Liâ€™s results (1998) are correct and robust. The Comment offers an interesting perspective from real-options theory, albeit misguided--from a resource-economics point of view. In particular, the lack of sensitivity in the optimal harvest trigger to biomass uncertainty is questionable. This reply begins with a general overview of resource-economic principles and the role of real options in them. A specific reply to each point raised in The Comment will then be addressed.Productivity Analysis, Research Methods/ Statistical Methods,

Li, Eric A.L.

English

Research Papers in Economics

83
Marine Resource Economics, Volume 24, pp. 83-87  0738-1360/00 $3.00 + .00
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A Reply to “Option Value of Harvesting” Revisited 
(The Comment)
ERiC A.L. Li
Eastern Mediterranean University
Abstract   Li’s results (1998) are correct and robust. The Comment offers an 
interesting perspective from real-options theory, albeit misguided--from a resource-
economics point of view. In particular, the lack of sensitivity in the optimal harvest 
trigger to biomass uncertainty is questionable. This reply begins with a general over-
view of resource-economic principles and the role of real options in them. A specific 
reply to each point raised in The Comment will then be addressed.
General
Let us review the standard case of the economically efficient harvester. He maximizes his 
harvest profit:
  Π(E) = revenue – cost = p H – c E.  (1)
E is the fishing effort (the control variable), p the fish price, and c the unit cost of effort. 
The total harvest, H, is given by the Gordon-Schaefer average surplus yield (G-S ASY):
    
                                                           ,                                    (2)
where k is the ceiling stock size, r the natural stock growth rate, and q the fixed catchabil-
ity coefficient. In equation (2) we used:
                                         ,  (3)
where X (sustainable stock size) is a state variable resulting from G-S ASY. Substituting 
equation (2) into equation (1) and maximizing over the control variable, E, will give us 
the profit-maximizing effort:
    
                                           .  (4)
  We show E′ in figure 1. It also shows the MSY (maximum sustainable/surplus yield) 
effort, E*. In general, E′ is more conservationist, while MSY E* is not economically ef-
ficient. The parallel lines indicate how the efficient E′ is obtained. The upper (lower) line 
represents the slope of the revenue (cost) curve. When the two slopes are equal (parallel 
lines), it implies the derivative of (1) with respect to E is set to zero. This is for profit 
maximization (and economic efficiency).
Eric A.L. Li is an associate professor in the Department of Economics, Eastern Mediterranean University, Mers-
in 10, TURKEY (email: Eric.Li@emu.edu.tr).
1
qE
H qEX qEk
r
  = = −  
 
1
qE
X k
r
  = −  
 
' 1
2
r c
E
q pqk
 
= −  
 Li 84
Next, we introduce the option value of harvesting into the profit equation (1):
  Π(E) = revenue – cost – value of option forgone.   (5) 
In other words, harvesting is justified only when revenue exceeds the cost of fishing and 
the value of the harvest option used. If we follow Li (1998), the harvest option value is 
(equation 8, p. 137):
    
                                                   (6)
X is the stock size that justifies harvesting, and β1 is a bigger-than-1 constant. In equation 
(6) we expressed X in terms of E using the G-S ASY.
  At this stage, we are free to set E at any value that will maximize total profit, Π(E). 
At the same time, stock sustainability is guaranteed through G-S ASY. Carrying out 
the maximization will give us the optimal effort Eop (Li 1998, equation 11, p. 138). The 
optimal harvest trigger, Xop, and CPUE follow immediately from equations (3) and (2), 
respectively.
  The crucial point is that E is not constant. It can be set at any value to achieve eco-
nomic efficiency. Before the harvest trigger is reached, E = 0. When the harvest trigger 
is reached, E is set at the value that maximizes total fishing profit (revenue – cost – op-
tion value). X cannot be controlled (directly) by the harvester. Substituting E out by X 
(equation 4 in The Comment) will remove the harvester’s ability to maximize profit for 
economic efficiency. Furthermore, the derivative with respect to X in the smooth-pasting 
condition (equation 10 in The Comment or Li 1998, equation 7, p. 137) should not in-
Figure 1. Economic Efficiency vs. Maximum Sustainable Yield
1 1
1 ,
pqEX pqkE qE
β β r
     
