OPTIMAL GROWTH, COORDINATION AND SUSTAINABILITY Jeroen C.J.M. van den Bergh and Peter Nijkamp E-mail: jbergh@econ.vu.nl or pnijkamp@econ.vu.nl Abstract
Until recently, the interaction between environmental quality, economic activity and growth is predominantly considered in an a-spatial context. Traditional neoclassical growth theor y following Solow/Swan has mainly addressed questions about environmental and resourc e limits to growth. Recently, much attention is also devoted to the environment-growth interface from an endogenous growth perspective. However, an important element that is missing i n both the neoclassical and endogenous growth approaches is a multiregional perspective. Th e present theoretical study tries to present one of the first attempts to fill this gap. The analysi s is based on a model of two interactive regions, with possible interaction between the regiona l environments via the global environment. The implications of this type of analysis ar e manifold. One is that endogeneity of growth is not only due to technology and knowledg e formation but also to the effects of trade, resource scarcity and environmental degradation .
Another implication is that a coordinated environmental policy of regions should addres s long-ter m sustainability, and take account of the positive technological and negativ e environmental dynamic externalities.
1. Introduction
The issue of environmental quality and long-run economic growth in a spatial or multi - regional context has been largely ignored by economists. A first attempt to analyze thi s problem is offered here, based on the use of dynamic descriptive and optimizatio n frameworks. First, an overview is given of economic approaches providing relevant inputs for a more comprehensive study of the relation between multiregional growth, trade, environment and policy coordination. Subsequently, a prototype dynamic two-country model is presente d which can address some of the main questions in this context.
The relationship between environmental quality and spatial economic processes i s extremely important, but has until now not received very much focused attention from eithe r regional or environmental economics. A general introduction is provided by Siebert (1985 and 1996). The lessons for environmental policy makers from the existing - mainly theoretical - literature on spatial environmental externalities include mainly adjustments or extensions o f Pigouvia n charges and taxation. These corrections are motivated by endogenous location , imperfect markets, international trade power, international coordination of policy (fo r national, transboundary and global problems) and long-term sustainability (see, e.g., Anderson and Blackhur st, 1992; Markusen et al., 1993; Motta and Thisse, 1994; Carraro, 1994). Thes e corrections go in different directions. In the case of imperfect markets, trade power an d endogenou s locations, the standard Pigouvian tax should be corrected downwards. In the cas e of transport related externalities, endogenous technology and sustainability a tighter policy i s needed. An integrated treatment of these issues has not received a great deal of attention s o Few models have been able to simultaneously address these various aspects. In the context of a static spatial equilibrium model the link between environmental regulation, trade , transport and spatial economic structure is addressed by Verhoef and van den Bergh (1996 ) and Verhoef et al. (1997). They derive policy rules under first- and second-best conditions , the latter case relating to a transport tax regime which compensates for disoptimal productio n taxes in regions that have not implemented an optimal environmental policy. Van den Berg h and Nijkamp (1995) perform numerical simulations with a dynamic descriptive model o f growth, trade and environment. They consider different scenarios, including symmetric an d asymmetric regions, and the latter case also with technology diffusion. It is shown that growth and technology formation may either stimulate or harm long-run sustainability, as the y enhance the capacity for resource-efficient and resource-intensive growth. The present pape r aims to improve upon this analysis, based on a more consistent formulation of trade as well as investments, i.e. the engine of growth, in each region, as well as the consideration o f coordination versus non-coordination growth problems.
In Section 2 the interdependence between environmental sustainability on a regional scal e and regional economic growth, and the interpretation of endogenous growth in a n environment-space context are discussed in more depth. A general framework fo r multiregional growth and environment is considered in Section 3. A prototype model i s presented in Section 4. It integrates multiregional growth, environmental processes an d multiregional trade. Furthermore, it considers the cases of coordinated and independen t regional environment-growth policy in an optimization framework. Some numerical dynami c simulation results obtained with a specific variation of this model are discussed in Section 5 .
Section 6 offers concluding remarks.
