Appendix 3. Choosing optimal inputs.

Consider a decision maker who wishes to calculate the input rate that maximizes the net economic return. The decision maker knows the current state Xt-1, the distribution of disturbances to input, and the parameter distribution of a forecasting model (Appendix 2). For any specified input a, the variance of disturbances is

 var(a) = a2 [exp(ς2) - 1]. (A.3.1)

Forecasts (Appendix 2) follow a t distribution, which approaches a normal distribution if the variance discounting parameter D is sufficiently close to 1.0, and the system has been observed for a sufficient number of time steps. The prediction of the next state has mean Xt and variance var(Xt) where

 Xt = Xt-1 + a + R(Xt-1) (A.3.2) var(Xt) = Q + a2 [exp(ς2) - 1] (A.3.3)

where Q is the variance of a prediction from R (Appendix 2).

The utility derived from a particular input rate a is calculated as follows. The utility of activities that pollute the lake is

 UL(at) = α a. (A.3.4)

That is, benefits from polluting activities rise directly with the level of pollution. The utility of ecosystem services is inversely related to pollutant levels in the lake according to

 UP(X) = 1 - β Xt2. (A.3.5)

This relationship follows from the inverse linear relationship of marginal utility to water quality, which implies dUP / dX = - k X.

The net utility is simply the sum

 V(a) = E[UP(Xt)] + UL(at) (A.3.6)

where E denotes the expectation operator. To find the input rate that maximizes expected net utility, we define a discrete mesh for X. For any given a value, we calculate the distribution of Xt, and then estimate

 E[UP(Xt)] = Σ p(Xt) (1 - β Xt2 ) (A.3.7)

where p(Xt) is the probability density associated with a mesh interval for Xt and the summation is taken over all mesh intervals. We then calculate V(a). The value of a that yields the largest V(a) is the optimal input rate one step ahead.

More generally, the decision maker will wish to calculate the optimal input rate from the present to infinite time, given present knowledge of the state of the system and the distribution of the forecasting parameters (Carpenter et al. 1999). The problem is to choose a sequence of inputs, a, that will maximize the net present value of polluting activities and ecosystem services.

A monetized stream of benefits is calculated as in equations A.3.4 - A.3.6. The net present value of a sequence of states Xt resulting from a sequence of inputs at is

 V(at) = Σ δt[UP(Xt) + UL(at)] (A.3.8)

where the summation is from t = 0 to t = infinity. The discount factor δ (0 < δ £ 1) is used to adjust future utilities to the present utility.

The policy that maximizes V(at) also maximizes

 V(Xt) = [UP(Xt) + UL(at)] + δ E[V(Xt+1)] (A.3.9)

where the expectation is over possible future values of X. The function a(X) is computed iteratively as described by Carpenter et al. (1998). The result is a curve that gives the input target, at, that will maximize V, as a function of Xt.

We assume that the decision maker repeats this calculation at each time step. To avoid the need to repeat this tedious calculation with each cycle, the Market Manager and Governing Board models use a table of input targets, at, on a discrete mesh of Xt and B1,t. Uncertainty in B is handled by computing the weighted average value of a, where the weights are probability masses for B on each mesh point.