Production and Costs

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Based on NCERT Economics Class XII (Introductory Microeconomics), Chapter 3.

What happens when you hire the tenth tailor?

A tailor has one sewing machine. Hire a second tailor and output more than doubles, because they share the work and the machine never sits idle. Hire a third and it still helps. By the tenth tailor, they queue for the machine and bump into each other, so the extra output is small. Nothing in the shop changed except the number of workers. That simple story is the law of variable proportions, and it drives every cost curve you will meet in this chapter. Costs are just production seen through the price of inputs.

Production, cost and profit

Production is the process by which inputs are transformed into output. It is carried out by producers (firms). A tailor uses a sewing machine, cloth, thread and labour to produce shirts. A farmer uses land, labour, a tractor, seed and water to produce wheat. The output may be consumed or used by other firms.

The chapter makes two simplifying assumptions. Production is instantaneous, so no time passes between combining inputs and getting output. And production and supply are used as synonyms. The money paid to acquire inputs is the cost of production. Selling the output earns revenue. The difference between revenue and cost is profit, and the firm's objective is to earn the maximum profit it can.

The production function

A production function is the relationship between inputs used and the output produced. For any quantities of inputs, it gives the maximum quantity of output that can be produced. Because it is a maximum, it deals only with efficient use of inputs: it is not possible to get more output from the same inputs.

A simple farm example: the farmer uses land $K$ (hectares) and labour $L$ (hours a day), and $q = K \times L$. With 1 hectare and 2 hours of labour the maximum output is 2 tonnes of wheat. For any $L$ and $K$ there is only one $q$. A production function is defined for a given technology. If technology improves, the maximum output for each input combination rises, and we have a new production function.

The inputs a firm uses are called factors of production. With two factors, labour $L$ and capital $K$:

$$q = f(L, K)$$

The table below shows part of an illustrative production function, with capital from 1 to 4 units. Each cell is the maximum output for that combination.

Reading the table: 1 unit of labour with 1 of capital gives at most 1 unit of output. 2 of labour and 2 of capital give 10. 3 of labour and 2 of capital give 18. If either input is zero, output is zero. More of any input raises output.

Isoquant. An isoquant is the set of all combinations of the two inputs that give the same maximum output. It is the production function's equivalent of an indifference curve. Output 10 can be made with (4 labour, 1 capital), (2 labour, 2 capital) or (1 labour, 4 capital). Output 50 can be made with (6 L, 3 K), (4 L, 4 K) or (3 L, 6 K). Isoquants slope downward: when marginal products are positive, using more of one input means the same output needs less of the other.

Beyond the textbook. For the exam: the production function gives the maximum output, so every firm is assumed to be efficient. In reality: many firms produce below the maximum because of poor organisation or idle resources. Why it matters: the production function is the benchmark of what is technically possible, so a firm below it is wasting inputs, not breaking the theory.

Short run and long run

Notice the last row. The long run is a longer period than the short run, but for different production processes it may be different lengths of time. Define the period by whether all inputs can be varied. In the example above, if capital is fixed at 4 units, the column for $K = 4$ shows the short run output of the firm as labour varies.

Total, average and marginal product

When one input varies and all others stay fixed, the relationship between the variable input and output is its total product (TP). Average and marginal product describe each worker's contribution.

$$AP_L = \dfrac{TP_L}{L}, \qquad MP_L = \dfrac{\Delta TP_L}{\Delta L} = TP_L - TP_{L-1}$$

TP is the sum of the marginal products of all units, and AP is the average of all marginal products up to that level. The marginal product of the first unit equals the average product of the first unit. MP is undefined at zero input.

Here is the short run schedule from the table with capital fixed at 4.

The curves in words. TP rises with labour. Its rate of increase is the MP, which first rises, so the TP curve steepens up to the third worker, and then falls, so the TP curve flattens. MP rises up to the third worker (16), then falls. This gives an inverse U-shaped MP curve. AP also rises at first, peaks at the third worker (13.33), and then falls, so it is inverse U-shaped too. The MP curve starts at the same point as AP and cuts AP from above at AP's maximum.

Why MP first rises. The tendency of MP to first rise and then fall is the law of variable proportions (the law of diminishing marginal product). It comes from changing factor proportions, the ratio in which two inputs are combined. Suppose a farmer has 4 hectares of land. With one worker, there is too much land for the worker to cultivate alone. As he adds workers, labour per unit of land rises, the proportions become more suitable, and each worker adds more than the last. This is the rising phase. When the fourth worker arrives, the land becomes crowded. Each worker now has insufficient land to work efficiently, so the output added by each extra worker is smaller.