2. Growth, environment and trade
The potential conflict between environmental quality and economic growth is a topic whic h has attracted a lot of interest and generated much debate among economic growth theorists .
For instance, neoclassical economic growth theory has frequently been applied to issues o f renewable resource limits, long run pollution effects and sometimes even to a combination o f these (for an early survey, see Kamien and Schwarz, 1982; for recent contributions, se e Tahvonen and Kuuluvainen, 1991 and 1993). Analyses in this flavour have focused on the role of substitution between man-made capital and natural resources materials in production , technological improvements in materials efficiency of production, and backstop technologie s (see Dasgupta and Heal, 1979). The issue of sustainability has also been explicitly treated , although there is no agreement as to its theoretical interpretations (see, e.g., Toman et al., 1994). Conservation-preservationist's and exploitationist's views are linked to the distinctio n between strong and weak sustainability (Pearce and Turner, 1990), with the Hartwick rule o f investing resource rents in man-made capital as an extreme case of the latter (Hartwick, 1977; Gutés, 1996). A survey of alternative approaches to modelling for sustainable development i s given in van den Bergh and Hofkes (1997), including attention for endogenous growth , evolutionary change, integrated models, general equilibrium models, macroeconomic models , The issue of sustainability in a multiregional context has attracted little attention. On e approach is to consider regions as open systems, both in terms of economic and environmental processes (e.g., van den Bergh and Nijkamp, 1991). Another approach is to describ e multiregional systems completely and explicitly. Regional economics provides a limite d perspective for theory regarding multi-regional dynamics (see Anderson and Kuenne, 1986 ; and Walz, 1997). Spatial price equilibrium theory offers one approach (see Verhoef and va n den Bergh, 1995), while spatial general equilibrium models offer a more complete bu t complex alternative (Roson, 1994; van den Bergh et al., 1996). Finally, game-theoreti c models may be used to deal with specific questions of environmental policy coordinatio n (e.g., Van der Ploeg and De Zeeuw, 1992; Batabyal, 1996).
The extension of the Solow-Swan growth framework with various types of endogenou s learning and technology mechanisms by Stokey, Romer and Lucas and others has generated a wealth of literature on endogenous growth since the end of the 1980s. Various recent article s and books provide good overviews (Romer 1994; Mankiw, 1995, Pasinetti and Solow, 1994 ; and Barro and Sala-i-Martin, 1995). Different contributions to the new growth theories focu s on the role of various externalities related to technological change, specialisation and trade , monopoly rents from innovation and "creative destruction", human capital and governmen t policy. Although economic growth theory with environment and resources is dominated b y the Solow-Wan tradition (see Toman et al., 1995), there have been several attempts already to model and endogenise technical progress within models designed to address environmenta l issues and sustainability (e.g., Gradus and Smulders, 1993; van den Bergh and Nijkamp, 1994; Bovenberg and Smulders, 1995). These models do not simultaneously address spatia l disaggregation, interregional trade and interaction between local and global environmenta l quality. The focus in the economic models nowadays is usually on endogenous technology and knowledge formation, mixing such positive externalities with negative environmenta l externalities, and arriving at more optimistic conclusions than in neoclassical growth model s with exogenous technology. The influence of pollution and preferences on technology choic e is important in this respect. These results are often stated against conclusions obtained wit h global "limits to growth" models (Meadows et al., 1972 and 1992). These lack advance d technology, price mechanisms, producers' and consumers' behaviour, and interregional trad e submodels, and have therefore been heavily attacked by economists (e.g., Nordhaus, 1973 and 1992). Alternative approaches to the relationship between technology, growth an d environment have stressed its disequilibrium, uncertainty and evolutionary/Schumpeteria n character (e.g., Dosi et al., 1988; and in an environmental context; Faber and Proops, 1990 ; Erdman, 1993; Clark et al., 1995). Whereas economists have been much concerned with th e potential conflict between economic dynamic efficiency and environmental sustainability , several other approaches focus on controllability (Perrings, 1991) and stability, the latter often based on ecological theory or metaphors (Holling, 1986 and 1994). Common and Perring s (1992) provide an interesting and systematic comparison of the alternative views.
