Question

A car manufacturer has the following total cost function: TC = 200 + 50 Q + 2Q2

a) Is this a short run or long run cost function? Why?
b) Compute the average total cost of producing cars, AC, and show that it is U-shaped.
c) If the marginal cost function is MC = 50 + 4Q, determine the minimum point of the AC curve.

Answer 1

The function provided is typical of short-run costs as it includes fixed and variable costs, but context is needed to be certain.

Step-by-step explanation:

The total cost function given by TC = 200 + 50 Q + 2Q2 can represent either short-run or long-run costs depending on the context, but typically short run because it includes fixed and variable costs. Fixed costs are usually associated with the short run as in the long run all costs are variable.

To find whether this function represents short-run or long-run costs, we would need additional information about the flexibility of the cost components.

To compute the average total cost (AC), we divide the total cost (TC) by the quantity (Q): AC = (200 + 50Q + 2Q2)/Q. Simplifying this gives AC = 200/Q + 50 + 2Q. This equation indeed forms a U-shape because as Q increases, initially the AC decreases due to the spreading out of the fixed cost (200/Q), reaches a minimum, and then increases as the variable term (2Q) becomes dominant.

To determine the minimum point of the AC curve, we set the derivative of AC with respect to Q equal to the marginal cost (MC). Therefore, d(AC)/dQ = MC. When AC is minimized, MC equals AC; hence, we set 50 + 4Q equal to 200/Q + 50 + 2Q and solve for Q to find the minimum point.