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Faustmann in the Sea Optimal Rotation Time in Aquaculture By Atle G. Guttormsen Researcher Agricultural University of Norway

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Alternative title: To Kill or Not to Kill -Decision Problems in Aquaculture

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Outline 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

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Background Aquaculture becomes more and more important Little research done on management issues/decision problems A lot to learn from other industries

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Motivation As fish farm enterprise gets larger and the industry more competitive, Optimal production planning and efficient management practice becomes key factors for success.

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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.

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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.

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When to harvest n A problem very similar to the historical Faustmann (forestry) problem Market normal Market in short supply Wait with the decision Slaughter and sell The slaughtering decision Market in high supply

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Related problems The tree-cutting problem Wicksell, Faustmann, Samuelson From agricultural economics 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

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Previous research on Optimal Harvesting of Farmed “Fish” Bjørndal (1988 and 1990) Arnason (1992) Heaps (1993 and 1994) Hean (1994). Mistiaen & Strand (1998) 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)

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The Objective Maximize NPV of the Pen/Pond Gives harvesting/rotation time Gives value of the pen/pond

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”early” conclusion 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.

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Without rotation Bjørndal 1 Bjørndal 2 (with cost) where l.h.s is marginal revenue, and r.h.s is marginal cost

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Faustmann in the sea + Continuous Release + Constant p’(w) + Constant p(w) All rotation periods of equal length = The Faustmann Formula Gives in the discrete case S{t}=net fish value V{0}= the capitalized value of the pond/pen immediately prior to releasing new juvenile fish (site value)

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The Problematic Assumptions underlying “Faustmann” 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

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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

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Relative price relationship (example salmon 1992-1997)

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Example The salmon Rotation Problem

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My Extended Faustmann model Makes the problem discrete Formulate it as a dynamic programming problem Solve it numerically with Matlab

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No analytical solutions, must be solved numerically

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Examples 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

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Life Cycle for salmon 2 - 3,5 years from roe to foodfish OctoberJanuary Egg hatches Aug-OctMarch-April Smolt release Oct slaughtering 2-10 kg

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Results the Faustmann model Typical ”spring”-smolts (150 gram) with ”April” growth function. Slaughter at 19 months Weight 5.54 kg Typical ”fall”-smolts (50 gram) with ”September” growth function. Slaughter at 23 months Weight 5.65 kg

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Results The Extended Model Both ”spring” and ”fall” smolts Release possible in March, April, May, August, September and October Harvest (month released, weight and kilo) August, 21 months, 4.7 kg September, 23 months, 5.3 kg October, 23 months, 5.6 kg March, 25 months, 6.2 kg April, 24 months, 6.0 kg May, 23 months, 5.7 kg

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Further development Make more realistic examples Make examples for different species Include more costs Include more constraints Feeding quotas Density regulations Etc.

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