Endogenous growth in a spatial-environmental context allows for more interpretations an d elaborations as compared with that in a non-environment and a-spatial setting. A fundamenta l point to note as a start is that there do in fact not exists "real" exogenous factors of production.
One can easily see that all production factors are the output of some economic process. S o labour is produced with exergy (useful energy), materials, capital and labour; exergy i s "produced" with exergy, materials, capital and labour; capital is produced with exergy , materials , capital and labour. Materials are "produced" with exergy, materials, capital an d labour. Knowledge is here assumed to be incorporated in capital and labour. Resource scarcity and environmental regulation may act as incentives to promote specific technological progress and growth (see Faber and Proops, 1990).
In a spatial context trade can be considered as a motor of growth (Grossman and Helpman, 1991). Also transport can be included in the production function on an aggregate level , representing a unique, essential and explicit reference to the spatial dimension (e.g., Walz , 1997). Endogenous growth theory in a spatial-environmental context could then pay attentio n to the various important (policy) questions. First, there are questions related to the rate o f growth: can diminishing returns to regional investment be overcome due to multiregiona l trade, and is regional growth slowed down due to multiregional environmental factors an d regulation? Second, there are questions relating to the direction of growth: do enviromenta l feedbacks to the economy cause a shift to more trade, to more domestic production, to mor e dirty or clean production, and - associated with this - to dirty or clean export and import ? Some of these question can be addressed by the models presented in the following sections.
3. A general framework for analysis of multiregional growth and environment
A logical s tarting point in the context of growth and environment is the existence of a finit e natural carrying capacity (CC), measured in physical resource use and/or emissions (or othe r types of environmental disturbance, like land use) which would act as a limiting factor to th e scale of the economy. Some possible patterns of economic scale and CC over time are show n in Figures 1 and 2. The standard density dependent (versus exponential) growth pattern , restrained by some limiting factor (resource), defining the respective carrying capacity, i s represente d in Figure 1. It shows a monotonic convergent pattern, and a cyclic pattern. Th e latter can occur when processes are non-deterministic, or in open systems like regions. Whe n one, as in our case, is interested in two-way interactions between economic and ecologica l systems, variations in the carrying capacity should clearly be allowed for. These have only a meaning if they go along with an endogenization of the CC, namely by making it dependen t on the scale of the economy. This is illustrated in Figure 2. A feedback process between th e CC and the econo mic scale can then give rise to a variety of patterns of the economic scal e indicator: monotonic and stable or unstable; or cyclical (not all these are shown in Figure 2).
Figure 2. Fixed carrying capacity with two possibl e Figure 2. Variable carrying capacity with two-wa y interaction between economy and environment. This simple picture provides a general conceptual framework for studying the interactio n between growth and environment on global and regional scales. In the latter context, it has, i n addition, to be realized that one may envision two types of CC, namely one relating to a closed, and one to an open region. Trade beyond the regional boundaries causes the ope n region's carrying capacity to differ from the closed region's one (in fact, to be larger). Thi s would then allow the open region's economy to attain a larger scale than the closed one's .
However, when opening up a region, the closed region's CC level may be exceeded due t o trade. This may cause destruction of the components of the closed region's CC. This in tur n will also negatively affect the open region's CC, which is partly based on the factor s supporting the closed region's CC, i.e. the region-internal factors or resources. Subsequently , this may negatively impact upon the the regional economy. The model in Figure 2 can in fac t also be regarded as relevant to illustrate the latter unfortunate process.
