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METHODS OF MODELING CONSUMPTION SATURATION WITH UNCERTAINTY IN OPTIMAL STOCKPILE PROBLEMS J. M. LLOYD ∗ and G. G. L. MEYER Whiting School of Engineering Johns Hopkins University, 3400 North Charles Street, Baltimore, MD E-mail: [email protected], [email protected]

Abstract. Economic problems in the optimal management of strategic resource stockpiles can be rigorously studied and solved by formulating them as optimal control problems in continuous time. In these optimal control problems, the proper description of the control constraints and objective function are critical to reflecting a realistic economic model. Existing work in this area often ignores fundamental saturation effects in the economic systems under scrutiny, and the following paper introduces and compares several methods for correcting this common modeling simplification in both deterministic and stochastic contexts. Key Words: Optimal stockpiling, saturation models, optimal control, uncertainty, minimum consumption. 1. Introduction. 1.1. Background. Dating back to at least the 18th century, producers, economists, and policy makers have been concerned with the issue of optimal resource management. Indeed, questions concerning resource economics motivated some of the earliest quantitative work in economics (e.g., Hotelling [1931]) and classical works in the area of game theory, although it was not referred to as such at that time (e.g., Cournot [1838]). While the essential question of scarcity is the fundamental issue in economics, the more specific concern of the optimal usage and consumption of various scarce resources lends itself to examination through methods traditionally employed in mathematics and engineering. While early work addressed basic questions, the mathematical tools necessary for rigorous analysis on the topic were not widely accessible until the middle of the 20th century with the advent of control theory and differential game theory. Although classical findings in the Calculus of Variations were already available by the 19th century, it was not until engineering applications motivated the development of that theory that the mathematical foundations for examining sophisticated resource economics problems were mature.

∗ Corresponding author. J. M. Lloyd; Whiting School of Engineering, Johns Hopkins University, 3400 North Charles Street Baltimore, MD, e-mail: [email protected] Received by the editors on 8t h January 2015. Accepted 18t h June 2015.

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METHODS OF MODELING CONSUMPTION SATURATION

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Of particular interest to government planners, problems in the optimal management of sovereign resources during periods of embargo are relatively new to the economics literature. The unifying question of research on this topic is: what is the optimal strategy for building and consuming a resource stockpile to maximize societal welfare in the face of possible periods of catastrophic scarcity? These “optimal stockpiling” problems are the focus of this paper. Motivated by the oil embargos, some significant studies examined the question in the late 70s and early 80s, but later papers have also contributed to the field. Among the first modern researchers investigating optimal stockpiling, Dasgupta and Heal [1974] published several papers on stockpile management, including one of the first works to examine the role of technology development in optimal stockpiling. Another important pioneering group investigating numerical techniques to solve optimal stockpiling problems, Ascher and Wan [1980] published initial work in the numerical solution of optimal consumption of a finite, multigrade exhaustible resource. Other early researchers included Hillman and Long [1983] who contributed work on the optimal pricing and depletion of a resource in the presence of embargos and a colluding production oligopoly, and Oren and Powell [1985] who extended previous results on resource production and management under the assumptions of a backstop technology. More recent research in the area was produced by Lindsey [1990] who proposed methods of estimating embargo probabilities and calculated the optimal stockpiling and consumption responses in that context. Gaudet [2007] sought to augment the classical findings of Hotelling [1931] with more complex economic factors and to match the improved result to real economic trends. Additional work by Gaudet and Lasserre [2011] developed optimal consumption, extraction, and stockpiling results factoring in uncertainties in resource stockpile risk and quality. Closely related to optimal stockpiling problems, another area of economics research, optimal saving problems, also informs the subject and predates its beginning. For instance, Leland [1968] conducted an early study of optimizing the saving of income for future consumption in the presence of uncertainty in a two-stage problem. His highly cited work is similar in key modeling aspects to the optimal consumption of a general resource and was the foundation of many subsequent efforts. Similarly, Sandmo [1970] studied a two-stage model of optimal saving to examine classical notions of income uncertainty and their impacts on an optimal result. Again, Sandmo’s work readily extends to problems in stockpile management, where the resource is general and not necessarily monetary. 1.2. Research overview. The typical analysis approach to optimal stockpile problems is to model them as optimal control problems and then apply the now well-known and rich family of solution techniques that exist for such problems (e.g., Bryson and Ho [1969]). One frequent simplification in the present optimal stockpiling literature is the implicit assumption of a non-negative minimum consumption (control input). Because of the objective and dynamics of a standard stockpiling

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model, this constraint is often automatically satisfied, but the explicit imposition of the constraint is important for two reasons, one technical and one theoretical. First, for optimal stockpile models with complicated dynamics and welfare functions, a non-negative minimum consumption condition might not be satisfied automatically at optimality. Second, it is desirable in many policy-related problems to impose a nonzero minimum consumption as a critical threshold. Frequently, questions of political economy have been ignored in the development of optimal stockpile models, which consider resource consumption from the sole perspective of economic efficiency. More generally, this paper seeks to examine stockpile optimality from a sustainability viewpoint with additional constraints imposed by political imperatives. Clearly, for typical free-market models, it is both economically and dynamically possible for the optimal consumption path of a sovereign stockpile to fall below given thresholds. For a policy maker interested in avoiding widespread civil unrest during a national emergency, however, it is desirable to have models that offer an optimal strategy that guarantees sustained consumption. In this respect, the following models address the issue of political economy and consumption sustainability through the imposition of saturation effects, effectively modeling a quasi-command economy. Unlike some of the previously mentioned literature, which addresses both issues of stockpile building and consumption, this study is intended to offer optimal solutions presuming an arbitrary existing stockpile and parameterized embargo scenarios. In many practical cases of sovereign stockpile management, the issue of optimal stockpile building is not relevant because all stockpiles are legacy reserves and no financial or political means exist to augment them. Because of this and the known practical interests of various sovereign entities, this paper focuses on stockpile consumption problems. In the future, the extension of these results to stockpile build up problems is merited. The first part of this paper examines various approaches to modeling a minimum consumption threshold, or control saturation. The standard method to model saturation is to impose an absolute constraint on inputs and/or states. In addition to this standard approach, we propose two alternatives that act through the objective function formulation. One, the shifted welfare approach, defines a problem objective that through its form inherently prohibits consumption levels below a predefined minimum. The second, the sigmoidal welfare function approach, allows for consumption levels below a critical threshold but strongly penalizes them. Of the three methods, the sigmoidal approach is proposed as the most intuitively correct, although it does not admit a closed-form solution. Each saturation model has a related economic interpretation, but they can also be perceived as falling along a continuum of modeling simplification. The second part of this study combines the development of the alternative forms of a consumption saturation model to previously reported results in modeling embargo


