Optimal Control Prof. Lutz Hendricks Econ720

October 3, 2017

1 / 45

Topics

Optimal control is a method for solving dynamic optimization problems in continuous time.

2 / 45

Example: Growth Model A household chooses optimal consumption to Z T

max

e−ρt u[c(t)]dt

(1)

0

subject to ˙ = rk(t) − c(t) k(t)

(2)

c (t) ∈ [0,¯c]

(3)

k(0) = k0 ,given

(4)

k(T) ≥ 0

(5)

3 / 45

Generic Optimal control problem Choose functions of time c (t) and k (t) so as to Z T

max

v[k(t), c(t), t]dt

(6)

0

Constraints: 1. Law of motion of the state variable k (t): ˙ = g[k(t), c(t), t] k(t)

(7)

2. Feasible set for control variable c (t): c (t) ∈ Y (t)

(8)

3. Boundary conditions, such as: k(0) = k0 ,given

(9)

k(T) ≥ kT

(10) 4 / 45

Generic Optimal control problem

I

c and k can be vectors.

I

Y (t) is a compact, nonempty set.

I

T could be infinite. I

I

Then the boundary conditions change

Important: the state cannot jump; the control can.

5 / 45

A Recipe for Solving Optimal Control Problems

A Recipe Step1. Write down the Hamiltonian H(t) = v(k, c, t) + µ(t)g(k, c, t) | {z }

(11)

˙ k(t)

µ is essentially a Lagrange multiplier (called a co-state).

Step 2. Derive the first order conditions which are necessary for an optimum: ∂ H/∂ c = 0

(12)

∂ H/∂ k = −µ˙

(13)

7 / 45

A Recipe

Step 3. Impose the transversality condition: I

I

for finite horizon: µ (T) = 0

(14)

lim H(t) = 0

(15)

for infinite horizon: t→∞

This depends on the terminal condition (see below).

8 / 45

A Recipe

Step 4. A solution is the a set of functions [c(t), k(t), µ(t)] which satisfy I

the FOCs

I

the law of motion for the state

I

the boundary / transversality conditions

9 / 45

Intuition ∂ H/∂ c = 0: Maximize Hamiltonian w.r.to control. I

v (k, c, t) picks up current utility of c

I

µ (t) is marginal value of additional “future” k.

I

µ (t) g (k, c, t) picks up change in continuation value (change in k˙ times value of future k)

˙ ∂ H/∂ k = −µ: I

˙ think of this as [∂ H/∂ k] /µ = −µ/µ

I

∂ H/∂ k is the value of additional k

I

[∂ H/∂ k] /µ is like a rate of return (give up c now, how much future value do you get?) ˙ µ/µ is the growth rate of marginal utility

I

10 / 45

Example: Growth Model

Z ∞

max

e−ρt u (c (t)) dt

(16)

0

subject to k˙ (t) = f (k (t)) − c (t) − δ k (t)

(17)

k (0) given

(18)

11 / 45

Growth Model: Hamiltonian

H (k, c, µ) = e−ρt u (c (t)) + µ (t) [f (k (t)) − c (t) − δ k (t)]

(19)

Necessary conditions: Hc = e−ρt u0 (c) − µ = 0 Hk = µ f 0 (k) − δ = −µ˙

12 / 45

Growth Model Substitute out the co-state: µ˙ = e−ρt u00 (c) c˙ − ρ µ

c˙ =

µ˙ + ρ µ e−ρt u00 (c)

= − f 0 (k) − δ − ρ

(20)

(21) u0 (c) u00 (c)

(22)

Solution: ct , kt that solve Euler equation and resource constraint, plus boundary conditions.

13 / 45

Details

First order conditions are necessary, not sufficient. They are necessary only if we assume that 1. a continuous, interior solution exists; 2. the objective function v and the constraint function g are continuously differentiable. Acemoglu (2009), ch. 7, offers some insight into why the FOCs are necessary.

14 / 45

Details If there are multiple states and controls, simply write down one FOC for each separately: δ H/δ ci = 0 ∂ H/∂ kj = −µ˙ j

There is a large variety of cases depending on the length of the horizon (finite or infinite) and the kinds of boundary conditions. I

Each has its transversality condition (see Leonard and Van Long 1992).

15 / 45

Next steps

Typical useful next things to do: 1. Eliminate µ from the system. Obtain two differential equations in (c, k) . 2. Find the steady state by imposing c˙ = k˙ = 0.

