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We cross to the other side of the market. A firm turns inputs into output — and every unit of output has a cost. Out of those costs, Chapter 4 will build the supply curve.
Starting from a consumer's preferences and her budget, we derived the market demand curve — one blade of the scissors.
Chapter 3 — how inputs become output, and what that output costs.
Chapter 4 — how a profit-maximising firm turns those costs into a
supply curve.
Chapter 5 then crosses the two blades and cuts out the price.
How inputs map to output. Isoquants. Short run vs long run.
TP, AP and MP — and the law of variable proportions.
What happens when all inputs change together.
TFC, TVC, TC · AFC, AVC, SAC · SMC, and why they are U-shaped.
LRAC and LRMC, and how returns to scale shape them.
Every "marginal vs average" rule you met in Chapter 2 returns here — in a new costume.
Production is the process by which inputs are transformed into output.
A firm is the unit that carries out production — it acquires inputs, produces output, and sells it.
The inputs a firm uses are called factors of production. To keep everything drawable we use just two: labour (L) and capital (K).
The production function of a firm is the relationship between the inputs used and the output produced. For various quantities of inputs, it gives the maximum quantity of output that can be produced.
Because the function reports the maximum obtainable output, it already assumes the inputs are used efficiently. Efficiency means it is not possible to get any more output from the same level of inputs.
A lazy or badly organised firm is not "on" its production function.
Technological knowledge is what determines the maximum output obtainable from any input combination. If technology improves, we get a new production function — the old one does not stretch, it is replaced.
| Labour ↓ / Capital → | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 3 | 7 | 10 | 12 | 13 |
| 2 | 0 | 3 | 10 | 18 | 24 | 29 | 33 |
| 3 | 0 | 7 | 18 | 30 | 40 | 46 | 50 |
| 4 | 0 | 10 | 24 | 40 | 50 | 56 | 57 |
| 5 | 0 | 12 | 29 | 46 | 56 | 58 | 59 |
| 6 | 0 | 13 | 33 | 50 | 57 | 59 | 60 |
1 labour + 1 capital → at most 1 unit.
2 labour + 2 capital → at most 10 units.
3 labour + 2 capital → at most 18 units.
The highlighted column (K = 4) is the one we will live in for the next several slides.
An isoquant is the set of all possible combinations of the two inputs that yield the same maximum possible level of output. Each isoquant is labelled with that output.
Output of 10 can be produced three ways:
(4L, 1K), (2L, 2K), (1L, 4K).
All three lie on the same isoquant, labelled q = 10.
An isoquant is the producer's version of an indifference curve. Both are downward sloping, both join points that are "equally good", both come in families — and on both, the slope is a rate of substitution.
Why downward sloping? When marginal products are positive, keeping output constant while using more of one input requires less of the other.
Using Table 3.1, identify all the input combinations that lie on the isoquant q = 50.
Scan the table for every cell containing 50:
(3L, 6K) · (4L, 4K) · (6L, 3K)
All three produce exactly 50 units, so all three lie on the isoquant labelled q = 50.
Notice the trade-off along it: moving from (3L, 6K) to (6L, 3K), the firm substitutes 3 more labour for 3 less capital and output is unchanged. That is the isoquant's downward slope in numbers.
A student writes: "Table 3.1 shows that 2 labour and 2 capital produce 10 units. So if the firm employs 2 labour and 2 capital but produces only 7 units because the manager is careless, then q = 7 is also a point on the production function."
Is the student right? Explain, and say what the diagram of this situation would look like.
The student is wrong, and the error is in overlooking the word maximum.
The production function reports, for each input combination, the maximum output obtainable. It is therefore defined only over efficient production. Producing 7 units from (2L, 2K) is certainly possible — but it is not on the production function, because 10 is obtainable from the same inputs.
The parallel with Chapter 1 is exact. A point inside the PPF was attainable but inefficient; the PPF itself was the boundary of what is possible. Here, (2L, 2K, 7 units) is attainable but inefficient, and it lies below the production function.
PPF (Ch 1) : boundary of attainable output combinations for a society
Production function (Ch 3) : boundary of attainable output for a firm
Both are frontiers. Inefficiency puts you inside; it never moves the frontier.
Why economists build it this way: if the function reported every possible output
including wasteful ones, then every quantity from 0 to 10 would "belong" to (2L, 2K)
and f would not be a function at all — one input combination would map to
many outputs. Taking the maximum is what makes the relationship well defined.