In van den Bergh (1993) a simple model is considered that can represent the process i n Figure 2. One set of assumptions represents a case where both internal (behavioural) an d external environmental feedbacks to economic growth are operational. It turns out to lead t o one interior stable equilibrium. Furthermore, three general types of pattern characterise al l possible behaviour far from equilibrium. These are dependent on the inclusion and intensity of the feedback mechanisms between the economy and the environment. The present study aim s to provide for a more detailed model consistent with the basic framework just sketched, an d formali zing the relation between endogenous growth, endogenous environmental change an d Incorporati ng these three main elements in a single model is a very difficult task an d clearly calls for heroic assumptions as well as alternative models and specifications. A two - regional setting is chosen here, which allows us to deal with imports and exports in one region causing unsustainable patterns in the respective region or in the other region, via interregiona l trade. This may be referred to as "importing sustainability and exporting unsustainability" .
The model allows for tracing the mutual impact of changes in the global and local or regiona l environment, and the repercussions of these in terms of consumption, production an d interregional trade. It may therefore be regarded as a prototype model of larger and mor e complex integrated climate-change/social-impact models. The model incorporates th e production and consumption processes in the regional economies, and interregional trade; regional environmental processes interact through the global environment, and are a lin k between generation and reception of environmental externalities; endogenous technology allows for more efficient use of material resources in production; there is diffusion of technology from one region to the other (having the benefit o f avoiding full symmetry as well as limiting model complexity); regional welfare is a function of indicators for consumption, and for regional and globa l 4. Optimal multiregional growth with environment
Consideri ng optimal growth in a multiregional setting from a standard perspective, i.e. base d on maximizing a discounted stream of utility over time, leads to a two sub-problems. In th e first one each region strives for optimal growth given the decisions made in the other region , in other words, a dynamic Nash equilibrium results. A social optimum however would require coordination, as there are two types of externalities: negative externalities if the regiona l environments are connected (we have not spoken about cross-boundary flows, but these might be implicit in the global-regional environment connections); furthermore, there are positiv e externalities due to diffusion of technological innovations from region A to region B.
In each region (j=A,B) relevant control variables, i.e. investment in capital (in bot h regions) and technological innovation (only in region A), I j and I consumption goods Cj and Cj , are chosen so as to maximize a discounted (with discount rat e s ) flow of utility over time (where T can approach infinity): This maximization is subject to conditions representing the system of economic an d environmental processes and interactions. These include the following.
Production in region j may be represented as: Accounting equations for distribution of output in regions A and B between investment an d consumption of domestic and imported goods are as follows: Capital accumulation and depreciation is a standard equation, where investment is considered a regional activity, and is restricted from above by regional production.
Investment in technological innovation in region A is represented as: Diffusion of technological innovation to region B, with some time delay equal to d, is: Regional environmental dynamics are either linear (e=0) or logistic (e=1) functions with a carrying capacity M j and possibly based on interactions between the regional environment s (dum=0) or not (dum=1). Negative impacts of the economy on the environment are include d via the last term, based on waste per unit of output w j, decreasing in the state of technology T j, dEj/dt = rj*(dum Ej + (1-dum)*Sqrt[E A*EB])*(1-e*E j/Mj) - wj(Tj)*Qj And finally the following initial conditions apply: Note that all variables have a time index and are non-negative.
To comple te the non-coordination model, one may assume that for region A E B and CA , and are exogenous, reflecting a limited potential for control o f regions. The two region-specific commodities considered are assumed to be different an d essential in the utility function. In other words, there is complete specialization, so as not t o complicate the model too much (one of the unavoidable heroic assumptions). This implies that there will be trade between regions. The interregional trade occurs via the following tw o market behaviour conditions (i.e. optimal utility conditions): Here the objective function is for simplicity considered as the discounted flow of the sum o f unweighted utilities of the two regions.
{UA[CA ,CA ,EA] + UB[CB ,CB ,EB]}*exp[-s t]dt subject to (2)- (8), since all variables are endogenous now.
For an elaboration of this optimization problem we refer to the annex. Comparing th e optimality conditions for the coordination and non-coordination cases, the following remark s Under coordination the rate of growth will be slower when the environmental externalitie s dominate the technological externalities, and vice versa.
Multireg ional externalities may imply unsustainability for the global system and eac h region; regional sustainability without global sustainability is not possible.
