Choose plant scale and distinguish industry cost patterns

Choose plant scale and distinguish industry cost patterns

Question and evidence

Does the cheapest small plant remain cheapest at every output?

A fictional firm can choose plant S with total cost 20+6 Q or plant L with cost 60+2 Q. For each output it may choose either plant before committing. An industry-level scenario then raises both plants’ input costs when many firms expand.

The numerical case is a teaching model, not a measured real-world effect.

Mechanism

Long-run costs allow all inputs to vary; short-run costs hold at least one input fixed. A cost envelope selects the cheapest available plant at each output. Economies of scale concern average cost as a firm scales up; increasing-cost industries concern input prices as the entire industry expands. Neither concept automatically implies the other.

Worked calculation

At Q=5, S costs 50 and L 70, so choose S. At Q=15, S 110 and L 90, so choose L. They cross at 20+6 Q=60+2 Q, giving Q=10 and total cost 80. Average cost there is 8.

Challenge and revision

The lower envelope of the two plant costs is a long-run choice across plants, not an extra plant with zero fixed cost. Industry expansion changes external input prices, while a single firm’s scale choice changes its own plant. These are different sources of cost change.

Plot both total-cost lines and trace the lower envelope. Then explain which line changes when industry expansion raises a shared wage or land price.

Revise your answer to “Does the cheapest small plant remain cheapest at every output?” using both source IDs. Separate the checked calculation from a judgment, and identify evidence that could reverse your judgment.

Vocabulary

Long-run cost envelope: The least-cost combination among feasible plant or input choices at each output.

Assumption: A condition accepted for a model or argument; it needs justification before applying the result elsewhere.

Independent extension

Independently extend the investigation “Choose plant scale and distinguish industry cost patterns” with a documented real dataset or scholarly source. Preserve definitions, recreate one calculation or evidence table, compare a rival explanation and report why the teaching example cannot establish the real-world conclusion.

Worked graph

Segments show the stated relationships; the table identifies every series without relying on color. Axes start at zero.

Series	Output	Total cost (credits)
Small plant	0	20
Small plant	5	50
Small plant	10	80
Small plant	15	110
Small plant	20	140
Large plant	0	60
Large plant	5	70
Large plant	10	80
Large plant	15	90
Large plant	20	100

Choose a plant, then trace average cost

This separate original fictional cost schedule offers three plants and eight integer output choices for one period. Choose one plant before committing; do not combine plants or interpolate fractional outputs. Costs include all stated economic costs. Prices of inputs are fixed within each scenario.

Predict: Can the least-cost average fall, stay constant and then rise as output changes? Is the largest plant always cheapest? Write the ranges you expect before checking the envelope.

Prediction and proposed graph

Writing is temporary; copy the report before leaving. Existing lesson experiments and completion records are unchanged.

Required output (units/period)12345678 Medium plant available before commitment Separate common input-cost scenarioNo added unit costAdd 2 credits per output unit to every plant

At Q = 4, compare total costs: S: 40 credits; M: 32 credits; L: 40 credits. Least cost = 32 credits; eligible best plant: M. Average cost = total/Q = 8 credits per unit. A tie retains both choices.

Read the envelope, not just one plant

Least available average costs for Q=1 through 8: 12, 9, 8, 8, 8, 9, 10, 12. The envelope minimizes the available plants at each specified quantity. It does not say the firm can instantly switch a committed short-run plant, or that these three choices exhaust real technology.

Adjacent average-cost changes: 1→2: decreasing; 2→3: decreasing; 3→4: constant; 4→5: constant; 5→6: increasing; 6→7: increasing; 7→8: increasing. Under the baseline menu and fixed input-price assumptions, falling average cost illustrates economies of scale, the flat Q=3–5 portion illustrates constant average cost, and the rise after Q=5 illustrates diseconomies. Physical returns to scale require a production/input account; cost numbers alone do not establish that mechanism.

Inspect total and average costs

Total costs are nondecreasing within each plant; no lower-cost higher output is hidden as a disposal option. There is no Q=0 average cost: total divided by zero is undefined, not zero.

Each cell: total credits / average credits per unit, current menu/scenario

Q	S	M	L	Envelope TC / AC	Best plant(s)
1	12 / 12	20 / 20	36 / 36	12 / 12	S
2	18 / 9	22 / 11	38 / 19	18 / 9	S
3	27 / 9	24 / 8	39 / 13	24 / 8	M
4	40 / 10	32 / 8	40 / 10	32 / 8	M
5	60 / 12	45 / 9	40 / 8	40 / 8	L
6	84 / 14	66 / 11	54 / 9	54 / 9	L
7	112 / 16	91 / 13	70 / 10	70 / 10	L
8	144 / 18	120 / 15	96 / 12	96 / 12	L

Transfer: remove a feasible choice

Remove the medium plant at Q=4

Predict what happens to the Q=3–5 flat segment when M is unavailable. Identify the new Q=4 choices and explain why a changed menu can alter the shape even if no remaining plant's technology changed. Do not call the old envelope a physical plant.

Then restore M and add the common 2-credit/unit cost. Each total becomes TC+2Q, so each positive-output average becomes AC+2. Compare plant rankings and shape before concluding anything about scale. This surcharge is stipulated; it is not evidence that a real industry's expansion raises input prices by this amount.

Revised cost graph, transfer and scope explanation

Compare with the reasoning guide

Baseline envelope AC is 12,9,8,8,8,9,10,12. S is cheapest at Q=1–2, M at 3–4 and L at 5–8. Without M, Q=3 costs 27 and AC 9; at Q=4, S and L tie at TC 40 / AC 10; at Q=5, L costs 40 / AC 8. Thus the former flat range disappears. Nothing here establishes a smooth tangent or every real scale option.

A common 2-credit/unit addition shifts every AC point up 2 and leaves all minimum-cost plant choices and adjacent differences unchanged. Distinguish this stipulated input-price change from changing a firm's output at unchanged prices, and from diminishing marginal productivity when a particular input is fixed. The finite schedule supports cost comparisons, not an empirical explanation of management, productivity or industry structure.

Method source and model provenance

OpenStax Principles of Microeconomics 3e, section 7.5 explains long-run plant choice and decreasing, constant and increasing average costs. All numerical schedules, diagram and transfer here are original fictional teaching material, not observations or copied textbook examples. The finite-menu envelope is a bounded illustration; real assessment requires an unfamiliar construction and review.

Copy your plant-choice report

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Reset clears writing and restores Q=4 / all plants / no surcharge; guidance exposure remains for this mount and clears on reload. Open guidance and ungraded writing are not independent mastery.

Reset plant envelope