all factors are variable. R is the correct explanation of why LRAC is U-shaped, but it is not the explanation of why there are no fixed costs in the long run. The two statements are unrelated logically: A is about the absence of fixity; R is about the curvature of LRAC.

3.12 NCERT Exercises — All 30 Questions with Model Answers

Click "Show Answer" beside each question to reveal a complete CBSE-style model answer. Q1–21 are theory questions; Q22–30 are numerical problems.

EXERCISE 1

Q1. Explain the concept of a production function.

A production function is a relationship between inputs used and the maximum quantity of output that can be produced from those inputs, given a specific technology. Symbolically, with two inputs labour L and capital K, it is written q = f(L, K). It tells us the largest output obtainable for each combination of inputs and is therefore concerned only with the efficient use of inputs. A change in technology produces a new production function: the same inputs yield a higher maximum output.

EXERCISE 2

Q2. What is the total product of an input?

When a single input is varied while all other inputs are held constant, the resulting relationship between that variable input and the output is called the Total Product (TP) of that input. It is also called the total physical product or total return to that variable input. For example, if capital is fixed at K = 4, then the output values for L = 0, 1, 2, 3, … (i.e. 0, 10, 24, 40, …) form the TP-of-labour schedule.

EXERCISE 3

Q3. What is the average product of an input?

Average Product (AP) of an input is the output per unit of that variable input. With labour as the variable input and capital fixed: AP_L = TP_L ÷ L. It is computed by dividing the total product at any L by the level of labour at that point. AP is also called the average physical product or average return.

EXERCISE 4

Q4. What is the marginal product of an input?

Marginal Product (MP) of an input is the change in total output per unit change in that input, with all other inputs held constant. With labour as the variable input: MP_L = ΔTP_L ÷ ΔL = (TP at L units) − (TP at L − 1 units). MP is undefined at zero level of input. It is also called the marginal physical product or marginal return.

EXERCISE 5

Q5. Explain the relationship between the marginal products and the total product of an input.

For any level of input employment, the sum of marginal products of all preceding units of that input gives the total product at that level. Symbolically: TP at L units = MP₁ + MP₂ + MP₃ + … + MP_L. Geometrically, this means the area under the MP curve up to L equals the height of the TP curve at L. Verifying with NCERT Table 3.2 at L = 4: MP₁ + MP₂ + MP₃ + MP₄ = 10 + 14 + 16 + 10 = 50, which equals TP at L = 4.

EXERCISE 6

Q6. Explain the concepts of the short run and the long run.

In the short run, at least one factor of production cannot be varied — it is the fixed factor. The firm changes output only by changing the variable factor. In the long run, all factors of production can be varied; there is no fixed factor. The long run is generally a longer time period than the short run, but the cut-off in calendar time differs across industries — NCERT explicitly says the short and long runs are not defined in terms of days, months or years; they are defined by whether all inputs can be varied or not.

EXERCISE 7

Q7. What is the law of diminishing marginal product?

The Law of Diminishing Marginal Product states that the marginal product of a variable input rises initially with the level of employment of that input, but after a certain level of employment it starts falling. The reason is that the factor proportions become "more suitable" up to a point, after which the production process becomes too crowded with the variable input.

EXERCISE 8

Q8. What is the law of variable proportions?

The Law of Variable Proportions and the Law of Diminishing Marginal Product are the same law expressed in two different ways. It states that as we hold one factor fixed and increase the other, the marginal product of the variable input first rises (because factor proportions improve) and then falls (because the production process becomes crowded). The "variable proportions" name highlights that the ratio of the two factors changes as the variable input is changed.

EXERCISE 9

Q9. When does a production function satisfy constant returns to scale?

A production function satisfies Constant Returns to Scale (CRS) when a proportional increase in all inputs leads to an increase in output by the same proportion. Mathematically, with two inputs: f(t · x₁, t · x₂) = t · f(x₁, x₂) for any t > 1. Doubling both labour and capital exactly doubles the output.

