Consider a fishery where the harvest rate is related to effort:

H= a/b E - 1/b E^2

and total cost of effort is: TC = cE

E relates to the stock level, S, in the following way: E= a - bS

Let a = 20, b = 0.2 and c = $20. Assume a competitive market price of fish to be $10 per ton.

a) Find the open-access equilibrium value of effort and harvest. 

b) Find the fishing effort that maximizes resource rent, EMEY, and the corresponding harvest, HMEY.

c) Find the fishing effort that maximizes sustainable yield, EMSY. 

d) Under what circumstances would a price increase lead to an elimination of the stock in this fishery? Explain. 

e) Suppose a management council decides to regulate this fishery for the first time. The management council is considering two alternative policies: a tax on the fish catch, and freely allocated individual transferable quotas (ITQs). Compare the impact of the two policies on (i) total social surplus generated by the fishery and (ii) profits to owners of the fishing enterprises in the fishery.

Answer:

a) The open access equilibrium level of effort is 19.6 and stock level is 2 tons of fish.

b) The fishing effort that maximizes the resources rent is 9.8 and the EMEY is 9.8. The corresponding harvest level is HMEY=499.8

c) The fishing effort which maximizes the sustainable yield is 10 and the EMSY or the highest level of harvest is 500.

d) If the equilibrium effort is greater than the maximum sustainable yield, an increase in price would eliminate the stock level. here , effort has to be greater than 10 as it is the maximum sustainable level of effort. E has to be 20>10

e) Policy of imposing tax on catches:

i) Social surplus will rise as social equilibrium will be achieved

ii) Profit will fall because the cost of fishing has increased. So, profit will reduce.

Policy of individual transferable quotas(ITQ):

i) Social surplus will rise as by targeting fishing with quotas, governments want to achieve social equilibrium. 

ii) The profit will rise for those who can buy rights at a cheaper price as by receiving quotas, industries can exercise fishing and raise profit. But those who cannot afford it end up with less amount of fishing.

In both the policies, social surplus will rise whereas in tax imposition, profit will reduce and in quotas, profit may or may not rise.

Step-by-step explanation

The harvest rate is given as a function of effort. Which is H=a/bE-1/bE^2 where E is effort.

Total cost is TC=cE

The stock level is given by, E=a-bS

a) The open access equilibrium value of effort and harvest can be obtained by the condition fishery profit equals to zero.

So, at revenue - cost=0, we get the open access equilibrium. 

The revenue is P*H where P is the price level or the competitive market price per ton of fish. 

Hhence , P(a/bE-1/bE^2) and

E=a-bS

P[a/b(a-bS)-1/b(a-bS)^2]

p=$10, a=20 and b=0.2 by substituting these values we get,

R=P[20/0.2(20-0.2S)-1/0.2(20-0.2S)^2]

=10[100(20-0.2S)-5(20-0.2S)^2]

=10[(2000-20S)-5(20-0.2S)^2]

And cost C=cE

C=20(20-0.2S)=400-4S

So, equilibrium will be, 

R-C=0

10[(2000-20S)-5(20-0.2S)^2]-(400-4S)=0

2S^2-204S+400=0

=>(S-100)(S-2)=0.

By solving, we get the stock levels as S= 100 or S=2. 

If S=100, the effort becomes zero as E=a-bS and E=20-0.2*100=0

But for S=2, we get positive effort, E=20-0.2*2=19.6. 

So, open access equilibrium value of effort will be 19.6 as it cannot be 0 because then harvest will be 0 which is not possible. 

And the corresponding harvest value will be H=a/bE-1/bE^2

H=20/0.2(19.6)-1/0.2(19.6)^2

H=39.2

b) The resource rent is the excess revenue or simply the difference between revenue and cost. This is the profit. 

Hence, profit=TR-TC and taking derivatives with respect to E or effort, we get the effort level which maximizes resource rent.

So, the condition is MR=MC.

TR= P(a/bE-1/bE^2)=10(100E-5E^2)=1000E-50E^2

And TC=20E

So, MR=1000-100E

MC=20

And 1000-100E=20

=>E=9.8 

So, effort level 9.8 maximizes resource rent.

Maximum resource rent is also known as maximum economic yield(MEY).

The EMEY is 9.8 as it is the fishing effort for maximum economic yield.

And HMEY is the harvest level at the maximum economic yield.

At MEY we got E=9.8, 

The corresponding harvest is H=100E-5E^2

H=100*9.8-5(9.8)^2

H=980-480.2

H=499.8

 

c) The maximum sustainable yield refers to the highest level of harvest that can be made in the fishing industry. So, the fishing effort which maximizes the sustainable yield is,

The first order condition is ,

dH/dE=d(a/bE-1/bE^2)/dE=a/b-2/bE

And the 2nd order condition is,

d^2(H)/dE^2 =-2/b<0 so the condition shows maximum level of harvest. 

And the value of effort is =a/b-2/bE=0

2/bE=a/b

2E=a

Or,  E=a/2=20/2 as a=20 so we get,

E=10. 

 and EMSY= a/bE-1/bE^2

EMSY=highest level of harvest

EMSY=a/bE-1/bE^2

=100*10-5*(10)^2

=500

d) When price increases, then to eliminate fishing stock, the equilibrium level of fishing effort has to be higher than the maximum sustainable yield. Because only then biological overfishing compared to economic overfishing. So, here E=a-bS and if S=0 then E=a, E=20 which is higher than the maximum sustainable yield which was 10. E has to be 20 >10. 

e) The social surplus is the maximum sustainable yield (MSY) and the profit is the maximum economic yield(MEY). 

Policy of imposing tax on catches:

i) When the government imposes tax on catch, tax is imposed on the harvest. then social surplus will be increased  as the government will receive the tax revenue and distribute the revenue evenly. Also, the entire society is benefited because when taxes are imposed, the fish catching moves towards social equilibrium and the negative externality is reduced.  

ii) Harvest will reduce because tax is imposed on the catches so the total revenue will decrease, therefore profit will fall. The profit will also fall due to the increase in cost of fishing. 

Policy of individual transferable quotas (ITQ):

i) As the quotas are given, firms have limited fishing rights and they trade those rights. The limit is according to the social optimal level of fishing. So, under quota, the social surplus will be increased because the fishing industry is moving towards the social equilibrium.

ii) When individual transferable quotas (ITQ) are given to the fisheries freely, the most cost effective fisheries get the access of quotas. By receiving the quotas 

According to the need of the harvesting quantities the fisheries profit depends on the price of permits charged by the industry. It may rise for those who can get the rights at a cheaper rate and maximize their profits and profit may fall for those who can not afford to buy the rights and thus, end up fishing less.

Hence in both cases, social surplus will rise as fishing creates negative externality and by targeting fishing the government wants to  achieve social equilibrium. But for tax imposition, profit will fall and for quotas, profit may or may not increase.