Question

3. Sunnyvale Orchards is one of many small, perfectly competitive firms growing plums for the U.S. market. The forecasted price of plums in 2018 is $24 a crate. The management of Sunnyvale Orchards estimates its short run cost function to be: TC = 3000 + 244 -0.045q? + 0.00005q', where q is output per week Find the profit maximizing output. How much profits will Sunnyvale make? Output = . Profits = $ Market conditions deteriorate. Sunnyvale should shut down if the price of plums falls below $ Then they make losses of $

Answer 1

MC = dTC/dq = 24 - 0.09q + 0.00015q²

(a) Profit is maximized when P = MC.

24 - 0.09q + 0.00015q² = 24

0.00015q² = 0.09q

Assuming q is non-zero, dividing by q:

0.00015q = 0.09

q = 600

Revenue (TR) = P x q = 24 x 600 = 14,400

TC = 3,000 + (24 x 600) - (0.045 x 600 x 600) + (0.00005 x 600 x 600 x 600) = 3,000 + 14,400 - 16,200 + 10,800

= 12,000

Profit = TR - TC = 14,400 - 12,000 = 2,400

(b) Shut-down price is the minimum value of AVC.

TVC = 24q - 0.045q² + 0.00005q³

AVC = TVC/q = 24 - 0.045q + 0.00005q²

AVC is minimized when dAVC/dq = 0.

dAVC/dq = - 0.045 + 0.0001q = 0

0.0001q = 0.045

q = 450

Shut-down price = AVC_minimum = 24 - (0.045 x 450) + (0.00005 x 450 x 450) = 24 - 20.25 + 10.125 = 13.875

In this case,

TR = 13.875 x 450 = 6,243.75

TVC = AVC x q = 13.875 x 450 = 6,243.75

TC = 3,000 + TVC = 3,000 + 6,243.75 = 9,243.75

Loss = TC - TR = 9,253.75 - 6,243.75 = 3,000