Question

3. (4 points) Lake Iznik can be freely accessed by independent fishermen. The cost of taking a boat on the lake is 10 TL. When b boats are taken out on the lake, f(b) = 90 b1/3 kgs fish are caught in total (so that each boat catches f(b)/b kgs. of fish). The price of fish is 1 TL/kg, which is unaffected by the level of the catch. a) What is the equilibrium number of boats that are taken out on the lake? b) What is the efficient number of boats that should be taken out on the lake? c) What per-boat fishing tax would restore efficiency? d) Suppose that the lake is instead owned by a single individual who can choose how many boats to send out. What level would this owner choose?

ı think it is maximation problem with constraints,which can be done by using partial derivatives but ı don't know that how it is carried out

Answer 1

QA) at market equilibrium

Average Revenue (AR) = MC

TR = P* f(b)

TR = 90b^1/3

So AR = TR / b = 90/(b^{​​​​2/3})

So at eqm, 90*b^{​​​​​​-2/3} = 10

Thus b* = 27 boats

b) at efficient equilibrium, MR = MC

MR = ( 1/3)*90*b^{​​​​​​-2/3} = 30b^-2/3

so at eqm, 30b^-2/3 = 10

3 = b^{​​​​​​2/3}

b` = 3^1.5 = 5 boats

c) now tax increase MC , so at efficient equilibrium level,

AR = new MC

Now AR = 90(b`)^-2/3 = 90/3 = 30.

So new MC = 30

Thus per boat tax = 30-10 = 20

d) now the single owner will want to hire boats to maximize the profits , so hire b = 5 boats