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Natural Resource EconomicsAcademic year: 2015-2016
Prof. Luca Salvatici [email protected]
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
4Natural Resource Economics - a.a.2015/16
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:.
5Natural Resource Economics - a.a.2015/16
6 Economia delle risorse naturali a.a.
2008/09
Dynamic vs. Static solution
6Natural Resource Economics - a.a.2015/16
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
Natural Resource Economics - a.a.2015/168
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
Natural Resource Economics - a.a.2015/16
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
14Natural Resource Economics - a.a.2015/1614
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
15Natural Resource Economics - a.a.2015/16
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
Natural Resource Economics - a.a.2015/1617
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
19Natural Resource Economics - a.a.2015/16
