Question

A local microbrewery has total costs of production given by the equation C(Q) = 500 + 10Q + 5Q2. The market demand for beer is given by the equation Q = 105 – (1/2)P. The firm operates in a perfectly competitive market.

a. Write the equations showing the brewery's average total cost and average variable cost, average fixed cost, and marginal cost, each as a function of quantity (Q). b. What is the break-even price and break-even quantity for this firm in the short run? c. What is the shut-down price and shut-down quantity for this firm in the short run? d. If the market price of the output is $50, how many units will this firm produce? e. Given a market price of $50, how many firms are in this market?

Answer 1

(a)

ATC = C/Q = (500 / Q) + 10 + 5Q

TVC = 10Q + 5Q², so AVC = TVC / Q = 10 + 5Q

TFC = 500, so AFC = TFC / Q = 500 / Q

MC = dC/dQ = 10 + 10Q

(b)

In break-even, P = MC = ATC

(500 / Q) + 10 + 5Q = 10 + 10Q

5Q = 500/Q

Q² = 100

Q = 10

P = MC = 10 + 10 x 10 = 10 + 100 = 110

(c)

Shut-down price = Minimum value of AVC

AVC is minimum when Q = 0 (shut-down quantity).

Shut-down price = AVC = 10

(d)

Setting P = MC,

10 + 10Q = 50

10Q = 40

Q = 4

(e)

From market demand, Q = 105 - (50/2) = 105 - 25 = 80

Number of firms = Market quantity / Firm quantity = 80 / 4 = 20