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length uncertainty and minimum consumption uncertainty (Bahel [2011], Lloyd and Meyer [2014]). As evidenced by existing research and known economic phenomena, uncertainty plays a significant role in the accurate treatment of stockpiling problems. Two forms of uncertainty, embargo length uncertainty and consumption saturation uncertainty, are examined for all three alternative models. The resulting optimal stockpiling and consumption solutions are then compared for these cases and the corresponding implications are explored. 1.3. The optimal strategic reserve problem. To maintain consistency with existing work and to facilitate comparisons between solutions, the following problems are developed in continuous time using simple system structures and solved via applications of Pontryagin’s Maximum Principle (PMP). For two of the models, this enables an analytical, closed-form solution, while in the third case a tractable numerical solution is possible. 1.4. Optimal consumption with finite embargo length. A large number of the published models in optimal strategic reserve research are founded on variants of the basic finite-horizon, discounted, continuous time optimal control system shown in equations (1).

Tf

J(q) =

e−r t (U (q(t)))dt,

0

x˙ = −q, x(0) = X0 , (1)

x(Tf ) = Xf .

In this formulation, J(.) is the objective function, x is the resource stockpile level, q is the consumption rate of the resource, X0 is the initial stockpile level, Xf is the final stockpile level, and r is the discounting rate. Additionally, U (q) is the social welfare of consumption and is a model-specific function that quantifies the economic benefit of the consumption of a resource to a society. One distinction of note between this basic model and existing literature is in the treatment of the problem horizon. In Hotelling’s classic paper, which was derived in the research context of optimal resource extraction, the optimal control problems are formulated with a free end-time. Many subsequent publications employ this same approach. Contrasting this method, the models in this paper are treated as fixed end-time problems. As a matter of practicality, policy makers often estimate a most likely embargo period based on expert analyses and intelligence sources for planning purposes. Furthermore, it is assumed that at the conclusion of an embargo period, residual stockpile quantities are useless. Better suited for natural resource

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extraction problems, free end-time models decouple the definition of optimality from defined scenario planning, with government policy makers typically unwilling to accept “open ended” solutions. Because of these issues, and also because the development of stochastic horizon models progresses from the deterministic, fixed end-time case, a fixed end-time problem definition is used. Assuming a constant relative risk aversion (CRRA) welfare function form, the societal welfare can be defined as in equation (2), where σ is a parameter corresponding to the risk aversion of the consuming entity. Furthermore, a welfare function of this form is isoelastic, with an elasticity of 1 − σ and coefficient of relative risk aversion σ. (2)

q 1−σ . 1−σ

U (q) =

Given this functional form, the partial derivative of the isoelastic welfare function with respect to consumption can be expressed as (3)

Uq (q) =

1 . qσ

The objective of the optimal strategic resource problem is to find an optimal consumption policy (optimal control), q ∗ (t), that maximizes the objective function subject to the dynamic and endpoint constraints in (1). (4)

q ∗ (t) = argmax J (q (t)) . q

The optimal consumption policy corresponds to an optimal stockpile level, x∗ (t). Furthermore, a negative stockpile or a negative consumption rate are impossible in this context, so two assumptions in (5) are implicit but not imposed. q ∗ (t) ≥ 0, (5)

x∗ (t) ≥ 0.

The problem defined in equations (1) and (4) admits a closed-form analytical solution provided the welfare function is of a suitable form. References documenting the methods to solve a variety of optimal control problems through the application of PMP can be found in Bryson and Ho [1969] and Athans and Falb [1966]. To solve the basic optimal stockpile consumption problem, PMP is applied by first forming the Hamiltonian, H. (6)

H = e−r t U (q) − λq,


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where λ is the costate. Next, the costate necessary conditions for optimality are solved, yielding the fact that the costate is a constant. Note that subscripts indicate the partial derivative of a function with respect to the subscript variable (e.g., Hq = ∂∂Hq ). (7)

Hx = 0 ⇒ λ (t) = λ0 .

Continuing, the necessary condition for maximization is solved for the optimal control. If the welfare function U (.) is a convex function, the Hamiltonian is also convex, and the PMP necessary conditions are sufficient for optimality. λ0 −r t ∗ ∗ −1 Hq = e Uq (q ) − λ0 = 0 ⇒ q = Uq (8) . e−r t Applying the definition of the welfare function to the solution for the optimal control in (8) and expanding yields the optimal consumption as a function of the costate. (9)

∗

q =

e−r t λ0

σ1 .

Finally, the state equation is integrated after substituting in the optimal consumption solution to solve for the costate constant value. The boundary conditions of the state equation are chosen such that Xf = 0 and X0 is an arbitrary, non-negative constant. t ∗ x (t) = X0 + −q ∗ (t) dt, 0

Xf = X 0 +

Tf

−q ∗ (t) dt = 0,

0

(10)

1 1 −σ e−r T f σ σ 1 σ + , X0 = r λ0 r λ0 σ σ rTf σ λ0 = . 1 − e− σ X0 r

Substituting this expression for the costate into the solution for the optimal consumption in (9) yields the solution for optimal consumption as a function of time and the initial stockpile level: rt

(11)

q ∗ (t) =

X re− σ 0 . rTf − σ σ 1−e

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From the form of (11), it can be seen that q ∗ (t) satisfies the non-negativity constraint in (5). This solution is well known in the literature, particularly for the case where Tf → ∞. Due to the dynamics and boundary conditions of problem (1), the optimal consumption solution and its corresponding optimal stockpile level naturally satisfy the inequality constraints in (5). This result, however, is not generally true for nonzero inequality constraints on consumption. For a more descriptive model, a nonzero minimum consumption constraint is needed to address concerns of political economy. The nonzero constraint enforces a minimum consumption level below which politically unacceptable damage would occur to an economy during a period of resource shortages. There are several possible ways to model such an absolute consumption floor, each posing advantages and disadvantages, embodying different interpretations, and resulting in differing solutions. These approaches are explored in the following sections. 2. Minimum consumption problem and solutions. 2.1. Minimum consumption levels: hard constraint. One method of solving the optimal consumption problem with a nonzero minimum consumption limit is the explicit imposition of a control constraint in the problem formulation (12):

Tf

J(q) =

e−r t (U (q(t))) dt,

0

x˙ = −q, −q ≤ −¯ q, x (0) = X0 , (12)

x (Tf ) = Xf ,

where q¯ is an arbitrary non-negative minimum consumption level. In this formulation, there is a critical consumption level at which the demand transitions from isoelastic demand with elasticity constant σ1 to purely inelastic demand. Applying this insight, this model represents a discontinuous jump in the price elasticity of demand, which is a plausible piecewise modeling approach for certain goods with highly nonlinear demand elasticity functions. Moreover, in this model open market price movements would not impact demand at the consumption limit. In the case of a constrained control, the Hamiltonian couples the objective function, system dynamics, and control constraint equations:

(13)

˜ = e−r t U (q) − λq + ν (¯ q − q) . H


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Applying PMP to this new problem yields the necessary conditions for optimality in the form of equations (14). The fourth equation is an additional complementarity condition and arises from the explicit control constraint. ˜x , λ˙ = −H x˙ = −q, ˜ q = 0, H (14)

ν (t) (¯ q − q (t)) = 0.

