Question

Suppose that a firm has a technology represented by the following production function: \[ Y(K, L) = K^x L^y \] where \( K \) denotes capital, \( L \) denotes labour, \( Y \) denotes the maximum output that is possible to produce using capital \( K \) and labour \( L \). \( x \) and \( y \) are two positive real numbers. It is also known that the production function satisfies constant returns to scale. Then which of the following is true?

• A. \( x + y = 0.5 \)
• B. \( x + y = 1 \)
• C. \( x + y = 1.5 \)
• D. \( x + y = 2 \)

Hint:

For constant returns to scale, the sum of the exponents of capital and labour in the production function must equal 1.

Answer 1

The Correct Option is B

Step 1: Understanding constant returns to scale.
The condition for constant returns to scale is that if all inputs are scaled by a constant factor \( \lambda \), the output should also scale by the same factor. Mathematically, for the production function \( Y(K, L) = K^x L^y \), we need: \[ Y(\lambda K, \lambda L) = \lambda Y(K, L) \] Substituting the production function into this equation: \[ (\lambda K)^x (\lambda L)^y = \lambda \cdot K^x L^y \] Simplifying: \[ \lambda^x K^x \lambda^y L^y = \lambda K^x L^y \] \[ \lambda^{x+y} = \lambda \] For this equation to hold true for all values of \( \lambda \), we must have: \[ x + y = 1 \] Step 2: Conclusion.
Thus, the correct answer is (B) \( x + y = 1 \).