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Optimal savings and the value of population

Contributed by Kenneth J. Arrow, September 10, 2007 (received for review December 17, 2006)
Abstract
We study a model of economic growth in which an exogenously changing population enters in the objective function under total utilitarianism and into the state dynamics as the labor input to the production function. We consider an arbitrary population growth until it reaches a critical level (resp. saturation level) at which point it starts growing exponentially (resp. it stops growing altogether). This requires population as well as capital as state variables. By letting the population variable serve as the surrogate of time, we are still able to depict the optimal path and its convergence to the longrun equilibrium on a twodimensional phase diagram. The phase diagram consists of a transient curve that reaches the classical curve associated with a positive exponential growth at the time the population reaches the critical level. In the case of an asymptotic population saturation, we expect the transient curve to approach the equilibrium as the population approaches its saturation level. Finally, we characterize the approaches to the classical curve and to the equilibrium.
Arrow et al. (1) have formulated a dynamic model of the economy in which an exogenously changing population is one of the state variables of the system. Furthermore, they jettison the standard classical assumption of an exponentially growing population—an assumption that is obviously absurd, particularly in view of the dramatic reduction in birth rates throughout the world.
How does such an economy do systematic planning for the future? Arrow et al. study this question by formulating the problem of the economy as an optimal control problem. The focus of their analysis is to determine whether and to what extent the optimal policy results in a gain in aggregate welfare.
Arrow et al. provide an analysis of the role of varying population in the measurement of savings. This provides a criterion for improvement in welfare. This is accomplished by recognizing population as another form of capital and formulated as a state variable of the system in its optimal control formulation.
The objective function used is that of maximizing the integral of discounted utilities of per capita consumption weighted by the size of the population over time. This derives from the concept of total utilitarianism going back to Henry Sidgwick and Francis Edgeworth in the 1870s. The concept argues for representing the instantaneous wellbeing of each generation by the product of the population size and the utility derived from total consumption distributed equally among the population. This position is endorsed by Meade (2) and Mirrlees (3) among others. An alternative position called average utilitarianism bases the instantaneous wellbeing of each generation on just per capita consumption, and it has been shown by Dasgupta (4) to imply results that are ethically not defensible. Dasgupta suggests another ethically defensible alternative, where discounting arises due to a constant, exogenously specified rate of extinction. Arrow et al. also consider this later framework known as dynamic average utilitarianism.
The other state variable that Arrow et al. consider in their paper in addition to the population is one form of capital made of a good that can be used for consumption as well as capital formulation. Although the formulation of the problem and the measurement of savings can be extended to allow for other capital variables such as natural resources, human capital, and knowledge, most of their analysis is limited to only two state variables—capital and population.
The analysis of the resulting optimal control problem is carried out by using Pontryagin's maximum principle (5, 6). The application of the maximum principle results in a costate variable associated with the variable population. This can be interpreted as the shadow price or the accounting price of the population. It is important to remark that the formulation does not exclude a negative shadow price for population a priori.
Arrow et al. consider the twin roles of population: one in the objective function bringing in the concept of total utilitarianism represented by the sum of the utilities of per capita consumption for every one in the population and the other in the state equation as the labor component of the production function. The maximum principle also provides the dynamics of the shadow prices associated with capital and population over time.
Using these shadow prices, Arrow et al. derive an expression for per capita genuine savings in the commodity units. They show that the optimal consumption path in their model is sustainable if and only if the per capita genuine savings is strictly positive. They also derive a similar result for the criterion of dynamic average utilitarianism. Finally, Arrow et al. provide economic interpretations of several variables and expressions that arise in the course of their analysis.
In this paper, we take up the twostate model of Arrow et al. and analyze it much further. While the maximum principle could be used for our analysis, we find it convenient to use dynamic programming (DP), because our analysis focuses on the phase diagram, namely, the behavior of the value function and its gradient with respect to the state variables. The DP method allows us to do this directly, whereas it is much more cumbersome with the use of the Maximum Principle. Another reason we use DP is that it provides us with sufficient conditions for optimality. Nevertheless, we shall provide the transformation to the corresponding terms in Arrow et al. (1) for the convenience of the readers.
