Unit 2: Indifference curves & production function
Managerial Economics notes · PTU syllabus (BCOMGE 101-18)
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Unit summary
How does a consumer choose between goods, and how does a firm combine inputs? This unit covers indifference curve analysis and consumer equilibrium, the production function in the short run (law of variable proportions) and long run, isoquants and returns to scale.
After this unit you can
- Explain indifference curves, their properties and the budget line
- Determine consumer equilibrium with indifference curves
- Explain the law of variable proportions
- Explain isoquants, the least-cost combination and returns to scale
PTU syllabus topics
- Consumer equilibrium
- indifference analysis
- short and long run production function
- isoquants
- returns to scale
Stage I
Total product rises at an increasing rate
No: fixed factor underused
Stage II
Total product rises at a decreasing rate
Yes: rational zone
Stage III
Total product falls; MP is negative
No: too much variable factor
Topic 1
Indifference curve analysis
An indifference curve (IC) shows combinations of two goods that give the consumer the same level of satisfaction (Hicks and Allen — ordinal utility).
| Combination | Apples | Oranges | MRS (oranges given up per apple) |
|---|---|---|---|
| A | 1 | 12 | — |
| B | 2 | 8 | 4 |
| C | 3 | 5 | 3 |
| D | 4 | 3 | 2 |
| E | 5 | 2 | 1 |
- Marginal rate of substitution (MRSxy): amount of Y the consumer gives up for one more unit of X while staying equally satisfied; it diminishes.
Slope downward
More of one good means less of the other
Convex to the origin
Due to diminishing MRS
Never intersect
Intersection would be contradictory
Higher IC = higher satisfaction
Do not touch the axes (normally)
- Indifference map: a set of ICs; higher curves represent higher satisfaction.
Topic 2
Budget line and consumer equilibrium
The budget (price) line shows all combinations of two goods a consumer can buy with given income and prices: Px·X + Py·Y = M. Slope = −Px/Py.
- 1Budget line tangent to the highest attainable IC
- 2MRSxy = Px ÷ Py
- 3IC convex to the origin at the point of tangency
- Income effect: change in income shifts the budget line parallel — the income consumption curve (ICC).
- Price effect: change in price of one good rotates the budget line — the price consumption curve (PCC).
- Substitution effect: change in relative prices with real income constant. Price effect = Income effect + Substitution effect (Slutsky/Hicks).
Exam tip
For a Giffen good, the negative income effect outweighs the substitution effect — so demand rises with price.
Topic 3
Production function: short run
A production function shows the technical relationship between physical inputs and output: Q = f(L, K).
- Short run: at least one factor is fixed (capital); output changes by varying the variable factor (labour).
- Long run: all factors are variable.
Law of variable proportions
Total product (TP)
Total output with given inputs
Average product (AP)
TP ÷ Units of variable factor
Marginal product (MP)
Change in TP ÷ Change in variable factor
| Labour | TP | MP | AP | Stage |
|---|---|---|---|---|
| 1 | 10 | 10 | 10 | I — increasing returns |
| 2 | 24 | 14 | 12 | I |
| 3 | 36 | 12 | 12 | I ends (AP max) |
| 4 | 44 | 8 | 11 | II — diminishing returns |
| 5 | 48 | 4 | 9.6 | II |
| 6 | 48 | 0 | 8 | II ends (TP max) |
| 7 | 45 | −3 | 6.4 | III — negative returns |
- Stage I: TP rises at an increasing rate, then AP reaches maximum — under-use of fixed factor.
- Stage II: MP diminishes but positive; rational stage of production.
- Stage III: MP negative; TP falls — no rational producer operates here.
- Causes of diminishing returns: fixed factor, imperfect substitutability of factors, overcrowding.
Topic 4
Isoquants and least-cost combination
An isoquant (equal-product curve) shows combinations of two inputs (L and K) that produce the same output.
- Properties: slope downward, convex to the origin (diminishing MRTS), do not intersect, higher isoquants = higher output.
- MRTS(L,K): units of capital that can be replaced by one unit of labour keeping output constant = MPL/MPK.
- Isocost line: combinations of inputs with the same total cost: w·L + r·K = C.
Shows
Equal output from two inputs
Equal satisfaction from two goods
Measurement
Cardinal — output can be measured
Ordinal — satisfaction ranked
Slope
MRTS
MRS
- Producer's equilibrium (least-cost combination): where an isocost line is tangent to an isoquant — MRTS = w/r, or MPL/w = MPK/r.
- Expansion path: the locus of least-cost combinations as output expands.
Topic 5
Returns to scale (long run)
Returns to scale describe how output changes when all inputs are increased in the same proportion.
- 1Increasing returns
Output rises by a greater proportion (inputs ×2, output ×2.5)
- 2Constant returns
Output rises in the same proportion
- 3Decreasing returns
Output rises by a smaller proportion
- Causes of increasing returns: internal economies — specialisation, indivisibility of machines, managerial and technical economies.
- Causes of decreasing returns: diseconomies of large scale — coordination and management difficulties.
- Cobb–Douglas production function: Q = A·L^α·K^β; returns to scale increasing if α + β > 1, constant if = 1, decreasing if < 1.
Period
Short run
Long run
Inputs varied
One factor, others fixed
All factors in same proportion
Factor ratio
Changes
Constant
Key terms
- Indifference curve
- Combinations of two goods giving equal satisfaction
- Marginal rate of substitution
- Rate at which a consumer substitutes one good for another
- Budget line
- Combinations of goods affordable with given income and prices
- Isoquant
- Combinations of inputs giving the same output
- Returns to scale
- Change in output when all inputs change proportionately
Quick revision
- IC: downward sloping, convex, non-intersecting.
- Equilibrium: MRSxy = Px/Py at tangency.
- Price effect = income effect + substitution effect.
- Variable proportions: three stages; stage II is rational.
- Least cost: MRTS = w/r; Cobb–Douglas α + β tells returns to scale.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.Define an indifference curve.
- Q2.Why are indifference curves convex to the origin?
- Q3.State the conditions of consumer equilibrium.
- Q4.What is the law of variable proportions?
- Q5.Define an isoquant.
- Q6.Distinguish returns to a factor and returns to scale.
Long-answer questions
- Q1.Explain consumer equilibrium using indifference curve analysis.
- Q2.Explain the income, substitution and price effects.
- Q3.Explain the law of variable proportions with a table and diagram.
- Q4.Explain isoquants, the least-cost combination and returns to scale.
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