CA Foundation · Business Economics

Theory of Production and Cost: formula sheet

Full chapter guide

Key formulas

Production function: Q = f(L, K, Land, Entrepreneur ...); Q is the maximum output from the given inputs and technology.
Short-run production function: Q = f(L) with K held 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; No fixed factor. Scale of production can change.
Factor rewards: Land → Rent; Labour → Wages; Capital → Interest; Entrepreneur → Profit; Frequently asked as a matching question.
Total product: TP = AP × L; L is the number of units of the variable input. Also TP = sum of all MPs up to that unit.
Average product: AP = TP ÷ L; Divide TP at that level by the units of the variable input at that level.
Marginal product: MP = ΔTP ÷ ΔL = TPn − TPn−1; When input rises by exactly one unit, MP is simply the difference between successive TP values.
TP as sum of MP: TPn = MP1 + MP2 + … + MPn; Holds when TP is zero at zero input and input rises one unit at a time.
AP-MP rule: MP > AP → AP rises; MP = AP → AP at maximum; MP < AP → AP falls; MP curve cuts AP curve from above at AP's highest point.
TP-MP rule: MP > 0 → TP rises; MP = 0 → TP maximum; MP < 0 → TP falls; TP rises at an increasing rate while MP rises, and at a decreasing rate while MP falls but stays positive.
Total product: TP = Σ MP (sum of marginal products up to that unit); If MP is given for each unit, add them up to get TP.
Marginal product: MP = ΔTP ÷ ΔL = TPₙ − TPₙ₋₁; L is the variable input. ΔL is the change in the variable input.
Average product: AP = TP ÷ L; L is the number of units of the variable input.
Stage I: MP rising; TP rising at an increasing rate; In the MP-based textbook division, Stage I ends where MP is highest and starts to fall. Some texts instead end Stage I where AP is highest (MP = AP).
Stage II: MP falling but positive (MP > 0); TP rising at a decreasing rate; Ends where MP = 0 and TP is maximum. A rational firm operates here.
Stage III: MP < 0; TP falling; A firm avoids this stage.
Total cost: TC = TFC + TVC; TFC is constant in the short run. TVC is zero at zero output.
Average fixed cost: AFC = TFC ÷ Q; Always falls as Q rises.
Average variable cost: AVC = TVC ÷ Q; U-shaped.
Average total cost: ATC = TC ÷ Q = AFC + AVC; The gap between ATC and AVC is AFC, and it narrows as Q rises.
Marginal cost: MC = ΔTC ÷ ΔQ = TCn − TC(n−1); Same as ΔTVC ÷ ΔQ, since TFC does not change.
Sum of MC: TVC at Q units = sum of MC of the first Q units; Useful when only MC values are given.
Economic cost: Economic cost = Explicit cost + Implicit cost; Accounting cost counts only explicit cost.
AC and MC rule: MC < AC → AC falls; MC = AC → AC is at minimum; MC > AC → AC rises; The same rule holds for MC and AVC.
Total revenue: TR = P × Q; P is price per unit and Q is quantity sold.
Average revenue: AR = TR ÷ Q = P; AR is the same as price. The AR curve is the demand curve.
Marginal revenue: MR = ΔTR ÷ ΔQ = TRn − TRn−1; Use TRn − TRn−1 when quantity rises by one unit.
Long-run average cost: LRAC = LTC ÷ Q; LTC is long-run total cost. The curve is the envelope of SAC curves.
Break-even condition: TR = TC (profit = 0); Profit = TR − TC. Profit is positive above break-even output, if price and costs stay unchanged.
Profit-maximising condition: MR = MC, with MC rising through MR; MC must cut MR from below.
Revenue under constant price (perfect competition): AR = MR = P; The firm is a price taker, so each extra unit sells at the same price.

Quick revision

Production function links inputs to maximum output for a given technology.
Short run: at least one factor is fixed. Long run: all factors can vary.
AP = TP ÷ units of variable input. MP = change in TP ÷ change in input.
When MP is above AP, AP rises. When MP is below AP, AP falls. MP cuts AP at AP's maximum.
TP is maximum when MP is zero. TP falls when MP is negative.
Law of variable proportions has three stages: increasing, diminishing and negative returns. A rational producer works in stage II.
Returns to scale: increasing, constant or decreasing, depending on whether output rises by more than, equal to or less than the proportion of inputs.
An isoquant shows combinations of inputs that give the same output. Isoquants slope downward and do not intersect.
Total cost = total fixed cost + total variable cost. Fixed cost does not change with output.
AC = TC ÷ Q. MC = change in TC ÷ change in Q. MC cuts AC at AC's minimum.
AFC falls continuously as output rises. AC and AVC are U-shaped.
Long-run average cost curve is U-shaped and envelops the short-run average cost curves.
AR = TR ÷ Q. MR = change in TR ÷ change in Q. Under perfect competition, price = AR = MR.

Common mistakes

Treating short run and long run as fixed time spans like one year or five years. Fix: Define by flexibility of inputs. Short run has at least one fixed factor. Long run has none.
Saying a fixed factor is one that costs nothing. Fix: A fixed factor does not vary with output in that period, but it still has a cost (fixed cost).
Dividing TP by input to get MP. Fix: MP is a difference, not a ratio. MP = change in TP ÷ change in input. AP is the ratio.
Saying MP equals AP when TP is maximum. Fix: MP = 0 at maximum TP. MP = AP at maximum AP. These are different points.
Saying diminishing returns means TP falls. Fix: Diminishing returns means MP falls, not TP. TP keeps rising as long as MP is positive. TP falls only in Stage III.
Mixing up the law of variable proportions with returns to scale. Fix: Variable proportions is short run with one variable input and others fixed. Returns to scale is long run with all inputs changed in the same proportion.
Including fixed cost in MC. Fix: MC is the change in TC for one more unit. Fixed cost cancels out in the subtraction.
Saying implicit costs are not real costs. Fix: Implicit costs are the opportunity cost of the owner's own resources. They count in economic cost but not in accounting cost.
Saying fixed costs exist in the long run. Fix: In the long run all inputs are variable, so all costs are variable.
Thinking the LRAC curve passes through the minimum points of all SAC curves. Fix: LRAC is tangent to each SAC curve. Where LRAC is falling, the tangency is to the left of that SAC's minimum point. Where LRAC is rising, it is to the right. Only at the minimum of LRAC (optimal scale) do the minimum points of LRAC and that SAC coincide.

Exam tips

Expect direct definition questions: short run, long run, fixed and variable factors. Learn each in one line.
Reward matching is a favourite. Memorise the four pairs: Rent, Wages, Interest, Profit.
Watch for options with always or never. Most definition-based MCQs reward careful wording.
Link this topic to the Law of Variable Proportions and Returns to Scale, as questions often combine the period with the law.
Skip a question if two options still look right after elimination, because wrong answers lose 0.25 marks.
Most questions ask for one cell of a table. Compute only that cell.
Learn the three rules: MP > AP means AP rises, MP = AP means AP maximum, MP < AP means AP falls.
Remember that MP is zero at maximum TP, and negative when TP falls.