CB2400 Chap.8 Production, Isoquants and Cost Minimisation
Production, Isoquants and Cost Minimisation
Technology sets the feasible output
A production function gives the maximum output attainable from specified labour and capital under the existing technology. Marginal product changes one input while holding the others fixed.
Returns to scale instead changes all inputs by the same proportion, so the two ideas answer different questions.
Isoquants and isocosts mirror consumer geometry
An isoquant joins input bundles that produce one output level. Its slope magnitude is the marginal rate of technical substitution, MPL divided by MPK.
An isocost line joins combinations with the same expenditure, TC = wL + rK, and has slope magnitude w divided by r. Output chooses the isoquant; input prices choose the relevant isocost slope.
Least cost combines feasibility and slope
At a smooth interior cost minimum, MRTS = w/r, equivalently MPL/w = MPK/r. The target production equation supplies the scale of the input bundle and the tangency supplies its mix.
A candidate must satisfy both. A technology improvement shifts the required isoquant inward, while an input-price change rotates the isocost and can alter the chosen mix.
What this chapter covers
- 01
production functions
- 02
marginal products
- 03
returns to scale
- 04
isoquant slope
- 05
isocost equation
- 06
cost minimisation
Worked example · free
Choose the least-cost input bundle
- 3Derive MRTS from the two marginal products.
- 3Match MRTS to the input-price ratio.
- 4Combine the ratio with the output constraint and calculate cost.
Key terms
- Production Function
- The maximum output obtainable from stated input quantities under a given technology.
- Marginal Product
- The additional output from one more unit of an input while others stay fixed.
- Isoquant Curve
- A set of input bundles that can produce the same output.
- Technical Substitution Rate
- The input trade-off that holds output constant along an isoquant.
- Isocost Line
- A set of labour and capital bundles with the same total input cost.
Production, Isoquants and Cost Minimisation FAQ
What does an isoquant hold constant?
It holds output fixed while labour and capital change. Moving to an isoquant farther northeast normally represents greater output when more inputs raise production. Movement along one curve is input substitution, whereas movement between curves changes the production target or reflects a technology change.
Why are two equations needed for cost minimisation?
The slope condition chooses the least-cost input ratio, but many bundles can share that ratio at different scales. The production constraint places the firm on the isoquant for the required output. Solving them together fixes both the mix and quantities, which can then be audited through output and total cost.
What happens when the wage rises?
With labour on the horizontal axis, a higher wage makes the isocost line steeper because w divided by r rises. For a smooth interior technology, the firm generally substitutes away from labour and toward capital while still meeting output. The exact quantities require the production function and new tangency condition.
Exam move
Keep a two-column method: the left column is technology, with the target output and marginal products; the right is cost, with wL + rK and the input-price ratio. Do not merge returns to scale with diminishing marginal returns. End each calculation by checking output, MRTS, total cost and the direction implied by any price change.
Begin sketches with labour and capital labels, then draw one isoquant and several parallel isocosts. The least-cost interior line is the lowest one that reaches the required output. Algebraically, derive MPL and MPK, form MRTS and equate it to w/r. This produces an input ratio; combine it with the production target to obtain actual quantities.
Verify the candidate by substituting into the production function and by checking MPL/w equals MPK/r. If the technology permits boundaries or kinks, compare those feasible alternatives rather than assuming tangency. For input-price shocks, hold the target isoquant fixed and rotate the isocost. A higher wage usually reduces labour relative to capital when substitution is possible.
Keep returns-to-scale exercises separate: change every input by the same percentage and compare the output percentage. A one-input marginal-product experiment fixes other inputs, so it cannot by itself establish increasing or decreasing returns to scale. Recompute total cost at one nearby feasible bundle to see concretely why the tangent candidate uses fewer dollars for the same required production.
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