Natural Resource Economics Academic year: 2015-2016 Prof. Luca Salvatici luca.salvatici@uniroma3.it...

Natural Resource EconomicsAcademic year: 2015-2016

Prof. Luca Salvatici luca.salvatici@uniroma3.it

Lesson 24: Optimal (harvesting) effort

Outline

• Dynamic vs. Static solution• Dynamic models using E as control variable• “Optimal” extinction• «Micro-foundations» of the rent dissipation

Gordon-Schaefer model: dynamic version

Maximize rent using the catch as control variable:

fyfyACPyfACyPyH

ATxxyxfx

dtetyxACtPyyT

tyy

(.)^

)(,0)0(,)(

)()()(max

.

0

0

0,0

0,~0,

)(*

y

y

ty

3Natural Resource Economics - a.a.2015/16

Singular solution

Differentiating the maximum principle:

(1)

Co-state equation:

Then we eliminate the costate variable from (1)

yxfACxACACP xx

yACffyACH xxxxx ^

0)~()~()~()~(:~~*

xfxACxACPxfxyy

ACPfxfAC

yACACPfyxfAC

xx

xx

xxx

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Singular solution and arbitrage conditionFrom the equation for the singular stock:

Since the singular solution is a steady-state:(P - AC)*f(x) = R it’s a perpetuity: what is its present value? [(P - AC)f]/d Interpretation: an optimal solution implies that the instantaneous profit (P – AC) is equal to the present value of the change in the sustainable rent

cxx

xx

xx

rffpACHp

ACPdx

xfACPd

ACPxfACACPf

ACPfxfAC

ovvero ,,0:.

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6 Economia delle risorse naturali a.a.

2008/09

Dynamic vs. Static solution

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Maximizing the Present Value of Resource Rent in a Gordon-Schaefer Model

The classical Gordon–Schaefer model presents equilibrium revenue (TR) and cost (TC), including opportunity costs of labor and capital, in a fishery where the fish population growth follows a logistic function.Unit price of harvest and unit cost of fishing effort are assumed to be constants. In this case, the open access solution without restrictions (OA) is found when TR=TC and no rent (abnormal profit, P=TR-TC) is obtained. Abnormal profit (here resource rent) is maximized when TR'(X)=TC'(X) (maximum economic yield, MEY). Discounted future flow of equilibrium rent is maximized when P'(X)/d=p, where p is the unit rent of harvest and d is the discount rate. This situation is referred to as the optimal solution (OPT), maximizing the present value of all future resource rent. The open access solution and MEY equilibriums are found to be special cases of the optimal solution, when the discount rate is infinite or null, respectively.

Natural Resource Economics - a.a.2015/16

8

Control variable: E

Problem structure:

Are we going to have bang-bang solutions?

)()()((.)

)0(

)(

max

.

0)(0

xfEqxcPqxeqExxfEcPqxeH

Ax

qExxfx

dtcEPqxEe

tt

tEtE

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Singular solution I

From the maximum principle

(1)

From the costate equation

(2)

dt

dx

qx

c

qx

cPe

dt

d

qx

cPe

t

t

2

)(' xfqEPqEedt

d t

Natural Resource Economics - a.a.2015/16 9

Singular solution II

Using (1) e (2) substituting out l:

qx

cP

qx

xcf

qx

cPxf

qx

cP

qx

xcf

x

cExPfPqEPqE

x

cE

qx

xcf

xfqEqx

cPePqEeqExxf

qx

c

qx

cPe ttt

2

2

2

)()('

)(')('

)(

)(')(

Natural Resource Economics - a.a.2015/16 10

11 11

Bang-bang solutions

0)(~)(~

tExx

EtExx

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12

Optimal «extinction»: costs depending on the stock

TC(x) =>Extinction only with critical depensation: from the property rights distribution point of view, when is it more likely?

)0(AC

Natural Resource Economics - a.a.2015/1612

13

Optimal «extinction»: costs independent from the stock

AC(y) = c, if P > c what is going to be x*(T) with free access?Single owner with pure compensation:

Given that 2bx>0, what is going to happen if a<d?

bxadx

bxaxdACP

dx

xdfACPACP

dx

xfACPd2

)()(

2

Natural Resource Economics - a.a.2015/1613

Optimal «extinction»: depensation

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15

Rent dissipation: single owner (d = 0)Steady-state (singular solution of the optimal control) ==>Static solution = dynamic solution ==>

cb

Pqa

Pq

bE

cEb

Pq

b

Pqa

dE

cEEb

PqE

bPqa

d

dE

cExPqEd

dE

dR

2

2

22

2*

02~

)(1~0~~~~ 2

.

qEab

xxqExbxaxxx

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16

Rent dissipation: two owners (d = 0) Steady-state:

Solution firm 1:

Solution firm 2:

EEqab

xxEEqxbxax ^1~0~)^(~~ 2

.

Eb

EPqc

b

Pqa

Pq

bEE

cb

EPqE

b

Pq

b

Pqa

dE

dR

cEb

EEqaPqEcExPqEEERE

^2

1^

2

~*

0^2

)^(~)^,(max

2

2

22

EE2

1*^

Natural Resource Economics - a.a.2015/16

16

Economia delle risorse naturali a.a.2007/08 17

Nash equilibrium

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18

Rent dissipation: n firms (r = 0) In N:

n firms: steady-state

Optimal effort

3

2*

2

1*

4

3*

4

1

2

1*

2

1

2

1^

2

1*

EEEEEE

3

4^

3

2^ EEEE

EnEqab

xxEnEqxbxax )^1(1~0~))^1((~~ 2

.

)1(21

1*

nE

Natural Resource Economics - a.a.2015/1618

19

Rent dissipation: synoptic table n E nE 1 a a 2 (2/3)a (4/3)a ....... ........ ....... 10 (2/11)a (20/11)a ........ ........ ......... infinite 0 2 a

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