Form the current-value Hamiltonian
H(Q, p, λ) =ψ(p) +π(p) +λ[−f(p)], (10.39) where λsatisfies the relation
λ˙ =ρλ, λ(T)≥0, λ(T)Q(T) = 0, (10.40) which implies
λ(t) =
⎧⎪
⎨
⎪⎩
0 ifQ(T)≥0 is not binding, λ(T)eρ(t−T) ifQ(T)≥0 is binding.
(10.41)
To obtain the optimal control, the Hamiltonian maximizing condition, which is both necessary and sufficient in this case (see Theorem 2.1), is
∂H
∂p =ψ+π−λf = (p−λ)f−g = 0. (10.42) To show that the solution s(λ) for p of (10.42) actually maximizes the Hamiltonian, it is enough to show that the second derivative of the Hamiltonian is negative ats(λ).Differentiating (10.42) gives
∂2H
∂p2 =f−g+ (p−λ)f. Using (10.42) we have
∂2H
∂p2 =f−g+ g
ff. (10.43)
From the definition of Gin (10.32), we can obtain G = fg−gf
f3 , which, when substituted into (10.43), gives
∂2H
∂p2 =f−Gf2. (10.44)
328 10. Applications to Natural Resources The right-hand side of (10.44) is strictly negative because f < 0, and G ≥ 0 by assumption. We remark that ˆp = s(0) using (10.35) and (10.42), and hence the second-order condition for ˆp of (10.35) to give the maximum of H is verified. In Exercise10.17 you are asked to show thats(λ) increases from ˆpasλincreases from 0, and thats(λ) = ¯pwhen λ= ¯p−G(0).
Case 1: The constraintQ(T)≥0 is not binding. From (10.41),λ(t)≡0 so that from (10.42) and (10.35),
p∗ = ˆp. (10.45)
With this value, the total consumption of the resource is T f(ˆp), which must be ≤Q0 so that the constraint Q(T)≥0 is not binding. Hence,
T f(ˆp)≤Q0 (10.46)
characterizes Case 1 and its solution is given in (10.45).
Case 2: T f(ˆp) > Q0 so that the constraint Q(T) ≥0 is binding. Ob- taining the solution requires finding a value ofλ(T) such that
t∗
0 f(s[λ(T)eρ(t−T)])dt=Q0, (10.47) where
t∗ = min
T, T+1 pln
p¯−G(0) λ(T)
. (10.48)
The timet∗ , if it is less thanT,is the time at whichs[λ(T)eρ(t∗−T)] = ¯p.
From Exercise 10.17,
λ(T)eρ(t∗−T)= ¯p−G(0) (10.49) which, when solved for t∗,gives the second argument of (10.48).
One method to obtain the optimal solution is to define ¯T as the longest time horizon during which the resource can be optimally used.
Such a ¯T must satisfy
λ( ¯T) = ¯p−G(0), and therefore,
T¯ 0 f
' s
%{p¯−G(0)}eρ(t−T¯)
&(
dt=Q0, (10.50)
10.3. An Exhaustible Resource Model 329 which is a transcendental equation for ¯T . We now have two subcases.
Subcase 2a: T ≥T .¯ The optimal control is
p∗(t) =
⎧⎪
⎨
⎪⎩ s
'{p¯−G(0)}eρ(t−T¯) (
fort≤T ,¯
¯
p fort >T .¯
(10.51)
Clearly in this subcase,t∗= ¯T and
λ(T) = [¯p−G(0)]e−ρ( ¯T−T). A sketch of (10.51) is shown in Fig.10.8.
Figure 10.8: Optimal price trajectory forT ≥T¯ Subcase 2b: T <T .¯ Here the optimal price trajectory is
p∗(t) =s
%
λ(T)eρ(t−T)
&
, (10.52)
where λ(T) is to be obtained from the transcendental equation T
0 f '
s
%
λ(T)eρ(t−T)
&(
dt=Q0. (10.53) A sketch of (10.52) is shown in Fig.10.9.
In Exercise10.18 you are given specific functions for the exhaustible resource model and asked to work out explicit optimal price trajectories for the model.
330 10. Applications to Natural Resources
Figure 10.9: Optimal price trajectory forT <T¯
Exercises for Chapter 10
E 10.1 As an alternate derivation for the turnpike level ¯x of (10.12), use the maximum principle to obtain the optimal long-run stationary equilibrium triple {x,¯ u,¯ λ¯}.
E 10.2 Prove that ¯x ∈ (xb, X) and ¯u < U, where ¯x is the solution of (10.12) and xb is given in (10.4).
E 10.3 Show that ¯xobtained from (10.12) decreases asρincreases. Fur- thermore, derive the relation (10.15).
E 10.4 Obtain the turnpike level ¯xof (10.12) for the special caseg(x) = x(1−x), p= 2, c=q= 1,andρ= 0.1.
E 10.5 Perform the following:
(a) For the Schaefer model withg(x) =rx(1−x/X) andq = 1,derive the formula for the turnpike level ¯x of (10.12).
