Cost and Revenue Curves
A firm plays a dual role — as a producer it minimises cost, and as a seller it maximises revenue. This unit covers the cost function, various cost concepts (explicit, implicit, fixed, variable), short-run and long-run cost curves, the relationship between AC and MC, economies and diseconomies of scale, economies of scope, revenue concepts, and the relationship between price elasticity and revenue.
In this chapter
Cost Function
A cost function shows the relationship between the cost of production and the level of output. Cost is influenced by the level of output, the price of inputs, and technology. The general cost function is C = f(Q, P_f, T), where C is cost, Q is output, P_f is input prices, and T is technology. For simplicity, we assume cost is a function of output alone: C = f(Q).
Cost function
Different Concepts of Cost
Key Cost Concepts
- Explicit cost — actual money payments to outsiders (wages, rent, raw materials). Recorded in accounting.
- Implicit cost — the imputed value of self-owned resources used in production (owner's time, owned land). Not recorded in accounting.
- Economic cost = Explicit cost + Implicit cost. Used in economic decisions.
- Fixed cost (FC) — costs that do not vary with output (rent, salaries of permanent staff). Exists only in the short run.
- Variable cost (VC) — costs that vary with output (raw materials, hourly wages).
- Total cost (TC) = FC + VC. In the long run, all costs are variable: TC = VC.
Short-Run Cost Curves
- In the short run, some costs are fixed and some are variable. The key per-unit cost curves are: Average fixed cost (AFC) = FC/Q — continuously falling as output rises
- Average variable cost (AVC) = VC/Q — U-shaped
- Average cost (AC) = TC/Q = AFC + AVC — U-shaped
- Marginal cost (MC) = ΔTC/ΔQ — U-shaped, lies below AC when AC is falling and above when AC is rising. MC cuts both AVC and AC at their minimum points.
Per-unit cost formulas (short run)
Relationship Between AC and MC
AC–MC Relationship
- Both AC and MC are derived from total cost, and both are U-shaped.
- When AC is falling, MC is below AC and falls faster than AC.
- When AC is rising, MC is above AC and rises faster than AC.
- When AC is at its minimum, MC equals AC — MC cuts AC at its lowest point.
- The same relationship holds between AVC and MC.
Long-Run Cost Curves
In the long run, all factors are variable — there are no fixed costs. The long-run average cost (LAC) curve is the envelope of all short-run average cost curves. It is U-shaped due to economies and diseconomies of scale. The long-run marginal cost (LMC) curve is derived from the LAC and intersects it at its minimum point. The LAC is flatter than the short-run AC curves because the firm has more flexibility to adjust all inputs.
Economies and Diseconomies of Scale
Economies of scale are the decrease in long-run average cost as output increases. Internal economies arise from the firm's own growth — economies in production, marketing, management, and transport. External economies arise from the growth of the industry as a whole. Diseconomies of scale are the increase in long-run average cost as output grows further — caused by managerial inefficiency, labour problems, and communication difficulties in very large organisations.
Economies of scope — joint production is cheaper
Economies of Scope
Economies of scope exist when it is cheaper for one firm to produce two products jointly than for two separate firms to produce them independently: TC(Q_A, Q_B) < TC(Q_A) + TC(Q_B). Causes include shared production facilities, use of by-products, and common marketing/administration. Example: a university teaching both economics and management shares lecture halls, libraries, and admin staff.
Theory of Revenue
Revenue is the money a firm receives from selling its output. Total revenue (TR) = Price × Quantity = P × Q. Average revenue (AR) = TR/Q = P (under any market structure, AR equals price). Marginal revenue (MR) = ΔTR/ΔQ — the additional revenue from selling one more unit. Under perfect competition, the firm is a price taker: AR = MR = P (horizontal AR/MR curve). Under imperfect competition (monopoly, monopolistic competition, oligopoly), the firm faces a downward-sloping demand curve: AR falls, and MR lies below AR.
Total, average, and marginal revenue
Relationship: Elasticity, MR, AR, and TR
Relationship Between Price Elasticity and Revenue
| Elasticity | MR Sign | Effect of Price Cut on TR | Effect of Price Rise on TR |
|---|---|---|---|
| Elastic (E_p > 1) | MR > 0 | TR rises ↑ | TR falls ↓ |
| Unitary (E_p = 1) | MR = 0 | TR unchanged | TR unchanged |
| Inelastic (E_p < 1) | MR < 0 | TR falls ↓ | TR rises ↑ |
Relationship between MR and price elasticity
Profit Maximisation Rule
A firm maximises profit where MR = MC (and MC cuts MR from below). If MR > MC, producing more adds to profit. If MR < MC, producing more reduces profit. At MR = MC, the firm has no incentive to change output — this is the equilibrium.
Practice Problem
A firm has the total cost function TC = 100 + 20Q + Q². (a) Find FC, VC, AFC, AVC, AC, and MC. (b) If the firm sells in a perfectly competitive market at P = Rs 40, find the profit-maximising output and the level of profit.