 Reply to “Option Value of Harvesting” Revisited (The Comment) 85
volve E. E is the variable of optimization in the profit-maximization stage after the option 
value is derived.
  That E and X are unrelated in the option value stage is not an internal inconsistency 
with G-S ASY. When determining the option value, no harvesting is actually taking place 
(see Li 1998, line 26 (directly above equation 8), p. 137 – “..the level of profit (i.e., option 
value) required before harvest is initiated...”). Once harvesting starts, the option to harvest 
is killed. The harvester proceeds to set an E that maximizes his profit for economic effi-
ciency. At this stage, G-S ASY must be invoked for stock sustainability.
Specific (reference to The Comment)
1.  E was related to X by G-S ASY (lines 2-3, abstract).
Reply: E and X are not related in the option value stage (approaching boundary)    
since harvesting is not taking place. The option to harvest is retained.
2.  E and X were treated as unrelated to each other (lines 4-5, abstract).
Reply: E is the control variable for profit maximization. Hence, it is not affected by 
the stock-size derivative in the smooth-pasting condition (equation 10, The Com-
ment). E should not be substituted out by stock size, X. X should be substituted out 
by E during profit maximization.
3.  E and X are related to one another (lines 7-8, p. 77).
Reply: X is a state variable, which is the result of E (control variable). Therefore, X 
is a decreasing function of E at time of harvest.
4.  When deriving the option value, E is treated as if it is unrelated to X (lines 9-10, p.  
  77).
Reply: E and X are unrelated in the option value stage since no harvesting is tak-
ing place. The harvest production function, H = qEX, is only notional ex ante. Once 
harvesting begins, the option is killed. The harvester then proceeds to set E that max-
imizes his harvest profit. Now G-S ASY must be enforced for stock sustainability.
5.  Incorporate the G-S ASY relationship between E and X when deriving option value    
  and optimal harvest trigger (lines 11-12, p. 77).
Reply: The G-S ASY relationship between E and X need not be invoked when deriv-
ing the option value. There is no harvesting at this point. As a result, E should not be 
expressed as a function of X in the value-matching (VM) and smooth-pasting (SP) 
conditions. These conditions are used to derive the option value, not the optimal har-
vest trigger. Consider the following sequence adopted in Li (1998): leave E as free 
variable in VM and SP → solve for the option value (pqEX/β1) → insert option value 
in profit equation Π(E) → substitute X out by E using G-S ASY → maximize Π(E) 
over E → obtain optimal Eop → calculate optimal harvest trigger Xop. 
6.  The G-S ASF relationship was not enforced (lines 15-16, p. 77).
Reply: G-S ASF need not be enforced in the option value stage since no harvesting 
is actually occurring. Furthermore, it is not E but H (harvest yield) that is constrained Li 86
by the biological sustainability requirement. The harvest yield function H = qEX 
(equation 2, this reply) where X = k(1-qE/r) is the Gordon-Schaefer average surplus 
yield (G-S ASY). E is a free variable during profit optimization as long as G-S ASY 
is satisfied.
Without any constraints on E or H, option value would be higher, but H  might get so                                                                                                                                                            7. 
  large that the fish stock may be driven to extinction (lines 18-20, p. 77).
Reply: When the option value is higher, the optimal harvest trigger will be more con-
servative (larger). The harvester will less likely harvest. As a result, stock will less 
likely be driven to extinction.
E is related to X by the sustainable yield  model of Gordon-Schaefer (G-S ASY).    8. 
  (equation 4, p. 78).
Reply: E should stay a free variable at the VM and SP stage. Equation (4) (in The 
Comment) should express X as a function of E. That is, X is a result of exerting E. 
Furthermore, this step should occur at the profit-maximization stage, after VM and 
SP.
The option value satisfies the ordinary differential equation (ODE) using standard    9. 
  techniques (equation 5, p. 78).
Reply: The ODE is derived from the following economic reasoning. A small random 
change in stock size, dX, over dt results in a small expected option value change of 
E[F(X + dX)]. This can be discounted back to t using the discount rate ρ; i.e., F = e-ρdt 
E[F(X + dX)]. e-ρdt is approximately 1 - ρdt, and E[F(X + dX)] can be expanded by 
Ito’s lemma (Dixit and Pindyck 1994). That is, E[F(X + dX)] = E [F(X) + dX F’(X) 
+ 0.5 (dX)2 F’’(X)] = F(X) + α X F’(X) dt + 0.5 σ2 X2 F’’(X) dt. On rearranging, we 
have the ODE.
E should not be unrelated to the fish stock at harvest X* (lines 6-7, p. 79). 10. 
Reply: E is unrelated to X* at the option value stage since fish stock is not at harvest. 
In fact, X* (the optimal harvest trigger) is unknown at this stage. X* (a function of 
the optimal effort Eop) is obtained only after Eop is determined during the profit-maxi-
mization stage. See Reply (12) below. 
The optimal harvest trigger can be used to compute the optimal effort, optimal har-    11. 
  vest size, and resulting CPUE (lines 1-2, p. 80).
Reply: The optimal effort should be obtained first, then use it to obtain optimal har-
vest trigger, harvest size, and CPUE. See Reply (5) above.
This is the optimal harvest trigger, X 12.  op, from Li (1998) (equation 16, p. 80).
Reply: This is not Xop. Xop is not given in Li (1998). Li’s Xop can be obtained as fol-
lows:
            
