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| '''Wonderland''' is an integrated [[mathematical model]] used for studying phenomena in [[sustainable development]]. First introduced by
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| ([[#san94|Sanderson 1994]]), there are now several related versions of the model in use. Wonderland allows [[economists]], [[policy analysts]] and [[environmentalist]] to study the interactions
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| between the [[economic]], [[demographic]] and [[human impact on the environment|anthropogenic]] sectors of an idealized world, thereby enabling them to obtain insights transferable to the real world.
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|
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| == Introduction ==
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|
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|
| '''Wonderland''' is a compact model.
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| In total, there are only four continuous state variables, one each for the economic and demographic sectors and two for the anthropogenic sector; thus making Wonderland more compact and amenable to analysis than larger, more intricate models like [[World3]]. For this reason it is often used as an initial testing ground for new techniques in the area of policy analysis ([[#lpb03|Lempert, et al., 2003]]).
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| == Governing equations ==
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| Denote the four state variables as: <math>x(t)</math> – population, <math>y(t)</math> – per capita output, <math>z(t)</math> – stock of natural capital and <math>p(t)</math> – pollution flow per unit of output. Let <math>x,y\in[0,\infty)</math> and
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| <math>z,p\in[0,1]</math>, then the state variables evolve in [[discrete time]], according to the following [[recurrence relation]]s ([[#san94|Sanderson, 1994]]).
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| : <math>
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| \begin{align}
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| x(t+1) &= x(t)\left[1+b\Big(y(t),z(t)\Big) - d\Big(y(t),z(t)\Big)\right],\\
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| y(t+1) &=y(t)\left(1+\gamma-(\gamma+\eta)\Big[1-z(t)\Big]^{\lambda}\right),\\
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| z(t+1) &= \frac{g\Big(x(t),y(t),z(t),p(t)\Big)}{1+g\Big(x(t),y(t),z(t),p(t)\Big)},\\
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| p(t+1) &= p(t)(1-\chi),\\
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| \ &\ \\
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| \!\!\!\text{where,} \qquad &\ \\
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| \ &\ \\
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| b(y,z)&= \beta_0\left[\beta_1 - \left(\frac{e^{\beta y}}{1+e^{\beta y}}\right)\right],\\
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| d(y,z)&= \alpha_0\left[\alpha_1 - \left(\frac{e^{\alpha y}}{1+e^{\alpha y}}\right)\right]
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| \left[1+\alpha_2(1-z)^{\theta}\right],\\
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| g(x,y,z,p)&= \frac{z}{1-z}\,e^{\,\delta z^{\rho}-\omega f(x,y,p)}, \ \text{and}\\
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| f(x,y,p)&=xyp.
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| \end{align}
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| </math>
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| Altogether, these equations depend upon 15 parameters.
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| {| class="wikitable"
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| |-
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| ! Sector !! Parameter
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| |-
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| | Economic || <math>\ \gamma, \ \eta, \ \lambda</math>
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| |-
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| | Demographic || <math>\ \chi, \ \delta, \ \rho, \ \omega</math>
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| |-
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| | Anthropogenic || <math>\ \alpha, \ \alpha_0, \ \alpha_1, \ \alpha_2, \ \beta, \beta_0, \ \beta_1, \ \theta</math>
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| |}
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| : <math>b(y,z)</math> and <math>d(y,z)</math> represent the [[birth rate]] and [[death rate]] respectively. Both saturate as per capita output rises, in agreement with empirical studies ([[#jec95|Cohen, 1995]]). | |
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| The form of <math>f(x,y,p)</math> follows from the [[I = PAT]] hypothesis.
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| == System behavior ==
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| Using the [[Scenario analysis]] technique, Sanderson ([[#san94|1994]]) studied two possible futures for the idealized world described by Wonderland. One future entitled ''Dream'', held out the possibility of unending sustainable growth, while the other termed ''Horror'', ended in environmental collapse and eventual extinction of the population. Subsequent work ([[#gk06|Kohring, 2006]]) showed that the parameters of the model can be bisected into two sets, one which always produces sustainable futures and one which always ends in collapse and extinction. Additionally, the equations of Wonderland exhibit [[chaotic behavior]] ([[#gr96|Gröller, et al., 1996]], [[#wgp97|Wegenkittl, et al., 1997]], [[#hl98|Leeves and Herbert, 1998]]).
