A Multicriteria Macroeconomic Model with Intertemporal Equity and Spatial Spillovers
AA Multicriteria Macroeconomic Model withIntertemporal Equity and Spatial Spillovers
Herb Kunze ∗ Davide La Torre † Simone Marsiglio ‡ November 20, 2019
Abstract
We analyze a macroeconomic model with intergenerational equity considerations and spatial spillovers,which gives rise to a multicriteria optimization problem. Intergenerational equity requires to add in thedefinition of social welfare a long run sustainability criterion to the traditional discounted utilitarian cri-terion. The spatial structure allows for the possibility of heterogeneiity and spatial diffusion implies thatall locations within the spatial domain are interconnected via spatial spillovers. We rely on different tech-niques (scalarization, (cid:15) -constraint method and goal programming) to analyze such a spatial multicriteriaproblem, relying on numerical approaches to illustrate the nature of the trade-off between the discountedutilitarian and the sustainability criteria.
Keywords:
Multicriteria Optimization; Intergenerational Equity; Spatial Diffusion
After decades of debates a wide consensus on the effects of anthropogenetic activities on the environment hasfinally emerged, and even policymakers seem finally convinced that it is now time to act in order to ensurethe long term sustainability of economic activities. Sustainability is a complicated notion to define butin its most widely accepted terms it requires some respect of natural resources and some efforts to ensureintergenerational equity ([30]). While the former aspect can be easily accounted for in macroeconomicanalysis by adding an additional constraint to the standard optimization problem representing the society’splanning device, the latter is more problematic since the criterion generally used in the definition of such aproblem, the discounted utilitarianism, attaches less and less weight to future generations and so it cannotaccommodate for intergenerational issues ([13]). Different approaches to overcome such an issue have beenproposed, but probably the most commonly used consists of extending the objective function to include anadditional term representing somehow long term sustainablity considerations ([12]). The introduction of suchan additional term in the optimization problem makes it a multiple objective problem in which the societyneeds to balance two conflicting goals, represented by short term and long term objectives, respectively; suchan interpretation of a macroeconomic problem as a multicriteria problem allows to rely upon the operationsresearch methods to analyze macroeconomic issues, bridging somehow these two important but distinct fieldsof research ([21]; [22]).The goal of this paper is therefore to analyze from an operations research perspective a traditionalmacroeconomic model extended along two different directions. The first consists of allowing for long termsustainability considerations, in the form of intertemporal equity issues, which introduces a second criterionin the otherwise standard unicriterion macroeconomic optimization problem; several studies have analyzed ∗ Department of Mathematics and Statistics, University of Guelph, Guelph, Canada. Email: [email protected] † SKEMA Business School - Universit´ e Cˆ o te d’Azur, Sophia Antipolis Campus, France. Email: [email protected] ‡ Department of Economics and Management, University of Pisa, Pisa, Italy. Email: [email protected] a r X i v : . [ ec on . T H ] N ov n different contexts problems of this kind by focusing on pollution control and debt reduction settings ([17];[19]). The second consists of allowing for a spatial dimension in which different locations within the entireeconomy interact one another through the trade channel; several studies analyze problems of this kind,which generally are all unicriterion, by focusing on capital accumulation and environmental problems ([16];[18]). Specifically, we consider a simple setting with intergenerational equity (as in [13]) in which capitalevolves both over time and across space (as in [3]), and to the best of our knowledge our paper representsthe first attempt to analyze a spatial macroeconomic problem from an operational research point of view.Our paper relates thus to two very distinct literatures addressing the intertemporal equity issues as-sociated with sustainabiility and the existence of spatial spillovers in macroeconomic geography contexts,respectively. Several papers discuss the problems embedded in the use of discounted utilitarianism as a socialwelfare criterion by proposing alternative criteria and discussing their limits in terms of applicability andin terms of existence of optimal solution paths ([26]; [11]; [23]; [20]; [2]; [14]). Several other papers insteadmore recently discuss how the presence of a spatial dimension allows to characterize the possibility of spatialheterogeneity and spatial spillovers, along with their implications for macroeconomic outcomes ([6]; [3]; [7];[8]; [16]). We bridge these two branches of the economics literature by developing a spatial version of thesimplest intertemporal equity problem to show how an operations research approach can be used to informour analysis.The paper proceeds as follows. Section 2 recalls the basic definitions and properties of optimizationtheory with multiple objectives, discussing the main methods that we shall adopt in our analysis, thatare scalarization, (cid:15) -constraint method and goal programming. Section 3 presents our multicriteria problemshowing how it can be reformulated from the points of view of scalarization, (cid:15) -constraint method and goalprogramming. Section 4 determines the solution of the problem relying on scalarization illustrating thenature of the trade-off between the discounted utilitarian and the sustainability criteria and deriving thePareto-frontier in a specific model’s parametrization. Section 5 presents some further numerical experi-ments by analyzing the problem via the (cid:15) -constraint method and goal programming, under different model’sparametrizations. Section 6 as usual concludes. This section recalls some basic facts in Multiple Objective Optimization (MOP). In a very abstract setting,a finite-dimensional MOP problem (see [27]) takes the formmax x ∈ X J ( x ) (1)where ( X, (cid:107) · (cid:107) ) is a Banach space and J : X → R p is a vector-valued functional. As usual we suppose that R p is ordered by the classical Pareto cone R p + . A point x ∈ X is said to be Pareto optimal or efficient if J ( x ) is one of the maximal elements of the set of achievable values J ( X ). Thus a point x is Pareto optimalif it is feasible and, for any possible x (cid:48) ∈ X , J ( x ) ≤ R p + J ( x (cid:48) ) implies x = x (cid:48) . In a more synthetic way, apoint x ∈ X is said to be Pareto optimal if ( J ( x ) + R p + ) ∩ J ( X ) = { J ( x ) } . Because of its dimensionalityand the existing of conflicting criteria, a MOP model is usually difficult to be solved and the determinationof the entire or part of the Pareto frontier can be very complicated and computationally intensive. Inparticular this applies when the number of objectives is larger than two, leading to a higher-dimensionalPareto surface. To overcome this difficulties and reduce the model complexity, several techniques havebeen proposed in literature. The generation of the Pareto frontier can be accomplished through one of twopredominant techniques: scalarization and vectorization methods. Among the scalarization techniques, themost frequently applied are linear scalarization, the (cid:15) -constraint method, and goal programming. Thesetechniques will be used in the sequel of this paper. The vectorization algorithms, instead, tackle the MOPmodel directly without transforming it into some equivalent single criterion model.2 .1 Linear Scalarization The linear scalarization technique (or weighted sum) is probably the simplest and the most widely usedtechnique to solve MOP problems and it converts the MOP model into a family of parametric single criterionoptimization models. By using this approach, a multiple objective model can be reduced to a