Question

If a labour output function for laundry service is described by the following equation: \(O = (a^0 - 4L^0)^6\), where \(L\) denotes Labour and \(O\) denotes output. Then, output .........

• A. Increases as labor increases
• B. Decreases as labor increases
• C. Remains constant irrespective of the amount of labor
• D. None of the option is correct

Answer 1

The Correct Option is C

Given the labour output function for the laundry service:
\[ O = (a^0 - 4L^0)^6 \]
where \(L\) denotes labour and \(O\) denotes output.
1. \(a^0\) is always equal to 1 for any non-zero \(a\).
2. \(L^0\) is always equal to 1 for any non-zero \(L\).
Thus, the expression simplifies to:
\[ O = (1 - 4 \cdot 1)^6 \]
\[ O = (1 - 4)^6 \]
\[ O = (-3)^6 \]
\[ O = 729 \]
Therefore, the output \(O\) remains constant at 729, regardless of the amount of labour \(L\).
Correct answer is Option C : Remains constant irrespective of the amount of labor.

Answer 2

For any non-zero base, \(a^{0}=1\) and \(L^{0}=1\) regardless of the actual value of \(L\). Substituting:
\[O=(1-4\times1)^{6}=(-3)^{6}=729\]

Since \(L^{0}=1\) for every non-zero value of \(L\), the expression inside the bracket never changes as \(L\) varies, so the rate of change of output with respect to labour is:
\[\frac{dO}{dL}=0\]

1. Option A (Increases as labor increases): Would require \(\frac{dO}{dL} \gt 0\), which is false here.
2. Option B (Decreases as labor increases): Would require \(\frac{dO}{dL} \lt 0\), which is also false.
3. Option C (Remains constant irrespective of the amount of labor): Matches, since \(\frac{dO}{dL}=0\) and \(O=729\) always.
4. Option D (None of the option is correct): Ruled out since Option C is valid.

The output stays fixed at 729 no matter how much labour is used.

Hence, the correct answer is Option C: Remains constant irrespective of the amount of labor.