Question

Bob sells batteries for e-bikes. Bob’s short-run cost function for e-bikes is given by C(q) = q² + 25q + 144

a. If the market price is $75 per battery, how many batteries will Bob produce?

b. What is the price that will provide Bob zero profits?

c. If the price is below the level you found in part (b), will Bob shut down? If so, explain. If not, below what price will she shut down?

Answer 1

Answer)a) Let q be the quantity of batteries produced by bob.

Profit = Revenue (price of one battery * quantity of batteries produce) - Cost(Cost function)

Profit = 75q - (q² + 25q + 144)

Profit = 75q - q² - 25q - 144

d(profit)/d(q) = 75 - 2q - 25 - 0

For maximum profit, d(profit)/d(q) = 0

0 = 75 - 2q - 25

2q = 50

q = 50/2

q= 25

So, bob will produce 25 batteries.

Answer b) here, profit = 0

So, Average cost (AC) = Marginal cost (MC)

C(q) = q² + 25q + 144

MC = C(q)/q

MC = (q² + 25q + 144)/q

MC = q + 25 + 144/q

and, AC = d(C(q))/ dq

AC = d(q² + 25q + 144) / dq

AC = 2q + 25

Now, equating AC and MC , and solving for q

q + 25 + 144/q = 2q + 25

144/q = 2q + 25 - q -25

144/q = q + 0

144 = q²

q = 12

Now, substituting the value of q in question of MC or AC

AC = 2q + 25

AC = 2*12 + 25

AC = $49

So at $49/ battery will provide zero profit.

Answer c) No, bob will not shut down his shop.

He will shut down when the price is minimum of Average variable cost (AVC)

AVC = MC

AVC = 2q + 25

for minimum cost q = 0

AVC = $25

So bob will shut down at price $25