single criterionproblem by summing up all criteria with different weights. More precisely, by linear scalarization a MOPmodel boils down to: max x ∈ X p (cid:88) i =1 β i J i ( x ) (2)where β is a vector taking values in the interior of R p + , namely β ∈ int( R p + ). Since the Pareto optimalsolution depends on β , by modifying the weights β different points on the Pareto optimal set can be found.Linear scalarization can also be applied to problems in which the ordering cone is different from the Paretoone. In this case, we have to rely on the elements of the dual cone to scalarize the problem.If J is a vector-valued concave functional, namely each component J i is concave, then the linear scalarizedproblem (2) is also concave. This means that one can find Pareto optimal points of a concave problem bysolving a concave scalar optimization problem, and for each β ∈ int( R p + ), different Pareto optimal points canbe obtained. For concave problems the converse of this result is only partially true, since for each Paretooptimal point ¯ x , there is a nonzero ¯ β ∈ R p + such that ¯ x is a solution of the scalarized problem (2) with β = ¯ β . This is stated in the following theorem. Proposition 1. ([27]) Suppose that D is convex and J i are concave for all i = 1 ...p . Then for all Paretooptimal solutions ¯ x there exists β ∈ R p + such that ¯ x ∈ argmax x ∈ D (cid:40) p (cid:88) i =1 β i J i ( x ) (cid:41) . (3)If the linear scalarization method is used for non-concave problems, the Pareto frontier generated willbe incomplete and the Decision Maker (DM) will have a non-complete set of possible solutions. In thiscase, other scalarization methods can be found in literature and one which is worth to be mentioned is the Chebyshev scalarization model . (cid:15) -constraint method The second model that is proposed to solve the vector-valued problem is the (cid:15) -constraint method . In thismethod, one of the objective functions is selected to be optimized while the others are converted intoadditional constraints. The method is an hybrid methodology, in fact for the { J i } i (cid:54) = k , least acceptablelevels, (cid:15) i have to be set while the remaining objective function J k is optimized. Then the decision makerplays a crucial role in this setting, by choosing which objective function to optimize and the least acceptablelevels for the objective functions to add as constraints. Under this method, the original vector-valuedproblem can be now written as: max J k ( x ) (4)subject to: (cid:40) J i ( x ) ≥ (cid:15) i i (cid:54) = kx ∈ X (5)This method has the advantage of being theoretically able to identify Pareto optimal points also of non-convex problems. However, it also has two potential drawbacks: the identified optimal point is only grantedto be weakly Pareto optimal, and the problem might become unfeasible due to the additional constraints.3 .2 Goal Programming The
Goal Programming (or GP approach) is another widely used method to deal with vector-valued problems(see [9] [10]). With respect to other scalarization techniques, the idea behind this model is the determinationof the aspiration levels of an objective function. A GP model does not try to find an optimal solution butan acceptable one, as it tries to achieve the goals set by the DM rather than maximizing or minimizing theobjective functions. However, an optimization procedure is involved anyway. Within this formulation, onetries to minimize any possible deviation from the objective goals, either positive or negative. In fact, the GPmodel is a distance-function model in which the obtained optimal solution represents the best compromisebetween different objectives. Since the introduction of this methodology for MOP problems, many variantshave been presented in literature. Among them, the most popular one is the