Furthermore, we shall consider the model involving only population and one form of capital, even though the formulation and the concepts used in the paper can be extended to economies with additional forms of capital and multiple consumption goods and services including environmental amenities. Indeed, the stimulus for measuring the new concept of genuine savings has come from an increasing attention to the role of the ecological and environmental factors in general. In this vein, Asheim (7) and Pezzey (8) consider multiple capital forms and multiple consumption goods in the context of related concepts of sustainability and green net national product (GNNP).
Asheim (7) follows Arrow et al. and considers green national accounting when population is changing and instantaneous wellbeing or utility depends both on per capita consumption and population size. He shows that in an economy with multiple capital forms and multiple commodities, welfare gain can be measured by an expanded genuine savings measurement. Like Arrow et al., he shows that dynamic welfare is increasing at time t if and only if the expanded genuine savings is strictly positive at that time.
Pezzey (8) shows that if an economy with multiple consumption goods (including environment amenities) and constant population maximizes the present value of the utility stream of per capita consumption with constant discounting, then it is unsustainable at time t if the change in “augmented” GNNP is zero or negative at that time. Here augmented means that time is treated as a productive stock that includes value of future, exogenous changes in technology or terms of trade. He also finds adjustments that are needed for the measure when population is assumed to grow exponentially. Pezzey considers an economy to be sustainable at a time t, if and only if per capita consumption utility at that time does not exceed the maximum level of per capita consumption utility that can be sustained forever from t onward, given the capital stock at time t. We should note that this is a different definition of sustainability than the one used by Arrow et al.
However, the focus of refs. 7 and 8 is to derive some general propositions, and not to obtain optimal consumption path over time. We, on the other hand, provide a fairly complete solution of the problem, which we can do only in the case of two state variables. While some of our analysis can be extended to multiple capital forms and commodities, we believe that the result we obtain for the simple twostate variable system are representative of the economies with multiple capital forms of goods and services.
The plan of this paper is as follows. In Section 1, we develop the notation and the model. The state variables are aggregate capital and population. The control variable is consumption. The objective is to maximize the present value of the society's utility of consumption over time. The model is transformed to per capita variables at the end of Section 1. In Section 2, we use DP to study the problem. In Section 3, we derive partial differential equations (PDEs) for the adjoint variables. The classical model with the exponential population growth is discussed in Section 4. The classical model is usually analyzed with the help of a phase diagram. In this diagram, there is an optimal path, which we term as the classical curve. The general case is studied in Section 5. The idea for the analysis of the general case is to come up with a transient path that merges with the classical curve and then follows it subsequently. Here we assume that the population starts growing exponentially at some critical finite population level. In this case, the transient path joins the classical curve at the time when the population reaches the critical level. All of this can be shown with the help of an extended phase diagram. We do not need a threedimensional phase diagram, because we can show that the adjoint variable associated with the population level is related algebraically to the one associated with the capital state. In Section 5, we examine the properties of the phase diagram and address the computational issues. Results are illustrated through numerical examples. The special case of zero population growth beyond a saturated population level is treated in Section 6. Section 7 concludes the paper.
A set of online appendices supplement the paper. In supporting information (SI) Appendix A , we provide the maximum principle formulation of the problem. In SI Appendix B , we discuss the steady state analysis of the classical model. The detailed derivation of the PDEs involved are presented in SI Appendix C . The proofs of the formal results are collected in SI Appendix D . Additional numerical examples are analyzed in SI Figs. 5–8.
