Unit 4: Cost Theory and Estimation

Cost theory links a firm's production technology to the money value of resources it uses, and cost estimation fits that theory to real data so managers can price, plan output, and size plants. The economist's cost is opportunity cost, so every cost function below rests on the distinction between what is paid and what is forgone.

• Economic vs. accounting cost: Accounting cost records historical outlays only; economic cost adds implicit costs (owner's forgone salary, forgone interest on own capital).
• Explicit vs. implicit cost: Explicit costs involve actual cash payment (wages, rent); implicit costs are the value of self-owned resources in their next-best use.
• Time periods: The short run has at least one fixed input (usually capital, K); the long run has all inputs variable and no fixed cost.
• Cost as derived from production: Every cost curve is the money mirror of a production function; diminishing marginal returns in the short run and returns to scale in the long run drive the curve shapes.
• Relevant cost for decisions: Only incremental and avoidable costs matter; sunk costs are ignored.

II. Short Run Cost Functions

Short-run costs split output-independent (fixed) and output-dependent (variable) components, and their shapes follow directly from the law of diminishing marginal returns.

A. Total cost families

Total cost is the vertical sum of fixed and variable cost at each output level.

• Total fixed cost (TFC): Cost of fixed inputs; constant at all outputs, positive even at Q = 0 (e.g., a $500 lease).
• Total variable cost (TVC): Cost of variable inputs; zero at Q = 0, rising with output, first at a decreasing then an increasing rate.
• Total cost (TC): Their sum.

TC(Q) = TFC + TVC(Q)

• Symbols: Q = output; TFC = total fixed cost; TVC = total variable cost.

B. Average and marginal cost

Per-unit measures come from dividing totals by Q or differentiating them.

• Average fixed cost (AFC = TFC/Q): Falls continuously toward zero as fixed cost spreads over more units ("spreading overhead").
• Average variable cost (AVC = TVC/Q): U-shaped; falls while marginal product rises, rises when diminishing returns dominate.
• Average total cost (ATC = TC/Q = AFC + AVC): U-shaped, lying above AVC by the shrinking AFC gap.
• Marginal cost (MC = ΔTC/ΔQ = dTC/dQ = dTVC/dQ): Cost of one more unit; independent of TFC.

MC = dTC/dQ = w / MPL

• Symbols: w = wage of the variable input; MPL = marginal product of labour. MC and MPL move inversely, so rising MPL means falling MC.

C. Relationships among the curves

The curves intersect at points fixed by algebra, not coincidence.

• MC cuts AVC and ATC at their minima: When MC < average, the average falls; when MC > average, it rises; equality occurs at the trough.
• Cubic cost form: A common estimated specification is TC = a + bQ + cQ² + dQ³, giving a U-shaped MC and ATC when b, d > 0 and c < 0.
• Worked example: If TVC = 10Q − 0.5Q² + 0.02Q³, then MC = 10 − Q + 0.06Q²; at Q = 10, MC = 10 − 10 + 6 = 6.

III. Long Run Cost Curves

In the long run the firm chooses plant size freely, so long-run cost is an envelope of the best short-run option for each output.

A. Deriving the long-run curve

The long-run average cost curve (LRAC) traces the lowest attainable ATC for each output.

• Envelope property: LRAC is tangent to each short-run ATC curve, never crossing below any of them; it is not drawn through their minima except at the LRAC minimum.
• Planning function: Each point on LRAC identifies the optimal plant size for that output.
• Expansion path basis: LRAC derives from the least-cost input combination where the isoquant meets the isocost, i.e., where MPL/w = MPK/r.
  • Symbols: MPK = marginal product of capital; r = rental price of capital.

B. Shape of the LRAC

The typical LRAC is U-shaped, driven by returns to scale rather than diminishing returns.

• Falling segment: Economies of scale lower unit cost as output rises.
• Flat segment: Constant returns to scale give a range of minimum efficient plant sizes.
• Rising segment: Diseconomies of scale raise unit cost at large outputs.
• Long-run marginal cost (LRMC): Cuts LRAC at its minimum; lies below LRAC when LRAC falls, above when it rises.