Why MP cuts AP at the maximum of AP. AP is an average of marginal products. If the new marginal product is above the current average, it pulls the average up. If it is below, it drags the average down. So AP rises as long as MP > AP, falls when MP < AP, and is at its maximum where they meet. Think of a class average: a new student who scores above the class average raises it.

Worked example: building AP and MP from TP. TP of labour is 8, 20, 36, 49, 56, 59 for 1 to 6 workers. Find MP and AP, and identify where each peaks.

MP peaks at the 3rd worker (16), and diminishing returns begin from the 4th worker. AP peaks at the 4th worker (12.25). The 4th worker's MP (13) is above the AP of 12 at three workers, so AP rises to 12.25. At the 5th worker MP (7) is below AP (12.25), so AP falls. The MP curve cuts AP at its peak.

Common wrong step. Computing MP as TP ÷ L. That is average product. For the 5th worker, $56 \div 5 = 11.2$ is AP, but MP is $56 - 49 = 7$.

Worked example: working backwards from AP. The average product of labour is 3, 4, 5, 4.5 and 4 for 1 to 5 workers. Find TP and MP.

Common wrong step. Taking MP as the change in AP, or averaging AP values. MP always comes from the change in total product.

Pause and predict: at some level of employment, AP is 12 and MP is 10. Is AP rising or falling, and will the next worker's MP be above or below 12?

Check your answer: AP is falling, because MP is below AP. The next worker's MP could be anything, but until MP climbs back above AP, the average keeps falling.

Returns to scale

The law of variable proportions arises because one factor is fixed and proportions change. In the long run, both factors can change. The special case is when all inputs are scaled up by the same proportion $t > 1$.

Double all inputs. If output exactly doubles, it is CRS. More than doubles is IRS. Less is DRS.

For the Cobb-Douglas production function $q = x_1^{\alpha} x_2^{\beta}$, scaling both inputs by $t$ gives $q_1 = t^{\alpha + \beta} q_0$. So CRS holds if $\alpha + \beta = 1$, IRS if $\alpha + \beta > 1$ and DRS if $\alpha + \beta < 1$.

Worked example: classify the returns to scale. Test (a) $q = 5L^{1/2}K^{1/2}$, (b) $q = L^{0.6}K^{0.6}$, (c) $q = L^{0.3}K^{0.4}$, each by doubling both inputs.

For (a), check with numbers: at $L = K = 100$, $q = 5 \times 10 \times 10 = 500$, and at $L = K = 200$, $q = 5 \times \sqrt{200} \times \sqrt{200} = 5 \times 200 = 1000$. Output exactly doubles.

Common wrong step. Looking at one exponent alone, such as the 0.6 in (b). Returns to scale depend on the sum of the exponents, because both inputs rise together. Also, do not mix this up with variable proportions, where only one input rises.

Costs: from inputs to rupees

A given output can usually be made with many input combinations. In the table, 50 units can be made by (6 L, 3 K), (4 L, 4 K) or (3 L, 6 K). With input prices given, the firm picks the least expensive combination for each output level. The cost function gives the least cost of producing each level of output, given factor prices and technology.

Short run costs

In the short run, the cost of the fixed inputs is the total fixed cost (TFC): it does not change with output. The cost of the variable inputs is the total variable cost (TVC): it rises with output. Their sum is total cost.

$$TC = TFC + TVC$$

$$SAC = \dfrac{TC}{q}, \qquad AVC = \dfrac{TVC}{q}, \qquad AFC = \dfrac{TFC}{q}, \qquad SAC = AVC + AFC$$

$$SMC = \dfrac{\Delta TC}{\Delta q} = TC_q - TC_{q-1}$$

Since TFC cannot change in the short run, any change in TC is entirely a change in TVC. So SMC is also the increase in TVC for one more unit. The sum of marginal costs up to any level gives TVC at that level, and AVC is the average of all marginal costs up to that level. At zero output, SMC, AVC, AFC and SAC are undefined.

The standard cost schedule, with TFC of ₹20:

Totals:

Averages and marginal cost for the same schedule:

Together the two tables tell the full cost story. For instance, at output 5, SMC is $53 - 49 = 4$, that is ₹4. AVC at 5 units is $33 \div 5 = 6.6$, and $AFC + AVC = 4 + 6.6 = 10.6$, so SAC is ₹10.6.