Weighted Goal Programming ([1]) which reads as follows: Given a set of ideal goals g i , with i = 1 , . . . , p , chosen by the DM, solve thefollowing program: min p (cid:88) i =1 θ + i δ + i + θ − i δ − i Subject to: J i ( x ) + δ − i − δ + i = g i i = 1 , . . . , pδ − i , δ + i ≥ ∀ i = 1 , . . . , px ∈ X (6)where δ + i , δ − i are the positive and negative deviations (slack variables), respectively, and θ + i , θ − i are thecorresponding weights. Due to its simplicity, the GP model and its variants have been widely appliedto different areas such as accounting, marketing, human resources, production, and so on. A negativecounterpart of the GP model, that it is important to consider, is the ability of GP to produce solutions thatare not Pareto efficient. To overcome this difficulty, in order to produce Pareto optimal solutions the GPmodel is implemented within a two-steps algorithm: At first, the GP solution is tested for Pareto efficiencyand, if it is not efficient, a restoration or projection method is proposed to restore efficiency at the secondstep. The simplest macroeconomic setting to account for intergenerational equity ([21]; [22]) and spatial linksand spillovers ([6]; [3]) consists of an optimization problem in which the social planner, by considering theevolution of the capital stock K ( x, t ), tries to determine the optimal level of consumption C ( x, t ) over time t ∈ [0 , T ] and across space x ∈ Ω to maximize the vector-valued social welfare which is composed of twoterms. The first term represents the discounted utility stream of the representative individuals located indifferent venues within the entire spatial economy J = (cid:82) T (cid:82) Ω U ( C ( x, t )) e − ρt dxdt , where the utility functiondepends on consumption and the discount factor is ρ >
0. The second term represents the payoff of therepresentative individuals located in different venues within the entire spatial economy at the end of theplanning horizon, J = (cid:82) Ω K ( x, T ) where the payoff function depends on the final capital level. The society’soptimization problem can therefore be stated as a bi-criteria problem as follows:max C ( x,t ) W = (cid:18)(cid:90) T (cid:90) Ω U ( C ( x, t )) e − ρt dxdt, (cid:90) Ω K ( x, T ) dx (cid:19) (7)Subject to ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + f ( K ( x, t ) , C ( x, t )) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (8) f ( K ( x, t ) , C ( x, t )) with f K > f C <
0, and ∇ ( d ( x ) ∇ K ( x, t )) which quantifies the process of spatial diffusion between differentlocations within the spatial domain. Note that the function f ( K ( x, t ) , C ( x, t )) determines the nature ofthe trade-off between the two terms in (7). The first term represents the traditional criterion considered inmacroeconomic analysis, characterizing the short termism of policymaking which leads the society to careabout the wellbeing of the current generation. The second term represents instead a sustainability criterionintroduced to account for the wellbeing of future generations by allowing the society to determine today howmany resources not to exploit in order to preserve them for the future. Clearly, a higher consumption todayincreases the first criterion and through its subtraction of resources to capital accumulation decreases thesecond one. Therefore, the society needs to balance these two conflicting criteria determining the optimaldynamic path of consumption.The above maximization has to be understood in a Pareto sense and with respect to the Pareto or-der in R : Given a, b ∈ R , a ≤ b if and only if a i ≤ b i , i = 1 ,
2. So, in other words, a feasiblepair ( ¯ K ( x, t ) , ¯ C ( x, t )) that solves (15) is optimal if there is no other feasible pair ( K ( x, t ) , C ( x, t )) suchthat( J ( K ( x, t ) , C ( x, t )) , J ( K ( x, t ) , C ( x, t ))) dominates ( J ( ¯ K ( x, t ) , ¯ C ( x, t )) , J ( ¯ K ( x, t ) , ¯ C ( x, t ))). This canbe rewritten by stating that there exists no feasible ( K ( x, t ) , C ( x, t )) such that( J ( ¯ K, ¯ C ) , J ( ¯ K, ¯ C )) ∈ ( J ( K, C ) , J ( K, C )) + int( R ) . Using the three approaches presented in the previous section to reduce a multiple objective problem to asingle objective model, we can define the following three different single-criterion formulations.
