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Intermediate Microeconomics Unit 6: Cost Minimization and Cost Curves

Firms choose the cheapest input mix for each output level; doing that for every output gives the cost function, whose short-run and long-run shapes drive supply decisions.

Covers: Economic vs accounting and sunk costs, Isocost lines and cost minimization, Conditional factor demands and cost functions, Short-run cost curves, MC, AVC and ATC relationships, Long-run costs and economies of scale, Multi-plant cost allocation

Study guide

Economic cost counts every opportunity cost, including implicit costs such as the owner's time and capital, so it can exceed accounting cost. Sunk costs, which have already been paid and cannot be recovered, are irrelevant to forward-looking choices. To produce output q at least cost, a firm chooses inputs on the isoquant for q that touch the lowest isocost line wL + rK = C. With smooth isoquants and both inputs used, the solution is a tangency: MRTS = MPL/MPK = w/r. Equivalently, MPL/w = MPK/r, so the last dollar spent on each input adds the same output. If an input price rises, the firm moves along the isoquant toward the other input.

The cost-minimizing inputs, written as functions of w, r and q, are the conditional factor demands, and plugging them in gives the cost function C(w, r, q). For q = K^0.5 L^0.5, the tangency gives K/L = w/r, and the cost function is C = 2q√(wr). Shephard's lemma says differentiating the cost function with respect to an input price recovers that input's conditional demand. Fixed-proportions technologies cost the price of the required input bundle per unit; perfect-substitute technologies use only the input that delivers output most cheaply, comparing w/MPL with r/MPK.

In the short run some inputs are fixed, so total cost splits into fixed cost (FC) and variable cost (VC). Average fixed cost FC/q falls steadily; average variable cost and average total cost are usually U-shaped because of diminishing marginal returns. Marginal cost, dTC/dq, is the same whether or not fixed costs are included. MC crosses both AVC and ATC at their minimum points: when MC is below an average, the average falls, and when it is above, the average rises. These relationships let you find minimum AVC or ATC by setting the derivative to zero or by setting MC equal to the average.

In the long run all inputs are variable, so the firm picks the best plant for each output. Long-run average cost is the lower envelope of the short-run ATC curves, each touching LRAC at the output for which that plant is optimal. Falling LRAC shows economies of scale, often from increasing returns, specialization or spreading fixed setup costs; rising LRAC shows diseconomies of scale. The output where LRAC first reaches its minimum is minimum efficient scale. A firm with several plants minimizes cost by allocating output so marginal costs are equal across plants.

Key ideas

• Economic cost includes implicit opportunity costs; sunk costs should be ignored.
• Cost minimization requires MRTS = w/r, or MPL/w = MPK/r.
• Conditional factor demands give the cheapest inputs for each output; they fall as their own price rises.
• MC crosses AVC and ATC at their minimum points.
• Short-run cost is never below long-run cost; LRAC is the envelope of short-run ATC curves.
• Economies of scale mean falling LRAC; constant returns give constant LRAC.
• Multi-plant firms equalize marginal cost across plants.

Formulas and rules

Common mistakes

• Including sunk costs when deciding whether to keep producing.
• Setting MPL = MPK instead of MPL/w = MPK/r.
• Forgetting fixed cost when computing short-run total or average total cost.
• Judging economies of scale from total cost instead of average cost.
• Splitting output evenly across plants instead of equalizing marginal cost.

Before the test, make sure you can

• Distinguish economic, accounting and sunk costs and use only avoidable costs in decisions.
• Solve for cost-minimizing inputs using the tangency condition.
• Derive cost functions for Cobb-Douglas, fixed-proportions and perfect-substitute technologies.
• Calculate FC, VC, ATC, AVC and MC from a cost function.
• Find the minimum of AVC or ATC and confirm it with MC.
• Compare short-run and long-run costs and explain the envelope relationship.
• Allocate output across plants by equalizing marginal costs.

Words to know

Economic cost

The full opportunity cost of resources used, including implicit costs.

Sunk cost

A cost that has been paid and cannot be recovered; it should not affect future choices.

Isocost line

All input combinations with the same total cost: wL + rK = C.

Cost minimization condition

MRTS = w/r, so each input gives the same extra output per dollar.

Conditional factor demand

The cost-minimizing quantity of an input as a function of input prices and output.

Cost function

The minimum cost of producing each output level at given input prices.

Shephard's lemma

The derivative of the cost function with respect to an input price equals that input's conditional demand.

Fixed cost

Cost that does not vary with output in the short run.

Variable cost

Cost that changes with the level of output.

Marginal cost

The extra cost of producing one more unit, dTC/dq.

Average total cost

Total cost divided by output, equal to AFC + AVC.

Average variable cost

Variable cost divided by output.

Long-run average cost

The lowest average cost at each output when all inputs can vary; the envelope of short-run ATC curves.

Economies of scale

Long-run average cost falls as output rises.

Diseconomies of scale

Long-run average cost rises as output rises.

Minimum efficient scale

The smallest output at which long-run average cost reaches its minimum.

Expansion path

The set of cost-minimizing input bundles as output rises with input prices fixed.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.