Solving the costate equations reveals that the costate is a constant: (15)

˜ x = 0 ⇒ λ (t) = λ0 . H

H is convex, so the necessary conditions for optimality are also sufficient. In contrast to the basic consumption problem in equations (1), the solution of the PMP equations is in two parts, one corresponding to the interior solution and one corresponding to the solution at the control boundary. The solution in equations (16) also provides the corner condition for t, which is used to solve the dynamic constraint equations for λ0 . The first step is to calculate the corner condition for t to determine the transition time of q ∗ from the interior solution to the boundary of the feasible set. ˜ q = e−r t Uq (q) − λ0 − ν = 0, q ∗= q¯ ⇒ ν ≥ 0 ⇒ H ν = e−r t Uq (q) − λ0 ≥ 0, e−r t Uq (q) − λ0 ≥ 0 ⇒ e−r t ≥ (16)

λ0 , Uq (q)

1 λ0 0 ≤ t ≤ − ln = t¯. r Uq (¯ q)

After solving for the corner condition, the complementarity condition is applied and solved for the interior solution. ˜ q = e−r t Uq (q ∗ ) − λ0 = 0, q > q¯ ⇒ ν = 0 ⇒ H λ0 , e−r t λ0 ∗ −1 q = Uq . e−r t

Uq (q ∗ ) = (17)

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Combining the piecewise solutions for q ∗ and the partial derivative of the welfare function yields the complete solution for q ∗ . ⎧ 1 ⎪ ⎨ e−r t σ t ≤ t¯ q∗ = (18) λ0 ⎪ ⎩ q¯, t¯ ≤ t. Finally, we apply the dynamics constraints and boundary conditions (19) to obtain a value for λ0 . x∗ (t) = X0 +

t

−q ∗ (t) dt,

0

(19)

Xf = X 0 +

Tf

−q ∗ (t) dt = 0,

0

t¯

1 Tf e−r t σ dt + q¯dt, λ0 t¯ 0 1 1 −σ e−r t¯ σ σ 1 σ X0 = + + q¯ (Tf − t¯) . r λ0 r λ0 X0 =

(20)

Substituting in (16) for t¯ in (20) yields an equation relating the initial stockpile level, X0 , to λ0 :

(21)

q¯ σ q σ λ0 ) + X0 = Tf q¯ + ln (¯ r r

1 1

(λ0 ) σ

− q¯ .

Unfortunately, equation (21) is a transcendental equation and must be solved numerically for λ0 given values for q¯, Tf , σ, and X0 . 2.2. Minimum consumption levels: Shifted welfare function. An alternative to the direct approach for constraining the minimum consumption input is to “shift” the welfare function used in equation (1). Equation (22) expresses the shifted isoelastic welfare function, while Figure 1 illustrates the shifted welfare function alongside its unshifted counterpart.

(22)

U (q) =

(q − q¯)1−σ . 1−σ


265

14 Welfare Shifted Welfare

12

Welfare, U(q)

10 8 6 4 q 2 0

0

1

2

3

4 5 6 Consumption, q

7

8

9

10

FIGURE 1. A shifted welfare function.

Corresponding to the new definition for the shifted welfare function, there is a revised expression for the partial derivative of welfare with respect to consumption.

(23)

Uq (q) =

1 . (q − q¯)σ

Shifting the welfare function increases the welfare at a given consumption level relative to the unshifted welfare function, and because the isoelastic welfare function infinitely decreases as it approaches the minimum consumption level, the minimum consumption is “softly” constrained above a minimum. Formulating the problem in this context yields the new problem statement in equations (24):

Tf

J(q) = 0

x˙ = −q, x (0) = X0 , (24)

x (Tf ) = Xf .

e−r t (U (q(t))) dt,

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Based on (24), we can form the Hamiltonian and systematically apply PMP. H = e−r t U (q) − λq.

(25)

Applying the necessary conditions for optimality to the shifted welfare problem initially generates a solution similar to the one for problem (1). However, the unique form of the welfare function has implications for the structure of the optimal control. As with the basic problem case, the costate is a constant. Again, H is concave, so we can find the maximum via the first-order conditions, but there are now no complementarity requirements and the optimal consumption result is somewhat more simple.

(26)

−r t

Hq = e

∗

∗

Uq (q ) − λ0 = 0 ⇒ q =

Uq−1

λ0 e−r t

.

Substituting (23) into the solution for the optimal control in (26) results in the analytical solution for the optimal control in terms of the costate.

(27)

q∗ =

e−r t λ0

σ1 + q¯.

Similar to the basic optimal consumption problem, the optimal control solution is substituted into the system dynamic equations, which are then integrated to produce the solution for the costate. x∗ (t) = X0 +

t

−q ∗ (t) dt,

0

Tf

Xf = X 0 +

−q ∗ (t) dt = 0,

0

(28)

X0 = Tf q¯ +

1 (λ0 )

1 σ

−σ −r T f e σ r

+

σ 1

(λ0 ) σ r

Algebraically rearranging equation (28) to isolate λ0 yields ⎛ (29)

λ0 = ⎝

−r T f σ

⎞σ

σ − σe ⎠ . rX0 − rTf q¯

.


267

Finally, substituting the solution for the costate (29) into the analytical solution for the optimal consumption (27) yields the complete solution for the optimal consumption. (30)

∗

q =

rX0 − rTf q¯ σ − σe

−r T f σ

e

−r t σ

+ q¯.

