Short-run cost curves: shapes and relationships

Detail

1. The basic building blocks

• Short run: a period in which at least one input is fixed (for example, machines or a factory building). Only the variable input (for example, labour or raw material) can be changed.
• Total Fixed Cost (TFC): the cost of fixed inputs. It stays the same at every output level, even at q = 0. Examples are rent and the interest on machine loans.
• Total Variable Cost (TVC): the cost of variable inputs. It is 0 at q = 0 and rises as output rises.
• Total Cost (TC): TC = TFC + TVC.
• Short-run marginal cost (SMC): the extra cost of producing one more unit of output.

2. Formulas (learn these exactly)

Cost	Formula	Meaning in plain words
Short-run average cost	SAC = TC/q	total cost per unit
Average variable cost	AVC = TVC/q	variable cost per unit
Average fixed cost	AFC = TFC/q	fixed cost per unit
Identity	SAC = AVC + AFC	the per-unit cost splits into two parts
Short-run marginal cost	SMC = ΔTC/Δq	extra cost of one more unit

• Since TFC does not change, ΔTC = ΔTVC. So SMC = ΔTVC/Δq as well.
• All of these are undefined at q = 0, because you cannot divide by zero.
• Worked example: TFC = ₹20 and TVC = ₹39 at q = 6.
• TC = ₹59
• AFC = 20/6 = ₹3.33
• AVC = 39/6 = ₹6.5
• SAC = 59/6 = ₹9.83, which equals 3.33 + 6.5

3. AFC: the curve that always falls

• AFC falls continuously. The same fixed cost is shared over more and more units.
• Its curve is a rectangular hyperbola. This is a curve where x × y stays constant. Here AFC × q = TFC, a constant.
• As q gets close to zero, AFC becomes very large. As q grows, AFC gets close to zero.
• The curve comes close to both axes but never touches them. (Such lines are called asymptotes.)
• Example (TFC = ₹20):
• q = 1 → AFC = 20
• q = 5 → AFC = 4
• q = 10 → AFC = 2

• AFC × q = 20 every time.

• Exam trap: AFC is the only short-run cost curve that is not U-shaped.

4. Geometry: reading costs off graphs

• AFC = tanθ. This is the slope of a ray (a straight line) from the origin to the TFC line. At output q₀ it equals Aq₀/Oq₀.

• TFC is a horizontal line. As q₀ moves right, the ray gets flatter, so AFC falls.

• AVC = the slope of a ray from the origin to the TVC curve, which is Eq₀/Oq₀.

• AVC is lowest where this ray is tangent to the TVC curve, meaning it just touches it.

• SMC = the slope of the TC curve (or of the TVC curve, which has the same slope).

• Areas under the curves:
• TFC = area of rectangle OFCq₁ under the AFC curve (height AFC × base q₁)
• TVC = area of rectangle OVBq₀ under the AVC curve (height AVC × base q₀)

• TVC = area under the SMC curve. Adding all the marginal costs gives the total variable cost.

• Check with Table 3.3: ΣSMC from q = 1 to q = 10 = 95, which is TVC at q = 10.

5. Why the curves have these shapes: each mirrors the Law of Variable Proportions (LVP)

• LVP: when more of one variable input is added to a fixed input, its marginal product (MP) first rises and then falls.
• MP is the extra output from one more unit of the input.

• AP (average product) is output per unit of the input.

• Cost curves are the mirror image of product curves. When product rises, cost falls, and the reverse.

SMC is U-shaped

• While MP is rising:
• each extra unit of output needs less of the variable input

• so SMC falls

• Once MP starts falling:

• each extra unit of output needs more of the variable input
• so SMC rises

Beyond NCERT: the exact link between cost and product

• MC = w/MP_L and AVC = w/AP_L, where w is the wage per worker and L is labour.
• So maximum MP ↔ minimum MC, and maximum AP ↔ minimum AVC.
• Worked example (w = ₹100):
• The 3rd worker has MP = 16 units, so MC = 100/16 ≈ ₹6.25 per unit.
• The 6th worker has MP = 1 unit, so MC = 100/1 = ₹100 per unit.
• Falling MP pushes MC sharply upward.

