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The Global Environment, Natural Resources, and Economic Growth$

Alfred Greiner and Willi Semmler

Print publication date: 2008

Print ISBN-13: 9780195328233

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780195328233.001.0001

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(p.194) Appendix: Three Useful Theorems from Dynamic Optimization

(p.194) Appendix: Three Useful Theorems from Dynamic Optimization

Source:
The Global Environment, Natural Resources, and Economic Growth
Publisher:
Oxford University Press

In this book, we have presumed that economic agents behave intertemporally and perform dynamic optimization. In this appendix, we present some basics of the method of dynamic optimization using Pontryagin's maximum principle and the Hamiltonian.

Let an intertemporal optimization problem be given by

(A.1)
max u ( t ) W ( x ( 0 ) , 0 ) , W ( ) 0 e - ρ t F ( x ( t ) , u ( t ) ) d t ,
subject to
(A.2)
d x ( t ) d t x ˙ ( t ) = f ( x ( t ) , u ( t ) ) , x ( 0 ) = x 0 ,
with x(t) ∈ ℝn the vector of state variables at time t and u(t) ∈ Ω ∈ ℝm the vector of control variables at time t and F : ℝn × ℝm → ℝ and f : ℝn × ℝm → ℝn. ρ is the discount rate and e t is the discount factor.

F(x(t), u(t)), fi(x(t), u(t)), and ∂fi(x(t), u(t))/∂xj(t), ∂F(x(t), u(t))/∂xj(t) are continuous with respect to all n + m variables for i, j = 1, …, n Further, u(t) is said to be admissible if it is a piecewise continuous function on [0,∞) with u(t) ∈ Ω.

Define the current-value Hamiltonian ℋ(x(t)), u(t), λ(t), λ0) as follows:

(A.3)
( x ( t ) , u ( t ) , λ ( t ) , λ 0 ) λ 0 F ( x ( t ) , u ( t ) ) + λ ( t ) f ( x ( t ) , u ( t ) ) ,
with λ0 ∈ ℝ a constant scalar and λ(t) ∈ ℝn the vector of co-state variables or shadow prices. λj(t) gives the change in the optimal objective functional W 0 resulting from an increment in the state variable xj(t). If xj(t) is a capital stock, λj(t) gives the marginal value of capital at time t. Assume that there exists a solution for (A.1) subject to (A.2). Then, we have the following theorem.

Theorem A.1 Let u (t) be an admissible control and x 0(t) is the trajectory belonging to u 0(t). For u 0(t) to be optimal, it is necessary that there (p.195) exists a continuous vector function λ(t) = (λ1(t),…,λn(t) with piecewise continuous derivatives and a constant scalar λ0 such that

  1. a. λ(t) and x 0(t) are solutions of the canonical system

    x ˙ 0 ( t ) = λ H ( x 0 ( t ) , u 0 ( t ) , λ ( t ) , λ 0 ) , λ ˙ ( t ) = ρ λ ( t ) - x H ( x 0 ( t ) , u 0 ( t ) , λ ( t ) , λ 0 ) .

  2. b. For all t ∈ [0,∞) where u 0(t) is continuous, the following inequality must hold: H(x 0(t), u 0(t), λ(t), λ0) ≥ H(x 0(t), u(t), λ(t), λ0),

  3. c.0, λ(t) ≠ (0,0) and λ0 = 1 or λ0 = 0.

Remarks:

  1. 1. If the maximum with respect to u(t) is in the interior of Ω, ∂ℋ(·)/∂u(t) can be used as a necessary condition for a local maximum of ℋ(·).

  2. 2. It is implicitly assumed that the objective functional (A.1) takes on a finite value, that is, 0 e - ρ t F ( x 0 ( t ) , u 0 ( t ) ) < . if x 0 and u 0 grow without an upper bound F(·) must not grow faster than ρ.

Theorem A.1 provides only necessary conditions. The next theorem gives sufficient conditions.

Theorem A.2 If the Hamiltonian with λ0 = 1 is concave in (x(t), u(t)) jointly and if the transversality condition limt→∞ e -ρ tλ(t)(x(t - x 0(t)) ≥ 0 holds, conditions a and b from theorem A.1 are also sufficient for an optimum. If the Hamiltonian is strictly concave in (x(t), u(t)) the solution is unique.

Remarks:

  1. 1. If the state and co-state variables are positive the transversality condition can be written as stated in the foregoing chapters, that is, as limt→∞ e -ρtλ(t) x 0(t) = 0.1

  2. 2. Given some technical conditions, it can be shown that the transversality condition is also a necessary condition.

Theorem A.2 requires joint concavity of the current-value Hamiltonian in the control and state variables. A less restrictive theorem is the following.

Theorem A.3 If the maximized Hamiltonian

H 0 ( x ( t ) , λ ( t ) , λ 0 ) = max u ( t ) Ω H ( x ( t ) , λ ( t ) , λ 0 )
(p.196) with λ0 = 1 is concave in x(t) and if the transversality condition limt→∞ e -ρtλ(t)(x(t) - x 0(t)) ≥ 0 holds, conditions a and b from theorem A.1 are also sufficient for an optimum. If the maximized Hamiltonian0(x(t), λ(t), λ0) is strictly concave in x(t) for all t, x 0(t) is unique (but not necessarily u 0(t)).

Because the joint concavity of ℋ(x(t), u(t), λ(t), λ0) with respect to (x(t), u(t)) implies concavity of ℋ0(x(t), λ(t, λ0 with respect to x(t), but the reverse does not necessarily hold, theorem A.3 may be applicable where theorem A.2 cannot be applied.

The three theorems demonstrate how optimal control theory can be applied to solve dynamic optimization problems. The main role is played by the Hamiltonian (A.3). It should be noted that in most economic applications, as in this book, interior solutions are optimal so that ∂ℋ(·)/∂u(t) = 0 can be presumed. For further reading and more details concerning optimal control theory, we refer to the books by Feichtinger and Hartl (1986) or Seierstad and Sydsaeter (1987).

Notes:

(1) Note that in the book we did not indicate optimal values by0.