EXERCISE 10

Q10. When does a production function satisfy increasing returns to scale?

A production function satisfies Increasing Returns to Scale (IRS) when a proportional increase in all inputs leads to an increase in output by a larger proportion. Mathematically, f(t · x₁, t · x₂) > t · f(x₁, x₂) for t > 1. Doubling both inputs more than doubles output. Reasons: specialisation, division of labour, and indivisibilities of certain capital equipment.

EXERCISE 11

Q11. When does a production function satisfy decreasing returns to scale?

A production function satisfies Decreasing Returns to Scale (DRS) when a proportional increase in all inputs leads to an increase in output by a smaller proportion. Mathematically, f(t · x₁, t · x₂) < t · f(x₁, x₂) for t > 1. Doubling inputs less than doubles output. Reasons: managerial diseconomies, coordination problems, and non-replicable resources at very large scale.

EXERCISE 12

Q12. Briefly explain the concept of the cost function.

The cost function describes the least cost of producing each level of output, given the prices of the factors of production and the available technology. Since a particular output can typically be produced by many input combinations, the firm chooses the combination that costs the least. The mapping from output to that minimum cost is the cost function.

EXERCISE 13

Q13. What are the total fixed cost, total variable cost and total cost of a firm? How are they related?

Total Fixed Cost (TFC) is the cost of employing the fixed inputs in the short run; it does not change with output. Total Variable Cost (TVC) is the cost of employing the variable inputs; it rises as output rises. Total Cost (TC) is the sum: TC = TFC + TVC. At zero output, TVC is zero and TC equals TFC. As output rises, TFC stays constant while TVC and TC both rise.

EXERCISE 14

Q14. What are the average fixed cost, average variable cost and average cost of a firm? How are they related?

Average Fixed Cost (AFC) = TFC ÷ q. Average Variable Cost (AVC) = TVC ÷ q. Short-run Average Cost (SAC, or ATC) = TC ÷ q. They are related by the identity SAC = AFC + AVC. AFC falls continuously as q rises (rectangular hyperbola); AVC and SAC are both U-shaped, with SAC lying above AVC by exactly the value of AFC at every q.

EXERCISE 15

Q15. Can there be some fixed cost in the long run? If not, why?

No, there cannot be any fixed cost in the long run. By definition, in the long run all factors of production are variable — the firm can adjust every input. There are no fixed factors, so no fixed cost. As a consequence, in the long run total cost (TC) equals total variable cost (TVC); the long-run average cost equals the long-run total cost divided by output.

EXERCISE 16

Q16. What does the average fixed cost curve look like? Why does it look so?

The AFC curve is downward sloping throughout — specifically, it is a rectangular hyperbola. AFC = TFC ÷ q. Since TFC is a positive constant and q is positive, as q rises AFC falls; as q approaches zero AFC becomes very large; as q approaches infinity AFC approaches zero. Multiplying any q with its corresponding AFC always gives the constant TFC. This product-equal-to-constant property is precisely the definition of a rectangular hyperbola.

EXERCISE 17

Q17. What do the short run marginal cost, average variable cost and short run average cost curves look like?

All three curves are U-shaped. The SMC curve falls initially, reaches its minimum and then rises. The AVC curve also falls, reaches a minimum (at a higher output than SMC's minimum) and then rises. The SAC curve too is U-shaped; it lies above the AVC curve by the value of AFC at every q, and its minimum lies to the right of AVC's minimum (because AFC continues to fall after AVC has begun rising). SMC cuts AVC at AVC-min and cuts SAC at SAC-min, both from below.

EXERCISE 18

Q18. Why does the SMC curve cut the AVC curve at the minimum point of the AVC curve?

As long as AVC is falling, SMC must be less than AVC — only a marginal value below the average can pull the average down. When AVC is rising, SMC must be greater than AVC. Therefore at the exact output where AVC transitions from falling to rising — that is, at AVC's minimum — SMC must equal AVC. Geometrically, SMC crosses AVC from below at AVC-min.

