IAI Actuarial Core Principles · Business Economics

Production, costs, revenue and profit in price and output decisions: formula sheet

Full chapter guide

Key formulas

Production function: Q = f(L, K); Q is output, L labour, K capital. In the short run K is fixed.
Average product: AP = TP ÷ L; Output per unit of the variable input.
Marginal product: MP = ΔTP ÷ ΔL; Extra output from one extra unit of the variable input.
MP and AP relationship: MP > AP ⇒ AP rises; MP < AP ⇒ AP falls; MP = AP at maximum AP; MP cuts AP from above at AP's peak.
Maximum total product: TP is maximum when MP = 0; If MP is negative, TP falls.
Returns to scale test: Scale all inputs by k (k > 1). Compare f(kL, kK) with k × f(L, K); Greater: increasing. Equal: constant. Smaller: decreasing.
Cobb-Douglas returns to scale: Q = A × L^a × K^b: increasing if a + b > 1, constant if a + b = 1, decreasing if a + b < 1; Holds for this form with A > 0 and positive a, b.
Total cost: TC = FC + VC; FC is constant in the short run. At Q = 0, TC = FC.
Average total cost: ATC = TC ÷ Q = AFC + AVC; Valid for Q > 0.
Average fixed and variable cost: AFC = FC ÷ Q; AVC = VC ÷ Q; AFC falls continuously as Q rises.
Marginal cost: MC = ΔTC ÷ ΔQ = ΔVC ÷ ΔQ; with a cost function, MC = dTC/dQ; Fixed cost drops out of MC.
MC and average rule: MC < average → average falls; MC > average → average rises; MC = average at the minimum of the average; Applies to both AVC and ATC.
Economic cost: Economic cost = explicit cost + implicit cost; Implicit cost is the opportunity cost of owner-supplied resources.
Scale effects: LRAC falls: economies of scale; LRAC constant: constant returns; LRAC rises: diseconomies of scale; Long-run concept; all inputs variable.
Total revenue: TR = P × Q; Price times quantity sold.
Average revenue: AR = TR ÷ Q = P; AR is the price, so the AR curve is the demand curve.
Marginal revenue (discrete): MR = ΔTR ÷ ΔQ; Use for tables. For a one-unit step, MR = TR(n) − TR(n−1).
Marginal revenue (continuous): MR = dTR/dQ; Differentiate TR as a function of Q.
Perfect competition: AR = MR = P; Horizontal demand curve for the individual firm.
Linear demand: P = a − bQ gives TR = aQ − bQ² and MR = a − 2bQ; MR has twice the slope of demand and the same price intercept.
MR and elasticity: MR = P(1 + 1/E) = P(1 − 1/|E|), where E is the (negative) price elasticity of demand; MR > 0 if |E| > 1, MR = 0 if |E| = 1, MR < 0 if |E| < 1.
Profit: π = TR − TC = (AR − AC) × Q; With AR = price. Positive means supernormal profit.
Profit-maximising rule: MR = MC, with MC cutting MR from below; Second condition: slope of MC is greater than slope of MR at that output.
Marginal revenue and cost: MR = ΔTR ÷ ΔQ; MC = ΔTC ÷ ΔQ; With calculus: MR = dTR/dQ, MC = dTC/dQ.
Perfect competition: P = AR = MR, so profit maximum at P = MC; Firm is a price taker, with a horizontal demand curve.
Normal profit (break-even): AR = AC, so TR = TC; Economic profit is zero. Owner earns exactly the opportunity cost.
Supernormal profit and loss: AR > AC: supernormal profit; AR < AC: loss; Total profit is the gap (AR − AC) multiplied by Q.
Short-run shutdown rule: Shut down if P < AVC; Continue if AVC ≤ P < AC, because the loss is smaller than fixed cost.
Second-order condition (calculus): d²π/dQ² < 0; Equivalent to MC rising faster than MR at the optimum.

Quick revision

Short run: at least one input is fixed. Long run: all inputs can vary.
Diminishing returns concern one variable input with others fixed; returns to scale concern changing all inputs together.
Returns to scale: increasing if output rises by a greater proportion than inputs, constant if equal, decreasing if smaller.
TC = FC + VC; AC = TC ÷ Q; AVC = VC ÷ Q; AFC = FC ÷ Q.
MC = change in TC ÷ change in Q, and it is not affected by fixed cost.
MC cuts AVC and AC at their lowest points.
TR = P × Q; AR = TR ÷ Q, which equals price; MR = change in TR ÷ change in Q.
For a price taker, P = AR = MR and the demand curve is horizontal.
For a firm with a downward-sloping demand curve, MR lies below AR.
Profit is maximised where MR = MC and MC is rising through MR.
Economic profit = TR − TC, where TC includes opportunity cost, so normal profit is included in costs.
In the short run, a firm should keep producing if price covers AVC, and shut down if price is below AVC.

Common mistakes

Saying diminishing returns means output falls. Fix: Say that MP falls, while TP still rises as long as MP is positive.
Mixing up diminishing returns and decreasing returns to scale. Fix: Diminishing returns: short run, one input varied, others fixed. Returns to scale: long run, all inputs varied in proportion.
Including fixed cost in marginal cost. Fix: MC is the extra cost of one more unit. FC does not change, so it drops out. TC ÷ Q is average cost.
Saying MC cuts ATC at the minimum of MC. Fix: MC cuts ATC and AVC at the minimum of ATC and AVC, not at the minimum of MC. MC reaches its own minimum earlier.
Saying MR = P for a monopolist. Fix: Only a price taker has MR = P. With downward-sloping demand, MR < P because the price cut applies to all units.
Using the wrong slope for the MR curve, such as the same slope as demand. Fix: Differentiate TR. For P = a − bQ, MR = a − 2bQ, twice as steep.
Reading the price off the MC or MR curve at the optimum output. Fix: After finding Q, go up to the demand (AR) curve. That height is the price. Then compare it with AC at the same Q.
Using MR = P for a firm with a downward-sloping demand curve. Fix: P = MR only for a price taker. For P = a − bQ, MR = a − 2bQ. Check the market structure first.

Exam tips

Read the first line for the time frame. One fixed input means short run and diminishing returns. All inputs variable means long run and returns to scale.
In MCQs, watch for options that say output falls. Diminishing returns means MP falls, not necessarily TP.
In tables, compute MP first. Most marks are for correct differences and for naming the point where MP starts to fall.
In written answers, define the term, give the numbers, then state the conclusion in one sentence.
Remember that MP cuts AP at AP's maximum. Examiners like this as a reasoning check.
Draw MC, AVC, ATC and AFC on one diagram and label the crossing points. Examiners reward correct crossings.
Always name the reason for a curve shape: diminishing marginal returns for the short run, economies and diseconomies of scale for the long run.
In numerical questions, show the formula, the working and the unit. Mark schemes award method marks.