AVC is U-shaped

• SMC and AVC start at the same point for the first unit, because for q = 1, SMC = TVC = AVC. (Table 3.3: both are ₹10.)
• AVC is the average of all the MCs so far.
• While SMC < AVC, each new unit pulls the average down.
• Once SMC > AVC, each new unit pulls the average up.

• Think of a cricket batting average: a score below your average lowers it, and a score above your average raises it.

• Real-world view: the AVC curve is often a flat-bottomed U, and the MC curve falls faster and rises more steeply than AVC [2].

• Why the U, in practical terms [2]:
• At very low output, the plant is under-used and workers do not have enough work, so average cost is high.
• As output rises, average cost falls.
• As output nears the plant's capacity, crowding in the plant causes waste, so average cost rises quickly.

SAC is U-shaped

• At first: falling AFC is stronger than rising AVC, so SAC falls.
• Later: rising AVC is stronger than falling AFC, so SAC rises.

6. The key relationships between the curves

• SMC cuts AVC and SAC from below, at their minimum points.
• At the lowest point of AVC: SMC = AVC.

• At the lowest point of SAC: SMC = SAC.

• Minimum SAC lies to the right of minimum AVC.

• Just past the lowest point of AVC, AVC starts rising slowly.
• AFC is still falling, so SAC keeps falling for a while.

• SAC starts rising only once the rise in AVC is bigger than the fall in AFC.

• The vertical gap between SAC and AVC equals AFC.

• AFC keeps falling, so the gap narrows as q grows.

• AFC is never zero, so the gap never closes. The two curves never meet.

• Order of the lowest points: SMC → AVC → SAC, from left to right.

7. Table 3.3 (TFC = ₹20): the pattern in numbers

q	TVC	TC	AFC	AVC	SAC	SMC
1	10	30	20	10	30	10
3	24	44	6.67	8	14.67	6
5	33	53	4	6.6	10.6	4 (min)
6	39	59	3.33	6.5 (min)	9.83	6
7	47	67	2.86	6.7	9.57 (min)	8
8	60	80	2.5	7.5	10	13
10	95	115	2	9.5	11.5	20

• The lowest points come in this order: SMC (q = 5) → AVC (q = 6) → SAC (q = 7).
• SMC below AVC: at q = 6, SMC is 6 and AVC is 6.5. AVC is at its lowest here.
• SMC above AVC: at q = 7, SMC is 8, above AVC, so AVC rises to 6.7.
• SAC still falling: at q = 7, SAC falls to 9.57 because AFC is still dropping (3.33 → 2.86).
• SAC rising: at q = 8, SMC is 13, above SAC (9.57), so SAC rises to 10.
• The gap shrinks: SAC − AVC = 20 at q = 1, but only 2 at q = 10.
• ΣSMC for q = 1 to 10 = 95 = TVC at q = 10.

8. Practice drills (NCERT exercises)

Ex. 25: TFC = ₹10 (TFC equals TC at q = 0)

• TVC (q = 0 to 6) = 0, 20, 35, 45, 60, 80, 110
• AVC = 20, 17.5, 15, 15, 16, 18.33 (lowest at q = 3 and q = 4)
• SAC = 30, 22.5, 18.33, 17.5, 18, 20 (lowest at q = 4, which is to the right of the AVC minimum)
• SMC = 20, 15, 10, 15, 20, 30 (lowest at q = 3)

Ex. 26: AFC at q = 4 is ₹5, so TFC = 5 × 4 = ₹20

• TVC = 30, 45, 55, 75, 110, 165
• SAC = 50, 32.5, 25, 23.75, 26, 30.83
• SMC = 30, 15, 10, 20, 35, 55
• Method: first find TFC from AFC × q. Then use TC = TFC + TVC, and SMC = the change in TC.

