06/06/2026, 16:18 Firm Theory — ECON2291 Revision Guide
ECON2291 · DURHAM UNIVERSITY · REVISION GUIDE
Firm Theory
Production functions, isoquants, cost minimisation, elasticities, and the CES family
Contents
1. The Production Function 7. Elasticity of Substitution
2. Forms of Production Function 8. Returns-to-Scale Elasticity
3. Isoquants and MRTS 9. Cost Minimisation
4. Returns to Scale 10. CES Production Function
5. Homogeneous Functions 11. Past Paper Questions
6. Expansion Path & Homotheticity 12. Practice Questions
PART I
Core Theory
1. The Production Function
A firm transforms inputs into output. The production function is the
technological relationship that describes the maximum output attainable from
any combination of inputs.
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DEFINITION — PRODUCTION FUNCTION
A production function q = f(k, l) maps quantities of capital k and labour l to a
maximum level of output q. Capital is treated as fixed in the short run; both
inputs are variable in the long run.
SIX CORE PROPERTIES (NICHOLSON–SNYDER, CH.9)
# PROPERTY FORMAL STATEMENT
1 Constant returns to scale λq = f(λl, λk) for all λ > 0
2 Output increases in inputs ∂q/∂l > 0, ∂q/∂k > 0
3 Diminishing MPL ∂²q/∂l² < 0
4 Diminishing MPK ∂²q/∂k² < 0
5 MPL ↑ when k ↑ ∂²q/∂l∂k > 0
6 TFP shifts production q = Af(l,k), A = Total Factor Productivity
Short run vs long run: In the short run capital is fixed at k*, so q = f(l, k*) and the
slope of this curve at any l is the marginal product of labour, MPl = ∂q/∂l.
Diminishing returns means this slope decreases as l rises.
2. Forms of Production Function
2.1 Linear (Perfect Substitutes): σ = ∞
DEFINITION
q = f(k, l) = ak + bl, where a, b are constants.
Capital and labour are perfectly substitutable at a constant rate. Isoquants are
straight lines with slope −a/b, and MRTSlk = a/b everywhere. Because MRTSlk is
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constant, the capital–labour ratio can change infinitely with no change in MRTS,
so σ → ∞.
2.2 Leontief (Fixed Proportions): σ = 0
DEFINITION
q = f(k, l) = min{al, bk}, where a, b are positive constants.
Inputs are used in a fixed ratio b/a. The optimal input mix at any output level q*
is l* = q*/a and k* = q*/b. Isoquants are L-shaped; any additional unit of one input
beyond the kink contributes nothing to output. Because the ratio k/l never
changes optimally, MRTS is undefined except at the vertex, and σ = 0.
WORKED EXAMPLE — LEONTIEF
Let q = min{l/2, k}, with w = 4, r = 2, and target q = 20.
Optimality condition: al = bk ⟹ l/2 = k ⟹ l = 2k
For q = 20: l/2 = 20 ⟹ l = 40; k = 20
Total cost: C = wl + rk = 4(40) + 2(20) = 160 + 40 = 200
2.3 Cobb-Douglas: σ = 1
DEFINITION
q = f(k, l) = kalb, where a, b > 0.
The marginal products are MPl = bkalb−1 and MPk = aka−1lb. The MRTS is
therefore:
MRTS_lk = MP_l / MP_k = (bk^a l^(b-1)) / (ak^(a-1) l^b) = bk / al
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ECON2291 · DURHAM UNIVERSITY · REVISION GUIDE
Firm Theory
Production functions, isoquants, cost minimisation, elasticities, and the CES family
Contents
1. The Production Function 7. Elasticity of Substitution
2. Forms of Production Function 8. Returns-to-Scale Elasticity
3. Isoquants and MRTS 9. Cost Minimisation
4. Returns to Scale 10. CES Production Function
5. Homogeneous Functions 11. Past Paper Questions
6. Expansion Path & Homotheticity 12. Practice Questions
PART I
Core Theory
1. The Production Function
A firm transforms inputs into output. The production function is the
technological relationship that describes the maximum output attainable from
any combination of inputs.
file:///C:/Users/tyala/Downloads/firm_theory_revision_guide (2).html#part3 1/27
,06/06/2026, 16:18 Firm Theory — ECON2291 Revision Guide
DEFINITION — PRODUCTION FUNCTION
A production function q = f(k, l) maps quantities of capital k and labour l to a
maximum level of output q. Capital is treated as fixed in the short run; both
inputs are variable in the long run.
SIX CORE PROPERTIES (NICHOLSON–SNYDER, CH.9)
# PROPERTY FORMAL STATEMENT
1 Constant returns to scale λq = f(λl, λk) for all λ > 0
2 Output increases in inputs ∂q/∂l > 0, ∂q/∂k > 0
3 Diminishing MPL ∂²q/∂l² < 0
4 Diminishing MPK ∂²q/∂k² < 0
5 MPL ↑ when k ↑ ∂²q/∂l∂k > 0
6 TFP shifts production q = Af(l,k), A = Total Factor Productivity
Short run vs long run: In the short run capital is fixed at k*, so q = f(l, k*) and the
slope of this curve at any l is the marginal product of labour, MPl = ∂q/∂l.
Diminishing returns means this slope decreases as l rises.
2. Forms of Production Function
2.1 Linear (Perfect Substitutes): σ = ∞
DEFINITION
q = f(k, l) = ak + bl, where a, b are constants.
Capital and labour are perfectly substitutable at a constant rate. Isoquants are
straight lines with slope −a/b, and MRTSlk = a/b everywhere. Because MRTSlk is
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, 06/06/2026, 16:18 Firm Theory — ECON2291 Revision Guide
constant, the capital–labour ratio can change infinitely with no change in MRTS,
so σ → ∞.
2.2 Leontief (Fixed Proportions): σ = 0
DEFINITION
q = f(k, l) = min{al, bk}, where a, b are positive constants.
Inputs are used in a fixed ratio b/a. The optimal input mix at any output level q*
is l* = q*/a and k* = q*/b. Isoquants are L-shaped; any additional unit of one input
beyond the kink contributes nothing to output. Because the ratio k/l never
changes optimally, MRTS is undefined except at the vertex, and σ = 0.
WORKED EXAMPLE — LEONTIEF
Let q = min{l/2, k}, with w = 4, r = 2, and target q = 20.
Optimality condition: al = bk ⟹ l/2 = k ⟹ l = 2k
For q = 20: l/2 = 20 ⟹ l = 40; k = 20
Total cost: C = wl + rk = 4(40) + 2(20) = 160 + 40 = 200
2.3 Cobb-Douglas: σ = 1
DEFINITION
q = f(k, l) = kalb, where a, b > 0.
The marginal products are MPl = bkalb−1 and MPk = aka−1lb. The MRTS is
therefore:
MRTS_lk = MP_l / MP_k = (bk^a l^(b-1)) / (ak^(a-1) l^b) = bk / al
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