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| == Avoiding collapse ==
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| In the basic model it is impossible to avoid or recover from the environmental collapse seen in the ''Horror'' scenario without changes to the model itself. Two such changes have been studied:
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| pollution ''abatement'' and pollution ''avoidance''.
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| === Pollution abatement ===
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| {{see also|Pollution control|Environmental mitigation}}
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| Abating the effects of pollution draws funds from other sources to pay for cleaning up the environment ([[#san94|Sanderson, 1994]]). This decreases the value of <math>y</math> entering into the equations for birth, <math>b</math>, and death, <math>d</math>:
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| :<math> | |
| \begin{align}
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| y^{\prime}=y-\phi(1-z)^{\mu}y
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| \end{align}
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| </math>
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| The time evolution of <math>y(t)</math> is unaffected because those goods and services needed for pollution abatement must also be considered part of the overall output. The impact of these changes on the environment is expressed by changes to <math>f</math>:
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| : <math>
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| \begin{align}
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| f(x,y,p)=xyp - \kappa\frac{e^{\epsilon \phi(1-z)^{\mu}yx}}{1+e^{\epsilon \phi(1-z)^{\mu}yx}}
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| \end{align}
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| </math>
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| These changes introduce three new parameters into the model:
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| {| class="wikitable"
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| |-
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| ! Sector !! Parameter
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| |-
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| | Policy Levers || <math>\ \phi, \ \mu, \ \kappa</math>
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| |}
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| By adjusting the policy levers, it is possible to clean up a polluted environment and recover from the collapse seen in the ''Horror'' scenario. However, the recovery is only temporary, after a brief time of robust growth the system again collapses, leading to endless cycles of collapse followed by recovery. Abating pollution does not alter the fundamental division of the parameters into the two sets of sustainable and unsustainable futures ([[#gk06|Kohring, 2006]]).
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| === Pollution avoidance ===
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| {{see also|Pollution prevention}}
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| Pollution avoidance aims to prevent pollution from entering into the environment, by making its production unprofitable. This is modeled by means of a pollution tax ([[#hl98|Herbert and Leeves, 1998]], [[#lpb03|Lempert, et al., 2003]]):
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| :<math>
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| \begin{align}
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| y(t+1) &= y(t)\left(1+\gamma-\left(\gamma+\eta\right)\Big[1-z(t)\Big]^{\lambda}-\gamma_0\,\frac{\tau}{1-\tau}\right),\\
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| p(t+1) &= p(t)\left(1-\chi-\chi_0\frac{\tau}{1+\tau}\right).\\
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| \end{align}
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| </math>
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| The new parameters for the pollution avoidance model are:
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| {| class="wikitable"
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| |-
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| ! Sector !! Parameter
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| |-
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| | Policy Levers || <math>\ \gamma_0, \ \chi_0, \ \tau</math>
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| |}
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| With these changes, it is possible to raise the tax rate, <math>\tau</math>, such that the system never collapses and the horror scenario is avoided altogether. Regardless of the other parameters, it is always possible to increase <math>\tau</math> in order to avoid collapse thereby enabling unending sustainable growth ([[#gk06|Kohring, 2006]]).
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| == Variations ==
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| === Production function ===
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| Instead of the relatively simple economic growth equation used for <math>y(t)</math> some researchers use a [[Cobb–Douglas]] production function instead ([[#lh02|Leeves and Herbert, 2002]]).
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| === Multiple countries ===
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| The standard form of the Wonderland model contains a single, homogeneous entity. Herbert et al. ([[#hbll05|2005]]) extended Wonderland to a multi-country model by allowing the different entities to use different sets parameters and assuming the outputs are coupled through trade flows.
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| === Differential equations ===
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| Originally developed in terms of discrete time, finite difference equations, it is often recast as a set of continuous time [[differential equations]] ([[#gr96|Gröller, et al., 1996]])
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| == References ==
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| <div class="references-small"> | |
| *{{cite book
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| | ref = jec95
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| | last = Cohen | first = J.E.
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| | authorlink = Joel E. Cohen
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| | title = How Many People can the Earth Support?
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| | publisher = W. W. Norton & Company
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| | place = New York
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| | year = 1995 }}
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| *{{cite encyclopedia
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| | ref = fh09
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| | last1 = Frigg | first1 = R.
| |
| | last2 = Hartmann | first2 = S.