A careless manager does not change the production function. He moves the firm off it — and in Chapter 4 that shows up as a cost higher than the cost function says it should be.
The short run is a period in which at least one factor of production cannot be varied and therefore remains fixed.
The long run is a period in which all factors of production can be varied. In the long run there is no fixed factor.
The factor that cannot be varied in the short run. Typically capital — a factory building, a plot of land, a machine.
The factor the firm can vary in the short run. Typically labour, raw materials, power.
Short run and long run are not defined in days, months or years. It is not advisable to define them by calendar time at all.
The only test: can all inputs be varied? If no → short run. If yes → long run.
For a roadside tea stall the long run may be a few weeks; for a steel plant it may be a decade. Same economics, different clocks.
Two firms are described:
Firm A — a software company. It can hire or release programmers within a week and rent additional cloud servers within an hour.
Firm B — a nuclear power company. It can hire engineers within a month, but a new reactor takes eleven years to build.
For a planning horizon of one year, is each firm in the short run or the long run? Now explain why the same one-year horizon gives different answers, and what this implies about drawing "the" short-run cost curve for an industry.
Firm A is in the long run. Within a year it can vary every input — labour and servers alike. Nothing is fixed, so by definition it is in the long run.
Firm B is in the short run. Labour is variable within a year, but the reactor — its capital — is not. At least one factor is fixed, so it is in the short run.
Why the same year gives different answers: because the short run/long run distinction is not a property of time. It is a property of the relationship between a firm's technology and the horizon being considered. The question "is one year long?" is meaningless on its own — long relative to what? A year is an eternity for a software firm's capital and an instant for a reactor.
Short run is not "soon". Long run is not "later".
Short run = some input is stuck. Long run = nothing is stuck.
What it implies for cost curves: a firm does not have one short-run cost curve — it has a different one for every level of the fixed factor. Firm B's short-run average cost curve with one reactor is a different curve from its SAC with two reactors. The long-run curve is the one the firm can reach once it is free to choose the reactor count as well.
This is exactly why LRAC lies below every SAC curve. In the long run the firm has strictly more freedom — it can pick the best plant size for the output it wants, instead of being stuck with whatever plant it happens to have. More choice can never make you worse off, so long-run cost can never exceed short-run cost.
Hold capital fixed at K = 4 and vary labour. The highlighted column of Table 3.1 becomes our whole world.
Total Product (TP) of a variable input is the relationship between that input and output, all other inputs held constant — the total output produced at each level of the variable input.
Average Product (AP) is output per unit of the variable input.
Marginal Product (MP) is the change in output per unit change in the variable input, all other inputs held constant.
| Labour | TP | MPL | APL |
|---|---|---|---|
| 0 | 0 | — | — |
| 1 | 10 | 10 | 10 |
| 2 | 24 | 14 | 12 |
| 3 | 40 | 16 | 13.33 |
| 4 | 50 | 10 | 12.5 |
| 5 | 56 | 6 | 11.2 |
| 6 | 57 | 1 | 9.5 |
At L = 2: MP = 24 − 10 = 14 ✓
AP = 24 / 2 = 12 ✓
1. TP is the sum of the MPs.
10 + 14 + 16 + 10 + 6 + 1 = 57 ✓
2. AP is the average of all MPs up to that level.
At L = 3: (10 + 14 + 16)/3 = 13.33 ✓
MP is undefined at zero input — inputs cannot be negative, so there is no "previous" unit to compare against.
Also note MP₁ = AP₁ = 10. For the first unit they must be equal, since the average of one number is that number.
| When MP is… | TP does… | In Table 3.2 |
|---|---|---|
| positive and rising | rises at an increasing rate | L = 1 to 3 |
| positive but falling | rises at a decreasing rate | L = 4 to 6 |
| zero | is at its maximum | just beyond L = 6 |
| negative | falls | beyond that point |
In Chapter 2, for Total Utility and Marginal Utility. Swap the words and the logic is untouched:
Ch 2: MU > 0 → TU rises · MU = 0 → TU max · MU < 0 → TU falls
Ch 3: MP > 0 → TP rises · MP = 0 → TP max · MP < 0 → TP falls
This is not a coincidence and not a memory trick. A total is the running sum of its marginals — so the marginal is the rate of change of the total. Whenever you add a positive number the total grows; add zero and it stands still; add a negative and it shrinks.
