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1 Introduction

The artificial enrichment of lakes and rivers with residual nutrients from eco-
nomic activity such as agriculture can lead to the transformation of these habi-
tats from clear waters that provide a high level of ecosystem services into turbid
waters containing an overabundance of aquatic plant life and often leads to the
development of toxic algal blooms.

Certain bodies of water, and in particular shallow lakes, present a hysteresis
in their response to phosphorous loading. That is, a lake will remain in an olig-
otrophic state over long periods of time with gradual increases in phosphorous
loading to a point at which it suddenly flips to an altnernate, eutrophic state.
Once the flip has occurred, the lake then remains eutrophic despite decreases
in phosphorous loading. The threshold point at which the lake changes state is
known as a Skiba point. It denotes the flip point between alternative basins of
attraction and it is unique, as establised by Wagener (2003).

Several authors have integrated the dynamics of the shallow lake into eco-
nomic analysis, in particular, S. R. Carpenter, D. Ludwig, W.A. Brock, W. D.
Dechert, S. I. O’Donnell, L. Grüne, M. Kato, W. Semmler, K.-G. Maler, A.
Xepapadeas, A. De Zeeuw and F.O.O. Wagener. Carpenter et al. (1999) pose
a lake dynamic equation with respect to phosphorous such that it can be used
for economic analysis. They perform a dynamic stochastic analysis to show
that models random shocks prescribe more conservative levels of phosphorous
loading than deterministic models. Dechert and Brock (2000) first pose the
problem as a dynamic game of communities each maximizing its welfare in its
use of the lake and identify the presence of Skiba points when there are more
than two communities around the lake. Gruüne et al. (2005) use dynamic pro-
gramming to solve the problem. O’Donnell and Dechert (2004); Dechert and
O’Donnell (2005) use stochastic programming to derive Nash equilibrium re-
sults when the phosphorous loading into the lake is subject to rainfall as the
random shock. Müaler et al. (2003) propose a tax as the optimal policy to induce
a Pareto-optimal solution to the game.

It is assumed that society as a whole benefits from a body of water acting
as a waste sink for agriculture but also providing a source of clean water for
consumption, other production and recreational activities. Therefore, commu-
nities that share the use of a lake will often have the same relative preference
for the lake as a waste sink to other uses that require a clean lake. In short,
this means that each community around the lake will have the same welfare
function for alternative uses of the lake. As shown by Müaler, Xepapadeas and
De Zeeuw, a tax on phosphorous loading can achieve the Pareto-optimal state
for the lake when the communities each seek to maximize their welfare in a
non-cooperative manner.

When different communities benefit in different proportions from phospho-
rous loading to other benefits provided by the lake, however, it is not possible
to model the welfare of all communities with the same function. In this case,
the Pareto-optimal level of phosphorous loading will be different than when the
welfare function is the same across all communitie. In addition, each community
acting to maximize its welfare will lead to a different Nash equilibrium. This



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