In contrast to the hard consumption constraint problem, the solution for the shifted welfare technique is a closed-form, analytical solution. A comparison of the solutions is discussed later. 2.3. Sigmoid welfare. Both the method of an explicit consumption constraint and a shifted welfare function result in absolute floors to the minimum consumption. We hypothesize that a more accurate theoretical model of an economic system during embargo, however, would reflect a sharp drop off in welfare at a critical consumption level, but allow for significantly reduced welfare at levels of consumption below the critical one. Contrasting the previous two models, in this context lower levels of consumption have a defined, nonzero welfare. One possible welfare function that exhibits this quality is the sigmoidal welfare function, shown in equation (31). Researchers have previously advanced the notion that the sigmoid form is a more accurate reflection of individual welfare (e.g., van Praag and Frijters [1999]) as well as overall welfare, although this is a point of rigorous debate (e.g., Seidl [1994]). Interpreting the sigmoidal welfare function in conjunction with a short-run production function, however, theoretically justifies its use in this study. Equation (31) defines the sigmoid welfare function, where b is a parameter that influences the curvature of the sigmoid: (31)

U (q) =

1 . 1 + e−b(q −¯q )

Figure 2 depicts the sigmoid welfare function for b = 1 and q¯ = 5. Clearly, the function exhibits a near-constant level of welfare above a certain consumption threshold, below which the welfare sharply falls down to a nominal level. Previously, it has been assumed that societal welfare is derived from direct consumption of a natural resource. More realistically, the welfare of consumption is a function of consuming a generic good, z, as opposed to direct consumption of a raw material, q. Again, an isoelastic welfare function is assumed for the welfare of consumption of a generic good. (32)

U (z) =

z 1−σ . 1−σ

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1 0.9 0.8

Welfare, U(q)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4 5 6 Consumption, q

7

8

9

10

FIGURE 2. Sigmoidal welfare function, b = 1, q¯ = 5.

An unknown production function is assumed to relate a consumer good and a raw resource, upon which the production of the consumer good is heavily dependent. For simplification of analysis, let the production function be an exclusive function of a single resource: (33)

z = f (q) .

Substitution of the production function definition in (33) into the left-hand side of (32) relates the welfare of consumption of a raw material to the welfare of consumption of a generic good. (34)

U (z) = U (f (q)) = U (q) .

Combining equations (34), (32), and (31) allows for the solution of the implied production function for the generic good: (35)

U (q) =

1 1 + e−b(q −¯q )

=

(f (q))1−σ ⇒ f (q) = 1−σ

1−σ 1 + e−b(q −¯q )

1 1 −σ

.

This production function is sigmoidal in shape and assumes the well-known form of a short-run production function. A short-run production function is appropriate for an embargo model, where limited capacity will exist to expand production


269

capability quickly. Furthermore, this demonstrates that a sigmoidal welfare function for resource consumption is mathematically compatible with an isoelastic welfare for finished good consumption and a typical short-run production function. Thus, a sigmoidal welfare function for material consumption is adopted in the sequel. Following the definition for sigmoid welfare, the equation for the partial derivative of the welfare with respect to q is then

Uq (q) =

(36)

be−b(q

∗ −¯ q)

1 + e−b(q ∗ −¯q )

2 .

Repeating the PMP procedure employed in the first two cases, the costate is determined to be a constant. Next, solving the first-order maximization conditions yields an equation relating the optimal consumption solution to the costate. Hq = e−r t Uq (q ∗ ) − λ0 = 0, ∗

be−r t e−b(q −¯q ) 2 − λ0 = 0. 1 + e−b(q ∗ −¯q )

(37)

Equation (37) is transcendental in q ∗ and must be solved numerically for a given value of λ0 . The value of λ0 is not, however, arbitrary and must be selected so that q ∗ also satisfies the boundary conditions through the system dynamics: x∗ (t) = X0 +

t

−q ∗ (t) dt,

0

(38)

Xf = X 0 + 0

Tf

∗

−q (t) dt = 0 ⇒ X0 =

Tf

q ∗ (t) dt.

0

Therefore, the system of equations (37) and (38) must be solved simultaneously to obtain the optimal solutions for q ∗ and the costate λ0 . Correspondingly, noclosed-form optimal consumption solution exists for the sigmoidal welfare function, although widely available software allows for convenient calculation of the solution for arbitrary parameters. 3. Deterministic minimum consumption model results. A common set of values were chosen for the model parameters X0 , q¯, Tf , r, b, and σ and the solutions for each modeling technique were calculated and compared. Figure 3 contrasts the optimal consumption solutions for the hard consumption constraint and shifted welfare models, while Figure 4 compares their corresponding optimal stockpile levels.

270


80 Shifted Welfare Model Hard Constraint Model

70

Consumption, q(t)

60 50 40 30 20 10 0

0

0.5

1

1.5

2

2.5 3 Time, t

3.5

4

4.5

5

FIGURE 3. Comparison of optimal consumption strategy of two consumption floor approaches.

Examining these figures, it is clear that for an identical value of σ, the shifted welfare solution results in a slower initial consumption rate and a more gradual decrease in stockpile levels. Furthermore, the shifted welfare solution asymptotically approaches the consumption limit, while the hard constrained consumption solution takes the value of the minimum consumption at the corner condition and retains that value for the remainder of the solution interval. Both solutions are relatively similar over the problem horizon. The sigmoid welfare solution, however, differs significantly from the first two cases and is shown in Figures 5 and 6. The optimal consumption solution for the sigmoidal case is very nearly constant and the stockpile level decreases almost linearly. The optimal consumption level also remains well above the absolute consumption limit, q¯ = 10. Dictated by b, a sharper curvature of the sigmoid welfare function causes the optimal consumption solution to approach the value of q¯. While the shifted welfare modeling approach facilitates a closed-form solution, it is perhaps the most idealized model of the three alternatives, as welfare, price, and demand elasticity are undefined below the critical consumption level. The hard constraint model, on the other hand, admits no analytical solution but does correspond to the intuitive microeconomic interpretation of a discontinuous demand elasticity transition. In actual systems, however, this discontinuity might be unrealistic. Of the three models, the sigmoid welfare form is the most mathematically complicated but also retains the most intuitive microeconomic interpretation.


271

80 Shifted Welfare Model Hard Constraint Model

70

Stockpile Level, X(t)

60 50 40 30 20 10 0

0

0.5

1

1.5

2

2.5 3 Time, t

3.5

4

4.5

5

FIGURE 4. Comparison of optimal stockpile path of two consumption floor approaches.