Perfectly symmetric regions may imply multiple equilibria; non-linear environmenta l processes may imply absence of an equilibrium.
Technological positive externalities require coordination investment policies regardin g Coordination of what growth/investment policy ? trade policy? environmental policy? No coordination may imply free rider benefits which are increasing over time, until th e global-regional feedbacks become excessive.
5. A numerical example
It may be interesting to consider a dynamic descriptive counterpart of the model presented i n the previous section, which focuses on the structure of the model conditions, and not o n optimality from a social perspective. There are then more degrees of freedom. Th e assumption is then that investment is not chosen to optimize some objective - along with other controls - but that it is examined how ex-ante (or intrinsic) growth of man-made productio n factors results via environmental feedback in ex-post effective growth of consumption an d welfare. The question is thus not what determines growth, but what is the long ter m environmental impact of a positive rate of growth, given that there exist mechanisms o f technological progress and diffusion, consumption and international trade, and interactio n between global and regional environmental quality.
The different parts of the model are presented below (i denotes region i; i=A,B) in so fa r as they differ or are more specific than the general model formulated in the previous sectio n (see for more details on the model of this section van den Bergh and Nijkamp (1995). Th e functiona l specifications and numerical values adopted here are just illustrative, hopefull y reflecting as much as possible choices in a range consistent with empirical variations.
Each region uses two regional inputs to produce output Q i according to a Cobb-Dougla s type of production function. The regional environment is one input (E i). The environmenta l inputs are externalities, as they cannot be directly chosen or influenced by the producers.
1. The technology impact as noted in equation (2) will be included via resourc e In order to allow for endogenous formation of trade patterns, regional production price s (Pj) are dete rmined per unit of output, based on the costs of the two inputs. The price o f capital pK =1 (numeraire) and the price of resource materials (R j) is p , depending o n environmental taxation (exogenous). The unit cost price condition for the commodity price P j, based on the assumption of constant returns to scale production, is as follows.
The demand trade system is in reduced form: It is assumed here that the value of imports equals that of export, while the terms of trade i s endogenous. Furthermore, prices clear commodity markets and capital investment equa l savings. These are features of an equilibrium model. In addition, resource supply is no t explicitly modelled as a market, but instead it is assumed to have an exogenous (and possibl y fixed) price. This can be regarded to include its public good nature - either as a strict physica l resource or environmental resource like clean water or good soil quality. Alternatively, on e may regard th is as either reflecting a large absolute resource supply - focusing on pollutio n problems - or as a relatively small demand, when only part of the world is modelled .
Whatever interpretation, it implies that a kind of disequilibrium approach, with stron g equilibrium features is adopted. It should be noted that consumption and production are no t the outcome of optimization of utility and profits. Instead, the model is fuelled by scenario s that drive investments, which in turn - together with the trade model and the "memory" fo r capital and environmental quality - determine the consumption of each good for both regions .
These features are characteristic for a macro disequilibrium approach, casted now in a n A large value of v refers to a more closed system, a small one (<2) means a very ope n system, 1 means that imports equal domestic consumption. The fixed value of v, although no t realistic perhaps for all countries, can be motivated for many countries by observing historical patterns. As the specific value is not important, v is set equal to 2 here.
Technological progress reaches region B through diffusion (T B) with a certain time delay .
This is set equal to 4 periods since its value is not essential, only that there is such a phenomenon. The parameter dr is a dummy variable indicating the rate of diffusion. Becaus e of the time delay a historical pattern is necessary as an initial condition.
TB(t) = dr TA(t-4), TB(s)=0, s=1,2,3.
The development of technology allows for more efficient resource use in production with a n upper limit, given an monotonically increasing function f RB = f (QB)*((TB + 100)/(4 T B + 100)).
The limit of the right hand sides (for very large value of technology indicators) is 0.25*f Ri(i=A or B), which indicates a maximum improvement (limit) of 400 % relative to the initia l period. This may be regarded as reflecting thermodynamic limitations.