1. Model Description
We consider a onesector economy in which the stock of capital K(t) and population N(t) are two state variables. We do not distinguish between population and labor force for convenience in exposition. The output rate F(K, N) of the economy depends on the capital stock K and the population, or labor force, N. The production function F(K, N) is strictly concave with constant returns to scale (CRS). Let c(t) be the rate of individual consumption, assumed to be same for all with u′(c) > 0, u″(c) < 0, and u′(0) = ∞. We will refer to it simply as the per capita consumption rate. Then the capital stock dynamics is It is important to note that the population N enters the dynamics in a nontrivial way.
As for the evolution of population over time, we assume that it is independent of economic conditions and that it is not affected by any control variable. Specifically, the population N is assumed to grow at the rate of ν(N). Thus the population growth equation is For each individual in the society, the rate of utility for consuming c units per unit time is u(c). In the tradition of total utilitarianism, which argues for treating people more or less equally, the objective becomes one of maximizing the total utility of the society given by Note that in Eq. 3 , we have weighted people by their futurity (discounting) but not according to number of their contemporaries.
The problem is to select the per capita rate of consumption c(t) ≥ 0, t ≥ 0, so as to maximize J(c(·)), subject to the condition that K(t) ≥ 0, t ≥ 0. Let the per capita capital stock k = K/N. Then Eq. 1 can be written as where, on account of the CRS assumption, we have We use DP for our analysis. As is standard, we shall let the initial k(0) = k and N(0) = N. Then we can write the value function as In the classical exponential growth case ν(N) = ν, a constant, the condition r > ν is required for the value function to be finite. In the absence of this condition, the discount rate is less than or equal to the rate of the population growth, and the value function v(k, N) becomes infinite for k > 0, N > 0. The generalization of the condition r > ν in our case is the condition that where N(t) is the solution of Eq. 2 .
2. Bellman Equation
The DP equation corresponding to the optimal control problem Eqs. 2 , 4 , and 5 is where v _{k} = ∂v/∂k and v _{N} = ∂v/∂N. Let us define the per capita value function Then we have Using Eqs. 8 and 9 , we can rewrite Eq. 7 as Let ĉ denote the consumption along an optimal path. Then ĉ satisfies the firstorder optimality condition Because we expect W _{k} to be finite, we have ĉ > 0 on account of our assumption that u′(0) = ∞; see, e.g., Karatzas et al. (9) for an explanation. In turn, we expect k(t) > 0. Note, however, that if k(0) = 0, then the optimal solution is ĉ(t) = 0, t ≥ 0.
Let the marginal value W _{k} of the capital stock be denoted by Then Eq. 11 can be written as Condition 13 equates the marginal utility of consumption to the marginal value of capital, all expressed in per capita terms.
Define also and which Arrow et al. interpret as value of life. Furthermore, the function Ψ plays a role in defining sustainability of an optimal path; see refs. 1 and 8.
Using Eqs. 12 , 14 , and 13 in Eq. 10 , we obtain the Bellman equation This equation is a PDE in W(k, N). We shall solve this equation and interpret various terms that arise in the course of our analysis.
3. Derivation of PDEs for P and Ψ
To solve Eq. 16 , it is convenient to first study PDEs for P(k, N) and Ψ(k, N) that can be derived from Eq. 16 . We differentiate Eq. 16 with respect to k and substitute for Ψ_{k} from Eq. 14 to obtain a PDE for P(k, N) as where f′(k) = df(k)/dk. Similarly, we can derive a PDE for Ψ(k, N) as where ν′(N) = dν(N)/dN. The details of these derivations are in SI Appendix B . These PDEs necessitate boundary conditions, which will be specified in Section 5.
Remark 1:
We proceed formally with differentiation. A rigorous approach would be to consider finite differences first and then make a limit argument. However, the equations we obtain are well posed, so the formal argument that we use is justified.
