Chapter 4 The CAPRI fish model
4.4 Behavioral model for fish supply and feed demand
an assumption is made that the diet is uniform. The feed formulation chosen for crustaceans refers to shrimp feed. Mollusks are a filter non-fed seafood category and therefore have no feed demand. Pelagic, demersal, and other marine fish are mostly cultured in the ocean and are assumed to consume the same feed.
Freshwater and diadromous fish is an important but heterogeneous CAPRI fish category which accounts for the largest part (47%) of total aquaculture production. This category includes herbivorous and omnivorous fish such as carp, barbells and tilapia, and carnivorous fish such as sturgeon, eel, salmon, trout, smelts, and shad. According to Tacon and Metian (2008, 2015), the feed conversion ratio of herbivorous fish such as carp and tilapia ranges between 1.5 and 2, whereas the ratio of carnivorous fish like trout and salmon is between 1.3 and 1.5. Furthermore, about 30% of the farmed fish belong to non-fed filter-feeding species such as silver carp, bighead carp and invertebrates (FAO, 2018). This metric is accounted for by reducing the feed conversion ratio accordingly for this fish group.
assumptions of a given scenario. Therefore, the considerations of the following methodology relate exclusively to aquaculture supply.
The aim of analyzing producers’ behavioral functions is to explore how farmers make the decision to produce the types of fish as discussed above and to use inputs in the production to obtain an optimal operating result under the assumption of competition. In other words, which combination of FIML, FIOL, different fish feed ingredients (agricultural crops) and FIOT should be used for the fish farmers. The decision-making process in determining optimal seafood supply and feed demand can be assessed by either the profit function or the cost function starting from the dual approach of the production theory (Fuss & McFadden, 2014). Profit function means the producers’ profit as a function of input and output prices, while the cost function represents farmers’ economic behavior which strives to minimize the producing cost subject to the given output quantities and input prices.
Both profit maximizing and cost minimizing approaches should generate the same results with a suitable nesting, and the preference of either approach depends on the type of disaggregation of the behavioral functions in the specific model (Colman, 1983). As a complex market representation of seafood is being applied, we must disaggregate the full producer’s problem of profit maximization with prices of seafood and feed ingredients, which
are the variables that determine supply and demand quantities under market equilibrium conditions.
Next, the methodological considerations concerning aquaculture production technology are detailed to specify the underlying profit function.
Aquaculture supply
The profit function, determined by input and output prices, can be formulated depending on production margins subject to a quadratic cost function. In other words, fish farmers determine the optimal mix of the six CAPRI fish species, each valued at the corresponding net revenues. Net revenues basically assume the role of output prices due to our assumption of a fixed FCR. The calculation of profit maximizing aquafeed demand for individual feed ingredients is divided into two levels as shown in Figure 4-2.
The total feed quantity is technically linked to profit maximizing aquaculture supply, allowing for total fish feed demand to be disaggregated into two feed demand stages.
1st level: Profit Maximization Equation 14
max π = ∑ fi∙ Pi− ∑ fcri∙ fi∙ Wi− qV
i i
Equation 15
s. t. α0+ ∑ αi∙ fi+ ∑ ∑ αij∙ fi∙ fj= V
j i i
i = CRUS(1), MOLS(2), FFIS(3), DFIS(4), PFIS(5), OFIS(6)
q: input price vector, V: input vector, fi: production quantity of fish type i, Pi: producer price of fish type i, fcri: feed conversion rates for fish type i, Wi: prices of formulated feed
for fish type i, fj: alias of fi
The demand of total feed quantity (TFQ):
TFQ = ∑ fcri∙ fi i
The demand of total feed quantity (TFQ) of fish type i:
TFQi = fcri∙ fi
Supply of FIML, FIOL and FIOT
FIML and FIOL are by products processed from fresh fish (typically from capture) as well as from fish waste from human food consumption. In general, one ton of forage fish can be extracted to 225 kilograms of FIML and 50 kilograms of FIOL. Thereby, the supply of FIML&FIOL is technically determined, given processing use of feed fish and human consumption of food fish. The variable 𝑃𝑅𝐷𝐻𝐶𝑂(𝑓𝑚𝑜𝑙) represents the
production of FIML or FIOL from human consumption waste (Equation 16).