The local environmental quality in each region is described by a logistic growth curve. The carrying capacity (C ) is assumed constant (set equal to 180). The intrinsic regeneratio n (growth) rate of the renewable environment (r(.)) is positively related to a global environmen t ). Resource extraction in region j (R j) causes the environmental quality in th e index represents the quality of the global environment, which is obtained b y summing additive and multiplicative aggregation functions of the regional environmenta l indicators 1. The aim is here to capture at least two effects. First, when only one region has a low environmental quality, this has severe implications for the global environment. Second , only when both regions collapse, the global environment will do so.
Finally, welfare is calculated for each region, based on the consumption of each commodity , regional environmental quality, and the state of the global environment.
In simulation experiments one can investigate the implications of different scenarios. Th e above model allows for a great number of scenarios to be studied. Each of these represent s then a specific combinations of values for all parameter and initial state variables .
Recogniz ing this large potential of choices, several scenarios can be considered. Fo r illustrative purposes we focus here on a symmetric economic system, i.e. identical productio n functions for both regional economies: a =b =0.5, i=1,2. Furthermore, the ex ante net growth is assumed to equals 3%: g=0.03, and positive outlays on research and development: I A =5.
With fixed investment in environmental technology in region A, interregional differentia l impacts are found in our spatial economic system. Figure 3 shows that this strategy ha s positive impacts on the production in region A relative to region B. Considering Figure 4, on e can observe that the environmental quality in region A is improving relative to region B. Th e global environment is a sort of average, and improves initially. However, the environment i n region B would be worse off without region A investing in technological innovation. Becaus e of the technology strategy in region A, its initial welfare is lower than in region B, as is shown in Figure 5. However, after time period 12 the positions are reversed, and a maximu m difference is reached in time period 38. Finally, Figure 6 shows that the trade pattern i s 1 It is noted that this should not be confused with taking averages of the two terms (which is just as arbitrary as any other specification).
As one referee remarked, there are three additive terms so that the coefficients sum up to 1.5. For the qualitative character of the results it does not matter much whether this or a slightly different specification is used.
asymmetric and irregular. Domestic consumption of commodity 2 in region B (C B ) is decreasing from time 0 on, and is substituted by importing commodity 1 (C B ). This is th e consequence of changes in relative prices of the commodities. In region A the domesti c consump tion of commodity 1 (C ) can increase because of the relatively favourabl e development of environmental quality, acting as a comparative advantage, while th e moderate ly increasing import of commodity 2 (C A ) is the net result of the sharp increase o f buying power of region A and the increase in the relative price of commodity 2.
[ INSERT Figure 3. Capital and production ] [ INSERT Figure 4. Regional and global environmental quality ] [ INSERT Figure 5. Regional welfare indexes ] [ INSERT Figure 6. Consumption and trade ] These results are just exemplatory, aiming to give a flavour of what dynamic patterns on e can obtain with a numerically specified descriptive model of growth, trade and environment.
6. Conclusions
The models in Sections 4 and 5 are illustrative of the type of questions to be addressed in th e context of spatial sustainable economic development, based on the use of formal modellin g procedures. The aim was, rather than to show doomsday or optimistic scenarios - which ca n never be tested anyway - to show the impact of spatial disaggregation and trade in the contex t of econom ic growth, given technological progress and diffusion, and environmental taxation .
Many issues ha ve not been dealt with yet, but can in principle be included in the simulatio n approach, though not so easily in the optimization approach. This is also indicated by the lac k of literature on completely specified (and solved) optimal optimal equilibrium growth model s with a spatially disaggregated economy and environment. Clearly, introducing such a multi - regional setting in an already complicated dynamic optimization framework leads to a n enormous increase of complexity. It should be noted that choosing a social evaluatio n criterion is more complicated here than in a non-spatial setting, which is caused by the fac t that one has to undertake then also an aggregation of regional welfare into some measure o f multireg ional social welfare. Theoretical generalizations would of course still be useful, bu t these are - like the two-sector models in standard growth theory - a difficult topic fo r Giving more attention to spatial processes can involve explicit treatment of the lin k between trade, transport and environment. Dealing with heterogeneity of regions can involv e asymmet ric versus symmetric characteristics, or developed versus developing countries .
