Given the production function Q = 0.4K0.5 L0.5 with fixed capital at 100 units, what is the firm's short-run production function when the price of labor is $4?

To determine the firm's short-run production function when capital (K) is fixed at 100 units, we start with the original production function given by ( Q = 0.4K^{0.5}L^{0.5} ). In the short run, one input (capital, in this case) is held constant while the other input (labor) can vary.

When we substitute the fixed value of capital (K = 100) into the production function, we can simplify the equation. First, we calculate ( K^{0.5} ):

[ K^{0.5} = 100^{0.5} = 10 ]

Now we substitute this value back into the production function:

[ Q = 0.4 \cdot 10 \cdot L^{0.5} ]

This simplifies to:

[ Q = 4L^{0.5} ]

The value ( 4 ) represents a scaling factor derived from the fixed amount of capital and the production function.

This matches option B, which defines the short-run production function as ( Q = 4L^{0.5} ). Thus, since capital is fixed and we are varying labor