First, the human consumption quantity of each fish category is multiplied by the WR to calculate the possible amount of food fish waste that might be used in FIML&FIOL production. Then the computed quantity is multiplied by the 𝑅𝑅(𝑓𝑚𝑜𝑙) which is the reduction ratio to obtain the final quantity of fishmeal or fish oil.
Equation 16
𝑃𝑅𝐷𝐻𝐶𝑂(𝑓𝑚𝑜𝑙) = ∑ 𝐻𝐶𝑂𝑀(𝑓𝑖𝑠ℎ) ∙ 𝑊𝑅(𝑓𝑖𝑠ℎ) ∙ 𝑅𝑅( 𝑓𝑚𝑜𝑙)
𝑓𝑔
With 𝑓𝑚𝑜𝑙 = 𝑓𝑖𝑠ℎ𝑚𝑒𝑎𝑙, 𝑓𝑖𝑠ℎ 𝑜𝑖𝑙
The major source of raw materials for FIML&FIOL production is the small pelagic forage fish catch in the reduction fisheries as mentioned above.
Therefore 𝑃𝑅𝐷𝑅𝐸𝐷(𝑓𝑚𝑜𝑙) refers to the production quantity of FIML or FIOL from the reduction of fish (PRCM) as shown in Equation 17.
Equation 17
𝑃𝑅𝐷𝑅𝐸𝐷(𝑓𝑚𝑜𝑙) = ∑ 𝑃𝑅𝐶𝑀(𝑓𝑖𝑠ℎ) ∙ 𝑅𝑅(𝑓𝑚𝑜𝑙)
𝑓𝑔
The total production of FIML&FIOL is derived from the aggregation of two sources, fish from human food waste and fish from reduction fisheries (Equation 18).
Equation 18
𝑀𝐴𝑃𝑅(𝑓𝑚𝑜𝑙) = 𝑃𝑅𝐷𝐻𝐶𝑂(𝑓𝑚𝑜𝑙) + 𝑃𝑅𝐷𝑅𝐸𝐷(𝑓𝑚𝑜𝑙)
Fish waste (FIOT) in the CAPRI fish model is the aggregate of heterogeneous and not statistically recorded material such as animal bone powder and shrimp head meal which are added in the fish feed formulation.
Due to the lack of sufficient information, FIOT is assumed to be non-traded and unusable for the agricultural sector. Therefore, an ad-hoc semi-log form supply function (Equation 19) for FIOT is introduced in the CAPRI fish model with a low producer price (𝑃𝑃𝑅𝐼𝐹𝐼𝑂𝑇). In addition, the assumption of a standard supply elasticity (𝑆𝑈𝑃𝐸𝐿𝐴𝑆𝐹𝐼𝑂𝑇= 1)) is made.