4. Case of Exponential Population Growth
In this section, we take up the special case ν(N) = ν > 0. This assumption gives us the well studied classical case of a onesector economy with exponential population growth (5). In this case, we may have P _{N} = Ψ_{N} = 0 and define p̄ _{k} = P(k, N) and ψ̄(k) = Ψ(k, N). As derived in SI Appendix C , the steady state relations are given by There exists one and only one solution of these equations, and this solution, denoted as (k _{∞}, c _{∞}, p _{∞}, ψ_{∞}), represents the steady state values of the per capita capital, the per capita consumption rate, the marginal valuations of the capital, and the marginal value of the population, respectively. It should be observed that System 19 is the same as the steady state conditions in the literature for the classical case, except that there the equation for ψ_{∞} is not needed, and therefore not derived.
It is also derived in SI Appendix C that when ν(N) = ν > 0, the PDE 17 reduces to the ordinary differential equation (ODE) We show in SI Appendix C that there is an optimal path p̄(k) starting from any initial capital stock k _{0} that converges to the steady state. This analysis leads to the optimal path shown as the solid curve in Fig. 1 given in Section 5. We shall refer to this curve as the classical curve.
5. Back to the General Case ν (N)
In this section, we treat a general case with the following specification for the rate of population growth: We assume ν > 0 for simplicity in exposition. This is the case when the population grows exponentially for N ≥ N̄, and it grows in an arbitrary specified manner for N < N̄. This case is more general than the case when population reaches a saturation level at N̄. That is why we first study the case ν > 0. The special case of ν = 0 will be treated in Section 6. Note further that the important case is a special case of Eq. 21 with ν = 0 and ν (N) = α(N̄ − N), N ≤ N.
We can now define the boundary conditions related to PDEs 17 and 18. With ν (N) as defined in Eq. 21 , the steady state equations are the same as Eq. 19 , defining k _{∞}, c _{∞}, p _{∞}, and ψ_{∞}. The boundary conditions to Eqs. 17 and 18 are with p̄(k) and ψ̄(k) satisfying the equations Observe that these are precisely the equation obtained by using the Pontryagin maximum principle for our analysis. The first two equations are the adjoint equations associated with capital and population, respectively, and the last one is the Hamiltonian maximization condition.
By the method of characteristics for solving the PDE 17, we introduce the following system of ODEs for k(N) and p(N): For convenience in exposition, we shall refer to this path defined by Eqs. 22 , 23 , and 24 as the transient curve.
The solutions p(N) and k(N) of Eqs. 22 and 23 satisfy To see this, let π(N) = P(k(N), N). Then π(∞) = p(∞). Moreover, from Eqs. 17 and 22 , we have Hence, π(N) = p(N). Because p(N) = P(k(N), N), we get which defines the value of the solution of the PDE 17 at any point (k _{0}, N _{0}).
Similarly, let ψ(N) = Ψ(k(N), N). Then ψ(N̄) = ψ̄(k(N̄)) = Ψ(k(N̄), N̄). Also, from Eqs. 18 , 22 , and 25 , we have and ψ(N _{0}) = Ψ(k _{0}, N _{0}).
Approach to the Classical Curve.
Now we analyze the positioning of the transient curve in relation to the classical curve defined by Eqs. 19 and 20 in two special cases. We focus on proving the result when ν(N) > ν for N _{0} < N < N̄. A similar approach can be used to examine the case when ν(N) < ν for N _{0} < N < N̄. Special Case (i). ν (N) > ν for N _{0} < N < N̄: Let p(N) and k(N) solve Eqs. 22 , 23 , and 24 , and let p̄(k) be the curve defined by Eqs. 19 and 20 . Define p̃(N) = p̄(k(N)). We establish that p(N) > p̃(N) through a series of lemmas.
Note that p̃(N̄) = p(N̄). Also From Eq. 27 , we obtain
Lemma 1.
Assume k _{0} ≤ k _{∞}. Then f(k _{0}) − k _{0}ν − u′ ^{−1}(p _{0}) ≥ 0.
Lemma 2.
Assume k _{0} ≤ k _{∞}. Then p _{0} = p(N _{0}) > p̃(N _{0}).