Asymmet ry may result from unique regional production structure, growth capacity, natura l resource endowments, environmental resilience, preferences, and initial levels of physica l capital, technology and human capital. Finally, multi-dimensional elaborations do not see m immedi ately necessary for increasing our understanding of spatial sustainability, as th e potential to learn from two-region studies is clearly not yet exhausted.
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Figure 4. Regional and global environmental quality. Figure 5. Regional welfare indexes. Annex 1. Optimal coordination problem
The problem is mathematically one of optimal control. A current value Hamiltonian wa s formulated after resubstituting the equations 2-8. The following conditions resulted next (wit h partial derivatives of functions indicated by subindexes): Dynamic equations for costate variables l j (related to state variables ij; i=K,T,E; j refers to the dlA /dt = -[ lAK*(FAK- A ) - lA *w *FA ] dlA /dt = -[ lA *FA -lA * A - lA *(wA *FA+wA*FA ) + lB *FB - lB *(wB *FB+wB*FB ) ] dlA /dt = -[ UA + lA *FA + lA *(rA*(dum+(1-dum)*0.5*Sqrt[E B/EA])*(1-e*E A/MA) - rA(e/MA)*(dum*E A+(1-dum)*Sqrt[E A*EB]) - wA*FA ) + lB *(rB*(1-dum)*0.5*Sqrt[E B/EA] dlB /dt = [ UB + lA *rA*(1-dum)*0.5*Sqrt[E A/EB]*(1-e*EA/MA) + lB *FB + lB *(rB*(dum+(1-dum)*0.5*Sqrt[E A/EB])*(1-e*E B/MB) - rB*(e/MB)*(dum*E B + (1-dum) Total derivation w.r.t. time t of these optimality conditions lead then to: dlA /dt = UA *dCA /dt + UA *dCA /dt + UA *dEA/dt dlB /dt = UA *dCA /dt + UA *dCA /dt + UA *dEA/dt dlA /dt = UB *dCB /dt + UB *dCB /dt + UB *dEB/dt dlB /dt = UB *dCB /dt + UB *dCB /dt + UB *dEB/dt From these and the co-state conditions the following expression is derived, lB = -(( (- A + A +FA -FA )*UA - FB *UA - FA *lA *wA + lA *(FA *wA+FA*wA )) /(FB *wB + dlB /dt = -(wB *FB *dTA/dt + wB *(FB *dKB/dt+FB *dTA/dt+FB *dEB/dt) + wB *FB*dTA/dt + wB*(FB *dKB+FB *dTA/dt+FB *dEB/dt))*(UA *(FA -FA - A + A ) - UA *FB + lA *(-wA*FA +wA *FA+wA*FA ))/((wB *FB+wB*FB )2) + ((UA *dCA /dt+UA *dCA /dt + UA *dEA/dt)*(FA -FA - A - A ) + UA *((FA +FA )*dKA/dt + (FA +FA )*dTA/dt + (FA +FA )*dEA/dt) - (UA *dCA /dt + UA *dCA /dt + UA *dEA/dt)*FB - UA *(FB *dKB/dt+FB *dTA/dt+FB *dEB/dt) + dlA *(-wA*FA +wA *FA+wA*FA ) + lA *((-wA *FA +wA *FA+wA *FA )*dT /dt + (-wA*FA +wA *FA +wA*FA )*dKA/dt + (-wA*FA +wA *FA +wA*FA )*dTA/dt + (-wA*FA +wA *FA +wA*FA )*dEA/dt)) / The mathematical problem is now such that expressions for changes in the consumption variables are possible. These are extremely tedious expressions, which will not be repeated here. Together with the state variables numerical simulations based on optimal control conditions are in principle possible.

Source: http://papers.tinbergen.nl/97104.pdf


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