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| | title = Models in Science
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| | encyclopedia = The Stanford Encyclopedia of Philosophy
| |
| | editor-last = Zalta | editor-first = E.N.
| |
| | year = 2009
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| | url = http://plato.stanford.edu/archives/sum2009/entries/models-science}}
| |
| *{{cite journal
| |
| | ref = gr96
| |
| | last1 = Gröller | first1 = E.
| |
| | last2 = Wegenkittl | first2 = R.
| |
| | last3 = Milik | first3 = A
| |
| | last4 = Prskawetz | first4 = A.
| |
| | last5 = Feichtinger | first5 = G.
| |
| | last6 = Sanderson | first6 = W.C.
| |
| | title = The geometry of wonderland.
| |
| | journal = Chaos, Solitons & Fractals
| |
| | volume = 7
| |
| | year = 1996
| |
| | pages = 1989–2006}}
| |
| *{{cite journal
| |
| |ref = hl98
| |
| | last1 = Herbert | first1 = R.D.
| |
| | last2 = Leeves | first2 = G. D.
| |
| | title = Troubles in Wonderland
| |
| | journal = Complexity International
| |
| | volume = 6
| |
| | year = 1998
| |
| | pages = 1–20 }}
| |
| *{{cite conference
| |
| |ref = hbl05
| |
| | last1 = Herbert | first1 = R.D.
| |
| | last2 = Bell | first2 = R.D.
| |
| | last3 = Leeves | first3 = G. D.
| |
| | last4 = Lewis | first4 = B.G.
| |
| | contribution = Economic and Environmental Impacts of Pollution Control in a Multi-Country Model.
| |
| | editor-last = Zerger | editor-first = A.
| |
| | editor-last2 = Argent | editor-first2 = R.M.
| |
| | title = MODSIM05 Proceedings
| |
| | pages = 1035–1041
| |
| | publisher = MODSIM2005
| |
| | date = December 12–15, 2005
| |
| | year = 1994 }}
| |
| *{{cite journal
| |
| | ref = gk06
| |
| | last = Kohring | first = G.A
| |
| | title = Avoiding Chaos in Wonderland
| |
| | journal = [[Physica A]]
| |
| | volume = 368
| |
| | year = 2006
| |
| | pages = 214–224 }}
| |
| *{{cite journal
| |
| |ref = lh02
| |
| | last1 = Leeves | first1 = G. D.
| |
| | last2 = Herbert | first2 = R.D.
| |
| | title = Economic and environmental impacts of pollution control in a system of environment and economic interdependence
| |
| | journal = Chaos, Solitons & Fractals
| |
| | volume = 13
| |
| | year = 2002
| |
| | pages = 693–700 }}
| |
| *{{cite book
| |
| | ref = lpb03
| |
| | last1 = Lempert | first1 = R.J.
| |
| | last2 = Popper | first2 = S.W.
| |
| | last3 = Bankes | first3 = S.C.
| |
| | title = Shaping the Next One Hundred Years: New Methods for Quantitative, Long-Term Policy Analysis
| |
| | publisher = Rand
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| | place = Santa Monica
| |
| | year = 2003 }}
| |
| *{{citation
| |
| |ref = san94
| |
| | last = Sanderson
| |
| | first = W.C.
| |
| | contribution = Simulation Models of Demographic, Economic, and Environmental Interactions
| |
| | editor-last = Lutz
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| | editor-first = W.
| |
| | title = Population, Development, Environment: Understanding Their Interactions in Mauritius
| |
| | pages = 33–71
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | place = Berlin
| |
| | year = 1994 }}
| |
| *{{cite journal
| |
| | ref = wgp97
| |
| | last1 = Wegenkittl | first1 = R.
| |
| | last2 = Grõller | first2 = E.
| |
| | last3 = Purgathofer | first3 = W.
| |
| | title = Visualizing the Dynamical Behavior of Wonderland
| |
| | journal = [[IEEE Computer Graphics and Applications]]
| |
| | volume = 17
| |
| | year = 1997
| |
| | pages = 71–79}}
| |
| </div> | |
| | |
| [[Category:Economics models]] | |
| [[Category:Environmental economics]]
| |
| [[Category:Modeling and simulation]]
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| [[Category:Systems theory]]
| |
| [[Category:Sustainability]]
| |
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