Compared to existing models posed in the literature, the optimal consumption trajectories of the models with saturation effects are larger in magnitude, resulting in quicker exhaustion of resource stockpiles. The implication for policy planning is clear: the consideration of saturation effects necessitates larger initial stockpile levels if the problem horizon remains constant. For nonsaturation models, optimal consumption solutions and stockpile planning policies could easily result in undesirable real-world effects, such as shortages and widespread civil unrest. 3.1. Numerical solution fidelity. Two of the three saturation models considered admit no closed-form solution and require numerical techniques to arrive at an optimal solution. Fortunately, the numerical problems that must be solved are relatively simple boundary value problems, and commonly available mathematical software packages with robust zero-finding algorithms can be brought to bear on them. Solving these problems essentially reduces to the application of Newton’s method, and as long as the problems are well conditioned, accurate solutions are recoverable. For validation purposes, several of the optimal solutions were numerically perturbed to check for local optimality, which was confirmed. Because of the convexity of the problems, local optimality is also sufficient for global optimality. This result was also true for the uncertainty models explored in the sequel. 4. Minimum consumption models with saturation uncertainty. With the development of three alternative models for consumption saturation in optimal

272


16.1 16.08

Consumption, q(t)

16.06 16.04 16.02 16 15.98 15.96 15.94 15.92 15.9

0

0.5

1

1.5

2

2.5 3 Time, t

3.5

4

4.5

5

FIGURE 5. Optimal consumption path for sigmoidal welfare.

stockpiling problems, one research extension is to consider the impact of each form on the solutions to stochastic versions of the standard optimal consumption problem. In the previous section, deterministic models were examined, but building on prior work reported in Lloyd and Meyer [2014], the following sections contrast each model when the critical consumption level, either modeled as a hard or soft limit, is a random variable. Frequently, policy planners and decision makers are unable to precisely identify a critical level of consumption below which politically or societally unacceptable damage is done to an economy. Rather, these experts may provide a statistical distribution of possible minimum consumption limits. Understanding the implications of stochastic saturation effects in stockpiling decisions is an important extension to the prior deterministic results. One approach to estimating the optimal solution to the basic stockpiling problem with an uncertain minimum consumption level is to calculate the deterministic solution for several selected values of q¯. In this respect, the range of potential optimal solutions may be bounded by using the maximum and minimum values of the distribution of the minimum consumption parameter, assuming it has finite support. Furthermore, the expected value of the minimum consumption can be used in the deterministic analysis to achieve an optimal solution estimate. An example of this type of first-order analysis is presented in Figures 7 and 8. For the hard saturation limit model, the optimal consumption solution in equation (18)


273

80 70

Stockpile, X(t)

60 50 40 30 20 10 0

0

0.5

1

1.5

2

2.5 3 Time, t

3.5

4

4.5

5

FIGURE 6. Optimal stockpile solution for sigmoidal welfare.

was calculated using (21) for three values of q¯: a maximum value q¯ = qmax = 10, a minimum value q¯ = qmin = 0, and an expected value q¯ = qavg = 5. Examination of these figures suggests that the optimal stockpile solution is a smooth, monotonic function of the minimum consumption parameter, q¯. Furthermore, the optimal deterministic solution for the expected minimum consumption seems to be roughly approximated as the average of the maximum and minimum solutions. As further development will demonstrate, in the hard limit model the optimal solution for the uncertain minimum consumption problem (with an uniform distribution) is, in fact, the deterministic solution using the expectation of the consumption limit. In order to formulate the exact optimal solution for the stochastic minimum consumption problem, the formal PMP approach is again used. Following the PMP procedures employed in the previous section as well as those detailed in Lloyd and Meyer [2014], the necessary conditions for optimality are derived for the hard constraint model, the shifted welfare model, and the sigmoid welfare model, all with uncertain minimum consumption thresholds. Results from the solutions to the necessary conditions are examined in later sections. 4.1. Hard constraint model. To factor saturation uncertainty into the hard constraint model, the minimum consumption constraint, q¯, is treated as a random variable. In this work, the distribution of q¯ is assumed to be continuous and uniform. A uniform distribution is assumed for two reasons. First, subject matter experts

274


240 q

220

q

200

q

Consumption, q(t)

180

bar bar bar

=q =q =q

avg min max

160 140 120 100 80 60 40 20 0

0

1

3

2

5

4

Time, t FIGURE 7. Optimal consumption paths for mean and bounds of the uniform consumption limit distribution.

90 q 80

q q

Stockpile, X(t)

70

bar bar bar

=q =q =q

avg min max

60 50 40 30 20 10 0

0

1

2

3

4

5

Time, t FIGURE 8. Optimal stockpile levels for mean and bounds of the uniform consumption limit distribution.


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and policy makers are often unable to justify more complex statistical distributions in estimating saturation ranges. Typically, upper and lower bounds for critical consumption limits are more easily elicited, and analysts are usually comfortable with assuming a uniform probability distribution between these bounds. Second, the assumption of a more complicated distribution reduces or eliminates the analytical tractability of the necessary conditions. Taking these points into account, the consumption saturation limit is modeled as in equation (39). (39)

Pr (¯ q ) = U (¯ q | a, b) .

The continuous uniform probability distribution function is defined over the closed interval [a, b] according to (40). (40)

U (¯ q | a, b) =

1 . b−a

Similar to the deterministic case for the hard constraint model, a Hamiltonian is formed from the system dynamics, the consumption constraint, and the objective function. (41)

q − q) . H = e−r t U (q) − λq + ν (¯

In contrast to the deterministic case, where the application of the PMP necessary conditions requires the direct maximization of the Hamiltonian, the stochastic problem version requires the maximization of the expected value of the Hamiltonian, as in equation (42). Additional details on the following derivation are available in Lloyd and Meyer [2014], where an analogous procedure is followed but with the assumption of a truncated normal distribution for the consumption saturation limit: (42)

max E [H] . q ∈U

The only stochastic term present in (41) is the uncertain consumption saturation limit. Therefore, prior to taking the expectation of the Hamiltonian, the expectation of the consumption limit, ˆq¯, is calculated. The distribution of the random consumption limit is supported over the interval from 0 to qmax . Accordingly, the parameters for the uniform distribution PDF are defined as a = 0 and b = qm ax . ˆq¯ = E [¯ q] qm a x q¯U (¯ q | 0, qm ax ) d¯ q = 0

(43)

qm ax . = 2

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Substituting (43) and explicitly calculating the expected Hamiltonian yields E [H] = E −e−r t U (q) − λq + νC (q) = e−r t U (q) − λq + νE [C (q)]

(44)

= e−r t U (q) − λq + ν (E [¯ q ] − q) −r t = e U (q) − λq + ν ˆq¯ − q .

Using the expression for the expected Hamiltonian in (44), the solution procedure is similar to the deterministic case. The costate variable is determined to be a constant, and the corner condition is calculated from the complementarity condition and optimality condition. (45)

ˆt¯ = 1 ln λ0 . −r Uq ˆq¯

Furthermore, the optimal consumption profile prior to the corner condition is calculated from the optimality condition and combined with the expected saturation limit to form the general, piecewise optimal control solution. ⎧ 1 ⎪ ⎨ e−r t σ t ≤ ˆt¯ (46) q∗ = λ0 ⎪ ⎩ ˆt¯ ≤ t. ˆq¯, Finally, the endpoint constraints and system dynamics are used to derive the relationship between the costate, boundary conditions, and optimal control.

