Question

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.
• B.
• C.
• D.

Hint:

For Cobb-Douglas production functions, first solve for the labour input \( L \) in terms of output \( q \), then calculate total cost, marginal cost, and average variable cost. Ensure to apply the fixed capital condition carefully in short-run analysis.

Answer 1

The Correct Option is A, B, C

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 \( w = 12 \):
\[ TC = \frac{12q^2}{4K} + rK = \frac{3q^2}{K} + rK \]