Question

Consider the following Cobb-Douglas production function:

• A. \(\frac{\partial Q}{\partial L}\) increases as \(L\) increases.
• B. \(\frac{\partial Q}{\partial L}\) decreases as \(L\) increases.
• C. As \(L\) increases, \(Q\) increases but at a slower rate.
• D. As \(L\) increases, \(Q\) decreases but at a slower rate.

Hint:

In a Cobb-Douglas production function with exponent of labour between 0 and 1, output increases with labour, but marginal product of labour decreases. This shows diminishing marginal returns.

Answer 1

The Correct Option is B, C

Step 1: Identify the given production function.
The given Cobb-Douglas production function is
\[ Q(K,L)=AK^{0.5}L^{0.5} \] where \(A>0\), \(K>0\), and \(L>0\).
Since \(K\) is assumed to be a positive constant, the production function depends mainly on labour \(L\).
So, we can treat \(AK^{0.5}\) as a positive constant.
Let
\[ AK^{0.5}=C \] where
\[ C>0 \] Thus, the production function becomes
\[ Q=CL^{0.5} \]

Step 2: Find the marginal product of labour.
The marginal product of labour is obtained by differentiating output \(Q\) with respect to labour \(L\).
\[ \frac{\partial Q}{\partial L} = \frac{\partial}{\partial L}\left(CL^{0.5}\right) \] Using the power rule of differentiation,
\[ \frac{d}{dL}(L^n)=nL^{n-1} \] we get
\[ \frac{\partial Q}{\partial L} = C \cdot 0.5L^{-0.5} \] Therefore,
\[ \frac{\partial Q}{\partial L} = \frac{1}{2}CL^{-0.5} \] Substituting \(C=AK^{0.5}\),
\[ \frac{\partial Q}{\partial L} = \frac{1}{2}AK^{0.5}L^{-0.5} \] or,
\[ \frac{\partial Q}{\partial L} = \frac{AK^{0.5}}{2\sqrt{L}} \]

Step 3: Examine what happens to marginal product as \(L\) increases.
The marginal product of labour is
\[ \frac{\partial Q}{\partial L} = \frac{AK^{0.5}}{2\sqrt{L}} \] Here, \(A\), \(K^{0.5}\), and \(2\) are positive constants.
The only changing term is \(\sqrt{L}\) in the denominator.
As \(L\) increases, \(\sqrt{L}\) also increases.
When the denominator increases while the numerator remains constant, the value of the fraction decreases.
Therefore,
\[ \frac{\partial Q}{\partial L} \] decreases as \(L\) increases.
Hence, option (B) is correct and option (A) is incorrect.

Step 4: Check whether output increases or decreases as labour increases.
The production function is
\[ Q=CL^{0.5} \] Since \(C>0\) and \(L^{0.5}>0\), output \(Q\) is positive.
Also, as \(L\) increases, \(L^{0.5}\) increases.
Therefore, output \(Q\) increases when labour \(L\) increases.
This means production does not decrease with an increase in labour.
Hence, option (D) is incorrect.

Step 5: Check whether output increases at an increasing rate or a decreasing rate.
Although output increases as labour increases, the marginal product of labour decreases.
This means each additional unit of labour adds less and less extra output than the previous unit.
In mathematical terms, the slope of the production function with respect to labour is positive but decreasing.
This shows diminishing marginal returns to labour.
So, \(Q\) increases, but at a slower rate as \(L\) increases.
Therefore, option (C) is correct.

Step 6: Final conclusion.
For the given Cobb-Douglas production function, when capital \(K\) is constant, labour has a positive but diminishing effect on output.
The marginal product of labour decreases as labour increases.
At the same time, total output continues to increase, but at a slower rate.
Hence, the correct statements are
\[ \boxed{(B)\text{ and }(C)} \]