The Economic Value of Ecosystem-Based-Management
       from Optimal Control Theory
     By C. Costello, C. Kappel, J. Sanchirico, and D.Siegel

1  Background & Premises
  • EBM has 2 important features at its core:

    – ecosystem linkages
    – management (i.e. interventions by humans to achieve some ob-
     jective(s))

  • Together these features imply a (dynamic) control problem for us hu-
   mans with 3 key features:

    – a set of state and control variables
    – a description of how state variables change as functions of time,
     states, and controls (technically these are constraints)
    – a clearly defined objective function

  • EBM could be viewed as a control problem where ecosystem linkages
   and values, once swept under the rug or ignored altogether, are explic-
   itly included

Objective: The purpose of this paper is to derive an approach for calculat-
ing the importance of EBM (as opposed to, say, single species management)
as measured by the difference in economic value. The approach will be
applied to a stylized 3 species food web model (kelp-urchin-otter)




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2  Sketch of the Model
  • State Variables Consider a control problem of the following form: A
   collection of species interact in a ecosystem (i.e. a web). Taking into
   account meta-populations, human populations, etc, we will denote by
   Xt the vector of all relevant state variables in this system. The infor-
   mation contained in the vector Xt provides a complete characterization
   of the state of the system at any point in time, t.

  • Control Variables Now, humans exerts some influence on this sys-
   tem. Those influences can either be endogenous to the model (e.g.
   direct harvest of one of the species) or exogenous to this system (e.g.
   emit carbon and warm the earth). The exogenous influences can be
   captured, but we will assume (by definition) that those influences can-
   not be altered, and so they are taken as given. Denote by Ht the
   collection of endogenous influences - it is the list of control variables
   that constitute the ”management” part of Ecosystem Based Manage-
   ment.

  • State Transitions Control variables are chosen (perhaps subject to
   certain constraints) and are expected to affect state variables. State
   variables can also affect each other. These dynamical relationships are
   captured by the vector function below:

                Xt+1 = F (Xt , Ht )            (1)

  • Objective Function Per-period payoff is π(Xt , Ht ), and the objective
   function is:
                     T
                 max      π(Xt , Ht )        (2)
                 {Ht }
                     t=1




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3  Value of EBM
Solving this kind of problem is a non-trivial task. Essentially, the goal is to
find a function: H ∗ (Xt ) that determines the optimal control at any point
in time as a function of the entire state vector. A general solution concept
(dynamic programming) can be used to solve problems of this type. The
Bellman Equation is:

         Vt (Xt ) = maxHt π(xt , Ht ) + δVt+1 (Xt+1 )       (3)

Then V0 (X0 ) gives the value of any initial conditions (X0 ), provided that
the ecosystem is managed optimally.
  We assume that without EBM, the objective function is unchanged as
are the state and control variables, but that the planner does not recognize
the dynamical linkages in the ecosystem. Only the dynamics of the target
species are recognized. Other species are recognized, but their populations
are assumed to be constant. The DP approach can also be used to calculate
the policy function under this assumption. Let J0 (X0 ) be the value of initial
conditions provided that the ecosystem is managed without EBM.
  The value of EBM is simply V0 (X0 ) − J0 (X0 ). This value cannot be
negative, by Bellman’s principle of optimality.




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4  Example
To illustrate the application of this framework, we have constructed a 3
species food web between kelp, urchins, and otters. We make the simplifying
assumptions that humans value only urchin harvest, at a constant price of
1. The dynamics are as follows:

          xt+1 = axt (1 − xt /K) − bxt yt + xt          (4)
              = −cyt + dxt yt − eyt zt − ht
          yt+1                          (5)
                 = −f zt + gyt zt
          zt+1                          (6)

Here x is kelp, y is urchins, and z is otters. Urchin harvest is h and the
objective is simply:
                    T
                 max     ht              (7)
                  ht
                    t=1

  The policy function from optimal management is: h∗ (xt , yt , zt ), which
gives value V (x0 , y0 , z0 ) when applied given initial conditions. The policy
function from sub-optimal management is h0 (xt , yt , zt ), which gives value
J(x0 , y0 , z0 ). The value of EBM is V − J and the associated trajectories can
be compared.




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