In this first fomulation the two criteria are combined together through scalarization weights. The DM decidesthe value of a trade-off parameter Θ ∈ (0 , J Φ ( C, K ) := (1 − Φ) J ( C, K ) + Φ J ( C, K )Subject to ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + f ( K ( x, t ) , C ( x, t )) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω where J ( C, K ) = (cid:90) T (cid:90) Ω U ( C ( x, t )) e − ρt dxdt and J ( C, K ) = (cid:90) Ω K ( x, T ) dx If we replace Θ = Φ1 − Φ then the objective function boils down to:max J Θ ( C, K ) := J ( C, K ) + Θ J ( C, K )We will see in the following section dedicated to a numerical simulation that, by varying over Θ ∈ (0 , + ∞ ),it is possible to reconstruct the Pareto frontier. In fact the objective function is concave in ( C, K ) for anyΘ ∈ (0 , + ∞ ). 5 .2 Model II In this second formulation we proceed by using the (cid:15) -constraint method. Within this approach there areessentially two different formulations that can be proposed, once again this decision being dependent onthe relative importance of each criterion for the DM. If DM considers the intertemporal utility as the maincriterion and supposes that a certain level (cid:15) of the criterion has to be attained, the model boils down to:max J (cid:15) ( C, K ) := (cid:90) T (cid:90) Ω U ( C ( x, t )) e − ρt dxdt (9)Subject to (cid:82) Ω K ( x, T ) dx ≥ (cid:15), ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + f ( K ( x, t ) , C ( x, t )) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (10)
By varying (cid:15) it is possible to obtain different points of the Pareto frontier. If, instead, the DM is moreinterested in the sustainability criterion, the model can be written asmax J (cid:15) ( C, K ) := (cid:90) Ω K ( x, T ) dx (11)Subject to (cid:82) T (cid:82) Ω U ( C ( x, t )) e − ρt dxdt ≥ (cid:15), ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + f ( K ( x, t ) , C ( x, t )) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (12)
In this third formulation we rely on a GP approach. Let us suppose that g and g are the two goalsassociated with the two criteria J and J , respectively. Within this technique, the DM is not interested inthe maximization process tout court, but instead in the achievement levels of the two criteria J and J .The previous model, reformulated using the GP technique, reads asmin J g ,g := (cid:88) i =1 θ + i δ + i + θ − i δ − i Subject to (cid:82) Ω K ( x, T ) dx − δ +1 + δ − = g (cid:82) T (cid:82) Ω U ( C ( x, t )) e − ρt dxdt − δ +2 + δ − = g , ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + f ( K ( x, t ) , C ( x, t )) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (13)
In this section we present a numerical implementation of Model I introduced above and we determine anapproximation of the Pareto frontier. The next section, instead, will present other possible numerical im-plementations of Models II and III. Before doing that, we specify the components of this model and, inparticular, the form of the function f ( K ( x, t ) , C ( x, t )). It is a traditional Ramsey-type optimal controlproblem, extendeto account for intergenerational equity issues ([12]; [13]) along with spatial heterogene-ity and spillovers ([3]; [16]). The economy develops along a linear city and the social planner wishes tomaximize the vector-valued social welfare by choosing the level of consumption in each location, which in6urn determines the evolution of capital in each location and in the whole economy. Capital accumulationdepends on the difference between net (of depreciation, where the depreciation rate is δ K >
0) production, Y ( x, t ) = AK ( x, t ) α and consumption, augmented for the inflows of capital from other locations; these flowsare captured by a diffusion term ∇ ( d ( x ) ∇ K ( x, t )), where d ( x ) is the diffusion parameter. Given the initialcondition, K ( x,
0) = K ( x ), the problem can be summarized as follows:max (cid:18)(cid:90) T (cid:90) Ω U ( C ( x, t )) e − ρt dxdt, (cid:90) Ω K ( x, T ) dx (cid:19) (14)Subject to ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + AK ( x, t ) α − δ K K ( x, t ) − C ( x, t ) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (15)
