Given the production function Q = 4K0.5 L0.5, if the price of labor is $32 per unit, what is the total variable cost function?

To determine the total variable cost (TVC) function based on the given production function ( Q = 4K^{0.5}L^{0.5} ), we first need to express labor (L) in terms of output (Q) and any fixed input, which in this case appears to be capital (K).

Starting with the production function: [ Q = 4K^{0.5}L^{0.5} ]

We can rearrange this equation to solve for ( L ): [ L^{0.5} = \frac{Q}{4K^{0.5}} ] [ L = \left(\frac{Q}{4K^{0.5}}\right)^2 ]

Substituting this expression for L into the total variable cost function is essential. Since the price of labor is $32 per unit, the total variable cost function can be expressed as: [ TVC = \text{(Price of Labor)} \times \text{(Quantity of Labor)} ] [ TVC = 32L ]

Substituting our earlier expression for ( L ) into this equation gives: [ TVC = 32 \left(\frac{Q}{4