MP cuts AP from above, at AP's maximum.
Your batting average is 40. In the next innings — the marginal one — you score:
90 → above your average → average rises
40 → equal to it → average unchanged
10 → below it → average falls
A new value pulls an average towards itself. Always.
This is a purely arithmetic fact about averages and marginals — nothing to do with production. So it must hold everywhere the pair appears.
You will use it three more times in this book: SMC cuts AVC at min AVC, SMC cuts SAC at min SAC, and LRMC cuts LRAC at min LRAC. One idea, four applications.
Note the direction flips. MP cuts AP at its maximum (from above); MC cuts AC at its minimum (from below). The rule is the same — "marginal pulls average towards itself" — but product curves are inverse-U and cost curves are U.
The marginal product schedule of labour is given below, and TP is zero at zero labour. Calculate the total and average product schedules.
| L | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| MPL | 3 | 5 | 7 | 5 | 3 | 1 |
TP = running sum of MP · AP = TP ÷ L
| L | MPL | TPL | APL |
|---|---|---|---|
| 1 | 3 | 3 | 3.00 |
| 2 | 5 | 8 | 4.00 |
| 3 | 7 | 15 | 5.00 |
| 4 | 5 | 20 | 5.00 |
| 5 | 3 | 23 | 4.60 |
| 6 | 1 | 24 | 4.00 |
Check the relationship holds: MP peaks at L = 3, but AP peaks later — it is tied at its maximum of 5.00 at both L = 3 and L = 4, and MP = AP exactly at L = 4. For L = 2 and 3, MP > AP and AP rises (at L = 1 they are equal, as they always are for the first unit). From L = 5, MP < AP and AP falls. Exactly as the rule predicts.
The Law of Variable Proportions states that the marginal product of a factor input initially rises with its employment level, but after reaching a certain level of employment it starts falling.
NCERT gives this tendency two names for one law: "This tendency of the MP to first increase and then fall is called the law of variable proportions or the law of diminishing marginal product."
The glossary phrases the second name as: if we keep increasing the employment of an input with other inputs fixed, then eventually a point will be reached after which the marginal product of that input will start falling.
These are not two different laws, and the second is not a narrower "falling only" version. If a question asks "What is the law of diminishing marginal product?", your answer should describe the full rise-then-fall pattern, exactly as for the law of variable proportions. Describing only the falling phase risks an incomplete answer.
Factor proportions are the ratio in which the two inputs are combined to produce output.
It requires one factor fixed while another varies — that is what makes the proportions change. In the long run all factors vary together, so the proportions can be held constant, and this law does not apply. There we use returns to scale instead.
A farmer has 4 hectares of land — fixed — and chooses how much labour to use.
With only 1 worker there is far too much land for one person to cultivate. The factor proportions are badly out of balance.
As workers are added, land per worker falls towards a sensible ratio. Specialisation becomes possible. Each extra worker adds proportionally more — MP rises: 10, 14, 16.
When the 4th worker is hired the land begins to get crowded. Each worker now has insufficient land to work efficiently.
The output added by each additional worker is now proportionally less — MP falls: 10, 6, 1.
As you hold one factor fixed and keep increasing the other, the factor proportions change. First they improve towards the ideal ratio — MP rises. Then they deteriorate past it — MP falls.
The classic exam trap: "MP falls because the extra workers are less skilled." Wrong. All units of the variable factor are assumed homogeneous — every worker is identical. MP falls because of the changing ratio to the fixed factor, not because of any difference between the workers.
In Table 3.2, the marginal product of the 6th worker is +1 — still positive. A student concludes: "Since the 6th worker adds to total output, the farmer should definitely hire him."
(a) Is total product still rising at L = 6?
(b) Is average product rising or falling there?
(c) Is the student's conclusion sound? What information is missing?
(a) Yes. MP₆ = +1 > 0, so TP rises from 56 to 57. Output is higher with six workers than with five.
(b) Falling. AP goes 11.2 → 9.5. Consistent with the rule, since MP (1) is well below AP (9.5), and a marginal below the average drags the average down.
(c) The conclusion does not follow — the missing information is cost.
"Adds to output" answers a technical question.
"Should hire" is an economic question, and it needs prices.
The 6th worker produces 1 extra unit. Whether to hire him depends on whether that one unit is worth more than his wage:
If output sells for ₹50 and the wage is ₹300, the worker adds ₹50 of value and costs ₹300. Hiring him reduces profit by ₹250 — even though he raises output.