We first discuss how to solve numerically the scalarized problem:max J Θ ( C, K ) := J ( C, K ) + Θ J ( C, K )Subject to ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + AK ( y, t ) α dy − C ( x, t ) − δ K K ( x, t ) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω To determine an optimal policy result, let us define the current Hamiltonian function as H ( C, K, λ ) = U ( C ) + λ ( ∇ ( d ( x ) ∇ K ( x, t ) + AK ( x, t ) α − δK ( x, t ) − C ( x, t ))The following proposition provides the optimality conditions for an optimal solution of the problem above. Proposition 2.
Suppose that U ( C ) is a concave function of C . Then a pair ( ˜ C, ˜ K ) solves the above optimalcontrol model if and only if it is solution to the following Hamiltonian system: ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + AK ( x, y ) α − δ K K ( x, t ) − C ( x, t ) , ( x, t ) ∈ Ω × (0 , T ) ∂λ ( x,t ) ∂t = ρλ − ∇ ( d ( x ) ∇ λ ( x, t )) − λαAK α − ( x, t ) − δ K λ, ( x, t ) ∈ Ω × (0 , T ) U (cid:48) ( C ) = λ ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω d ( x ) ∂λ∂n ( x ) = 0 , x ∈ ∂ Ω λ ( x, T ) = Θ x ∈ Ω K ( x,
0) = K ( x ) x ∈ ΩSince analyzing explicitly the Hamiltonian system above is generally not possible (unless we introducerestrictive assumptions), we now proceed with numerical simulations to illustrate the optimal behavior ofcapital and consumption. If we use the dynamic constraint and plug it into the objective function we obtain: J ( K ) = (cid:90) T (cid:90) Ω U (cid:18) − ∂K ( x, t ) ∂t + ∇ ( d k ( x ) ∇ K ( x, t )) + AK ( y, t ) α − δ K K ( x, t ) (cid:19) e − ρt dxdt + Θ (cid:90) Ω K ( x, T ) dx Subject to − ∂K ( x,t ) ∂t + ∇ ( d ( x ) ∇ K ( x, t )) + AK ( x, t ) α − δ K K ( x, t ) ≥ , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω7he direction derivative of J along any feasible h is given by J (cid:48) ( K ; h ) = lim δ → J ( K + δh ) − J ( K ) δ = (cid:90) T (cid:90) Ω U (cid:48) ( − ∂K ( x, t ) ∂t + ∇ ( d ( x ) ∇ K ( x, t )) + AK α ( x, t ) − δ K K ( x, t )) ∗ (cid:18) − ∂h ( x, t ) ∂t + ∇ ( d ( x ) ∇ h ( x, t )) + Ah α ( x, t ) dx − δ K h ( x, t ) (cid:19) e − ρt dxdt + Θ (cid:90) Ω h ( x, T ) dx We propose an algorithm to determine an approximation of the optimal solution. At each step this algorithmdetermines the direction of growth h using the above calculated directional derivative J (cid:48) ( K ; h ). • Given the value of the state variable K n ( x, t ), solve the following problem − ∂h ( x,t ) ∂t + ∇ ( d ( x ) ∇ h ( x, t )) + Ah α ( x, t ) − δ K h ( x, t ) = (cid:104) U (cid:48) (cid:16) − ∂K n ( x,t ) ∂t + ∇ ( d ( x ) ∇ K n ( x, t )) + AK αn ( x, t ) − δ K K n ( x, t ) (cid:17)(cid:105) − (cid:0) − Θ ∂h∂t + 1 (cid:1) e ρt , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω h ( x,
0) = 0 . x ∈ Ω • Determine δ > J along the direction h • Update K n +1 = K n + δh • If | J ( K n +1 ) − J ( K n ) | < (cid:15) then stop otherwise go to the top.The following result shows that J is increasing along the sequence generated by the above algorithm.The implementation of the above algorithm generates a sequence of functions K n along which the objectivefunction is increasing. Proposition 3. If δ is small then J ( K n +1 ) ≥ J ( K n ) , ∀ n ≥ . Proof.
Computing we have: J ( K n +1 ) − J ( K n ) = δJ (cid:48) ( K n ; h ) + o ( δ )= δ (cid:18)(cid:90) T (cid:90) Ω (cid:18) − Θ ∂h∂t + 1 (cid:19) dxdt + Θ (cid:90) Ω h ( x, T ) dx (cid:19) + o ( δ )= δ (cid:18) − Θ (cid:90) Ω h ( x, T ) − h ( x, dx + T µ (Ω) + Θ (cid:90) Ω h ( x, T ) dx (cid:19) + o ( δ )= δ (cid:18) T µ (Ω) + o ( δ ) δ (cid:19) ≥ h ( x,
0) = 0. (cid:4)
We now apply the above algorithm to the following model where U ( C ) = [(1 + C ( x, t )) − T = 1,Ω = [0 , d k ( x ) = 1 − . x , α = 1, δ K = 0 . (cid:18)(cid:90) (cid:90) [(1 + C ( x, t )) − dxdt, (cid:90) K ( x, dx (cid:19) Subject to ∂K ( x,t ) ∂t = ∇ (cid:0) (1 − . x ) ∇ K ( x, t ) (cid:1) + K ( x, t ) − C ( x, t ) − . K ( x, t ) , ( x, t ) ∈ [0 , × (0 , − . x ) ∂K∂n ( x ) = 0 , x = 0 , K ( x,
0) = 1 + x. x ∈ [0 , J = 0 . J = 0 . J = 0 . J = 0 . .