16 / 45

Sufficient conditions

First-order conditions are sufficient, if the programming problem is concave. This can be checked in various ways.

17 / 45

Sufficient conditions I

The objective function and the constraints are concave functions of the controls and the states. I

The co-state must be positive.

This condition is easy to check, but very stringent. In the growth model: I

u (c) is concave in c (and, trivially, k)

I

f (k) − δ k − c is concave in c and k

I

µ = u0 (c) > 0

18 / 45

Sufficient Conditions II

(Mangasarian) First-order conditions are sufficient, if the Hamiltonian is concave in controls and states, where the co-state is evaluated at the optimal level (and held fixed). This, too is very stringent.

19 / 45

In the growth model

∂ H/∂ c = u0 (c) − µ ∂ H/∂ k = µ [f 0 (k) − δ ] ∂ 2 H/∂ c2 = u00 (c) < 0 ∂ 2 H/∂ k2 = µf 00 (k) < 0 ∂ 2 H/∂ c∂ k = 0 Therefore: weak joint concavity (because we know that µ > 0)

20 / 45

Sufficient Conditions III Arrow and Kurz (1970) I

First-order conditions are sufficient, if the maximized Hamiltonian is concave in the states.

I

If the maximized Hamiltonian is strictly concave in the states, the optimal path is unique.

Maximized Hamiltonian: Substitute controls out, so that the Hamiltonian is only a function of the states. This is less stringent and by far the most useful set of sufficient conditions.

21 / 45

In the growth model

Optimal consumption obeys u0 (c) = µ or c = u0−1 (µ) Maximized Hamiltonian: ˆ = u u0−1 (µ) + µ f (k) − δ k − u0−1 (µ) H

(23)

ˆ k > 0 and ∂ 2 H/∂ k2 = µf 00 (k) < 0. We have ∂ H/∂ ˆ is strictly concave in k. H Necessary conditions yield a unique optimal path.

22 / 45

Discounting: Current value Hamiltonian

Problems with discounting Current utility depends on time only through an exponential discounting term e−ρt . The generic discounted problem is Z T

max

e−ρt v[k(t), c(t)]dt

(24)

0

subject to the same constraints as above.

23 / 45

Applying the Recipe

ˆ g (k, c) H (t) = eρt v (k, c) + µ

(25)

∂H ˆt gc (kt , ct ) = 0 =⇒ e−ρt vc (kt, ct ) = −µ ∂ ct

(26)

∂H ˆt gk (kt , ct ) = −µ ˆ˙ t = e−ρt vk (kt , ct ) + µ ∂ kt

(27)

24 / 45

Applying the Recipe

Let ˆt µt = eρt µ

(28)

and multiply through by eρt : vc (t) = −µt gk (t)

ˆ. This is the standard FOC, but with µ instead of µ

25 / 45

Applying the Recipe

ˆt gk (t) = −eρt µ ˆ˙ t vk (t) + eρt µ

(29)

ˆ˙ t using Substitute out µ µ˙ t =

ˆt deρt µ ˆ˙ t = ρ µt + eρt µ dt

we have vk (t) + µt gk (t) = −µ˙ t + ρ µt This is the standard condition with an additional ρ µ term.

26 / 45

Shortcut We now have a shortcut for discounted problems. Hamiltonian (drop the discounting term): H = v (k, c) + µg (k, c)

(30)

∂ H/∂ c = 0

(31)

˙ ∂ H/∂ k = µ(t)ρ − µ(t) | {z } added

(32)

lim e−ρT µ(T)k(T) = 0

(33)

FOCs:

and the TVC T→∞

27 / 45

Equality constraints Equality constraints of the form h[c(t), k(t), t] = 0

(34)

are simply added to the Hamiltonian as in a Lagrangian problem: H(t) = v(k, c, t) + µ(t)g(k, c, t) + λ (t)h(k, c, t)

(35)

FOCs are unchanged: ∂ H/∂ c = 0 ∂ H/∂ k = −µ˙

For inequality constraints: h (c, k, t) ≥ 0; λ h = 0

(36) 28 / 45

Transversality Conditions

Finite horizon: Scrap value problems The horizon is T. The objective function assigns a scrap value to the terminal state variable: e−ρT φ (k(T)): Z T

max

e−ρt v[k(t), c(t), t]dt + e−ρT φ (k(T))

(37)

0

Hamiltonian and FOCs: unchanged. The TVC is µ (T) = φ 0 (k (T))

(38)

Intuition: µ is the marginal value of the state k.

