Given a production function Q = 4K0.5 L0.5 with fixed capital, what is the correct short run production function?

In the context of the production function ( Q = 4K^{0.5} L^{0.5} ), where ( K ) represents capital and ( L ) represents labor, analyzing the short-run scenario requires understanding what happens when capital is fixed. In the short run, at least one input (capital in this case) cannot be adjusted.

When capital is fixed, we denote it as a constant value (let's call it ( K_0 )). Consequently, we can substitute this fixed value into the production function to derive the short-run production function.

By substituting ( K_0 ) into the production function, we have:

[ Q = 4(K_0)^{0.5}L^{0.5} ]

Since ( K_0 ) is a constant, we can treat ( 4(K_0)^{0.5} ) as a new constant coefficient. Denoting this constant as ( A ), we rewrite the equation, focusing solely on ( L ):

[ Q = A L^{0.5} ]

Because ( A = 4(K_0)^{0.5} ) does not change as ( L \