Proposition 1.
p(N) ≥ p̃(N).
Special Case (ii). ν(N) < ν for N _{0} ≤ N ≤ N̄.
Proposition 2.
p(N) ≤ p̃(N).
Monotone Property and Computation of the Transient Curve.
Proposition 3.
Let k^{i}(N) and p^{i}(N) be the solutions to Eqs. 22 and 23 with the boundary conditions If k _{0} ^{1} > k _{0} ^{2}, then k ^{1}(N) > k ^{2}(N) and p ^{1}(N) < p ^{2}(N).
Proposition 3 says that if the system begins at a higher initial capital, then it joins the classical curve at a higher level of capital stock. Note that this result does not require any special structure for ν (N). The monotone property in Proposition 3 suggests that we can compute the transient curve (k(N), p(N)) for given values of k _{0} and N _{0} using the following procedure. That is, we can begin with two points on the classical curve and solve Eqs. 22 and 23 . We will then see where the given initial k _{0} lies in relation to the capital stocks obtained in the solutions at the initial population level N _{0}. This relation will tell us the direction in which we should change the previous guess for the points on the classical curve. This procedure will take us closer and closer to the point on the classical curve that corresponds to the initial k _{0}. More precisely, we start with arbitrary values of k _{N̄} ^{A} and k _{N̄} ^{B}, and compute the trajectories k ^{j}(N) and p ^{j}(N) that solve Eqs. 22 and 23 with the boundary conditions k ^{j}(N̄) = k _{N̄} ^{j} and p ^{j}(N̄) = p̄(k _{N̄} ^{j}), j = A and B. In each iteration, we revise the guess of k _{N̄} ^{A} or k _{N̄} ^{B}, and obtain the trajectories k ^{j}(N) and p ^{j}(N), j = A or B, for the corresponding boundary conditions on the classical curve. The algorithm picks a sequence of {k _{N̄} ^{A}, k _{N̄} ^{B}} so that the trajectories k ^{A}(N) and p ^{A}(N) (and the trajectories k ^{B}(N) and p ^{B}(N)) converge to the desired k(N) and p(N), respectively, for the given value of k _{0}. A detailed description of this algorithm is provided in SI Appendix D .
Proposition 4.
The above algorithm converges.
In the above procedure, we start from a candidate point k ^{A}(N̄) = k _{N̄} ^{A} and p ^{A}(N̄) = p(k _{N̄} ^{A}) from the classical curve, solve the ODEs 22 and 23, and compare k ^{A}(N _{0}) with the given value k _{0}. By Proposition 3, we can adjust the choice of k _{N̄} ^{A} to find the corresponding transient curve start with k _{0}. An alternate approach to obtain the transient curve is to start with a guess of p ^{A}(N _{0}) = p _{0} ^{A}. This is suggested by the following proposition.
Proposition 5.
Consider the solution {k ^{i}(N), p ^{i}(N)N _{0} ≤ N ≤ N̄} to the ODEs 22 and 23 with the boundary conditions If p _{0} ^{1} > p _{0} ^{2}, then p ^{1}(N) > p ^{2}(N) and k ^{1}(N) ≥ k ^{2}(N) for any N∈[N _{0}, N].
Denote k(N) and p(N) to be the transient curve defined in Eqs. 22 and 24 . Let k ^{A}(N) and p ^{A}(N) be the solution to Eqs. 22 and 23 with boundary condition k ^{A}(N _{0}) = k _{0} and p ^{A}(N _{0}) = p _{0} ^{A}. By Proposition 5, if p _{0} ^{A} < p(N _{0}), then p ^{A}(N̄) < p(N̄) and k ^{A} (N̄) ≤ k(N̄). Then, Thus (k ^{A}(N̄), p ^{A}(N̄)) is above the classical curve. Likewise, we can show that (k ^{A}(N̄), p ^{A}(N̄)) is below the classical curve if p _{0} ^{A} > p(N _{0}). Hence, Proposition 5 suggests that to compute the transient curve, we can start with a guess of p(N _{0}), and compare whether the resulting curve is below or above the classical curve at N = N̄. Then we adjust the guess of p _{0} ^{A} accordingly to obtain the transient curve that joins the classical curve at N = N̄.