30 / 45

Scrap value examples 1. Household with bequest motive Z T

U= 0

eρt u (c (t)) + V (kT )

(39)

with k˙ = w + rk − c. 2. Maximizing the present value of earnings Z T

Y=

e−rt wh (t) [1 − l (t)]

(40)

0

subject to h˙ (t) = Ah (t)α l (t)β − δ h (t) Scrap value is 0. TVC: µ (T) = 0.

31 / 45

Infinite horizon TVC

The finite horizon TVC with the boundary condition k (T) ≥ kT is µ (T) = 0. I

Intuition: capital has no value at the end of time.

But the infinite horizon boundary condition is NOT limt→∞ µ (t) = 0. The next example illustrates why.

32 / 45

Infinite horizon TVC: Example

Z ∞

max

[ln (c (t)) − ln (c∗ )] dt

0

subject to ˙k (t) = k (t)α − c (t) − δ k (t) k (0) = 1 lim k (t) ≥ 0

t→∞

c∗ is the max steady state (golden rule) consumption. No discounting - subtracting c∗ makes utility finite.

33 / 45

Infinite horizon TVC

Hamiltonian H (k, c, λ ) = ln c − ln c∗ + λ [kα − c − δ k]

(41)

Necessary FOCs Hc = 1/c − λ = 0 Hk = λ αkα−1 − δ = −λ˙

(42) (43)

34 / 45

Infinite horizon TVC We show: limt→∞ c (t) = c∗ [why?] Limiting steady state solves λ˙ /λ = αkα−1 − δ = 0 k˙ = kα − 1/λ − δ k = 0

Solution is the golden rule: k∗ = (α/δ )1/(1−α)

(44)

Verify that this max’s steady state consumption.

35 / 45

Infinite horizon TVC

Implications for the TVC... λ (t) = 1/c (t) implies limt→∞ λ (t) = 1/c∗ . Therefore, neither λ (t) nor λ (t) k (t) converge to 0. The correct TVC: lim H (t) = 0

t→∞

(45)

The only reason why the standard TVC does not work: there is no discounting in the example.

36 / 45

Infinite horizon TVC: Discounting With discounting, the TVC is easier to check. Assume: I

the objective function is e−ρt v [k (t) , c (t)]

I

it only depends on t through the discount factor

I

v and g are weakly monotone

Then the TVC becomes lim e−ρt µ (t) k (t) = 0

t→∞

(46)

where µ is the costate of the current value Hamiltonian. This is exactly analogous to the discrete time version lim β t u0 (ct ) kt = 0

t→∞

(47)

37 / 45

Example: renewable resource

Example: Renewable resource

Z ∞

max

e−ρt u (y (t)) dt

(48)

0

subject to

(49)

x˙ (t) = −y (t)

(50)

x (0) = 1

(51)

x (t) ≥ 0

(52)

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Example: Renewable resource

Current value Hamiltonian Necessary FOCs

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Example: Renewable resource

FOC Therefore: µ (t) = µ (0) eρt

(53)

y (t) = u0−1 [µ (0) eρt ]

(54)

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Solution

The optimal path has lim x (t) = 0 or Z ∞

Z ∞

y (t) dt = 0

u0−1 [µ (0) eρt ] dt = 1

(55)

0

This solves for µ (0).

42 / 45

Example: Renewable resource

TVC for infinite horizon case: lim e−ρt µ (0) eρt x (t) = 0

(56)

lim x (t) = 0

(57)

Equivalent to

43 / 45

Reading

I

Acemoglu (2009), ch. 7. Proves the Theorems of Optimal Control.

I

Barro and Martin (1995), appendix.

I

Leonard and Van Long (1992): A fairly comprehensive treatment. Contains many variations on boundary conditions.

44 / 45

References I

Acemoglu, D. (2009): Introduction to modern economic growth, MIT Press. Arrow, K. J. and M. Kurz (1970): “Optimal growth with irreversible investment in a Ramsey model,” Econometrica: Journal of the Econometric Society, 331–344. Barro, R. and S.-i. Martin (1995): “X., 1995. Economic growth,” Boston, MA. Leonard, D. and N. Van Long (1992): Optimal control theory and static optimization in economics, Cambridge University Press.

