Teddy Bear, Inc., a rapidly growing manufacturer of high fashion children's shoes, plans to open a new production facility in Gastonia. Based on information provided by the accounting department, the company estimates fixed costs of $250,000 per year. Its average variable cost is

AVC = $5 + $0.0001 Q
AVC is average variable cost in dollars and Q is pairs of shoes.
a. Find the total cost and average cost equations.
b. At what level of output is average total cost minimized? What is average total cost at that level of output?

a. To find the total cost equation, we need to sum fixed costs and variable costs. Fixed costs are given as $250,000 per year.

Variable costs (VC) can be found by multiplying the average variable cost (AVC) by the quantity of shoes produced (Q). The average variable cost equation is given as AVC = $5 + $0.0001Q.

Thus, VC = AVC * Q = ($5 + $0.0001Q) * Q = $5Q + $0.0001Q^2.

The total cost equation (TC) is the sum of fixed costs (FC) and variable costs (VC):

TC = FC + VC = $250,000 + ($5Q + $0.0001Q^2).

To find the average cost equation, we divide the total cost equation by the quantity Q:

AC = TC / Q = ($250,000 + $5Q + $0.0001Q^2) / Q.

b. The average total cost (ATC) is minimized when the slope of the average cost curve is equal to zero. To find the level of output at which this occurs, we need to differentiate the average cost equation with respect to Q and set it equal to zero:

d/dQ (AC) = (d/dQ) [(250,000 + 5Q + 0.0001Q^2) / Q] = 0.

To simplify the equation, we can use the quotient rule and then solve for Q when d/dQ (AC) = 0.

After differentiating and simplifying, the equation becomes:

(500 - 0.2Q) / Q^2 = 0.

Multiplying both sides by Q^2, we get:

500 - 0.2Q = 0.

Solving for Q, we have:

0.2Q = 500,
Q = 500 / 0.2,
Q = 2,500 pairs of shoes.

At this level of output, the average total cost can be found by substituting the value of Q into the average cost equation:

AC = ($250,000 + $5Q + $0.0001Q^2) / Q
= ($250,000 + $5 * 2,500 + $0.0001 * (2,500)^2) / 2,500
= ($250,000 + $12,500 + $6.25) / 2,500
= $268,750 / 2,500
= $107.50 per pair of shoes.