EXERCISE 19

Q19. At which point does the SMC curve cut the SAC curve? Give reason in support of your answer.

SMC cuts SAC at the minimum point of SAC, from below. Reason: as long as SAC is falling, SMC is less than SAC; when SAC is rising, SMC is greater than SAC. So at the exact output where SAC bottoms out (transitions from falling to rising), SMC must equal SAC. Note that SAC's minimum lies to the right of AVC's minimum because AFC keeps falling after AVC has begun to rise.

EXERCISE 20

Q20. Why is the short run marginal cost curve 'U'-shaped?

By the Law of Variable Proportions, the marginal product of the variable input first rises and then falls as the input is increased. With the input price given, when MP rises the input requirement per extra unit of output falls — so SMC falls. When MP falls the input requirement per extra unit of output rises — so SMC rises. The SMC curve therefore has a falling segment followed by a rising segment, i.e. it is U-shaped. SMC and MP are inversely related at every output level.

EXERCISE 21

Q21. What do the long run marginal cost and the average cost curves look like?

Both LRAC and LRMC are U-shaped. LRAC falls in the IRS region (because to raise output by a certain proportion, inputs need to rise less than that — so cost rises less), is at its minimum where CRS prevails, and rises in the DRS region (where inputs need to rise more than the increase in output). LRMC also falls then rises. LRMC cuts LRAC from below at LRAC's minimum point. As long as LRAC is falling, LRMC < LRAC; when LRAC is rising, LRMC > LRAC.

EXERCISE 22 — NUMERICAL

Q22. The following table gives the total product schedule of labour. Find the corresponding average product and marginal product schedules of labour.
L = 0,1,2,3,4,5; TP_L = 0, 15, 35, 50, 40, 48.

AP_L = TP_L ÷ L. MP_L = TP_L − TP_L−1.
L=0: TP=0, AP=—, MP=—.
L=1: TP=15, AP=15.00, MP = 15 − 0 = 15.
L=2: TP=35, AP=17.50, MP = 35 − 15 = 20.
L=3: TP=50, AP=16.67, MP = 50 − 35 = 15.
L=4: TP=40, AP=10.00, MP = 40 − 50 = −10.
L=5: TP=48, AP=9.60, MP = 48 − 40 = 8.
Note that MP becomes negative at L = 4 (Stage III), then turns positive again at L = 5 — this is a non-monotonic example given in NCERT to test computation skills.

EXERCISE 23 — NUMERICAL

Q23. The following table gives the average product schedule of labour. Find the total product and marginal product schedules. TP₀ = 0.
L = 1,2,3,4,5,6; AP_L = 2, 3, 4, 4.25, 4, 3.5.

TP_L = AP_L × L. MP_L = TP_L − TP_L−1.
L=1: TP = 2 × 1 = 2; MP = 2 − 0 = 2.
L=2: TP = 3 × 2 = 6; MP = 6 − 2 = 4.
L=3: TP = 4 × 3 = 12; MP = 12 − 6 = 6.
L=4: TP = 4.25 × 4 = 17; MP = 17 − 12 = 5.
L=5: TP = 4 × 5 = 20; MP = 20 − 17 = 3.
L=6: TP = 3.5 × 6 = 21; MP = 21 − 20 = 1.
Note: AP peaks at L = 4 (4.25); MP peaks at L = 3 (6); and MP > AP up to L = 3 while MP < AP from L = 4 onwards — exactly as theory predicts.

EXERCISE 24 — NUMERICAL

Q24. The following table gives the marginal product schedule of labour. TP₀ = 0. Calculate the total and average product schedules of labour.
L = 1,2,3,4,5,6; MP_L = 3, 5, 7, 5, 3, 1.

