A firm’s production function is q = x1ax2a. When both inputs are doubled, output becomes four times. What is the value of a, and what returns to scale does the production function show?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → a = 1; increasing returns to scale
In a Cobb-Douglas function, scaling both inputs by t scales output by t raised to the sum of the exponents. Here the sum is a + a = 2a.
\( \displaystyle q_1 = (2x_1)^a (2x_2)^a = 2^{2a} q_0 \)
\( \displaystyle 2^{2a} = 4 = 2^2 \)
\( \displaystyle 2a = 2, \text{ so } a = 1 \)
So a = 1, and the sum of the exponents is 2. A sum greater than 1 means increasing returns to scale: doubling inputs gives four times the output.
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.” “3.6 Returns to Scale When a proportional increase in all inputs results in an increase in output by a larger proportion, the production function is said to display Increasing Returns to Scale (IRS) Decreasing Returns to Scale (DRS) holds when a proportional increase in all inputs results in an increase in output by a smaller proportion. …” |