Numerical Examples.
We illustrate the results through some numerical examples. Consider the special case: In Fig. 1, we provide an example in which ν(N) > ν. The classical curve is the solid line, and the curves f ′(k) = r and f(k) − ν k − u′ ^{−1}(p) = 0 are shown as lines with alternating single dashes and single dots. The curves (shown as lines with alternating single dashes and two dots) are the transient curves for different choices of k _{0}. We observe that the transient curves are always above the classical curve, which is consistent with the result in Section 5, Approach to the Classical Curve. For illustration, consider the solution for k _{0} = 18. The optimal path goes along the bold dashed curve, then merges with the classical curve at the point (20.872, 0.306), and finally reaches the steady state (k _{∞}, p _{∞}) = (25, 0.27) along the classical curve. The corresponding optimal trajectories over time can be found in SI Fig. 5. An example with ν (N) < ν is also provided in SI Fig. 6.
In general, when ν(N) is not monotone, the transient curve and the classical curve may have multiple intersections. Fig. 2 provides such an example. The steady state solution (k _{∞}, p _{∞}) = (25, 0.27) and the classical curve do not change. In this example, for N ∈ [1, 2)∪(3, 5], α < 0 and ν(N) decreases in N. For N ∈ (2, 3), we have α > 0 and ν(N) increases in N. We illustrate a particular transient path with k _{0} = 21.7. This particular transient path switches twice at k = 18.01 and k = 23.95, and then joins the classical curve. Unlike in the previous examples, the transient curve intersects the classical curve twice. As shown in SI Fig. 8, the optimal capital path k(t) is neither monotone nor unimodal, since dk/dt defined in Eq. 4 depends explicitly on ν(N).
6. The Case of Saturated Population: ν = 0
This is a special case of the problem treated in Section 5, when ν = 0. In this case, N̄ denotes the saturation level of the population. In other words, the population stops growing when it reaches the level N̄. If the population reaches N̄ at a time t̄, then or, equivalently In this case, the steady state relations in Eq. 19 become When t ≥ t̄, equation Eq. 17 reduces to which together with the boundary value (k _{∞}, p _{∞}) in Eq. 31 defines the classical curve p̄(k).
When t ∈ [0, t̄), we have Ṅ(t) = ν(N)N > 0, and we can still use N ∈ [N _{0}, N̄] as the time index as suggested in the beginning of SI Appendix A . Thus, the transient curve is defined by If t̄ is finite, the optimal trajectory (k(N), p(N)) joins the classical curve Eq. 32 , along the transient path defined in Eqs. 33 and 34 , at time t̄. Thereafter, the optimal trajectory reaches the steady state (k _{∞}, p _{∞}) obtained in Eq. 31 , by sliding along the classical curve. It should be clear that this is a special case of the model treated in Section 5 with ν (N) ≥ ν = 0.
If there is no finite t̄ at which the population starting at N _{0} < N̄ reaches N̄, then we have a case of the population that approaches N̄ asymptotically. In this case, the transient path converges to the steady state as the population level approaches N̄. This result is stated below and its proof is omitted.
Theorem 1.
The optimal trajectory is the solution of Eqs. 33 and 34 with the boundary conditions That is, the optimal trajectory converges asymptotically to the steady state (k _{∞}, p _{∞}) as N → N̄.
In Fig. 3 the solid line corresponds to the classical curve. We note that the value of p(N _{0}) decreases with the choice of k _{0} as suggested by Proposition 3. The transient curves for different values of k _{0} converge to the steady state (k _{∞}, p _{∞}). Moreover, the approach of the transient curves to the steady state is tangential to the classical curve. Thus, all the transient curves have the same slope at (k _{∞}, p _{∞}).