TP_L = sum of MP up to L. AP_L = TP_L ÷ L.
L=1: TP = 3; AP = 3.00.
L=2: TP = 3 + 5 = 8; AP = 4.00.
L=3: TP = 8 + 7 = 15; AP = 5.00.
L=4: TP = 15 + 5 = 20; AP = 5.00.
L=5: TP = 20 + 3 = 23; AP = 4.60.
L=6: TP = 23 + 1 = 24; AP = 4.00.
AP peaks at L = 3 or L = 4 (both give 5.00) — and MP equals AP at this transition (MP = 5 at L = 4), confirming the rule "MP cuts AP at AP's max from above".

EXERCISE 25 — NUMERICAL

Q25. The following table shows the total cost schedule of a firm. What is the total fixed cost schedule of this firm? Calculate the TVC, AFC, AVC, SAC and SMC schedules of the firm.
Q = 0,1,2,3,4,5,6; TC = 10, 30, 45, 55, 70, 90, 120.

TFC = TC at Q = 0 = ₹10. So TFC schedule is constantly ₹10 at every Q.
TVC = TC − TFC: 0, 20, 35, 45, 60, 80, 110.
AFC = TFC/Q: —, 10.00, 5.00, 3.33, 2.50, 2.00, 1.67.
AVC = TVC/Q: —, 20.00, 17.50, 15.00, 15.00, 16.00, 18.33.
SAC = TC/Q: —, 30.00, 22.50, 18.33, 17.50, 18.00, 20.00.
SMC = ΔTC/ΔQ: —, 20, 15, 10, 15, 20, 30.
AVC reaches its minimum around Q = 3 to Q = 4 (15.00). SAC reaches its minimum at Q = 4 (17.50). SAC-min lies to the right of AVC-min — as theory predicts.

EXERCISE 26 — NUMERICAL

Q26. The following table gives the total cost schedule of a firm. AFC at Q = 4 is ₹5. Find the TVC, TFC, AVC, AFC, SAC and SMC schedules of the firm.
Q = 1,2,3,4,5,6; TC = 50, 65, 75, 95, 130, 185.

Step 1: AFC at Q = 4 is ₹5, so TFC = AFC × Q = 5 × 4 = ₹20. TFC is constant: ₹20 at every Q.
Step 2: TVC = TC − TFC: 30, 45, 55, 75, 110, 165.
AFC = 20/Q: 20.00, 10.00, 6.67, 5.00, 4.00, 3.33.
AVC = TVC/Q: 30.00, 22.50, 18.33, 18.75, 22.00, 27.50.
SAC = TC/Q: 50.00, 32.50, 25.00, 23.75, 26.00, 30.83.
SMC = ΔTC/ΔQ: 30 (Q=1, taking TC at Q=0 as TFC=20: 50−20=30), 15, 10, 20, 35, 55.
AVC minimum is at Q = 3 (18.33). SAC minimum is at Q = 4 (23.75). SAC-min is to the right of AVC-min, consistent with theory.

EXERCISE 27 — NUMERICAL

Q27. A firm's SMC schedule is shown. The total fixed cost of the firm is ₹100. Find the TVC, TC, AVC and SAC schedules of the firm.
Q = 0,1,2,3,4,5,6; SMC = —, 500, 300, 200, 300, 500, 800.

Step 1: TVC at Q = sum of SMC up to Q (since TVC = sum of marginal costs).
Q=0: TVC = 0; TC = 0 + 100 = 100.
Q=1: TVC = 500; TC = 600; AVC = 500.00; SAC = 600.00.
Q=2: TVC = 500 + 300 = 800; TC = 900; AVC = 400.00; SAC = 450.00.
Q=3: TVC = 800 + 200 = 1000; TC = 1100; AVC = 333.33; SAC = 366.67.
Q=4: TVC = 1000 + 300 = 1300; TC = 1400; AVC = 325.00; SAC = 350.00.
Q=5: TVC = 1300 + 500 = 1800; TC = 1900; AVC = 360.00; SAC = 380.00.
Q=6: TVC = 1800 + 800 = 2600; TC = 2700; AVC = 433.33; SAC = 450.00.
AVC minimum at Q = 4 (325.00). SAC minimum at Q = 4 (350.00). SMC is U-shaped (500 → 300 → 200 → 300 → 500 → 800) reaching its minimum at Q = 3 — as theory says, SMC-min lies to the left of AVC-min.