C. Minimum efficient scale

The smallest output at which LRAC reaches its minimum shapes market structure.

• Definition (MES): Lowest Q at which long-run average cost stops falling.
• Structural signal: A high MES relative to market demand supports few large firms; a low MES supports many small firms.

IV. Economies of Scale

Economies of scale are reductions in long-run average cost as the scale of operation expands, and their exhaustion produces diseconomies.

A. Returns to scale and cost

Cost economies mirror the production function's response to proportional input changes.

• Increasing returns to scale: Output more than doubles when all inputs double, so LRAC falls.
• Constant returns to scale: Output doubles exactly, LRAC flat.
• Decreasing returns to scale: Output less than doubles, LRAC rises.
• Output elasticity of cost: Measured as E_c = (dTC/TC)/(dQ/Q); E_c < 1 signals economies of scale.

B. Sources of economies of scale

Internal economies arise within the firm as it grows.

• Technical: Larger, more specialised equipment lowers unit cost; the "container rule" — volume rises faster than surface area, so a tank's capacity outpaces its material cost.
• Labour specialisation: Division of labour raises productivity at higher volumes.
• Managerial: Fixed administrative and R&D costs spread over more units.
• Financial: Large firms borrow at lower interest rates.
• Marketing and purchasing: Bulk buying secures input discounts.

C. Diseconomies of scale

Beyond MES, unit cost rises as size creates its own burdens.

• Managerial diseconomies: Coordination and communication grow complex; layers of hierarchy slow decisions.
• Motivational: Worker alienation in very large plants cuts productivity.
• Distinction from external economies: Internal economies depend on the firm's own size; external economies depend on industry size (shared suppliers, skilled labour pool) and shift the whole LRAC down.

V. Learning Curves

The learning curve captures how unit cost or labour input falls as cumulative output rises, reflecting experience rather than scale.

A. Concept and distinction

Learning economies come from doing, not from being large.

• Principle: Repetition improves worker skill, workflow, and tooling, so the labour time per unit declines with cumulative volume.
• Learning vs. scale: Economies of scale relate cost to the rate of output per period; learning relates cost to cumulative output over time. A firm can move along its learning curve without changing plant size.

B. The learning-curve formula

A constant-percentage relationship links unit cost to cumulative units produced.

Yₓ = a · x^(−b)
b = −log(learning rate) / log(2)

• Symbols: Yₓ = labour input (or cost) for the xᵗʰ unit; a = input for the first unit; x = cumulative units; b = learning coefficient.
• Learning rate: An 80% curve means each doubling of cumulative output cuts unit input to 80% of its prior value.
• Worked example: With a = 100 hours and an 80% curve, unit 1 = 100 hrs, unit 2 = 80 hrs, unit 4 = 64 hrs, unit 8 = 51.2 hrs — each doubling multiplies by 0.8.

C. Applications and limitations

The curve guides pricing, bidding, and capacity planning where production is repetitive.

• Pricing strategy: Firms may price near future (lower) cost to win share and ride down the curve — penetration pricing.
• Bidding and budgeting: Forecasting labour hours for later units in a contract avoids overpricing.
• Make-or-buy timing: A steep curve favours in-house production once experience accumulates.
• Limitations: Gains plateau as learning is exhausted; new products, process changes, or high labour turnover reset the curve; it applies mainly to labour-intensive, repetitive tasks, not automated lines.

D. Estimation of cost functions

Fitting theoretical cost curves to data lets managers quantify these relationships.

• Time-series regression: Regress a firm's cost on output over time; must adjust for inflation and technology change.
• Cross-section regression: Compare costs across firms of different sizes at one point in time to estimate LRAC.
• Engineering method: Build cost estimates from physical input requirements when historical data are thin or plant capacity is new.
• Survivor technique: Infer efficient scale from which firm sizes gain market share over time, on the logic that survivors have lower costs.