Question 3 OR (a)

Elasticity of substitution of a CES production function

9 marks

Time to write: About 18 minutes
Topic: Partial differentiation and production functions

Question

Question 3 OR (a) · 9 marks

Find the elasticity of substitution of the following function:

Q = 60(0.4 K^{-05} + 0.6 L^{-05})^{-20}

Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.

Before you write

Read off the question and its marks

18 minfor 9 marks

Read from the question

FindFormula first, then the working, then the answer with its unit (₹, %, units).

What the paper leaves open

As printed, the two inner exponents and the outer exponent are written without decimal points; they are kept in that form here. They are read with a decimal point after the first digit (the form of a constant-returns CES function); the elasticity of substitution does not depend on the outer exponent.

Write it in this order

Read the data; note exactly what is asked 1 min
1 Reading of the function and definition of σ 2 marks 4 min
2 Marginal products 3 marks 5 min
3 MRTS 2 marks 4 min
4 Elasticity of substitution and interpretation 2 marks 3 min
Check: arithmetic, units (₹, %), final answer underlined 1 min

1234

Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.

Solution

Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.

Reading of the function. Following the paper note, the exponents are read as −0.5 (inside the bracket) and −2.0 (outside it):

Q = 60(0.4K^(−0.5) + 0.6L^(−0.5))^(−2)

This is a constant elasticity of substitution (CES) function with substitution parameter ρ = 0.5 and distribution weights 0.4 and 0.6. MRTS and the elasticity of substitution are Unit 3 production-function topics.

Definition. The elasticity of substitution measures how fast the input ratio K/L changes in response to a change in the MRTS (the slope of the isoquant):

σ = d ln(K/L) / d ln(MRTS), where MRTS of labour for capital = MP_L/MP_K.

Marginal products. Write D = 0.4K^(−0.5) + 0.6L^(−0.5), so Q = 60D^(−2). By the chain rule and the power rule:

MP_K = ∂Q/∂K = 60 × (−2)D^(−3) × 0.4 × (−0.5)K^(−1.5) = 24D^(−3)K^(−1.5) MP_L = ∂Q/∂L = 60 × (−2)D^(−3) × 0.6 × (−0.5)L^(−1.5) = 36D^(−3)L^(−1.5)

Both are positive for K, L > 0.

MRTS. The common factor D^(−3) cancels:

MRTS = MP_L/MP_K = (36/24) × L^(−1.5)/K^(−1.5) = 1.5(K/L)^1.5.

Elasticity of substitution. Taking logarithms, ln(MRTS) = ln 1.5 + 1.5 ln(K/L), so d ln(MRTS) = 1.5 d ln(K/L) and

σ = d ln(K/L) / d ln(MRTS) = 1/1.5 = 2/3 ≈ 0.667.

This agrees with the general CES result σ = 1/(1 + ρ) = 1/(1 + 0.5). σ is a constant: it is the same at every input combination, and it involves neither the efficiency constant 60 nor the outer exponent, which fix only the scale and the degree of homogeneity (here ρ × 2 = 1, so the function has constant returns to scale).

Interpretation. Since σ = 0.667 < 1, labour and capital are less easily substituted for each other than in the Cobb-Douglas case (σ = 1): a 1% rise in the MRTS raises the capital-labour ratio by only about 0.67%.

Numerical check at K = 4, L = 1 and at K = 9, L = 1: MRTS = 1.5 × 4^1.5 = 12 and 1.5 × 9^1.5 = 40.5; the change in ln(K/L) is ln(9/4) = 0.8109 and the change in ln(MRTS) is ln(40.5/12) = 1.2164; 0.8109/1.2164 = 0.667.

The other reading (exponents taken literally as −5 and −20): ρ = 5 and σ = 1/(1 + 5) = 1/6 ≈ 0.167; the outer exponent again has no effect on σ.

Where the 9 marks sit

#	Component	Marks
1	Reading of the function and definition of σ	2
2	Marginal products	3
3	MRTS	2
4	Elasticity of substitution and interpretation	2
	Total	9

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