1. The values of the objec-tive functions after three iterations are given by: J = 0 . J = 0 . J = 0 . J = 0 . K in the Θ = 0 case (left panel) and Θ = 0 . . • For each Θ we solve the scalarized problem max J + Θ J , • Let K Θ ( x, t ) be the optimal solution, we plug it into the two separated criteria and get the pair ofvalues ( J , J ), • We plot the pair ( J , J ).The Pareto frontier is illustrated in Figure 2, from which we can clearly observe that it is bowed outwardas expected, give the nature of the trade off between the two criteria.Figure 2: Piecewise linear Pareto frontier under scalarization. In this section we discuss some further numerical experiments that can be implemented using different modelformulations based on the (cid:15) -constraint and the GP approaches.9 .1 Experiment I
We now discuss a model formulation that has been obtained by applying the (cid:15) -constraint method. In thisformulation we suppose that the DM maximizes his intertemporal utility and the level of physical capitalat the final horizon T is included in the set of constraints. Given a positive value of (cid:15) , the model can bewritten as: max (cid:90) T (cid:90) Ω U ( C ( x, t )) e − ρt dxdt (16)Subject to: (cid:82) Ω K ( x, T ) dx ≥ (cid:15), ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + AK ( x, t ) α − δ K K ( x, t ) − C ( x, t ) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (17)This model can be solved using the same approach and a slightly modified version of the algorithm usedin the linear scalarization case. If we use the dynamic constraint to express the level of consumption C ( x, t )as function of K ( x, t ), the model boils down to:max (cid:90) T (cid:90) Ω U (cid:18) ∂K ( x, t ) ∂t − ∇ ( d ( x ) ∇ K ( x, t )) − AK ( x, t ) α + δ K K ( x, t ) (cid:19) e − ρt dxdt (18)Subject to: (cid:82) Ω K ( x, T ) dx ≥ (cid:15), − ∂K ( x,t ) ∂t + ∇ ( d ( x ) ∇ K ( x, t )) + AK ( x, t ) α − δ K K ( x, t ) ≥ , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (19)As a numerical experiment, we now apply the above model when U ( C ) = [(1 + C ( x, t )) − T = 1,Ω = [0 , d k ( x ) = 1 − . x , α = 1, δ K = 0 .
01. The model is formulated as:max (cid:90) (cid:90) (cid:34)(cid:18) − ∂K ( x, t ) ∂t + ∇ (cid:0) (1 − . x ) ∇ K ( x, t ) (cid:1) + K ( x, t ) − . K ( x, t ) (cid:19) − (cid:35) dxdt Subject to (cid:82) K ( x, dx ≥ . − ∂K ( x,t ) ∂t + ∇ (cid:0) (1 − . x ) ∇ K ( x, t ) (cid:1) + K ( x, t ) − . K ( x, t ) ≥ , ( x, t ) ∈ [0 , × (0 , − . x ) ∂K∂n ( x ) = 0 , x = 0 , K ( x,
0) = 1 + x. x ∈ [0 , The application of the above algorithm provides J (1) = 0 . J (2) = 0 . J (3) = 0 . K is shown in Figure 3. In this model formulation we still use the (cid:15) -constraint method but we assume a linear utility function U ( C ) = C , and α = 1. In this context the spatial model can be reduced to a one-dimensional optimalcontrol model by introducing the average of consumption and physical capital at t , C M ( t ) and K M ( t ),defined as C M ( t ) = (cid:90) Ω C ( x, t ) dx K and K M ( t ) = (cid:90) Ω K ( x, t ) dx and the average amount of consumption per unit amount c M ( t ) as c M ( t ) = C M ( t ) K M ( t ) ∈ [0 , K, C ) solves the model:max C ( x,t ) ,K ( x,t ) (cid:90) T (cid:90) Ω C ( x, t ) e − ρt dxdt (20)Subject to: (cid:82) Ω K ( x, T ) dx ≥ (cid:15), ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + AK ( x, t ) α − δ K K ( x, t ) − C ( x, t ) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (21)Then, by easy calculations and taking the integral of the contraints, the pair ( K M , c M ) solves the followingone: max c M ( t ) ,K M ( t ) (cid:90) T c M ( t ) K M ( t ) e − ρt dt (22)Subject to ˙ K M ( t ) = AK M ( t ) − δ K K M ( t ) − c M ( t ) K M ( t ) , t ∈ (0 , T ) K M ( T ) ≥ (cid:15),K M (0) = (cid:82) Ω K ( x ) dx. (23)This is classical calculus of variations model with a bang-bang