Lemma 3.
For a given N^{a} ∈ [N_{0}, N̄], define ν^{a} = ν (N ^{a}). Denote p^{a}(N) and k^{a}(N) for N ∈ [N_{0}, N^{a}] as the solution to Eqs. 22 and 23 with the boundary conditions and p̄^{a}(k) defined by Then, Consider the population growth function Ṅ = Nν(N) with ν(N̄) = ν. Define ν(N) = ν + μ(N) with μ(N) → ν as N → N̄. For any given ν > 0, denote (k ^{ν} (N), p ^{ν} (N)) to be the transient path. The transient path joins the classical curve at (k ^{ν}(N̄), p ^{ν}(N̄)). Note that the first equation in 19 does not depend on ν, so that k _{∞} ^{ν} = k _{∞} ^{0} for any value of ν. Then Theorem 1 is equivalent to the following:
Theorem 2.
As ν → 0, the sequence k ^{ν} (N̄) converges to k _{∞} ^{0}.
In Fig. 4, we compute the optimal trajectories for k _{0} = 18 for three different values of ν. We observe that the classical curve shifts down as ν decreases. Moreover, as ν gets closer to zero, the point at which the transient curve merges with the classical curve becomes closer to the steady state (k _{∞} ^{v}, p _{∞} ^{ν}). Finally, when ν = 0, the merging point will be exactly the steady state (k _{∞} ^{0}, p _{∞} ^{0}).
Lemma 4.
The classical curve p̄^{ν}(k) decreases with ν.
7. Conclusion
We have studied a onesector model of an economy with a general population growth. We use DP for our analysis. We also show briefly how our analysis is related to the maximum principle. By showing that the costate of the population is only algebraically related to the costate of the capital stock, we are able to develop a twodimensional phase diagram of the problem. In the case when the population enters a constant exponential growth phase at some critical population level N̄, we show that the optimal path has a transient phase, which merges with the classical curve precisely at the time when the population reaches the level N̄. This approach to the classical curve is expected to be asymptotic if N̄ = ∞. However, when N̄ = ∞, the analysis leads to a PDE to be solved on an unbounded domain. The boundary conditions are then replaced with appropriate growth conditions. We have not done this in this paper to avoid a lengthy presentation.
We characterize the behavior of the transient curve depending on whether ν(N) > ν or ν(N) < ν. Furthermore, we develop an iterative algorithm that converges to an optimal solution. We use this algorithm to illustrate this behavior by computing optimal solutions in some special cases. We provide the optimal solution in an example when ν(N) > ν initially, then ν(N) < ν, and finally ν(N) ≥ ν converging to ν(N) = ν at N = N̄. Here we can see abrupt changes in the optimal trajectory at times when ν(N) goes from increasing to decreasing and vice versa. In the case when the population reaches a saturation level asymptotically, the optimal path converges to the steady state asymptotically as the population approaches its saturation level.
In this paper, we have modeled the population growth as an exogenous function. Note that our state transformation requires a monotonically increasing population. A topic for future research is to develop techniques to treat a more general population evolution path involving temporarily overshooting the steady state. It would also be interesting to examine the case when the population growth depends on the economic development. This can be done by making ν dependent on the per capita capital stock k.
Acknowledgments
Constructive comments from Gustav Feichtinger, Stefan Wrzaczek, and the reviewers are gratefully acknowledged.
Footnotes
 ^{†}To whom correspondence may be addressed. Email: arrow{at}stanford.edu or sethi{at}utdallas.edu

Author contributions: K.J.A. and S.P.S. designed research; K.J.A., A.B., Q.F., and S.P.S. performed research; and K.J.A., A.B., Q.F., and S.P.S. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/cgi/content/full/0708030104/DC1.
 © 2007 by The National Academy of Sciences of the USA
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