Question 3 (a)
Cobb-Douglas Q = 36L^(3/5)K^(2/5): marginal products, returns to scale and factor rewards
9 marks
- Time to write
- About 18 minutes
Question
Question 3 (a) · 9 marksA production function is given by Q = 36 L^{3/5} K^{2/5}, where L denotes the units of labour and K the units of capital.
Find how the marginal product of each factor behaves.
What is the nature of the returns to scale?
What total reward do labour and capital receive when each factor is paid a price that equals its marginal product?
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.
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Read off the question and its marks
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- Find…What is the…?What…?3 asks, each with its own marks below. Answer all 3.
- FindFormula 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 Marginal products and their behaviour 4 marks 7 min
- 2 Returns to scale 2 marks 4 min
- 3 Total reward product exhaustion 3 marks 5 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
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Solution
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A Cobb-Douglas production function, the form the DU guideline uses for this topic.
(i) Behaviour of the marginal products
MP_L = ∂Q/∂L = 36(3/5)L^(−2/5)K^(2/5) = (3/5)Q/L = 21.6 (K/L)^(2/5) > 0 ∂(MP_L)/∂L = 21.6 × (−2/5)L^(−7/5)K^(2/5) < 0
MP_K = ∂Q/∂K = 36(2/5)L^(3/5)K^(−3/5) = (2/5)Q/K = 14.4 (L/K)^(3/5) > 0 ∂(MP_K)/∂K = 14.4 × (−3/5)L^(3/5)K^(−8/5) < 0
Each marginal product is positive and diminishing: adding more of one factor, the other fixed, adds output at a falling rate. MP_L rises with K and MP_K rises with L (∂²Q/∂L∂K = 8.64 L^(−2/5)K^(−3/5) > 0): each factor is more productive when there is more of the other. Both marginal products depend only on the ratio K/L.
Q(tL, tK) = 36(tL)^(3/5)(tK)^(2/5) = t^(3/5 + 2/5) Q = tQ. The function is homogeneous of degree 1: constant returns to scale.
(iii) Total reward when each factor is paid its marginal product
L·MP_L + K·MP_K = (3/5)Q + (2/5)Q = Q.
Labour receives 3/5 (60%) of total output and capital 2/5 (40%); the total reward equals the total product, which is exactly exhausted (Euler's theorem for a function homogeneous of degree 1).
Illustration (L = 32, K = 1): Q = 36 × 32^(0.6) = 36 × 8 = 288; L·MP_L = 0.6 × 288 = 172.8 and K·MP_K = 0.4 × 288 = 115.2; 172.8 + 115.2 = 288.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Marginal products and their behaviour | 4 |
| 2 | Returns to scale | 2 |
| 3 | Total reward (product exhaustion) | 3 |
| Total | 9 |
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