EXERCISE 28 — NUMERICAL

Q28. Let the production function of a firm be Q = 5 · L^1/2 · K^1/2. Find out the maximum possible output that the firm can produce with 100 units of L and 100 units of K.

Substitute L = 100, K = 100 in the production function:
Q = 5 · (100)^1/2 · (100)^1/2
= 5 · 10 · 10
= 500.
The firm can produce at most 500 units of output. (Note: α + β = 0.5 + 0.5 = 1, so the function exhibits CRS.)

EXERCISE 29 — NUMERICAL

Q29. Let the production function of a firm be Q = 2 · L² · K². Find out the maximum possible output that the firm can produce with 5 units of L and 2 units of K. What is the maximum possible output that the firm can produce with zero unit of L and 10 units of K?

Part (a): L = 5, K = 2.
Q = 2 · (5)² · (2)² = 2 · 25 · 4 = 200 units.

Part (b): L = 0, K = 10.
Q = 2 · (0)² · (10)² = 2 · 0 · 100 = 0 units.
With zero labour the firm cannot produce anything — both L and K are necessary in this multiplicative production function. (Also: α + β = 2 + 2 = 4 > 1, so this function exhibits IRS.)

EXERCISE 30 — NUMERICAL

Q30. Find out the maximum possible output for a firm with zero unit of L and 10 units of K when its production function is Q = 5L + 2K.

Substitute L = 0, K = 10:
Q = 5 · 0 + 2 · 10 = 0 + 20 = 20 units.
Note: this is a linear (additive) production function — unlike the Cobb-Douglas multiplicative form, here the two factors are perfect substitutes and either factor alone can produce output. Q29 (multiplicative) gave 0 with zero labour; Q30 (additive) gives 20 with zero labour. The contrast highlights that production functions vary by the way inputs combine.

📌 NCERT Chapter 3 — Complete Summary

The production function q = f(L, K) gives the maximum output for any combination of inputs given technology.
In the short run, at least one factor cannot be varied; in the long run, all factors are variable. Definition is by variability, not calendar time.
Total Product (TP) = output from varying one input. AP = TP/L; MP = ΔTP/ΔL. TP is the sum of all preceding MPs.
Law of Variable Proportions: MP first rises, then falls — three stages (Stage I increasing returns, Stage II diminishing returns, Stage III negative returns). Rational producer operates in Stage II.
MP and AP curves are inverse-U shaped. MP cuts AP from above at AP's maximum.
Returns to Scale (long-run): proportional change in all inputs. CRS (= proportional output rise), IRS (more than proportional), DRS (less than proportional).
Cobb–Douglas: q = A · L^α · K^β. CRS if α + β = 1, IRS if α + β > 1, DRS if α + β < 1.
Cost function describes the least cost of producing each output level. TC = TFC + TVC.
SAC = AFC + AVC. AFC is a rectangular hyperbola (continuously falling). AVC, SAC, SMC are all U-shaped.
SMC cuts AVC at AVC-min and cuts SAC at SAC-min, both from below. SAC-min lies to the right of AVC-min.
In the long run, no fixed cost. LRAC and LRMC are U-shaped; LRMC cuts LRAC from below at LRAC's minimum. IRS → falling LRAC; CRS → flat (at minimum); DRS → rising LRAC.

Cost Function

Least cost of producing each output level given input prices and technology.

Total Fixed Cost (TFC)

Cost of fixed inputs in the short run; does not change with output.

Total Variable Cost (TVC)

Cost of variable inputs; rises as output rises. TVC = 0 at q = 0.

