Question

In Cobb-Douglas production function \( q = x_1^{\alpha} \times x_2^{\beta} \), if \( \alpha + \beta>1 \), then what does it show?

• A. Constant returns to scale
• B. Increasing returns to scale
• C. Decreasing returns to scale
• D. None of these

Hint:

Increasing returns to scale imply that as inputs increase, output increases by a greater proportion.

Answer 1

The Correct Option is B

Step 1: Understanding Cobb-Douglas production function.
The Cobb-Douglas production function is a commonly used functional form in economics. It shows the relationship between the inputs and outputs in a production process. The formula \( q = x_1^{\alpha} \times x_2^{\beta} \) represents the output (\( q \)) produced by two inputs (\( x_1 \) and \( x_2 \)) with respective output elasticities \( \alpha \) and \( \beta \).

Step 2: Analyzing the returns to scale.
- Increasing returns to scale occur when the sum of the exponents \( \alpha + \beta>1 \), meaning that doubling the inputs more than doubles the output. This leads to higher efficiency as output increases.
- Constant returns to scale would occur when \( \alpha + \beta = 1 \), where doubling the inputs results in a doubling of output. - Decreasing returns to scale would occur when \( \alpha + \beta<1 \), indicating that increasing the inputs leads to less than double the output.

Step 3: Conclusion.
When \( \alpha + \beta>1 \), the production function exhibits Increasing returns to scale. Thus, the correct answer is (B).