Question

The short-run production function of a firm is $Q = 200 + 0.2L^2 -0.0004L^3$. If wage rate equals Rs. 140 and the number of labours (L) is 100, then the Marginal Cost and the Average Variable Cost, respectively, are

• A. 5 and 7.78
• B. 6 and 7.78
• C. 5 and 6.68
• D. 6 and 6.68

Answer 1

The Correct Option is A

To find the Marginal Cost (MC) and the Average Variable Cost (AVC) of the firm, we need to first understand how these costs relate to the production function.

1. Given the short-run production function: \(Q = 200 + 0.2L^2 - 0.0004L^3\).
2. The Marginal Product of Labour (MPL) is given by the derivative of the production function with respect to \(L\): \(\text{MPL} = \frac{dQ}{dL} = 0.4L - 0.0012L^2\).
3. Marginal Cost (MC) is calculated using the wage rate (\(w\)) and the Marginal Product of Labour (MPL): \(\text{MC} = \frac{w}{\text{MPL}}\). Substituting the given wage rate and calculating MPL at \(L = 100\): \(\text{MPL} = 0.4 \times 100 - 0.0012 \times (100^2) = 40 - 12 = 28\).
4. Thus, \(\text{MC} = \frac{140}{28} = 5\).
5. Average Variable Cost (AVC) is calculated by dividing the total variable cost (TVC) by the quantity produced (Q). TVC is the total wages paid, i.e., \(w \times L = 140 \times 100\).
6. Substitute \(L = 100\) in the production function to find \(Q\): \(Q = 200 + 0.2 \times 100^2 - 0.0004 \times 100^3 = 200 + 2000 - 400 = 1800\).
7. Therefore, \(\text{AVC} = \frac{140 \times 100}{1800} = \frac{14000}{1800} \approx 7.78\).
8. Hence, the Marginal Cost is \(5\) and the Average Variable Cost is \(7.78\), which corresponds to the correct option: 5 and 7.78.