Faustmann in the Sea
Optimal Rotation Time in Aquaculture
By Atle G. Guttormsen
Researcher
Agricultural University of Norway
Alternative title:
To Kill or Not to Kill
-Decision Problems in Aquaculture
Outline
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Background and Motivation
The Problem
Previous Studies and Related Problems
The Faustmann Solution
Problems with the Faustmann Solution and an
Extended Faustmann Model
Applications on Salmon
Summary and Conclusions
Background



Aquaculture
becomes more and
more important
Little research done
on management
issues/decision
problems
A lot to learn from
other industries
Motivation
As fish farm enterprise gets larger and
the industry more competitive,
Optimal production planning and
efficient management practice
becomes key factors for success.
Decision Problems in Aquaculture
 When to release
juvenile fish
 How much and when
to feed
 When to harvest
 1-2 kg
 6-7 kg etc.
The Feeding Problem


”Not” a problem because it’s usually
never profitable to feed anything else
than either max (to saturation) or
nothing.
For salmon will feeding 70% of max
increase FCR substantially

Means:
70% feeding does not lead to 70% growth.
When to harvest
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A problem very similar to the historical
Faustmann (forestry) problem
Slaughter and sell
Market in high supply
The slaughtering
decision
Wait with the decision
Market normal
Market in short supply
Related problems
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The tree-cutting problem
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From agricultural economics
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Wicksell, Faustmann, Samuelson
Cow replacement
When to slaughter your pork/broiler
When to buy a new tractor
Traditional investment problems

Keep the old machine or buy a new one
Previous research on
Optimal Harvesting of Farmed “Fish”
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Bjørndal (1988 and
1990)
Arnason (1992)
Heaps (1993 and 1994)
Hean (1994).
Mistiaen & Strand
(1998)
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Karp, Sadeh and Griffin
(1986)
Leung (1986)
Leung & Shang (1989)
Leung, Hochman,
Wanitprapa, Shang and
Wang (1989)
Cacho (1990)
Hochman, Leung, Rowland,
and Wyban (1990)
Cacho, Kinnucan and Hatch
(1991).
Leung, Lee and Hochman
(1993)
The Objective
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Maximize NPV of the Pen/Pond
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Gives harvesting/rotation time
Gives value of the pen/pond
”early” conclusion
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The fish must be harvested when the
capital (fish in sea) gives a better return
than the opportunity cost.
Will always hold, however the problem
arise when we want to calculate the
opportunity cost.
Without rotation

Bjørndal 1
p '  w
w ' t 
w ' t  
 r  M
*
p  w
w t 
Bjørndal 2 (with cost)
*
*
*

p
w
w
t
 Ch 




p '( w) w '(t ) w '(t )



 r  M  

*
*

[ p( w)  Cs ]
w(t )


p( w) w(t )


C f F t * 
p( w) w  t * 
where l.h.s is marginal revenue, and
r.h.s is marginal cost
Faustmann in the sea

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Continuous Release
Constant p’(w)
Constant p(w)
Max  (t )  V (t )e
 rt
Gives in the discrete case
All rotation periods of equal
length
 V (t )e

2 rt
 V (t )e
3rt

V (t )
 ....   rt
e 1
r s t   V 0  rV t
S{t}=net fish value
V{0}= the capitalized value of the
pond/pen immediately prior to releasing
new juvenile fish (site value)
*
*

= The Faustmann Formula
The Problematic Assumptions
underlying “Faustmann”
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Possible to release juvenile fish to
seawater continuously during the year
One growth function (independent on
release time)
Constant relationship between prices for
different sizes of fish
What makes it difficult ?
 Ongoing process
 Rotation Problem
 Release of juvenile fish only possible during a
certain periods of the year
 Growth is a function of (among other) water
temperature
 Different growth functions for different “starting”
times
 Relationship between prices for different sizes of
salmon varies through the year
Relative price relationship
(example salmon 1992-1997)
135
(2-3 kilo/3-4 kilo)*100
(6-7 kilo/3-4 kilo)*100
125
115
105
95
85
sep-97
mai-97
jan-97
sep-96
mai-96
jan-96
sep-95
mai-95
jan-95
sep-94
mai-94
jan-94
sep-93
mai-93
jan-93
sep-92
mai-92
jan-92
75
Ju
l
y
y
O
ct
ob
er
Ja
nu
ar
y
Ap
ril
Ju
l
y
O
ct
ob
er
Ja
nu
ar
y
Ap
ril
Ju
l
y
O
ct
ob
er
Ja
nu
ar
y
Ap
ril
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
Ju
l
y
O
ct
ob
er
Ja
nu
ar
y
Ap
ril
Ju
l
il
Ap
r
Gram
Example
The salmon Rotation Problem
April
October
My Extended Faustmann model
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Makes the problem discrete
Formulate it as a dynamic programming
problem
Solve it numerically with Matlab
Vt ( wt , n)  max  pt  w, n  wt d t  c( f )  Vt 1 ( wt 1 , n  1)
dt , f ( t )
where
1 Harvest
dt  
0 Wait
wt 1  (1  m) g ( wt , ft ) wt (1  dt )
w  weight, n  week, p  price, c( f )  cost of feeding
  discount factor, m  natural mortality,
g ( wt , f t )  growth function
Vt (0, n)  max cr st  Vt 1 ( wt 1 , n  1)
st
where
1 Release
st  
0 Wait
wt 1  (1   t ) st wt
cr  release cost (includes cost of juvenile fish)
  first day/week death rate
No analytical solutions, must be
solved numerically
Examples
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Tabulated growth functions
Constant prices, costs, mortality and
interest rates
Includes only slaughtering costs
(i.e. no release nor feeding cost)
Applied on data for Salmon
Programmed and Solved in MatLab
Life Cycle for salmon
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2 - 3,5 years from roe to foodfish
Egg
hatches
October January
Smolt release
Aug-Oct
March-April
slaughtering
2-10 kg
Oct
Oct
Results the Faustmann model
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Typical ”spring”-smolts (150 gram) with
”April” growth function.
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Slaughter at 19 months
Weight 5.54 kg
Typical ”fall”-smolts (50 gram) with
”September” growth function.
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Slaughter at 23 months
Weight 5.65 kg
Results The Extended Model
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Both ”spring” and ”fall” smolts
Release possible in
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March, April, May, August, September and
October
Harvest (month released, weight and kilo)
March, 25 months, 6.2 kg
April, 24 months, 6.0 kg
May, 23 months, 5.7 kg
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August, 21 months, 4.7 kg
September, 23 months, 5.3 kg
October, 23 months, 5.6 kg
Further development
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Make more realistic examples
Make examples for different species
Include more costs
Include more constraints
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Feeding quotas
Density regulations
Etc.