45 / 45

October 3, 2017

1 / 45

Topics

Optimal control is a method for solving dynamic optimization problems in continuous time.

2 / 45

Example: Growth Model A household chooses optimal consumption to Z T

max

e−ρt u[c(t)]dt

(1)

0

subject to ˙ = rk(t) − c(t) k(t)

(2)

c (t) ∈ [0,¯c]

(3)

k(0) = k0 ,given

(4)

k(T) ≥ 0

(5)

3 / 45

Generic Optimal control problem Choose functions of time c (t) and k (t) so as to Z T

max

v[k(t), c(t), t]dt

(6)

0

Constraints: 1. Law of motion of the state variable k (t): ˙ = g[k(t), c(t), t] k(t)

(7)

2. Feasible set for control variable c (t): c (t) ∈ Y (t)

(8)

3. Boundary conditions, such as: k(0) = k0 ,given

(9)

k(T) ≥ kT

(10) 4 / 45

Generic Optimal control problem

I

c and k can be vectors.

I

Y (t) is a compact, nonempty set.

I

T could be infinite. I

I

Then the boundary conditions change

Important: the state cannot jump; the control can.

5 / 45

A Recipe for Solving Optimal Control Problems

A Recipe Step1. Write down the Hamiltonian H(t) = v(k, c, t) + µ(t)g(k, c, t) | {z }

(11)

˙ k(t)

µ is essentially a Lagrange multiplier (called a co-state).

Step 2. Derive the first order conditions which are necessary for an optimum: ∂ H/∂ c = 0

(12)

∂ H/∂ k = −µ˙

(13)

7 / 45

A Recipe

Step 3. Impose the transversality condition: I

I

for finite horizon: µ (T) = 0

(14)

lim H(t) = 0

(15)

for infinite horizon: t→∞

This depends on the terminal condition (see below).

8 / 45

A Recipe

Step 4. A solution is the a set of functions [c(t), k(t), µ(t)] which satisfy I

the FOCs

I

the law of motion for the state

I

the boundary / transversality conditions

9 / 45

Intuition ∂ H/∂ c = 0: Maximize Hamiltonian w.r.to control. I

v (k, c, t) picks up current utility of c

I

µ (t) is marginal value of additional “future” k.

I

µ (t) g (k, c, t) picks up change in continuation value (change in k˙ times value of future k)

˙ ∂ H/∂ k = −µ: I

˙ think of this as [∂ H/∂ k] /µ = −µ/µ

I

∂ H/∂ k is the value of additional k

I

[∂ H/∂ k] /µ is like a rate of return (give up c now, how much future value do you get?) ˙ µ/µ is the growth rate of marginal utility

I

10 / 45

Example: Growth Model

Z ∞

max

e−ρt u (c (t)) dt

(16)

0

subject to k˙ (t) = f (k (t)) − c (t) − δ k (t)

(17)

k (0) given

(18)

11 / 45

Growth Model: Hamiltonian

H (k, c, µ) = e−ρt u (c (t)) + µ (t) [f (k (t)) − c (t) − δ k (t)]

(19)

Necessary conditions: Hc = e−ρt u0 (c) − µ = 0 Hk = µ f 0 (k) − δ = −µ˙

12 / 45

Growth Model Substitute out the co-state: µ˙ = e−ρt u00 (c) c˙ − ρ µ

c˙ =

µ˙ + ρ µ e−ρt u00 (c)

= − f 0 (k) − δ − ρ

(20)

(21) u0 (c) u00 (c)

(22)

Solution: ct , kt that solve Euler equation and resource constraint, plus boundary conditions.

13 / 45

Details

First order conditions are necessary, not sufficient. They are necessary only if we assume that 1. a continuous, interior solution exists; 2. the objective function v and the constraint function g are continuously differentiable. Acemoglu (2009), ch. 7, offers some insight into why the FOCs are necessary.

14 / 45

Details If there are multiple states and controls, simply write down one FOC for each separately: δ H/δ ci = 0 ∂ H/∂ kj = −µ˙ j

There is a large variety of cases depending on the length of the horizon (finite or infinite) and the kinds of boundary conditions. I

Each has its transversality condition (see Leonard and Van Long 1992).

15 / 45

Next steps

Typical useful next things to do: 1. Eliminate µ from the system. Obtain two differential equations in (c, k) . 2. Find the steady state by imposing c˙ = k˙ = 0.

