Question

If \( Y = L^a K^b \) is a Cobb-Douglas production function with two factors labor and capital, then the presence of constant returns to scale exhibited by the production function can be tested using ...........

• A. only \( t \) statistic
• B. only \( F \) statistic
• C. both \( t \) and \( F \) statistic
• D. \( \chi^2 \) statistic

Hint:

To test for constant returns to scale in a Cobb-Douglas production function, use the \( F \)-test for joint hypotheses or the \( t \)-test for individual parameters.

Answer 1

The Correct Option is C

Step 1: Understand the Cobb-Douglas production function.
The Cobb-Douglas production function is widely used to model the relationship between inputs (labor and capital) and output. It is typically written as: \[ Y = L^a K^b \] where \( Y \) is the output, \( L \) is labor, \( K \) is capital, and \( a \) and \( b \) are the output elasticities of labor and capital, respectively.
Step 2: What does constant returns to scale mean?
Constant returns to scale imply that if all inputs are increased by the same proportion, output will increase by that same proportion. Mathematically, this condition is met when: \[ a + b = 1 \] If \( a + b>1 \), it indicates increasing returns to scale, and if \( a + b<1 \), it indicates decreasing returns to scale.
Step 3: Which statistical test to use?
To test for constant returns to scale, we need to test the hypothesis that \( a + b = 1 \). This can be tested using an \( F \)-test, which is used to test linear restrictions on parameters in econometric models. Additionally, we can also use a \( t \)-statistic for individual coefficient testing, especially when testing the individual parameters \( a \) and \( b \).
- Option (A) is incorrect because the \( t \)-statistic is used to test individual coefficients, not joint hypotheses like \( a + b = 1 \).
- Option (B) is correct because the \( F \)-statistic is used to test joint hypotheses, such as \( a + b = 1 \).
- Option (C) is correct as both the \( t \)-statistic and \( F \)-statistic can be used to test different aspects of the hypothesis (joint and individual parameters).
- Option (D) is incorrect because the \( \chi^2 \)-statistic is not used for testing the returns to scale in this case.
Final Answer: \[ \boxed{\text{both } t \text{ and } F \text{ statistic}} \]