The deeper point: a profit-maximising firm does not maximise output. Nothing in this chapter tells you how much to produce — Chapter 3 only describes what is technically possible and what it costs. The decision rule arrives in Chapter 4, and it is not "produce the most" but MR = MC — or, for hiring a factor, hire until wage = value of marginal product.
Notice too that the student's logic would justify hiring workers right up to the point where MP = 0 — maximum TP. A firm doing that would be pouring wages into workers whose last unit of output is worth almost nothing.
The law of variable proportions asked: what happens when one input rises and the factor proportions change?
Returns to scale asks: what happens when all inputs rise in the same proportion, so factor proportions stay constant? This can only happen in the long run.
A proportional increase in all inputs raises output by a larger proportion.
Double inputs → more than double output.
A proportional increase in all inputs raises output by the same proportion.
Double inputs → exactly double output.
A proportional increase in all inputs raises output by a smaller proportion.
Double inputs → less than double output.
For a production function q = f(x₁, x₂), scale both inputs by
t > 1:
CRS: f(tx₁, tx₂) = t · f(x₁, x₂)
IRS: f(tx₁, tx₂) > t · f(x₁, x₂)
DRS: f(tx₁, tx₂) < t · f(x₁, x₂)
| From (L, K) | Output | To (L, K) | Output | Inputs × | Output × | Verdict |
|---|---|---|---|---|---|---|
| (1, 1) | 1 | (2, 2) | 10 | 2 | 10 | IRS |
| (2, 2) | 10 | (4, 4) | 50 | 2 | 5 | IRS |
| (3, 3) | 30 | (6, 6) | 60 | 2 | 2 | CRS |
For q = x₁α · x₂β, scaling both inputs by t gives
tα+β times the output. So the exponents decide everything:
α + β = 1 → CRS · α + β > 1 → IRS · α + β < 1 → DRS
Example: q = 5L½K½ has α + β = 1 → CRS.
q = 2L²K² has α + β = 4 → IRS.
| Basis | Law of variable proportions | Returns to scale |
|---|---|---|
| Which inputs change | One input | All inputs |
| Factor proportions | Change | Stay constant |
| Time period | Short run | Long run |
| Measured by | MP of the variable factor | Proportional change in output |
A production function shows increasing returns to scale throughout.
A student argues: "If doubling all inputs more than doubles output, then the law of diminishing marginal product must be false for this firm — output is rising faster than inputs, so marginal product cannot be falling."
Show that the student is confusing two different things, using
q = 2L²K² as your example.
The two laws answer different questions and are perfectly compatible. One holds an input fixed; the other does not.
Returns to scale — vary both. For q = 2L²K², α + β = 4, so scaling both
inputs by t multiplies output by t⁴. Doubling inputs gives 16 times
the output. Strong IRS. ✓
Marginal product — hold K fixed. Set K = 1, so q = 2L²:
| L | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| q = 2L² | 2 | 8 | 18 | 32 |
| MPL | 2 | 6 | 10 | 14 |
Here MP is rising, not falling — so for this particular function the diminishing phase never arrives.
But that does not vindicate the student's reasoning. He inferred "MP cannot fall" from IRS. That inference is invalid, even though the conclusion happens to hold here.
The counter-example that settles it: take q = L½K½
scaled up — say q = L0.8K0.8. Then α + β = 1.6 > 1, so
IRS holds. But hold K fixed at 1 and q = L0.8, whose marginal
product falls as L rises (each extra unit of L adds less than the last).
So a firm can display IRS and diminishing MP simultaneously. They are not in conflict because:
Returns to scale is about the long run, all inputs moving together, factor
proportions constant.
Diminishing MP is about the short run, one input moving alone, factor proportions
changing.
The intuition: the reason MP eventually falls is crowding — too much labour per unit of fixed land. Scaling everything up adds land as well as labour, so no crowding ever occurs. Remove the fixed factor and you remove the cause.
The cost function describes, for every level of output, the minimum cost of producing it, given the prices of factors and the technology.
A given output can usually be produced by many input combinations. From Table 3.1, 50 units come from (6L, 3K), (4L, 4K) or (3L, 6K).
With input prices given, the firm picks the least expensive of them. So for every level of output the firm chooses the least-cost input combination — and the cost function records that cost.
"How to produce?" — choose the technique that produces the desired output at least cost, given factor prices. Chapter 1 stated it in words; this is the same idea with a name and a formula.