16 / 45

Sufficient conditions

First-order conditions are sufficient, if the programming problem is concave. This can be checked in various ways.

17 / 45

Sufficient conditions I

The objective function and the constraints are concave functions of the controls and the states. I

The co-state must be positive.

This condition is easy to check, but very stringent. In the growth model: I

u (c) is concave in c (and, trivially, k)

I

f (k) − δ k − c is concave in c and k

I

µ = u0 (c) > 0

18 / 45

Sufficient Conditions II

(Mangasarian) First-order conditions are sufficient, if the Hamiltonian is concave in controls and states, where the co-state is evaluated at the optimal level (and held fixed). This, too is very stringent.

19 / 45

In the growth model

∂ H/∂ c = u0 (c) − µ ∂ H/∂ k = µ [f 0 (k) − δ ] ∂ 2 H/∂ c2 = u00 (c) < 0 ∂ 2 H/∂ k2 = µf 00 (k) < 0 ∂ 2 H/∂ c∂ k = 0 Therefore: weak joint concavity (because we know that µ > 0)

20 / 45

Sufficient Conditions III Arrow and Kurz (1970) I

First-order conditions are sufficient, if the maximized Hamiltonian is concave in the states.

I

If the maximized Hamiltonian is strictly concave in the states, the optimal path is unique.

Maximized Hamiltonian: Substitute controls out, so that the Hamiltonian is only a function of the states. This is less stringent and by far the most useful set of sufficient conditions.

21 / 45

In the growth model

Optimal consumption obeys u0 (c) = µ or c = u0−1 (µ) Maximized Hamiltonian: ˆ = u u0−1 (µ) + µ f (k) − δ k − u0−1 (µ) H

(23)

ˆ k > 0 and ∂ 2 H/∂ k2 = µf 00 (k) < 0. We have ∂ H/∂ ˆ is strictly concave in k. H Necessary conditions yield a unique optimal path.

22 / 45

Discounting: Current value Hamiltonian

Problems with discounting Current utility depends on time only through an exponential discounting term e−ρt . The generic discounted problem is Z T

max

e−ρt v[k(t), c(t)]dt

(24)

0

subject to the same constraints as above.

23 / 45

Applying the Recipe

ˆ g (k, c) H (t) = eρt v (k, c) + µ

(25)

∂H ˆt gc (kt , ct ) = 0 =⇒ e−ρt vc (kt, ct ) = −µ ∂ ct

(26)

∂H ˆt gk (kt , ct ) = −µ ˆ˙ t = e−ρt vk (kt , ct ) + µ ∂ kt

(27)

24 / 45

Applying the Recipe

Let ˆt µt = eρt µ

(28)

and multiply through by eρt : vc (t) = −µt gk (t)

ˆ. This is the standard FOC, but with µ instead of µ

25 / 45

Applying the Recipe

ˆt gk (t) = −eρt µ ˆ˙ t vk (t) + eρt µ

(29)

ˆ˙ t using Substitute out µ µ˙ t =

ˆt deρt µ ˆ˙ t = ρ µt + eρt µ dt

we have vk (t) + µt gk (t) = −µ˙ t + ρ µt This is the standard condition with an additional ρ µ term.

26 / 45

Shortcut We now have a shortcut for discounted problems. Hamiltonian (drop the discounting term): H = v (k, c) + µg (k, c)

(30)

∂ H/∂ c = 0

(31)

˙ ∂ H/∂ k = µ(t)ρ − µ(t) | {z } added

(32)

lim e−ρT µ(T)k(T) = 0

(33)

FOCs:

and the TVC T→∞

27 / 45

Equality constraints Equality constraints of the form h[c(t), k(t), t] = 0

(34)

are simply added to the Hamiltonian as in a Lagrangian problem: H(t) = v(k, c, t) + µ(t)g(k, c, t) + λ (t)h(k, c, t)

(35)

FOCs are unchanged: ∂ H/∂ c = 0 ∂ H/∂ k = −µ˙

For inequality constraints: h (c, k, t) ≥ 0; λ h = 0

(36) 28 / 45

Transversality Conditions

Finite horizon: Scrap value problems The horizon is T. The objective function assigns a scrap value to the terminal state variable: e−ρT φ (k(T)): Z T

max

e−ρt v[k(t), c(t), t]dt + e−ρT φ (k(T))

(37)

0

Hamiltonian and FOCs: unchanged. The TVC is µ (T) = φ 0 (k (T))

(38)

Intuition: µ is the marginal value of the state k.

