Theory of Production: formula sheet

Key formulas

Production function

Q = f(L, K, ...)

Q is maximum output for given inputs and given technology. Add land and enterprise if the question lists them.

Short-run production function

Q = f(L) with K fixed

At least one input is fixed. Output changes only by changing variable inputs.

Long-run production function

Q = f(L, K) with all inputs variable

The firm can change plant size. Linked to returns to scale.

Fixed vs variable factor

Fixed factor: does not change with output. Variable factor: changes with output.

Rent of a factory building is fixed in the short run. Raw material is variable.

Rewards of factors

Land → rent; Labour → wages; Capital → interest; Enterprise → profit

Often asked as match the following.

Average product

AP = TP ÷ L

L is the number of units of the variable input.

Marginal product

MP = TPn − TP(n−1) = ΔTP ÷ ΔL

Use the change in TP when one more unit is added.

Total product from MP

TP = Σ MP

TP at any level is the sum of all MPs up to that level.

MP and AP link

MP > AP: AP rises; MP = AP: AP is maximum; MP < AP: AP falls

MP cuts AP at its highest point.

Stage boundaries

Stage I ends at maximum AP; Stage II ends at MP = 0 (TP maximum); Stage III has MP < 0

Stage II is where a rational producer operates.

Increasing returns to scale

% change in output > % change in all inputs

Output rises more than proportionately. If inputs ×k, output > ×k (k > 1).

Constant returns to scale

% change in output = % change in all inputs

Output rises exactly in proportion. If inputs ×k, output ×k.

Decreasing returns to scale

% change in output < % change in all inputs

Output rises less than proportionately. If inputs ×k, output < ×k (k > 1).

Percentage change

% change = (New − Old) ÷ Old × 100

Use this for both output and inputs before comparing.

Isocost line

C = w × L + r × K

C is total outlay, w is wage rate, r is price of capital, L is labour and K is capital.

Slope of isocost line

Slope = w ÷ r (ignoring sign)

This is the ratio of input prices. It is the vertical-axis intercept divided by the horizontal-axis intercept, when K is on the vertical axis and L on the horizontal.

Marginal rate of technical substitution

MRTS (L for K) = ΔK ÷ ΔL = MPL ÷ MPK

Taken as a positive number. It is the slope of the isoquant and it diminishes as you move down the curve.

Producer's equilibrium (least-cost condition)

MRTS = w ÷ r, or MPL ÷ w = MPK ÷ r

Isoquant is tangent to the isocost line. Also the isoquant must be convex to the origin at that point.

Properties of isoquants

Downward sloping; convex to origin; do not intersect; higher isoquant = more output

Isoquants are not necessarily parallel to each other. They do not touch the axes in the usual case of imperfect substitutes.

Average cost

AC = Total cost ÷ Output

Economies of scale mean AC falls as output rises. Diseconomies mean AC rises.

Internal vs external test

Source of benefit: own size = internal; industry size = external

Use this rule to classify any example in one step.

Long-run AC curve shape

Falling AC = economies; flat AC = constant; rising AC = diseconomies

The curve is usually U-shaped (or saucer-shaped) in textbooks.

Quick revision

• Production function shows the technical relationship between inputs and the maximum output they can give.
• Factors of production: land, labour, capital and enterprise, earning rent, wages, interest and profit.
• Short run: at least one input is fixed. Long run: all inputs are variable.
• Marginal product = change in total product from one more unit of the variable input.
• When marginal product is at its maximum, the total product curve has its steepest slope; when marginal product is zero, total product is at its maximum.
• Law of Variable Proportions: beyond a point, adding a variable input to fixed inputs gives diminishing marginal returns.
• Returns to scale: increasing, constant or decreasing, when all inputs change in the same proportion.
• An isoquant shows input combinations giving equal output; it slopes downward and is convex to the origin.
• Two isoquants never intersect, and a higher isoquant means more output.
• An isocost line shows input combinations that cost the same total amount.
• Producer's equilibrium: isocost line is tangent to an isoquant, so MRTS of labour for capital = MP_L ÷ MP_K = price of labour ÷ price of capital (w ÷ r).
• Internal economies arise within the firm; external economies arise from the growth of the industry.

Common mistakes

• Treating short run as a fixed period like one year. Fix: Define by inputs, not time. Short run means at least one factor is fixed.
• Calling money or a stock of cash the capital factor. Fix: In economics, capital in this context means man-made goods like machines and tools used to produce more goods.
• Saying Stage I ends where MP is maximum. Fix: MP peaks inside Stage I. Stage I ends where AP is maximum and MP = AP.
• Saying TP is maximum when MP is negative. Fix: TP is maximum when MP = 0. TP rises while MP is positive, even if MP is falling.
• Confusing returns to scale with returns to a factor. Fix: Ask whether all inputs change. All inputs, long run: returns to scale. One input with others fixed, short run: returns to a factor.
• Calling any rise in output increasing returns. Fix: Output always rises. Compare its proportion with the input proportion before naming the type.
• Saying isoquants can intersect or can slope upward. Fix: If two isoquants crossed, one point would give two different output levels, which is impossible. Downward slope arises because less of one input needs more of the other to hold output constant.
• Treating an isoquant as measuring satisfaction. Fix: An isoquant measures physical output and can be given numbers such as 100 units. An indifference curve shows ordinal satisfaction, so numbers on it have no cardinal meaning.
• Calling an industry-wide benefit an internal economy. Fix: Ask who benefits. If every firm in the industry gains, it is external.
• Confusing economies of scale with returns to scale. Fix: Returns to scale link inputs to physical output. Economies of scale link output to cost per unit.

Exam tips

• Expect direct MCQs on definitions, such as which statement correctly describes a production function.
• Learn the factor-reward pairs cold. They are quick marks.
• Read time-period questions for the word fixed. If one input cannot be changed, answer short run.
• Watch for options that say the long run is a period of a set number of years. Reject them.
• Remember technology is assumed constant when drawing a production function.
• Practise building MP and AP columns from a TP schedule quickly; most numerical MCQs need only one or two values.
• Memorise the three boundary markers: maximum AP, MP = 0, and MP negative.
• Watch for options that mix up the stages. Match each stage with TP, AP and MP behaviour before choosing.