Total Fixed Cost (TFC) is the cost a firm incurs to employ the fixed inputs. It does not change with the level of output — it is incurred even at zero output.
Rent on a factory, insurance, interest on a loan for machinery.
Total Variable Cost (TVC) is the cost a firm incurs to employ the variable inputs. It rises as output rises, and is zero at zero output.
Wages of casual labour, raw materials, power.
At q = 0: TVC = 0, so TC = TFC. This is how you find TFC when a question gives you only a TC schedule — read off the value at zero output.
TFC is a horizontal line; TVC and TC start apart and stay a constant distance apart — that vertical gap is TFC, at every level of output.
| q | TFC | TVC | TC | AFC | AVC | SAC | SMC |
|---|---|---|---|---|---|---|---|
| 0 | 20 | 0 | 20 | — | — | — | — |
| 1 | 20 | 10 | 30 | 20 | 10 | 30 | 10 |
| 2 | 20 | 18 | 38 | 10 | 9 | 19 | 8 |
| 3 | 20 | 24 | 44 | 6.67 | 8 | 14.67 | 6 |
| 4 | 20 | 29 | 49 | 5 | 7.25 | 12.25 | 5 |
| 5 | 20 | 33 | 53 | 4 | 6.6 | 10.6 | 4 |
| 6 | 20 | 39 | 59 | 3.33 | 6.5 | 9.83 | 6 |
| 7 | 20 | 47 | 67 | 2.86 | 6.71 | 9.57 | 8 |
| 8 | 20 | 60 | 80 | 2.5 | 7.5 | 10 | 13 |
| 9 | 20 | 75 | 95 | 2.22 | 8.33 | 10.56 | 15 |
| 10 | 20 | 95 | 115 | 2 | 9.5 | 11.5 | 20 |
Falls continuously — a fixed number spread over more units.
SAC = AVC + AFC …(3.10)
Also = ΔTVC / Δq, since TFC never changes.
Average Fixed Cost (AFC) is total fixed cost per unit of
output: AFC = TFC / q …(3.9)
TFC is a constant. So AFC × q = TFC always — the product of the two
axes is fixed. That is exactly the equation xy = c of a
rectangular hyperbola.
Geometrically: the area of the rectangle under any point on the AFC curve equals TFC, whichever point you pick.
AFC falls continuously and never rises — the same ₹20 spread over more and more units.
It never touches the horizontal axis. As output → ∞, AFC → 0, but TFC is still being paid, so AFC stays strictly positive.
The unitary elastic demand curve in Chapter 2 was also a rectangular hyperbola —
there, p × q was constant. Same curve, different variables.
Straight from the law of variable proportions. Initially MP rises, so each extra unit of output needs less and less of the variable factor — with the factor price given, SMC falls.
After a point MP falls, so each extra unit needs more and more of the factor — SMC rises.
SMC is the mirror image of MP. Where MP peaks, SMC bottoms out.
AVC is the average of all the marginal costs up to that output. While SMC is below AVC it drags AVC down; once SMC rises above AVC it pulls AVC up.
AVC mirrors AP, just as SMC mirrors MP.
SAC = AVC + AFC. Initially both components fall, so SAC falls.
Past min AVC, AVC starts rising while AFC is still falling. For a while the fall in AFC outweighs the rise in AVC, so SAC keeps falling. Only when the rise in AVC finally exceeds the fall in AFC does SAC turn upward.
That lag is exactly why min SAC (q = 7) lies to the right of min AVC (q = 6).
The batting-average rule again. While SMC < average, the average falls; while SMC > average, the average rises. So the average can only be at its lowest at the moment SMC crosses it — cutting it from below.
A firm's total cost schedule is given. The average fixed cost at 4 units is ₹5. Find TFC, TVC, AFC, AVC, SAC and SMC.
| Q | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| TC | 50 | 65 | 75 | 95 | 130 | 185 |
Step 1 — recover TFC. AFC = TFC / q, so at q = 4:
5 = TFC / 4 → TFC = ₹20 at every level of output.