Shapes of the cost curves in words.

The three relations to remember.

In the schedule above, AVC is lowest at 6 units (₹6.5): SMC at 6 units (₹6) is below it, and SMC at 7 units (₹8) is above it. SAC is lowest at 7 units (₹9.57): SMC at 7 units (₹8) is below it, and SMC at 8 units (₹13) is above SAC (₹10). The area under the SMC curve up to any output equals TVC at that output, and the area of the rectangle formed by AFC and output equals TFC.

Pause and predict: a firm's rent (a fixed cost) goes up by ₹100 a month. What happens to SMC and to AVC?

Check your answer: neither changes. SMC depends only on the extra variable cost of one more unit, and AVC depends only on TVC. TFC, AFC, TC and SAC all rise.

Worked example: building a cost table from total cost. The total cost of a firm is ₹40 at zero output, and ₹100, 150, 190, 245, 315 and 405 at outputs 1 to 6. Find TFC, TVC, AFC, AVC, SAC and SMC, and locate the minima.

Step 1: TFC equals total cost at zero output, ₹40, because nothing variable is used. Step 2: subtract it from TC to get TVC. Step 3: SMC is the change in TC (or TVC). Step 4: divide by $q$ for the averages.

Reading the minima: AVC is lowest at output 3 (₹50). SMC there is ₹40, below AVC, and the next SMC is ₹55, above the AVC of ₹50. That is the cut from below. SAC is lowest at output 4 (₹61.25), to the right of AVC's minimum. SMC at 4 is ₹55, below SAC, and SMC at 5 is ₹70, above SAC.

Common wrong step. Dividing TC by $q$ and calling it SMC. For output 4, $245 \div 4 = 61.25$ is SAC, but SMC is $245 - 190 = 55$. Another common slip is to leave out TFC when it must be read from TC at zero output.

Worked example: finding fixed cost from AFC. The AFC at 5 units is ₹8, and total costs are ₹70, 90, 105, 125 and 150 at outputs 1 to 5. Find TFC, TVC, SMC and AVC.

Common wrong step. Computing the first SMC as $70 - 0$, which treats TC at zero as zero. At zero output, TC equals TFC (₹40), so the first unit's SMC is $70 - 40 = 30$.

Beyond the textbook. For the exam: costs are the payments a firm makes to acquire inputs. In reality: economists also count implicit costs, such as the value of an owner's own time and capital, which an accountant would not list as a payment. Why it matters: it links this chapter to opportunity cost, and it explains why a business showing an accounting profit may still be a poor use of the owner's resources.

Long run costs

In the long run all inputs are variable, so there is no fixed cost and TC equals TVC. The long run average cost (LRAC) is TC divided by output. The long run marginal cost (LRMC) is the change in TC for one more unit of output.

$$LRAC = \dfrac{TC}{q}, \qquad LRMC = TC_{q} - TC_{q-1}$$

Returns to scale decide the shape of LRAC. Under IRS, inputs need to rise by a smaller proportion than output, so cost rises by a smaller proportion too, and average cost falls. Under DRS the opposite happens.

A typical firm is argued to have IRS at low output, then CRS, then DRS. So LRAC is U-shaped: the falling part is IRS, the rising part is DRS, and the minimum is where CRS is observed. LRMC starts at the same point as LRAC, is also U-shaped, and cuts LRAC from below at its minimum: when average cost is falling, marginal cost is below it, and when average cost rises, marginal cost is above it.

Beyond the textbook. For the exam: IRS makes LRAC fall and DRS makes LRAC rise. In reality: falling average cost usually comes from specialisation and buying inputs in bulk, and rising average cost from the difficulty of managing and coordinating a very large organisation. Why it matters: if a question asks why a large firm may become less efficient, think about coordination, not just input counts.

Exam traps

Formula sheet

60-second revision

Where this chapter connects

• Previous: Theory of Consumer Behaviour covers the consumer side, with indifference curves that isoquants mirror.
• Next: The Theory of the Firm under Perfect Competition uses these cost curves, especially SMC, to find the profit-maximising output.
• Later: Market Equilibrium combines firm supply with market demand.
• Foundation: Introduction explains the opportunity cost and technology ideas behind the production function.