Equation 19
𝑃𝑅𝑂𝐷𝐹𝐼𝑂𝑇
𝐷𝑎𝑡𝑎_𝑃𝑅𝑂𝐷𝐹𝐼𝑂𝑇 =𝑆𝑈𝑃𝐸𝐿𝐴𝑆𝐹𝐼𝑂𝑇∗ log( 𝑃𝑃𝑅𝐼𝐹𝐼𝑂𝑇
𝐷𝑎𝑡𝑎_𝑃𝑃𝑅𝐼𝐹𝐼𝑂𝑇) + 1
𝑃𝑅𝑂𝐷𝐹𝐼𝑂𝑇: Supply quantity, 𝑆𝑈𝑃𝐸𝐿𝐴𝑆𝐹𝐼𝑂𝑇: Supply elasticity of FIOT (𝑆𝑈𝑃𝐸𝐿𝐴𝑆𝐹𝐼𝑂𝑇 = 1), 𝐷𝑎𝑡𝑎_𝑃𝑅𝑂𝐷𝐹𝐼𝑂𝑇: original parameter of FIOT production, 𝐷𝑎𝑡𝑎_𝑃𝑃𝑅𝐼𝐹𝐼𝑂𝑇: original parameter of FIOT producer price
Fish feed demand
The decision of the distribution among FIML, FIOL and aggregated cereal is done at the second level of the producers’ problem, and among the individual cereals and FIOT at the third level. Here, demand quantities for individual feed ingredients are solved for implementing a cost minimization approach. The two levels are differentiated from the substitution elasticity assigned among the commodities as the feed ingredients are less flexible in substituting for each other at the second level than at the third level. The cost minimization responds to prices of FIML, FIOL and aggregated cereal at the second level and to the prices of individual cereals and FIOT at the third level. The cost minimizing demand quantity for fish feed used in aquaculture is given by Equation 23 and Equation 27. Both cost minimization problems rely on constant elasticity of substitution (CES) functions to represent the technical substitution possibilities, and the resulting demand quantities for feed ingredients are stated by the underlying cost minimization problem.
2nd level: Cost Minimization
Cost minimization for a given production level X, input quantities x and input prices w:
Equation 20
min C = ∑ xjwj
j
Equation 21
s. t. α(∑ σj jρ−1xjρ)
1
ρ≥ X (CES)
j: feed ingredients = FIML(1), FIOL(2) and aggregated plant-based ingredients(3) = SOYC, MAIZ, BARL, WHEA, RARI, RAPE, RAPO, RYEM, SOYO, SUNF, SUNO and FIOT X: aggregate feed quantity for fish type i, w: input prices, σ: distributing parameter
ρ = 𝜖 − 1 𝜖⁄ is the parameter related to the elasticity of substitution 𝜖 which are adopted in the CAPRI model (Britz and Witzke, 2012), α and σ are calibration parameters. Taking the first order conditions to optimize this problem gives the solution stated below.
Equation 22
xj∗= Xσj(αρ W
wj)
1
1−ρ for each j = 1…3
W∗= (∑ τjwj
−ρ 1−ρ j
)
1−ρ
−ρ
Here we can reparametrize the conditional demand equations using the elasticity of substitution, ε = 1
1−ρ. Equation 23
xj∗= Xσjαε−1(P wj)
ε
On the second level, we assign a relatively small elasticity coefficient (between 0.5 and 1) within the 3 major ingredient groups with respect to the proportion of those ingredients used to the formulated feed is rather fixed.
3rd level: Between other feed materials (simplification)
In this stage, we neglect the energy contained in the different crops and the processed products that are used as feed ingredients and simply assume they substitute for each other and assign a larger elasticity coefficient (between 5 and 10).
On the third level, we basically duplicate the cost minimization for a given production level X’ from the group 3, the aggregated plant-based feed use, with single input quantities of plant-based feed x’ to be chosen, given input prices w’:
Equation 24
min C′ = ∑ x′kw′k k
Equation 25
s. t. β(∑ σ′k kγ−1x′kγ)
1
γ≥ X′ (CES)
k: Plant-based ingredients = SOYC, MAIZ, BARL, WHEA, RARI, RAPE, RAPO, RYEM, SOYO, SUNF, SUNO and FIOT
X’: aggregate feed quantity for fish type i, w’: input prices, σ′: distributing parameter
Equation 26
x′k∗ = X′σ′k(βγ W′
w′k)
1
1−γ for each k = 1…12
W′∗= βγ−1(∑ σ′kw′k
−γ 1−γ k
)
1−γ
−γ
Here we can reparametrize the conditional demand equations using the elasticity of substitution, ε′ = 1
1−γ. Equation 27
x′k∗ = X′σ′kβε′−1(P′
w′k)
ε′