Step 2 — TVC = TC − TFC, then divide through.
| Q | TC | TFC | TVC | AFC | AVC | SAC | SMC |
|---|---|---|---|---|---|---|---|
| 1 | 50 | 20 | 30 | 20 | 30 | 50 | — |
| 2 | 65 | 20 | 45 | 10 | 22.5 | 32.5 | 15 |
| 3 | 75 | 20 | 55 | 6.67 | 18.33 | 25 | 10 |
| 4 | 95 | 20 | 75 | 5 | 18.75 | 23.75 | 20 |
| 5 | 130 | 20 | 110 | 4 | 22 | 26 | 35 |
| 6 | 185 | 20 | 165 | 3.33 | 27.5 | 30.83 | 55 |
Sanity checks: AVC + AFC = SAC throughout (e.g. at q = 4: 18.75 + 5 = 23.75 ✓). AVC bottoms at q = 3 (18.33) and SAC bottoms at q = 4 (23.75) — SAC's minimum to the right of AVC's, as it must be. SMC rises past both minima. ✓
In Table 3.3, look at output q = 7: SAC is at its minimum of ₹9.57.
A student says: "Minimum average cost means cost is lowest there, so the firm should always produce 7 units. Producing 8 units is wasteful because average cost rises to ₹10."
(a) Is it true that producing 8 units costs the firm more in total than 7?
(b) Is the student's advice correct? Under what circumstance would producing 8 be better?
(a) Yes in total, but that is not the point. TC rises from ₹67 to ₹80. Every extra unit costs something — total cost rises at every output level, on both sides of min SAC. That is true of the 2nd unit as much as the 8th, so it cannot by itself be an argument against producing more.
(b) The advice is wrong. It confuses minimising cost with maximising profit.
Minimum average cost is the point of greatest technical efficiency — most output per rupee. It is not the point of greatest profit.
The circumstance that decides it is the price. The 8th unit costs SMC = ₹13 to produce.
If the good sells for ₹20, the 8th unit brings in ₹20 and costs ₹13 — it adds ₹7 to profit. Refusing to produce it, to keep average cost "low", would throw away ₹7. The firm should certainly produce it, and consider the 9th too (SMC = ₹15, still below ₹20).
If the good sells for ₹10, the 8th unit costs ₹13 and earns ₹10 — it subtracts ₹3. Now the firm should stop earlier.
So the answer is not a quantity at all until you are told the price. This is precisely the gap Chapter 4 fills, with the rule P = MC: produce every unit whose marginal cost is covered by the price, and stop at the one where they are equal.
Where minimum SAC does matter: it is the break-even point — the lowest price at which the firm can cover all its costs. And minimum AVC is the shut-down point. Both are landmarks on the supply curve, but neither is a target output.
All inputs are variable, so there are no fixed costs. Consequently TC and TVC coincide, and there is no AFC to speak of.
Long Run Average Cost (LRAC) is cost per unit of output in the
long run: LRAC = TC / q …(3.13)
Long Run Marginal Cost (LRMC) is the change in total cost per
unit change in output: LRMC = TC(q₁) − TC(q₁−1) …(3.14)
| Phase | To raise output by a given %, inputs must rise by… | LRAC |
|---|---|---|
| IRS | less than that % → cost rises less than output | falls |
| CRS | the same % → cost rises in step with output | constant (its minimum) |
| DRS | more than that % → cost rises faster than output | rises |
MP cuts AP at max AP · SMC cuts AVC at min AVC · SMC cuts SAC at min SAC · LRMC cuts LRAC at min LRAC. If you learned the batting average once, you now have all four for free.
Explain why the following statement must be true for any firm:
"At every level of output, long-run average cost is less than or equal to short-run average cost — and they are equal at only one output level for a given plant size."
Then use this to explain why a question asking "can there be fixed costs in the long run?" has the answer it does.
The argument is about freedom of choice, not about arithmetic.
In the short run the firm is stuck with whatever plant it happens to have. To produce any output it must use that plant, however ill-suited.
In the long run the firm may choose any plant size — including the one it is currently stuck with. So the long-run choice set contains the short-run choice set.
Whenever you choose from a larger menu, you can never do worse. At worst you pick the same option you would have been forced into.
Therefore LRAC ≤ SAC at every output.
When are they equal? At exactly the output for which that plant is the optimal one. There, the long-run choice and the short-run constraint happen to coincide, so the curves touch. At every other output the firm would have preferred a different plant, so LRAC is strictly below. This is why LRAC is the "envelope" of all the short-run average cost curves — touching each at one point and lying below it everywhere else.
Now the fixed-cost question. "Can there be fixed costs in the long run?" No — and this is why. A cost is fixed precisely because the firm cannot vary the input causing it. But the long run is defined as the period in which all inputs can be varied. So a fixed cost in the long run would be a contradiction in terms: an input that is simultaneously variable (long run) and not variable (fixed).