(47)

ˆq¯ σ σ X0 = Tf ˆq¯ + ln ˆq¯ λ0 + r r

1 1

(λ0 ) σ

− ˆq¯ .

As with the deterministic case, equation (47) is transcendental in λ0 and must be solved numerically to provide the complete solution for the optimal consumption path. In contrast to the deterministic solution, (47) depends on the expected value of the saturation limit, ˆq¯. 4.2. Shifted welfare function. Parallel to the hard constraint case discussed in the prior section, the analysis of the shifted welfare function with minimum consumption uncertainty begins with evaluating the expectation of the Hamiltonian. In this case, the random variable is the shift to the isoelastic welfare function, q¯. Contrasting the hard constraint model, there is no control constraint and


277

the Hamiltonian takes a less complex form. Calculating the expected Hamiltonian yields equation (48). E [H] = E e−r t U (q) − λq = E e−r t U (q) − λq

qm a x ¯)1−σ −r t (q − q = e d¯ q − λq 1−σ 0 q m a x −e−r t (q − q¯)2−σ − λq = 1−σ 2−σ (48)

=

σ2

0

−e (q − qm ax )2−σ − q 2−σ − λq. − 3σ + 2 −r t

Applying the optimality conditions to the expectation of the Hamiltonian yields an expression relating the optimal control, q ∗ , and the costate, λ0 , as a function of time. (49)

E [H]q

q =q ∗

=

−e−r t ∗ (q − qmax )1−σ − q ∗1−σ − λ0 = 0. 1−σ

Again, from the costate condition of PMP, the costate variable is shown to be a constant. Equation (49) is transcendental in q ∗ and must be solved simultaneously with the system dynamics equation and boundary conditions (19) to yield a numerical solution for the optimal control. For a variety of system parameters, this calculation was performed using MATLAB. 4.3. Sigmoid welfare. Analysis of the sigmoid welfare model under minimum consumption uncertainty proceeds in a comparable manner to the shifted utility case. Once more, the key analytical step is to take the expectation of the system Hamiltonian. The stochastic variable, q¯, is the magnitude of the shift of the “center” of the sigmoid welfare function. Performing this calculation yields equation (50).

(50)

E [H] = E e−r t U (q) − λq = E e−r t U (q) − λq q max 1 −r t = e d¯ q − λq 1 + e−b(q −¯q ) 0

ln ebq + 1 ln ebq −bq max + 1 −r t − − λq. = −e qmax + b b

278


Next, applying the optimality condition to the expected Hamiltonian yields another implicit function of the optimal control, q ∗ . Once more, the optimality condition must be solved simultaneously with the system dynamic and boundary conditions to yield the complete optimal solution as a function of time. (51)

E [H]q

−r t

q =q ∗

=e

ebq −bq max ebq − ebq −bq max + 1 ebq + 1

− λ0 = 0.

The numerical solutions for the hard constrained, shifted welfare, and sigmoid welfare models are further explored and contrasted in the Section 6. Another critical dimension of uncertainty analyzed in this study is the duration of potential embargo scenarios. This effect is examined in the next section. 5. Minimum consumption models with embargo length uncertainty. The topic of modeling optimal stockpiling problems with uncertain embargo periods has been analyzed in the recent literature, including in papers by Bahel [2011] and Lloyd and Meyer [2014]. In Lloyd and Meyer [2014], a hard minimum consumption constraint was added to the feature of a stochastic time horizon and the sytem model was solved for an optimal consumption path. Those results are reviewed here, but with the addition of the shifted welfare and sigmoid welfare models to examine the impact of these alternative saturation models on the optimal solutions. 5.1. Hard constraint model. For clarity, the constrained minimum consumption model with an uncertain embargo length is restated in equations (52) using the form introduced by Lloyd and Meyer [2014]. Furthermore, to illuminate the derivations of the solutions for the shifted welfare and sigmoidal welfare cases, the solution procedure is repeated as in Lloyd and Meyer [2014].

Tf

J(q) =

−e−r t (U (q(t))) dt,

0

x˙ = −q, −q ≤ −q, ¯ Pr (Tf ) = T E (Tf | μ, σ, a, b) , x (0) = X0 , (52)

x (Tf ) = Xf .

In this model, random effects emerge in the form of a stochastic, finite horizon which corresponds to policy makers’ uncertainty in the embargo duration


279

1 0.9 0.75 0.5 0.25 0.1

0.9 0.8

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

2.5 Time

3

3.5

4

4.5

5

FIGURE 9. The truncated exponential distribution for various values of θ with support on the interval [0, 5].

probability. The probability distribution of the embargo duration is assumed to obey the truncated exponential distribution, (53). (53)

T E (x | θ, b) =

θe−θ x . 1 − e−θ b

The parameters θ and b affect the curvature and upper limit of support for the truncated exponential distribution, respectively. Figure 9 shows the truncated exponential distribution for selected values of θ with b = 5. As shown in the figure, smaller values of θ result in a more uniform distribution shape. For stochastic horizon problems of this type, the procedure for applying the PMP conditions is similar to the one for stochastic saturation problems, where the analysis proceeds from the expected Hamiltonian. In this case, the integral objective is the stochastic component of the Hamiltonian, so calculating the expected integral objective is sufficient to form the expected Hamiltonian. max J ⇒ max E [J] .

(54)

q ∈U

q ∈U

The details of the expectation calculation are treated in detail in Lloyd and Meyer [2014], but the ultimate result is shown in (55). (55)

E [J] = = 0

Tf

e−(r +θ )t − e−r t U (q(t)) dt. −1 + e−θ T f

280


Using equation (55), an expression for the expected Hamiltonian, EH, can be formed (56) and used to determine the optimality condition for consumption (57).

(56)

E [H] =

e−(r +θ )t − e−r t U (q(t)) − λq + ν (¯ q − q) , −1 + e−θ T f 1 − e−θ T f λ0 = 0. + Uq (¯ q)

(57)

−(r +θ ) t¯

e

−r t¯−θ T f

−e

Solving (57) for the optimal consumption path prior to reaching the saturation limit yields (58):

−λ0 1 − e−θ T f ∗ −1 q = Uq (58) . e−(r +θ )t − e−r t−θ T f Application of the complementarity and optimality conditions allows for the solution of the corner condition, t¯, in terms of the optimal control and the costate, λ0 , but the derivation is not included here. Combining the unsaturated optimal consumption path with the corner condition yields the full optimal control solution in terms of the costate. ⎧

1 ⎪ ⎨ e−(r +θ )t − e−r t−θ T f σ t ≤ t¯ (59) q∗ = −λ0 1 − e−θ T f ⎪ ⎩ q¯, t¯ ≤ t. Substituting the optimal control solution provided above into the system dynamics equation (19) and adding the boundary conditions results in an integral equation that must be satisfied by the costate, which the corner conditions and, accordingly, integrand limits depend on. Solving (60) numerically for the costate yields the solution for the optimal consumption path. t¯ (60)

X0 = 0

e−r t−θ T f − e−(r +θ )t −λ0 1 − e−θ T f

σ1

dt +

t¯

Tf

q¯dt.