(b) Allen (1973) and Clark (1976) estimated the parameters of the Schaefer model for the Antarctic fin-whale population as follows:
r = 0.08, X = 400,000 whales, and xb = 40,000. Solve for ¯x for ρ= 0,0.10,and ∞.
E 10.6 Obtainπ(x) from (10.13) and use it in (10.12) to derive (10.14).
Exercises for Chapter 10 331 E 10.7 Let π(x, u) = [p−c(x)](qux) in (10.3), where c(x) is a differ- entiable, decreasing, and convex function. Derive an expression for ¯x satisfying an equation corresponding to (10.12).
E 10.8 Show that extinction is optimal if∞> p≥c(0) andρ >2g(0) in Exercise 10.7.
Hint: Use the generalized mean value theorem.
E 10.9 Let the constant pricep in Exercise 10.7be replaced by a time dependent price p(t) which is differentiable with respect to t.Derive the equation ¯x corresponding to (10.12) for this nonautonomous problem.
Furthermore, find the turnpike level ¯x(t) satisfying the derived equation.
E 10.10 Letπ(x, u) of Exercise10.7 be
π(x, u) = [p−c(x)](qux) +V(x),
where V(x) with V(x) > 0 is the conservation value function, which measures the value to society of having a large fish stock. By deriving the analogue to (10.12), show that the new ¯x is larger than the ¯x in Exercise 10.7.
E 10.11 When c(x) = 0 in Exercise 10.9, show that the analogue to (10.12) reduces to
g(x) =ρ−p˙ p. Give an economic interpretation of this equation.
E 10.12 Find λ(t), t ∈ [0,∞), for the infinite horizon model of Sect.10.2.2.
E 10.13 Derive the second term inside the brackets of (10.27) by com- puting e−ρT(p−c) imp[¯x(T),0;T].
E 10.14 Derive (10.29) by using the first-order condition for maximizing J∗(T) of (10.27) with respect toT. Similarly, derive (10.31).
E 10.15 Forest Fertilization Model(N¨aslund1969). Consider a forestry model in which thinning is not allowed, and the forest is to be clearcut
332 10. Applications to Natural Resources at a fixed time T. Suppose v(t) ≥0 is the rate of fertilization at time t, so that the growth equation is
˙
x=r(X−x) +f(v, t), x(0) =x0,
where x is the volume of timber,r and X are positive constants, and f is an increasing, differentiable, concave function ofv.The objective is to maximize
J =−c T
0 e−ρtv(t)dt+e−ρTpx(T),
wherep is the price of a unit of timber andcis the unit cost of fertiliza- tion.
(a) Show that the optimal control v∗(t) is given by solving the equation
∂f
∂v = c
pe−(ρ+r)(t−T).
Check that the second order condition for a maximum holds for this v∗(t).
(b) If f(v) = (1 +t) ln(1 +v),then find explicitly the optimal control v∗(t) under the assumption that p/c > e(ρ+r)T. Show further that v∗(t) is increasing and convex int∈[0, T].
E 10.16 Show that ˆpdefined in (10.35) satisfiesp≤pˆ≤pm.
E 10.17 Show that s(λ),the solution of (10.39), increases from ˆp as λ increases from 0. Also show thats(λ) = ¯p,when λ= ¯p−G(0).
E 10.18 For the model of Sect.10.3, assume
f(p) =
⎧⎪
⎨
⎪⎩
¯
p−p forp≤p,¯ 0 forp >p,¯ G(q) =q2.
(a) Show that p∗ = 2¯p/3 ifT ≤3Q0/p.¯
Exercises for Chapter 10 333 (b) Show that ¯T satisfies ¯T +e−ρT¯/ρ= 1/ρ+ 3Q0/¯p.Moreover,
p∗(t) =
⎧⎪
⎨
⎪⎩
¯ p '
eρ(t−T¯)+ 2 (
/3 if t≤T ,¯
¯
p ift >T ,¯
forT ≥T ,¯ and
p∗(t) = 2¯p
3 + ρ[¯pT −3Q0] 3e−ρt(eρT −1) forT >T .¯
Chapter 11
Applications to Economics
Optimal control theory has been extensively applied to the solution of economic problems since the early papers that appeared in Shell (1967) and the works of Arrow (1968) and Shell (1969). The field is too vast to be surveyed in detail here, however. Several books in the area are:
Arrow and Kurz (1970), Hadley and Kemp (1971), Takayama (1974), Lesourne and Leban (1982), Seierstad and Sydsổter (1987), Feichtinger (1988), L´eonard and Long (1992), Van Hilten et al. (1993), Kamien and Schwartz (1992), and Dockner et al. (2000), and Weber (2011). We content ourselves with the discussion of three simple kinds of models.
In Sect.11.1, two capital accumulation or economic growth models are presented. In Sect.11.2, we formulate and solve an epidemic control model. Finally, in Sect.11.3we discuss a pollution control model.