30 / 45

Scrap value examples 1. Household with bequest motive Z T

U= 0

eρt u (c (t)) + V (kT )

(39)

with k˙ = w + rk − c. 2. Maximizing the present value of earnings Z T

Y=

e−rt wh (t) [1 − l (t)]

(40)

0

subject to h˙ (t) = Ah (t)α l (t)β − δ h (t) Scrap value is 0. TVC: µ (T) = 0.

31 / 45

Infinite horizon TVC

The finite horizon TVC with the boundary condition k (T) ≥ kT is µ (T) = 0. I

Intuition: capital has no value at the end of time.

But the infinite horizon boundary condition is NOT limt→∞ µ (t) = 0. The next example illustrates why.

32 / 45

Infinite horizon TVC: Example

Z ∞

max

[ln (c (t)) − ln (c∗ )] dt

0

subject to ˙k (t) = k (t)α − c (t) − δ k (t) k (0) = 1 lim k (t) ≥ 0

t→∞

c∗ is the max steady state (golden rule) consumption. No discounting - subtracting c∗ makes utility finite.

33 / 45

Infinite horizon TVC

Hamiltonian H (k, c, λ ) = ln c − ln c∗ + λ [kα − c − δ k]

(41)

Necessary FOCs Hc = 1/c − λ = 0 Hk = λ αkα−1 − δ = −λ˙

(42) (43)

34 / 45

Infinite horizon TVC We show: limt→∞ c (t) = c∗ [why?] Limiting steady state solves λ˙ /λ = αkα−1 − δ = 0 k˙ = kα − 1/λ − δ k = 0

Solution is the golden rule: k∗ = (α/δ )1/(1−α)

(44)

Verify that this max’s steady state consumption.

35 / 45

Infinite horizon TVC

Implications for the TVC... λ (t) = 1/c (t) implies limt→∞ λ (t) = 1/c∗ . Therefore, neither λ (t) nor λ (t) k (t) converge to 0. The correct TVC: lim H (t) = 0

t→∞

(45)

The only reason why the standard TVC does not work: there is no discounting in the example.

36 / 45

Infinite horizon TVC: Discounting With discounting, the TVC is easier to check. Assume: I

the objective function is e−ρt v [k (t) , c (t)]

I

it only depends on t through the discount factor

I

v and g are weakly monotone

Then the TVC becomes lim e−ρt µ (t) k (t) = 0

t→∞

(46)

where µ is the costate of the current value Hamiltonian. This is exactly analogous to the discrete time version lim β t u0 (ct ) kt = 0

t→∞

(47)

37 / 45

Example: renewable resource

Example: Renewable resource

Z ∞

max

e−ρt u (y (t)) dt

(48)

0

subject to

(49)

x˙ (t) = −y (t)

(50)

x (0) = 1

(51)

x (t) ≥ 0

(52)

39 / 45

Example: Renewable resource

Current value Hamiltonian Necessary FOCs

40 / 45

Example: Renewable resource

FOC Therefore: µ (t) = µ (0) eρt

(53)

y (t) = u0−1 [µ (0) eρt ]

(54)

41 / 45

Solution

The optimal path has lim x (t) = 0 or Z ∞

Z ∞

y (t) dt = 0

u0−1 [µ (0) eρt ] dt = 1

(55)

0

This solves for µ (0).

42 / 45

Example: Renewable resource

TVC for infinite horizon case: lim e−ρt µ (0) eρt x (t) = 0

(56)

lim x (t) = 0

(57)

Equivalent to

43 / 45

Reading

I

Acemoglu (2009), ch. 7. Proves the Theorems of Optimal Control.

I

Barro and Martin (1995), appendix.

I

Leonard and Van Long (1992): A fairly comprehensive treatment. Contains many variations on boundary conditions.

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References I

Acemoglu, D. (2009): Introduction to modern economic growth, MIT Press. Arrow, K. J. and M. Kurz (1970): “Optimal growth with irreversible investment in a Ramsey model,” Econometrica: Journal of the Econometric Society, 331–344. Barro, R. and S.-i. Martin (1995): “X., 1995. Economic growth,” Boston, MA. Leonard, D. and N. Van Long (1992): Optimal control theory and static optimization in economics, Cambridge University Press.

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