Both answers come from the same source. The long run is not "a lot of time" — it is the absence of constraints. That absence is what removes fixed costs, and it is what guarantees LRAC can never exceed SAC. One idea, two results.
The total cost schedule is given. Find TFC, and calculate TVC, AFC, AVC, SAC and SMC.
| Q | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| TC | 10 | 30 | 45 | 55 | 70 | 90 | 120 |
At q = 0, TVC = 0, so TC = TFC = ₹10 at every output.
| Q | TC | TFC | TVC | AFC | AVC | SAC | SMC |
|---|---|---|---|---|---|---|---|
| 0 | 10 | 10 | 0 | — | — | — | — |
| 1 | 30 | 10 | 20 | 10 | 20 | 30 | 20 |
| 2 | 45 | 10 | 35 | 5 | 17.5 | 22.5 | 15 |
| 3 | 55 | 10 | 45 | 3.33 | 15 | 18.33 | 10 |
| 4 | 70 | 10 | 60 | 2.5 | 15 | 17.5 | 15 |
| 5 | 90 | 10 | 80 | 2 | 16 | 18 | 20 |
| 6 | 120 | 10 | 110 | 1.67 | 18.33 | 20 | 30 |
Note SAC bottoms at q = 4 (17.5) while AVC bottoms at q = 3–4 (15) — SAC's minimum again lies at or to the right of AVC's. ✓
A firm's SMC schedule is shown. Total fixed cost is ₹100. Find TVC, TC, AVC and SAC.
| Q | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| SMC | 500 | 300 | 200 | 300 | 500 | 800 |
Key idea: the sum of marginal costs up to any output is the total variable cost at that output. So TVC is the running total of SMC.
| Q | SMC | TVC | TFC | TC | AVC | SAC |
|---|---|---|---|---|---|---|
| 1 | 500 | 500 | 100 | 600 | 500 | 600 |
| 2 | 300 | 800 | 100 | 900 | 400 | 450 |
| 3 | 200 | 1000 | 100 | 1100 | 333.33 | 366.67 |
| 4 | 300 | 1300 | 100 | 1400 | 325 | 350 |
| 5 | 500 | 1800 | 100 | 1900 | 360 | 380 |
| 6 | 800 | 2600 | 100 | 2700 | 433.33 | 450 |
Watch the crossings. AVC bottoms at q = 4 (325); SMC at q = 4 is 300, below it, and at q = 5 is 500, above it — so SMC cuts AVC between 4 and 5. Both averages turn upward once SMC has risen past them. ✓
Why do AVC and SAC both appear to bottom at q = 4 here? Because the data are
coarse. "Minimum SAC lies strictly to the right of minimum AVC" is a theorem about
smooth curves: since SAC = AVC + AFC and AFC is always falling,
SAC keeps being pulled down for a little longer after AVC has turned up.
With only whole-number outputs, that "little longer" can be less than one unit — so the two minima land in the same column. The theorem is not violated; the table is simply too coarse to show the gap.
28. Q = 5 L½ K½ — maximum output with L = 100, K = 100?
29. Q = 2L²K² — output with L = 5, K = 2? And with L = 0, K = 10?
30. Q = 5L + 2K — output with L = 0, K = 10?
28. Q = 5 × 100½ × 100½ = 5 × 10 × 10 = 500
29. Q = 2 × 5² × 2² = 2 × 25 × 4 = 200
With L = 0: Q = 2 × 0² × 10² = 0
30. Q = 5(0) + 2(10) = 20
Compare 29 and 30 carefully — this is the point of the pair.
In 29 the inputs are multiplied, so both are essential: with zero labour, output is zero no matter how much capital there is.
In 30 the inputs are added, so they are perfect substitutes: capital alone can produce output without any labour at all.
Note also that 29 has α + β = 4 → strong IRS, while 28 has α + β = 1 → CRS.
We now know what production costs. But Chapter 3 never told us how much to produce — deliberately, because that question needs a price. Chapter 4 supplies one, assumes the firm is a ruthless profit maximiser, and derives the supply curve from the cost curves we have just built.
Watch for the payoff: the firm's supply curve turns out to be nothing but the rising part of the marginal cost curve, and its two landmarks — the shut-down point and the break-even point — are min AVC and min SAC.