OPTIMAL DEPLETION OF RENEWABLE RESOURCES

Prof. Prabha PanthOsmania University,

Hyderabad

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Renewable resources

• Renewable resources can reproduce themselves.• E.g. forests, animals, birds, etc.• But they are also prone to be depleted if the

rate of economic exploitation > the rate of reproduction.

• Once depleted it is extremely difficult to bring them back to their original level.

• Optimal rate of exploitation = rate of regeneration (reproduction).

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Optimal rate of depletion of renewable resources

• Example: There is a lake with 1000 fishes, initial stock.

• Maximum possible rate of growth of fish = 10% p.a. Cet. Par.

• End of the year, 1000 + 100 = 1100 fishes are available in the lake.

• Rate of depletion (catch) = 100 fish, then remaining 1000 fishes are still available for future growth.

• This is called Sustainable Yield.

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• If 200 fishes are caught, then only 900 remain in the lake, (1100 – 200), and addition to stock at the end of the year @ of 10% growth = 90 fishes.

• Next year if 250 fishes are caught, then at the end of the year, the stock will be 740, (990 – 250).

• Economists want output to grow every year, leads to progressive depletion of the renewable resource.

• Also growth of the resource based on various other factors, such as age and health of the fish, water and climatic conditions, etc.

• If the stock becomes too small: biological limits: then this rate of reproduction cannot be maintained.

• The point where the population growth becomes unviable is called “Critical Minimum Size”.

• If population falls below this level, stocks will die out.Prabha Panth

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• If the rate of harvesting of the renewable resources > rate of its growth, there will be progressive depletion.

• When the renewable resource size or population falls to a very low level, it cannot regenerate itself.

• This is the biological limit, for instance if there are only 5-6 fishes remaining, then gth cannot be sustained at 10%.

• This point where the population becomes unviable is called “The Critical Minimum Size”.

• Below this level, the population cannot survive, and dies out.

• Left to itself, the resource will not grow forever. • There are natural checks which will regulate the

population and keep it within the ecological limits.Prabha Panth

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0

Growth rate

Maximum Sustainable Yield, MSY = G(x) max

MSY: Maximum growth that can be achieved

G(x1)

X mini X m X1 XCC= Carrying Capacity

g=0

Critical Minimum Size

Fish Population

Figure 1. Growth Pattern of a Renewable resource

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Growth pattern of a Renewable resource

• Figure 1 shows the growth and decline of a renewable resource – fish.

• Xmin is the point below which fish stock dies, and becomes zero.

• After this point, there is growth of the fish population.

• The fish population grows till it reaches Xm.

• Xm shows the highest rate of growth of fish, also called MSY = Maximum Sustainable Yield.

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Growth pattern of a Renewable resource

• After this point, rate of growth of fish starts declining, till it falls to XCC

• At this point, growth rate of fish = 0.• XCC is a Stable Equilibrium, if fish stock increases beyond it,

then death rate increases, bringing it back to XCC,

• If it is less, say at X1, then the population increases till it reaches XCC,

• XCC is known as the “Carrying Capacity” of the Ecosystem. • It is the maximum amount of the renewable resource that

can be supported by the ecosystem.

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• Economists assume that renewable resources can grow indefinitely.

• But this is not true, as the resource growth is constrained by biological and ecological limits.

• Economists feel that increasing the rate of harvesting of the renewable resource, will not affect its growth.

• But, if the resource is exploited up to the point of its Critical Minimum Size, then the resource will die out or even become extinct.

• It is difficult to revive the species once it reaches the critical minimum size.

• E.g. rhinos, tigers, pandas, lions, etc. are all in danger of reaching this critical level, due to over kill.

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Model of Optimal Level of Depletion of a Renewable resource

• How much of the resource should be harvested ?

• Conflict between ecologists and economists.Assumptions:1. Micro unit,2. Price of the resource (fish) is constant (P.C.).3. No discounting,4. MC of fishing is constant.