AC, AVC, AFC, and MC: Detailed Relationships
All four per-unit cost curves are derived from total cost. AFC (Average Fixed Cost = FC/Q) is a rectangular hyperbola — it falls continuously as output rises because the same fixed cost is spread over more units. AVC (Average Variable Cost = VC/Q) is U-shaped: initially falls due to increasing returns, then rises due to diminishing returns. AC (Average Cost = TC/Q = AFC + AVC) is also U-shaped but lies above AVC; the vertical gap between AC and AVC equals AFC, which shrinks as output rises. MC (Marginal Cost = ΔTC/ΔQ) is U-shaped and cuts both AVC and AC at their minimum points from below.
Cost Schedule Showing AC, AVC, AFC, MC
| Q | FC | VC | TC | AFC | AVC | AC | MC |
|---|---|---|---|---|---|---|---|
| 0 | 50 | 0 | 50 | — | — | — | — |
| 1 | 50 | 30 | 80 | 50 | 30 | 80 | 30 |
| 2 | 50 | 50 | 100 | 25 | 25 | 50 | 20 |
| 3 | 50 | 70 | 120 | 16.7 | 23.3 | 40 | 20 |
| 4 | 50 | 100 | 150 | 12.5 | 25 | 37.5 | 30 |
| 5 | 50 | 150 | 200 | 10 | 30 | 40 | 50 |
Long-Run Cost Curves: The Envelope Curve
The Long-Run Average Cost (LAC) curve is the envelope of all short-run average cost (SAC) curves. Each SAC corresponds to a particular plant size. The LAC touches each SAC at one point — tangency. For any output level, the firm chooses the plant size whose SAC is lowest at that output. The LAC is U-shaped due to economies of scale (falling portion) and diseconomies of scale (rising portion). The minimum point of the LAC is the minimum efficient scale — the smallest output at which long-run average cost is minimised.
Break-Even Analysis
The break-even point is the output level at which total revenue equals total cost — the firm earns zero economic profit (but covers all costs including normal profit). At break-even: TR = TC, or equivalently P × Q = FC + VC × Q. Solving for Q: Q_BE = FC ÷ (P − VC), where (P − VC) is the contribution margin per unit. Below Q_BE, the firm makes a loss; above it, a profit. Break-even analysis helps firms decide whether to enter a market, set prices, and plan production.
Break-even quantity
Real-Life Example: Nepali Tea Factory
A tea factory in Ilam has fixed costs of Rs 500,000 per year (machinery, rent). Variable cost per kg of processed tea is Rs 80 (labour, packaging). Selling price is Rs 200/kg. Break-even quantity = 500,000 ÷ (200 − 80) = 500,000 ÷ 120 = 4,167 kg/year. If the factory produces and sells 5,000 kg, it makes a profit of (5,000 − 4,167) × Rs 120 = Rs 99,960. If it sells only 3,000 kg, it loses (4,167 − 3,000) × Rs 120 = Rs 140,040.
Memory Aid: MC and AC Relationship
Think of MC as the newest student joining a class, and AC as the class average. If the new student (MC) scores above the class average (AC), the average rises. If below, the average falls. If equal, the average stays. MC always pulls AC toward itself — that is why MC cuts AC at its minimum point.
Key Terms and Definitions
- Cost function: Relationship between cost and output, C = f(Q) — लागत प्रकार्य: लागत र निर्गतबीचको सम्बन्ध।
- Fixed Cost (FC): Cost that does not vary with output (rent, salaries) — स्थिर लागत: निर्गतसँग परिवर्तन नहुने लागत।
- Variable Cost (VC): Cost that varies with output (raw materials) — परिवर्तनशील लागत: निर्गतसँग परिवर्तन हुने लागत।
- Marginal Cost (MC): Additional cost of producing one more unit, MC = ΔTC/ΔQ — सीमान्त लागत: एक थप एकाइ उत्पादनको थप लागत।
- Economies of Scale: LAC falls as output increases — स्केल अर्थव्यवस्था: निर्गत बढ्दा LAC घट्छ।
- Economies of Scope: Joint production cheaper than separate production — स्कोप अर्थव्यवस्था: संयुक्त उत्पादन छुट्टाछुट्टै भन्दा सस्तो।
- Total Revenue (TR): Total money received from sales, TR = P × Q — कुल आय: बिक्रीबाट प्राप्त कुल पैसा।
- Break-even point: Output where TR = TC (zero profit) — ब्रेक-इभन बिन्दु: TR = TC हुने निर्गत (शून्य नाफा)।
- Contribution margin: Price minus variable cost per unit, P − VC — योगदान मार्जिन: प्रति एकाइ मूल्य घटाउँदा परिवर्तनशील लागत।
- Envelope curve: LAC is the envelope of all SAC curves — खोप वक्र: LAC सबै SAC वक्रहरूको खोप हो।
Practice Problem
A bakery sells each cake for Rs 500. The fixed cost is Rs 10,000 per month. The variable cost per cake is Rs 300. (a) Calculate the break-even quantity. (b) If the bakery sells 80 cakes per month, what is the profit? (c) How many cakes must it sell to earn a target profit of Rs 5,000?