a) Diminishing Returns to Labour:

Explanation in Ordinary Language: Diminishing returns to labour means that as more workers are added to the production process (while keeping capital constant), the additional output produced by each new worker becomes smaller. In other words, adding more labour results in less and less increase in production as the workforce grows.

Does this production function exhibit diminishing returns to labour?

Let's check this using the production function:

$Q = 2L \cdot K$

The marginal product of labour (MPL) is the additional output produced by one more unit of labour, holding capital constant. To find the MPL, we take the partial derivative of $Q$ with respect to $L$:

$MPL = \frac{\partial Q}{\partial L} = 2K$

Since the marginal product of labour, $2K$, does not depend on $L$, it means that the additional output produced by each additional unit of labour does not diminish as more labour is added. In fact, the marginal product of labour is constant as long as capital $K$ is held constant.

Thus, this production function does not exhibit diminishing returns to labour. It actually has a constant marginal product of labour.

b) Constant Returns to Scale:

Explanation in Ordinary Language: Constant returns to scale means that if all inputs (both labour and capital) are increased by the same proportion, output will increase by that same proportion. For example, if both labour and capital are doubled, output will also double.

Does this production function exhibit constant returns to scale?

To check for constant returns to scale, we scale both inputs $K$ and $L$ by a factor of $t$, and see if output $Q$ also scales by a factor of $t$.

Let’s multiply $K$ and $L$ by $t$ in the production function:

$Q(tK, tL) = 2(tL) \cdot (tK) = 2tL \cdot tK = t^2 \cdot (2KL)$

So, output increases by a factor of $t^2$, not by $t$. Since output does not increase in the same proportion as inputs, this production function does not exhibit constant returns to scale. Instead, it exhibits increasing returns to scale because output increases by more than the proportional increase in inputs.

c) Profit-Maximizing Level of Output and Labour Usage:

To find the profit-maximizing level of output and labour usage, we need to calculate profit ($\pi$):

$\pi = \text{Revenue} - \text{Cost}$

Where:

• Revenue = Price per unit of output $\times$ Quantity of output ($Q$).
• Cost = (Cost of labour) + (Cost of capital).

Given:

• Price per unit of output = $400
• Quantity of output $Q = 2L \cdot K$
• Cost of each unit of labour = $120 per day
• Cost of capital is fixed: $8000 for 9 units of capital, so cost of capital = $\frac{8000}{9}$ per unit of capital
• The firm has already spent $8000 on 9 units of capital, which is fixed.

We can calculate the total cost of labour and capital:

$\text{Cost of labour} = 120L$ $\text{Cost of capital} = 8000 \quad (\text{fixed})$

Thus, the total cost is:

$\text{Total cost} = 120L + 8000$

The revenue is:

$\text{Revenue} = 400Q = 400 \cdot 2L \cdot 9 = 7200L$

Therefore, profit is:

$\pi = 7200L - (120L + 8000)$ $\pi = 7200L - 120L - 8000 = 7080L - 8000$

To maximize profit, we take the derivative of $\pi$ with respect to $L$ and set it equal to zero:

$\frac{d\pi}{dL} = 7080 = 0$

This shows that profit increases with $L$, meaning there is no critical point or maximum within this setup, so we might need to adjust our reasoning based on the marginal cost.

d) Maximum Profit:

In this case, since the optimal $L$ can be provided