5.2. Shifted welfare function. For the shifted welfare model, the expression for the expected Hamiltonian from the hard constraint derivation may be reused in


281

a modified form by eliminating the explicit constraint term. This yields the expected Hamiltonian for the shifted utility model (61). (61)

E [H] =

e−(r +θ )t − e−r t U (q(t)) − λq. −1 + e−θ T f

Taking the partial derivative of the expected Hamiltonian with respect to consumption yields the optimality condition. (62)

E [H]q =

e−(r +θ )t − e−r t−θ T f Uq (q ∗ ) − λ0 = 0. −1 + e−θ T f

Solving the optimality condition for the optimal consumption path while maintaining generality in terms of the utility function produces a utility-dependent expression for the optimal consumption path.

−θ T f 1 − e −λ 0 (63) . q ∗ = Uq−1 e−(r +θ )t − e−r t−θ T f For the shifted welfare function, the inverse function for Uq is shown in equation (64). −1 −1 Uq (y) = (y) σ + q¯. (64) Substituting (64) into (63) results in the optimal stockpile consumption for the shifted welfare case.

σ1 −r t−θ T f −(r +θ )t − e e (65) + q¯. q∗ = −λ0 1 − e−θ T f As in the previous solutions, the optimal consumption path depends on the costate, which is an unknown constant. To calculate the full solution, the optimal consumption equation must be solved simultaneously with the integrated system dynamics and boundary conditions.

σ1 T f −r t−θ T f Tf − e−(r +θ )t e X0 = (66) dt + q¯dt. −λ0 1 − e−θ T f 0 0 The coupled implicit equations (65) and (66) cannot be solved analytically, and therefore numerical methods must be applied to determine the exact optimal consumption solution. This is further explored in the results discussion.

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5.3. Sigmoid welfare. To calculate the optimal consumption solution for the sigmoid welfare model with an uncertain embargo duration, the first step is identical to the shifted utility case. Substituting the sigmoid welfare function into the previous expression for the expected Hamiltonian yields the expected Hamiltonian for a sigmoid welfare function. E [H] = =

(67)

e−r t − e−(r +θ )t U (q(t)) − λq −1 + e−θ T f e−r t − e−(r +θ )t 1 − λq. −1 + e−θ T f 1 + e−b(q −¯q )

Taking the partial derivative of the expected Hamiltonian results in the optimality condition for the consumption, q. Similar to the sigmoid welfare problem with an uncertain minimum consumption, the optimality condition cannot be solved analytically for the optimal consumption, and must instead be combined with the integrated dynamics and initial/final stockpile levels to yield a numerical problem. ∗

(68)

E [H]q =

be−b(q −¯q ) e−(r +θ )t − e−r t−θ T f 2 − λ0 = 0. −1 + e−θ T f 1 + e−b(q ∗ −¯q )

As with the previous models, the numerical problem can be solved via zero finding algorithms. 6. Stochastic model results. 6.1. Consumption saturation uncertainty model comparisons. Applying the results developed for modeling minimum consumption uncertainty in Section 4 to all three saturation models, their differences can be explored. As with the previous comparisons of results assuming a deterministic saturation limit, the common model parameters were selected to be X0 = 80, Tf = 5, r = 0.03, and qmax = 10, while the welfare function-specific parameters were designated to be σ = 0.1 and b = 1. Plotting the numerical solutions for the optimal consumption paths calculated using the hard constraint, shifted welfare, and sigmoid welfare models produced Figure 10. In the hard constraint case, the optimal consumption path converges to the expected minimum consumption level after initially taking high values. For the shifted welfare saturation model, a significant departure from the result of the hard limit is evident, with the optimal consumption path taking a much lower initial value and only gradually approaching the expectation of the minimum consumption. This difference is more pronounced than in the deterministic saturation limit case, with the shifted welfare model yielding an initially conservative consumption level but a


283

200 Hard Constraint Model Shifted Welfare Model Sigmoid Welfare Model

180

Consumption, q(t)

160 140 120 100 80 60 40 20 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time, t FIGURE 10. Optimal consumption solution comparison for three modeling approaches with minimum consumption uncertainty.

less conservative level in the later stages of the scenario. Of the three models, the sigmoid welfare approach results in the least dynamic optimal consumption over the problem horizon. A plot of the optimal stockpile levels in Figure 11 bears out the same trends as the optimal consumption solutions. Clearly, the hard minimum consumption limit results in the most quickly depleted stockpile, followed by the shifted welfare model and then the sigmoidal welfare model. Of course, within their model context, all three responses are optimal in the presence of a uniformly distributed random minimum consumption. One advantage to both the shifted welfare and sigmoidal welfare models is that their optimum consumption paths never drop below the maximum upper support of the uniform distribution of consumption uncertainty. Thus, these models, in addition to satisfying optimality, are mathematically consistent with an actual scenario where the deterministic realization of the random minimum consumption happens to be above its expected value. 6.2. Embargo uncertainty model comparisons. As with an uncertain minimum consumption level, the three saturation models can be contrasted assuming an uncertain embargo duration. Parameter settings for this set of modeling runs were X0 = 80, Tf = 5, r = 0.03, θ = 0.01, and qbar = 10, while the welfare functionspecific parameters were selected to be σ = 0.1, and b = 1. Interpreting these parameters, the maximum embargo length for the scenarios was 5 units, probabilistically

284



70

Stockpile, X(t)

60 50 40 30 20 10 0

0

0.5

1

1.5

2

2.5 3 Time, t

3.5

4

4.5

5

FIGURE 11. Optimal stockpile solution comparison for three modeling approaches with minimum consumption uncertainty.

distributed over an interval from 0 to 5 units. For the selected value of θ, the distribution of the embargo length probability is fairly even, approaching a uniform distribution. The value for the truncated exponential distribution was selected to emphasize differences in the optimal solutions of each model. Figure 12 shows the optimal consumption solutions for the three saturation models. Inspection of this figure leads to a few interesting observations. Similar to the results for the deterministic time horizon problems scrutinized in Section 3, the hard constraint model is less conservative than the shifted welfare model, with initial consumption for the former model being higher and converging more quickly to the minimum consumption limit. Relative to the deterministic time horizon model, however, both optimal consumption paths are initially steeper and converge more quickly to the minimum consumption, depleting the stockpile more quickly. This result corresponds to intuition, as the possibility that an embargo might end before the maximum specified time dictates that the optimal consumption strategy is to consume the resource slightly more quickly than if the embargo is definitively long. A similar trend holds for the sigmoidal welfare saturation model, which exhibits characteristics of the deterministic embargo duration case in its nearly constant consumption solution. In contrast to the deterministic embargo, case, however, the initial optimal consumption level for the sigmoidal welfare is higher, and then starts to taper off quickly at the limit of the embargo duration. This behavior is similar


