Envir onmental and Resource Economics 26: 603–624, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.603The Economics of Shallow LakesKARL-GÖRAN MÄLER1, ANASTASIOS XEPAPADEAS2andAART DE ZEEUW3,∗1The Beijer International Institute of Ecolo gical Economics;2Department of Economics,University of Crete;3Department of Economics and CentER, Tilburg University, The Netherlands,E-mail: [email protected] (∗Author for correspondence)Abstract. Ecological systems such as shallow lakes are usually non-linear and display discontinui-ties and hysteresis in their behaviour. These systems often also provide conflicting services as aresource and a waste sink. This implies that the economic analysis of these systems requires to solvea non-standard optimal control problem or, in case of a common property resource, a non-standarddifferential game. This paper provides the optimal management solution and the open-loop Nashequilibrium for a dynamic economic analysis of the model for a shallow lake. It also investigateswhether it is possible to induce optimal management in case of common use of the lake, by meansof a tax. Finally, some remarks are made on the feedback Nash equilibrium.Key words: ecological systems, non-linear differential games1. IntroductionThe purpose of this paper is to develop an economic analysis of the shallow lake.Lakes have been studied intensively and the shallow lake model is well tested anddocumented, so that the analysis has a direct meaning. However, the lake modelcan also be viewed as a metaphor for many of the ecological problems facingmankind today, so that the analysis developed in this paper will have a widerapplicability. The economic analysis is especially challenging because of the non-linear dynamics of the lake (which yields non-convex decision problems) and thegaming aspects related to the common property character of the lake.It has been observed that shallow lakes, due to a heavy use of fertilizers onsurrounding land and an increased inflow of waste water from human settlementsand industries, at some point tend to flip from a clear state to a turbid state with agreenish look caused by a dominance of phytoplankton (Carpenter and Cottingham1997; Scheffer 1997). The release of nutrients, especially phosphorus, into thelake stimulates the growth of phytoplankton and in addition to that, the resultingturbidity prevents light to reach the bottom of the lake so that submerged vegetationThis research was initiated at a meeting of the Resilience Network which had financial supportfrom the MacArthur Foundation. We are very grateful for the advice and comments of (in alpha-betical order) William Brock, Steve Carpenter, Davis Dechert, Marten Scheffer, Perry Shapiro, SjakSmulders, Robert Solow, David Starrett, Florian Wagener and the referees.[ 105 ]604 KARL-GÖRAN MÄLER ET AL.disappears. With the vegetation many species disappear such as waterfleas whichgraze on phytoplankton. It has also been observed that shallow lakes are hard torestore in the sense that the nutrient loads have to be reduced below the level wherethe flip occurred before the lake flips back to a clear state. The positive feedbackthrough the effect on the submerged vegetation is one explanation for this so-calledhysteresis effect.Ecological systems often display discontinuities in the equilibrium states of thesystem over time. A seminal paper in this area models the sudden outbreak of aninsect, called the spruce budworm, and the long time it takes before the budwormdensity jumps back to a low number again (Ludwig, Jones and Holling 1978).Technically, this hysteresis effect can be modelled by a non-linear differentialequation which has multiple steady-states with separated domains of attraction ina certain range of the exogenous variable. Other examples of ecological systemswith hysteresis, among which the lake model, are described in Ludwig, Walkerand Holling (1997).In the ecological literature, management of shallow lakes is mostly interpretedas preventing the lake to flip or, if it flips, as restoring the lake in its original state.However, this approach denies the economics of the problem in the sense of thetrade-offs between the utility of the agricultural activities, which are responsiblefor the release of phosphorus, and the utility of a clear lake. When the lake flips toa green turbid state, the value of the ecological services of the lake (e.g., the intakeof water and recreation) decreases but this situation corresponds to a high level ofagricultural activities. It depends, of course, on the relative weight attached to thesewelfare components whether it is better to keep the lake clear or not. Note that if itis better to keep the lake clear, it is very costly to let the lake flip first becauseof the hysteresis. A second economic issue is that lakes are usually commonproperty resources and therefore suffer from sub-optimal use, in the absence ofcoordination.The literature on the lake model is rapidly increasing. Carpenter, Ludwig andBrock (1999) focus on hysteresis and irreversibility issues. The paper that comesclosest to this one is by Brock and Starrett (1999). They consider the dynamics andthe optimal management of the lake and point out the occurrence of saddle-pointstable steady-states and Skiba points. This paper extends their analysis to Nashequilibria, for the game of common property, and to tax policies with the aim tointernalize the externalities (see below). Brock and de Zeeuw (2002) consider arepeated game version of the lake model. They show that the occurrence of “bad”Nash equilibria can in fact be beneficial because with these points as threats intrigger strategies, cooperation can be sustained for lower values of the discountfactor.In the first part of the paper, very simple welfare analysis is done on the possiblesteady-states of the lake model. Relative weights are chosen such that it is optimalto manage the lake in one of its clear states, called oligotrophic states. It is shown,however, that when the lake is shared by more than one community, two Nash[ 106 ]THE ECONOMICS OF SHALLOW LAKES 605equilibria occur: one in an oligotrophic state and one in a dirty state, called aeutrophic state. In the second part of the paper, intertemporal welfare is maximizedsubject to the dynamics of the lake. It is shown that in case the discount rate is lowenough, an optimal path for phosphorus loadings exists, from each initial conditionof the lake, which