A Cobb-Douglas type short-run production function is given by \[ q = 2\sqrt{L \overline{K}} \] where \( q, L \) and \( K \) are the output, labour and capital, respectively. \( K \) is fixed at \( \overline{K} \). The unit price of \( L \) is \( w \) and the unit price of \( K \) is \( r \). It is given that \( w = 12 \). Considering the above information, which of the following statements is/are CORRECT?
A Cobb-Douglas type short-run production function is given by \[ q = 2\sqrt{L \overline{K}} \] where \( q, L \) and \( K \) are the output, labour and capital, respectively. \( K \) is fixed at \( \overline{K} \). The unit price of \( L \) is \( w \) and the unit price of \( K \) is \( r \). It is given that \( w = 12 \). Considering the above information, which of the following statements is/are CORRECT?
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The Correct Option is A, B, C
Solution and Explanation
Step 1: Short-run Total Cost and Marginal Cost
The production function is \( q = 2\sqrt{LK} \), which gives us:
\[ L = \frac{q^2}{4K} \] The total cost function is the sum of the costs of labour and capital:
\[ TC = wL + rK \] Substituting \( L \) into the equation:
\[ TC = w \cdot \frac{q^2}{4K} + rK = \frac{wq^2}{4K} + rK \] Now, substitute \( w = 12 \):
\[ TC = \frac{12q^2}{4K} + rK = \frac{3q^2}{K} + rK \] 
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