Question

An e-scooter seller has found out that his total cost curve is given by the formula TC = 100 + b³ - 15b² + 85b where b is the number of e-scooters produce.

1. Describe this technology in terms of increasing, decreasing, or constant returns to scale.

2. What would be the e-scooters seller´s profit maximizing level of output and how much profit will he earn if the price of an e-scooter is 90 Dollar?

3. What would be his profit maximizing level of output and how much profit will he earn if the price of an e-scooter is 25 Dollar?

Answer 1

(1)

A cubic polynomial cost function means that ATC curve first decreases, then reaches a minimum and then starts increasing, giving this a U-shape. So, initially when ATC is decreasing, there is increasing returns to scale. When ATC reaches minimum point (or stays horizontal), there is constant returns to scale. When ATC is increasing, there is decreasing returns to scale.

(2)

MC = dTC/db = 3b² - 30b + 85

Setting P = MC,

3b² - 30b + 85 = 90

3b² - 30b - 5 = 0

Solving this quadratic equation,

b = 10.16 (other root is negative, so ignored as b >= 0)

TR = P x b = 90 x 10.16 = 914.4

TC = 100 + (10.16)³ - 15 x (10.16)² + 85 x 10.16 = 100 + 1048.77 - 1548.38 + 863.6 = 463.99

Profit = TR - TC = 914.4 - 463.99 = 450.41

(3)

Setting P = MC,

3b² - 30b + 85 = 25

3b² - 30b - 60 = 0

Solving this quadratic equation,

b = 11.7 (other root is negative, so ignored as b >= 0)

TR = P x b = 25 x 11.7 = 292.5

TC = 100 + (11.7)³ - 15 x (11.7)² + 85 x 11.7 = 100 + 1601.61 - 2053.35 + 994.5 = 642.76

Profit = TR - TC = 292.5 - 642.76 = - 350.26 (loss of 350.26)