Final answer:
The total variable cost function for the firm with the given production function and prices is TVC = 8Q², where variable costs include only the cost of labor as capital is fixed.
Step-by-step explanation:
The question involves finding the total variable cost function for a firm with a given production function Q = 4K·½ L·½, where capital (K) is fixed at 1 unit and the price of labor (PL) is $32 per unit. Since capital is fixed, we only need to consider the cost of labor (variable cost) for producing a certain level of output (Q).
The total variable cost (TVC) function represents the cost of the variable factors of production, which in this case is just labor. To compute the TVC, you multiply the quantity of labor (L) by the price per unit of labor (PL).
Given the production function, if we want to produce Q units of output, we solve for L from the production function: L = (Q / 4K·½)². Since K is fixed at 1, this simplifies to L = (Q / 4)². Therefore, TVC = PL × L = 32 × (Q / 4)² = 8Q².