Question

1. Suppose that each firm in a perfectly competitive market has a short-run total cost of TC = 75 + 500Q – 5Q2 + 0.5Q3, where MC = 500 – 10Q + 1.5Q2.

1. Calculate the output that minimizes the firm’s AVC.
2. What is the firm’s shutdown price?

Answer 1

(a) Total cost is the sum of fixed cost and variable cost:

$$TC=FC+VC.$$

In our case the variable cost of the firm takes form:

$$VC=500Q-5Q^2+0.5Q^3.$$

Let's find average variable cost (AVC). By the definition,

$$AVC=\dfrac{VC}{Q}=500-5Q+0.5Q^2.$$

Let's take the derivative of AVC equation:

$$\dfrac{dAVC}{dQ}=-5+Q=0,$$ $$Q=5.$$

The output of 5 units minimizes the firm’s AVC.

(b) The shutdown price occurs at the minimum of the average variable cost curve at a point where MC=AVC:

$$500-10Q+1.5Q^2=500-5Q+0.5Q^2,$$ $$Q^2-5Q=0,$$ $$Q(Q-5)=0,$$ $$Q=5.$$

In a perfectly competitive market P=MC, therefore:

$$P=500-10Q+1.5Q^2.$$

Let's substitute $Q$ into the previous equation and find the shutdown price:

$$P=500-10\cdot5+1.5\cdot(5)^2=\$412.5$$