Ex. 27: SMC = 500, 300, 200, 300, 500, 800; TFC = ₹100

• TVC is the running total of SMC: 500, 800, 1000, 1300, 1800, 2600
• TC = TVC + 100: 600, 900, 1100, 1400, 1900, 2700
• AVC is lowest at ₹325 when q = 4. SAC is lowest at ₹350 when q = 4.
• In this small table both lowest points fall on the same whole number.

• On smooth, continuous curves, the SAC minimum would still lie to the right of the AVC minimum.

• NCERT error: the column is headed "TC" but it actually lists SMC values. Read the headers carefully.

9. Beyond the textbook: capacity and costs in India

• The link to theory: the rising part of AVC and SMC shows up as a factory gets close to its full capacity [2].
• How India tracks this: the RBI runs the Order Books, Inventories and Capacity Utilisation Survey (OBICUS) every quarter.

• Capacity utilisation (CU) means the share of a factory's full capacity that is actually being used.

• Data from the 59th round (conducted in Q3:2022-23, covering Q2:2022-23) [3]:

• It surveyed 800 manufacturing companies, and 745 of them responded.
• CU rose to 74.0% in Q2:2022-23, up from 72.4% in the previous quarter.

• Seasonally adjusted CU was 74.5%, up 20 basis points (0.20 percentage points). Seasonally adjusted means corrected for normal ups and downs across the year.

• Why policymakers care:

• When CU is high, factories are on the rising part of their short-run cost curves.
• Producing more then pushes up per-unit costs.
• This encourages firms to invest in new capacity.

Prelims Hooks

• AFC is a rectangular hyperbola, because AFC × q = TFC, a constant. It is the only short-run cost curve that is not U-shaped, and it never touches either axis.
• SAC = AVC + AFC. The vertical gap between SAC and AVC is AFC. It narrows but never becomes zero.
• SMC cuts AVC and SAC from below, at their minimum points.
• Order of the lowest points: SMC → AVC → SAC (Table 3.3: q = 5, 6, 7).
• MC = w/MP_L; AVC = w/AP_L. Maximum MP matches minimum MC. Maximum AP matches minimum AVC.
• TVC = area under the SMC curve = ΣSMC. SMC is the slope of both TC and TVC, because TFC does not change.
• AFC = slope of a ray from the origin to the TFC line. AVC = slope of a ray from the origin to the TVC curve.
• At q = 1, SMC = AVC, because both equal TVC(1).
• Trap: the U-shape of short-run curves comes from the Law of Variable Proportions, not from returns to scale. Returns to scale are a long-run idea.
• OBICUS is a quarterly survey of capacity utilisation in manufacturing, run by the RBI (not MoSPI) [3].

Mains Points

• Cost structure shapes pricing and survival:
• A firm with high fixed costs (steel, telecom, power) has a high AFC at low output.
• So these firms push to raise output and use more of their capacity, which spreads fixed costs over more units.

• This explains price cuts in such sectors and pressure for consolidation (firms merging).

• Capacity utilisation and investment:

• The RBI's OBICUS reported CU of 74.0% in Q2:2022-23 [3].
• As CU nears full capacity, firms move onto the rising part of AVC and SMC [2].

• Rising unit costs then justify new private investment (capex), which is useful for GS-III answers on the investment cycle.

• Minimum AVC acts as a floor in the short run:

• If the price stays above AVC, a firm keeps producing even if it cannot cover its fixed costs.
• This helps explain why loss-making units, such as MSMEs during demand shocks, keep running in the short run.

• It also explains why relief aimed at variable costs (working capital, input subsidies) can keep them open.

• Productivity lowers costs:

• Since MC = w/MP, raising labour productivity (skills, technology) lowers the per-unit cost even if wages stay the same.
• This is the economic case for skilling programmes and technology upgrades in manufacturing competitiveness.