              (using G-S ASY)
 
1
op
op
qE
X k
r
 
= −  
 Reply to “Option Value of Harvesting” Revisited (The Comment) 87
(using the optimal effort, Eop, from Li 1998, equation 11, p. 138).
X* is not very sensitive to uncertainty (lines 23-24, p. 80). 13. 
Reply: In figure 1 (p. 81, The Comment), σ (sigma) represents stock size volatility 
in percentage terms. This is a result of the geometric Brownian motion. As the com-
menter’s harvest trigger, X*, suggests, it more or less fluctuates around 5E+05. At the 
same time, σ moves from 0 (no volatility) to 1 (a 100% increase or decrease in size). 
This lack of sensitivity to uncertainty is questionable. When stock could be totally 
wiped out in the next period, one would expect a much more conservative (higher) 
trigger than 5E+05. Ditto when stock could be doubled next period--harvesting the 
stock now is tantamount to killing the goose that lays the golden egg.
Harvesting effort is not constrained in Li (1998) (Conclusion, line 1; p. 80). 14. 
Reply: E need not be constrained at the option value stage since fish stock is not 
at harvest. Harvest size/yield, H, should be constrained by G-S ASY for biological 
sustainability in the profit-maximization stage. This is adopted in Li (1998) (equation 
10, p. 138).
The option aspect becomes less important when there are constraints on effort and    15. 
  harvest size (lines 3-4, p. 81).
Reply: The sustainability constraint (G-S ASY) is enforced on H (Li 1998, equa-
tion 10, p. 138) in the profit-maximization stage. The option aspect is important and 
becomes more so as X becomes more uncertain. This is because the larger-than-1 
option-value multiple β1/(β1 - 1) decreases fishing effort Eop (Li 1998, equation 11, p. 
138) and increases the optimal harvest trigger Xop (Reply 12). Furthermore, the larger 
the stock uncertainty, σ, the higher the option-value multiple, the lower the effort, 
Eop, and the higher the optimal trigger, Xop.
Conclusions
Li’s results (1998) are correct and robust. The Comment (“Option Value of Harvesting”   
Revisited) offers an interesting perspective from real-options theory, albeit misguided 
--from a resource-economic point of view. In particular, the lack of sensitivity in the opti-
mal harvest trigger to biomass uncertainty is questionable.
References
Dixit, A., and R. Pindyck. 1994. Investment Under Uncertainty. Princeton, NJ: Princeton 
University Press.
Li, E. 1998. Option Value of Harvesting – Theory and Evidence. Marine Resource Eco-
nomics 13:135-42.
1
1 1 1
1
2
β
c
β
k
pqk
   
            
   
 

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