Question

Consider the production function \(q = f(x_1, x_2)\) where the firm produces \(q\) amount of output using \(x_1\) amount of factor 1 and \(x_2\) amount of factor 2. The firm decides to increase the employment level of both factors by \(t\) (\(t>1\)). Identify the equation for decreasing returns to scale from the following:

• A. \(q = f(x_1, x_2)\)
• B. \(f(tx_1, tx_2) = t f(x_1, x_2)\)
• C. \(f(tx_1, tx_2)<t f(x_1, x_2)\)
• D. \(f(tx_1, tx_2)>t f(x_1, x_2)\)

Hint:

Use “less than sign” < for decreasing returns, “equal” for constant, and “greater than sign” > for increasing returns.

Answer 1

The Correct Option is C

Step 1: Recall definition of returns to scale.
- Constant returns to scale: Output increases proportionally to inputs, i.e., \(f(tx_1, tx_2) = t f(x_1, x_2)\).
- Increasing returns to scale: Output increases by more than proportionally, i.e., \(f(tx_1, tx_2)>t f(x_1, x_2)\).
- Decreasing returns to scale: Output increases by less than proportionally, i.e., \(f(tx_1, tx_2)<t f(x_1, x_2)\).
Step 2: Apply to question.
The question pertains to decreasing returns to scale, thus the defining condition is: \[f(tx_1, tx_2)<t f(x_1, x_2)\] Final Answer: \[\boxed{f(tx_1, tx_2)<t f(x_1, x_2)}\]