A firm’s production function is q = x10.4 x20.4. Input prices are given. Which chain of effects is correct as the firm expands output in the long run?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → α + β = 0.8 → decreasing returns to scale → cost rises by a larger proportion than output → LRAC rises
In a Cobb-Douglas function, add the powers: 0.4 + 0.4 = 0.8. This is less than 1, so the function shows decreasing returns to scale.
Under DRS, more output needs a larger rise in inputs, so cost rises faster than output and LRAC rises.
Option (1) α + β = 0.16 → decreasing returns to scale → fixed cost is spread over more output → LRAC falls — Incorrect. The powers are added, not multiplied. In the long run there is no fixed cost to spread.
Option (2) α + β = 0.8 → constant returns to scale → cost rises in the same proportion as output → LRAC stays constant — Incorrect. CRS needs α + β = 1. Here it is 0.8.
Option (3) α + β = 0.8 → decreasing returns to scale → cost rises by a larger proportion than output → LRAC rises — Correct. Each link holds.
Option (4) α + β = 0.8 → decreasing returns to scale → cost rises by a smaller proportion than output → LRAC falls — Incorrect. Under DRS, cost rises by a larger proportion than output, so LRAC rises.
NCERT (2026-27): “Cobb-Douglas Production Function … When α + β = 1, we have q1 = tq0. That is, the output increases t times. So the production function exhibits CRS. Similarly, when α + β > 1, the production function exhibits IRS. When α + β < 1 the production function exhibits DRS.” “Shapes of the Long Run Cost Curves … DRS implies that if we want to increase the output by a certain proportion, inputs need to be increased by more than that proportion. As a result, cost also increases by more than that proportion. So, as long as DRS operates, the average cost must be rising as the firm increases output.” |