Total Cost (TC)

Sum of fixed and variable costs: TC = TFC + TVC.

Average Fixed Cost (AFC)

TFC ÷ q. A rectangular hyperbola, falling continuously.

Average Variable Cost (AVC)

TVC ÷ q. U-shaped curve; SMC cuts it at its minimum.

Short-run Average Cost (SAC / ATC)

TC ÷ q = AFC + AVC. U-shaped; minimum to the right of AVC-min.

Short-run Marginal Cost (SMC / MC)

ΔTC ÷ Δq = ΔTVC ÷ Δq. U-shaped; cuts AVC and SAC at their minima.

Long-run Average Cost (LRAC)

TC ÷ q in the long run. U-shaped due to IRS → CRS → DRS sequence.

Long-run Marginal Cost (LRMC)

ΔTC ÷ Δq in the long run. U-shaped; cuts LRAC at LRAC's minimum.

Rectangular Hyperbola

A curve with the property AFC × q = TFC (a constant).

Marginal Cost

Additional cost of producing one more unit of output.

Average Cost

Cost per unit of output. ATC = AFC + AVC.

Frequently Asked Questions — Costs of Production — Short-Run, Long-Run Curves & Exercises

What is the difference between fixed cost and variable cost?

Fixed cost is the cost of fixed inputs in the short run — it does not vary with the level of output and must be paid even if output is zero (rent on factory, salaries of permanent staff). Variable cost is the cost of variable inputs and rises with output (raw materials, wages of casual labour). Total cost is TFC + TVC. NCERT Class 12 emphasises that fixed costs exist only in the short run.

What are AFC, AVC, ATC and MC?

AFC (average fixed cost) is TFC divided by output q — falls continuously as q rises. AVC (average variable cost) is TVC divided by q — first falls then rises (U-shaped). ATC (average total cost) is TC divided by q — also U-shaped, equal to AFC + AVC. MC (marginal cost) is the change in TC for a one-unit change in output, ΔTC / Δq. MC cuts both AVC and ATC at their minimum points.

Why is the AFC curve continuously falling?

Average fixed cost is TFC divided by output q. Since TFC is constant in the short run while q rises, AFC = TFC / q falls continuously as q rises. Geometrically the AFC curve is a rectangular hyperbola — it approaches the horizontal axis but never touches it. NCERT Class 12 uses this property to derive the U-shape of ATC from AFC + AVC.

Why does the marginal cost curve cut average variable cost and average total cost at their minimum?

When MC is below AVC or ATC, it pulls the average down. When MC equals the average, the average is at its minimum. When MC is above the average, it pulls the average up. So MC must cut AVC and ATC at their lowest points — this is a mathematical property of averages and marginals, not an economic assumption.

What is the long-run average cost (LRAC) curve?

The long-run average cost (LRAC) curve is the envelope of all short-run ATC curves, showing the lowest average cost at which the firm can produce each output level when all inputs are variable. It is typically U-shaped because of returns to scale — falling under increasing returns, flat under constant returns, and rising under decreasing returns. The minimum point gives the firm's most efficient long-run plant size.

How are cost curves related to the law of variable proportions?

Cost curves are the mirror image of product curves. When marginal product is rising, marginal cost is falling because each extra unit of output is produced with less labour. When MP reaches its peak, MC is at its minimum. When MP falls, MC rises. The U-shape of AVC, ATC and MC therefore comes directly from the inverted-U shape of MP under the law of variable proportions.

What types of NCERT exercises does Chapter 3 ask?

NCERT Class 12 Chapter 3 exercises ask students to define and derive total, average and marginal product, draw the law of variable proportions, distinguish short run and long run, write the production function in Cobb–Douglas form, distinguish returns to scale, compute TFC, TVC, AFC, AVC, ATC and MC from a TC schedule, and explain why short-run cost curves are U-shaped. All exercises are answered in this part.

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