Question

Meanwhile, in another part of the Empire, the different planets need to decide on how many space ships, s, they should send out to harvest space fish. A space ship costs 2 SEK and one unit of fish can be sold for a galactic market price of 1 (a constant price). Total fish production is given by: f(s) = 20√s. The Imperial senate has a space fish sub-committee which has the task of maximizing Imperial profits and therefore decides how many space ships the planets should build to fish with.

a) How many ships does the senate decide to build?

b) How large are total profits?

Due to political unrest caused by recent rebel activity, the Imperial senate loses its grip on power and now the planets decide individually how many space ships to build and send out fishing.

c) How many ships will now be fishing?

d) How large are total profits now?

e) Compare your answers in a) and b) to your answers in c) and d) and explain the differences, if any.

Answer 1

Q = f(s) = 20s^0.5

(a) Profit is maximized when (Output price x Marginal product) = Input price

Marginal product (MP) = dQ/ds = (20 x 0.5) / s^0.5 = 10 / s^0.5

Therefore,

1 x (10 / s^0.5) = 2

10 / s^0.5 = 2

s^0.5 = 5

Squaring,

s = 25

(b) When s = 25, Q = 20 x 5 = 100

Revenue (R) = Output price x Q = 1 x 100 = 100

Cost (C) = Input price x s = 25 x 2 = 50

Profit = R - C = 100 - 50 = 50

(c) In this case profit is maximized when MP = Input price.

10 / s^0.5 = 2

s^0.5 = 5

Squaring,

s = 25

(d) Since value of s is the same, Q, R and C will be the same and therefore, profit will equal 50.

(e) Since output price is 1, (Output price x Marginal product) = Marginal product, and therefore there is no difference in answers to the two parts.