control and it can be solved in closed-formto determine the optimal paths of C M ( t ) and K M ( t ). This third experiment is devoted to a different formulation that has been obtained by applying the GPapproach. Here we also assume a linear utility function U ( C ) = C , and α = 1. We also suppose that all11eights are equal and then normalized to 1 and that T is an integer number. Given two goals g and g forthe two criteria J and J , the model reads as:min δ +1 + δ − + δ +2 + δ − (24)Subject to: (cid:82) Ω K ( x, T ) dx − δ +1 + δ − = g (cid:82) T (cid:82) Ω C ( x, t ) e − ρt dxdt − δ +2 + δ − = g , ∂K ( x,t ) ∂t = ∇ ( d ( x ) ∇ K ( x, t )) + AK ( x, t ) − δ K K ( x, t ) − C ( x, t ) , ( x, t ) ∈ Ω × (0 , T ) d ( x ) ∂K∂n ( x ) = 0 , x ∈ ∂ Ω K ( x,
0) = K ( x ) . x ∈ Ω (25)To simplify the model, we take the integral of the PDE and reduce the analysis to the variables K M and C M introduced in the previous experiment. The model boils down to:min δ +1 + δ − + δ +2 + δ − (26)Subject to: K M ( T ) − δ +1 + δ − = g (cid:82) T C M ( t ) e − ρt dt − δ +2 + δ − = g , ˙ K M ( t ) = ( A − δ K ) K M ( t ) − C M ( t ) , t ∈ (0 , T ) K M (0) = (cid:82) Ω K ( x ) dx. (27)By discretizing the time, and introducing the discrete variables K M ( i ), C M ( i ), i = 0 ...T , the model can bewritten as min δ +1 + δ − + δ +2 + δ − (28)Subject to: K M ( T ) − δ +1 + δ − = g (cid:80) Ti =0 C M ( i ) e − ρi − δ +2 + δ − = g ,K M ( i + 1) − K M ( i ) = ( A − δ K ) K M ( i ) − C M ( i ) , i = 0 ...T − K M (0) = (cid:82) Ω K ( x ) dx. (29)This is a linear optimization model that can be solved by standard optimization solvers such LINGO orMATLAB. The introduction of intergenerational equity considerations associated with sustainability issues into a tradi-tional macroeconomic setting transforms the typical unicriterion macroeconomic problem into a bi-criteriaoptimization problem, which can be analyzed through the lenses of the multicriteria optimization techniquesdeveloped in the operations research literature. Recently traditional macroeconomic problems have beenextended to introduce a spatial dimension, allowing to consider the extent to which spatial heterogeneityand spatial spillovers affect economic outcomes not only over time but also across space. The goal of thispapers consists thus to merge these two different lines of research by analyzing a simple macroeconomicsetting to account for intergenerational equity and spatial spillovers from the operations research point ofview. In particular, we show that our macroeconomic problem can be reformulated as a multicriteria prob-lem by relying on different techniques (scalarization, (cid:15) -constraint method and goal programming), and suchdifferent formulations of the problem can be solved through numerical methods, which allow us to illustratethe nature of the trade-off between the two criteria and to derive the Pareto-frontier in some specific casesand parametrizations. 12ur paper represents one of the first attempts to bridge the economics and the operations researchliterature, but still much needs to be done in order to develop further the possible synergies existing be-tween these two different disciplines. In particular, apart from scalarization, (cid:15) -constraint method and goalprogramming approaches, several other techniques developed in the operations research literature can beapplied in similar macroeconomic contexts, especially in the context of stochastic or fuzzy multiple objectiveoptimization. It would also be worth exploring the use of vectorization algorithms and methods that tacklethe MOP model directly such as, for instance, genetic algorithms. Moreover, apart from the applicationsto macroeconomic questions, similar multicriteria approaches can be applied to other economic problemsarising in environmental economics, game theory and cost-benefit analysis.