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0

Revenue, Cost

MSY

B

R

Stock of Fish

NEffort = fishing

S0SCC

E maxE0

A

M

F

TC

TR

TC = TR

Fig.2. Optimal depletion of a renewable resource

Max profit

Note: Fig 1 is reversed

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• TR = p.Q, TC = ΣMC, Profit maximising condition = MC = MR, MC

• When Effort (fishing) = 0, then there is maximum carrying capacity, Scc.

• When Effort is maximum, fish stock is zero, So, E max.

• As effort increases TC increases, yield increases, and TR increases.

• Max profit = MC=MR, MC, at N. Max profit = AB.• But this is less than Max Sust Yield (MSY) at M.• If there is perfect competition, long run equilibrium is

at F, where TR=TC, • But now this is the declining part of fish growth, and

stocks will start depleting, and fall to So.

• Open access with unrestricted entry, leads to over exploitation, and the renewable resource is driven to extinction. Prabha Panth

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Discounting • The above model is a static analysis.• Taking ‘time’ into consideration we get:

– A) Time preference– B) Growth in capital value (interest). Leads to positive discount

rate.• If the fish is caught now, profits earned now (time

preference).• If left in pond, will reproduce in future. Also prices in future

may rise. More profits in future.• But the fisherman has to wait. • So harvesting the renewable resource now depends on:

Discount rate > = < Biological rate of growth + Growth in capital value (expected price).Prabha Panth

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• Example:Let growth rate of fish = 3%, discount rate = 10%, price

increases @ 5%Decision: To catch 100 tonnes fish now at Rs.100 per

tonne, or catch more in future @ Rs.105 per tonne.

Option 1. Fish now = Earn Rs.10,000Option 2. Wait for future, more fish, higher price. But

discounted at 10%, the present value = Rs.9,832 < Option 1.

Fish now Fish in future

Revenue Rs. 100 x 100 = 10000 100 (1.03) = 103 x105 =10,815

Discounted value Rs.

10,000 10,815/1.1 = Rs.9,831.82

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• If Discount rate > Biological rate of growth + capital appreciation = Harvest now.

• E.g. 10% > 3% + 5% in our former example.• If Discount rate < Biological rate of growth +

capital appreciation = Harvest later.• E.g. Discount rate 10% < Biological rate of

growth 4% + capital appreciation 8%• If both are equal, then the fisherman is

indifferent between present and future.

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• Therefore:o If discounted value of future P < Present price,

present is preferred. (Rs.9832 < Rs.10,000)o Otherwise wait for the future.

• In other words: P0(1+r) = P1(1+g), or P0/P1 = (1+g)/(1+r)

1) If r = g, P1 = P0 , indifferent

2) If r < g, P0 < P1, conserve for future,

3) If r > g, P0 > P1, exploit now.

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Dasgupta and Heal ModelFormer models assume fishing is free, i.e. there are no costs of

fishing.Assumptions:

1. Open access, e.g. sea fishing.2. Identical fishing firms, n,3. X = total number of fishing vessels, xi = composite input: labour, fishing equipment (variables).4. S = size of fishing area,5. Y = fish catch or output, YH(X,S)6. H = production function, CRS. H=f(X,S), since S is fixed, H = F(X)7. Diminishing returns to X, i.e. As fishing boats increase, congestion leads to loss of output.Prabha Panth

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• Let the firm own xi vessels, then its catch will be xi.(F(X)/X) = xi.A(X)

• Rental cost of each boat = r, (constant)• As number of boats increases, TC increases,

but AC = MC = r.• Over crowding with too many boats.• Leads to low catch, low profits.• Perfect competition, P = MC at D, leads to over

crowding and over exploitation.

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•AP and MP decline with increase in number of boats.•Initially equilibrium is at X2, with profits = EF.•But with perfect competition , and open access, more boats enter, levels of fish stock falls, and equilibrium is at D.•Normal profits at D.•If number of boats increases beyond D, then all will suffer losses.

Yield

Boats0

F

AC = MC

X2 X1

C

E D

AP

MP

Fig.3. Optimal depletion of renewable resource with costs

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