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100 90 Consumption, q(t)

80 70 60 50 40 30 20 10 0

0

0.5

1

1.5

2

2.5 3 Time, t

3.5

4

4.5

5

FIGURE 12. Optimal consumption paths for three alternative models with embargo length uncertainty. 80 Hard Constraint Model Shifted Welfare Model Sigmoid Welfare Model

70

Stockpile, X(t)

60 50 40 30 20 10 0

0

0.5

1

1.5

2

2.5 3 Time, t

3.5

4

4.5

5

FIGURE 13. Optimal stockpile levels for three alternative models with embargo length uncertainty.

to the initially less conservative consumption paths for the shifted welfare and hard constraint models. Plotting the corresponding optimal stockpile levels for the three modeling approaches produces Figure 13. These figures are consistent with the optimal consumption paths and the possibility of a shorter embargo based on the duration uncertainty. The curvature of the shifted welfare and hard constraint models is more pronounced than for the

286


deterministic embargo case, as the stockpiles are initially depleted at a higher rate. Due to the relative uniformity of the sigmoidal welfare model, the optimal stockpile curve for this case is relatively linear.

7. Conclusions. Three approaches to modeling a deterministic minimum consumption limit for a basic optimal consumption problem were developed and solved using a maximum principle solution technique. While the solutions of the three models are not equivalent, each model offers distinct qualities. While a hard constraint on the minimum consumption level is the most traditional optimal control approach, a shifted welfare approach results in a closed-form analytical solution. A sigmoidal welfare function technique is argued to offer the most realistic model of a consumption limit, but can only be solved via purely numerical means. After determining the optimal solutions to the deterministic versions of these three saturation models, the complications posed by a stochastic minimum consumption limit or stochastic embargo length were explored for each model. These model features were previously examined in other works, but not in combination with various approaches to modeling control saturation. For both the stochastic consumption limit and stochastic embargo period, the accompanying optimal consumption and stockpiling results were consistent with intuition. Of the three minimum consumption models considered, the sigmoidal welfare model is perhaps the least orthodox. Despite this, it is proposed as the most intuitive and least abstracted modeling approach. When considered as a macroeconomic tool for industrial base policy making, the sigmoidal welfare model has several attractive qualities. Modeling raw material consumption by a nation’s industrial base, it is theoretically attractive to presume that the welfare derived from critical material usage sharply decreases over a particular range of consumption, but that the welfare gained by lower consumption levels does not inexorably decrease to negative infinity. As an illustrative example, the societal gains derived from automobile production, which requires aluminum as an input, might drop significantly below a certain level of aluminum consumption because manufacturing can no longer be carried out at a large scale over certain critical feedstock input ranges. The derivation sections of this paper link this intuition to concrete microeconomic theory, demonstrating that sigmoidal welfare functions for raw material consumption are mathematically consistent with sigmoidal short-run production functions and isoleastic welfare functions for goods consumption. It is posited that a sigmoidal function for short-run production is a realistic and adaptable modeling approach for production, frequently reflective of empirical data. It should also be noted that other researchers have considered the evidence for sigmoidal representations of utility even at the individual consumer level (e.g., van Praag and Frijters [1999]). Even if a sigmoidal welfare approach might be the most intuitively and empirically attractive for industrial base resource consumption


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models, the shifted welfare and hard constraint models closely adhere to previous methods and both serve as reasonable objective formulations. The hard constraint model has an appealing interpretation as a discontinuous change in demand elasticity, while the shifted welfare model is analytically tractable. More research is needed to determine the ultimate accuracy of one model compared to another. These results are offered as a rigorous foundation for more complicated resource consumption models requiring constraints on the minimum consumption. Furthermore, they eliminate the previous limitations in the literature of zero-valued minimum consumption assumptions. Future work should seek to formally impose similar saturation limits on other quantities of interest, such as stockpile levels or other economic inputs. Any efforts along these lines, however, will result in the need for more sophisticated numerical solution techniques and will not allow analytical solutions. In this light, future research should seek to implement more sophisticated modern control methods of calculating optimal solutions in the presence of complicated stochastic effects, including correlated random variables, multiple random effects, and nonstandard uncertainty distributions. Acknowledgment. Justin Lloyd and Dr. Gerard Meyer thank Dr. Danielle Tarraf of Johns Hopkins University for technical insights and assistance. REFERENCES U. Ascher and F. Wan [1980], Numerical Solutions for Maximum Sustainable Consumption Growth with a Multi-Grade Exhaustible Resource, SIAM J. Sci. Statist. Comput. 1, 160–172. M. Athans and P. Falb [1966], Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York. E. Bahel [2011], Optimal Management of Strategic Reserves of Nonrenewable Natural Resources, J. Environ. Econ. Manage. 61, 267–280. A. Bryson and Y.-C. Ho [1969], Applied Optimal Control, Ginn and Company, Waltham, MA. A. Cournot [1838], Recherches sur les principes mathematiques de la theorie des richesses, Hachette, Paris. P. Dasgupta and G. Heal [1974], The Optimal Depletion of Exhaustible Resources. In The Review of Economic Studies. Vol. 41 of Symposium on the Economics of Exhaustible Resources, Oxford University Press, Edinburgh, pp. 3–28. G. Gaudet [2007], Natural Resource Economics Under the Rule of Hotelling, Canad. J. Econ. 40, 1033–1059. G. Gaudet and P. Lasserre [2011], The Efficient Use of Multiple Sources of a Nonrenewable Resource Under Supply Cost Uncertainty, Int. Econ. Rev. 52, 245–258. A. Hillman and N. V. Long [1983], Pricing and Depletion of an Exhaustible Resource When There Is Anticipation of a Trade Disruption, Quart. J. Econ. 98, 215–233. H. Hotelling [1931], The Economics of Exhaustible Resources, J. Polit. Econ. 39, 137–175. H. Leland [1968], Saving and Uncertainty: The Precautionary Demand for Saving, Quart. J. Econ. 82, 465–473. R. Lindsey [1990], Supply Management with Intermittent Trade Disruptions When the Probabilities Are Not Fully Known, J. Econ. Dyn. Contr. 14, 73–95.

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