Question 3 OR (a)

Q = [0.3/K + 0.7/L]⁻¹: returns to scale, MRTS, elasticity of substitution and Euler's theorem

9 marks

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

Question

Question 3 OR (a) · 9 marks

For the production function: Q = [0.3/K + 0.7/L]^{-1}

where L denotes Labour, K denotes Capital and Q denotes production.

Determine the following:

The nature of returns to scale.
The Marginal Rate of Technical Substitution.
The elasticity of substitution.
Whether the production function satisfies Euler's theorem.

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

(i)–(iv)4 items, each marked on its own: give each its share.
DetermineFormula first, then the working, then the answer with its unit (₹, %, units).

Write it in this order

Read the data; note exactly what is asked 1 min
1 Returns to scale homogeneity 2 marks 4 min
2 Marginal products and MRTS 3 marks 5 min
3 Elasticity of substitution 2 marks 4 min
4 Euler's theorem 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.

Write Q = 1/(0.3K⁻¹ + 0.7L⁻¹) = KL/(0.3L + 0.7K). (Homogeneity, Euler's theorem, returns to scale, MRTS and the elasticity of substitution are the Unit 3 production-function topics.)

(i) Returns to scale

Multiply both inputs by t > 0:

Q(tL, tK) = 1/(0.3/(tK) + 0.7/(tL)) = t × 1/(0.3/K + 0.7/L) = t Q(L, K).

The function is homogeneous of degree 1, so it shows constant returns to scale: doubling both inputs doubles output.

(ii) Marginal Rate of Technical Substitution

Let D = 0.3K⁻¹ + 0.7L⁻¹, so Q = D⁻¹. By the chain rule:

MP_L = ∂Q/∂L = −D⁻² × (−0.7L⁻²) = 0.7Q²/L² MP_K = ∂Q/∂K = −D⁻² × (−0.3K⁻²) = 0.3Q²/K²

MRTS of labour for capital = MP_L/MP_K = (0.7/0.3)(K/L)² = (7/3)(K/L)².

(Both marginal products are positive, and the MRTS falls as L rises relative to K, so the isoquants are convex.)

(iii) Elasticity of substitution

σ = d ln(K/L) / d ln(MRTS). From (ii), ln(MRTS) = ln(7/3) + 2 ln(K/L), so d ln(MRTS) = 2 d ln(K/L) and

σ = 1/2 = 0.5.

(This is a CES function with ρ = 1, for which σ = 1/(1 + ρ) = 0.5: substitution between labour and capital is less easy than in the Cobb-Douglas case, σ = 1.)

(iv) Euler's theorem

For a function homogeneous of degree 1, Euler's theorem requires L·MP_L + K·MP_K = Q:

L·MP_L + K·MP_K = 0.7Q²/L + 0.3Q²/K = Q²(0.7/L + 0.3/K) = Q² × D = Q² × (1/Q) = Q.

The theorem is satisfied: when each factor is paid its marginal product, the total product is exactly exhausted.

Numerical check at L = 10, K = 30: Q = 1/(0.01 + 0.07) = 12.5; MP_L = 0.7 × 156.25/100 = 1.09375; MP_K = 0.3 × 156.25/900 = 0.0520833; 10 × 1.09375 + 30 × 0.0520833 = 10.9375 + 1.5625 = 12.5 = Q.

Where the 9 marks sit

#	Component	Marks
1	Returns to scale (homogeneity)	2
2	Marginal products and MRTS	3
3	Elasticity of substitution	2
4	Euler's theorem	2
	Total	9

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