Environmental and
Resource Economics
Methods: Dynamics and Inter-temporal decisions
KLAUS EISENACK - RESOURCE ECONOMICS GROUP
HUMBOLDT-UNIVERSITÄT ZU BERLIN
WWW.RESOURCE-ECONOMICS.HU-BERLIN.DE
Mathematical tools to deal with
time, stocks and flows
Klaus Eisenack
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Stocks and flows, mathematically (recalled from intro)
• Flow 𝐹𝐹𝑡𝑡 in time 𝑡𝑡
• Stock 𝑆𝑆𝑡𝑡 with endogenous change rate 𝑓𝑓 𝑆𝑆 , described as a mathematical function
• Dynamics described by difference equations:
𝑆𝑆𝑡𝑡+1 = 𝑆𝑆𝑡𝑡 + 𝑓𝑓 𝑆𝑆𝑡𝑡 + 𝐹𝐹𝑡𝑡
f(S)
“fish stocknext year = fish stockthis year + fish growththis year – catchthis year ”
“atmospheric carbonnext year = atmospheric carbonthis year – decaythis year + emissionsthis year ”
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(example)
S
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A second way to describe change:
Differential equations (ordinary differential equations, ODE)
• Comparison to difference equations
xt +1 = xt + f ( xt , ut , t )
− While time evolves step-by-step (e.g. years, months, days) in difference equations, time evolved
continuously for differential equations
• Describe processes over time; applications in many scientific disciplines, e.g.
− Physics: mechanics, movement of bodies, radioactive decay
− Chemistry: course of chemical reactions
− Biology/ecology: population growth, ecosystem dynamics
− Demography/epidemiology: spreading of diseases, demographic change
− Economics: growth theory, resource economics
• Basic idea: the change rate of a system state over time dx/dt (derivative with
respect to time, „velocity “) at a given time t depends on exogenous parameters
u und the state x itself. Dynamics starts at time t0 with in state x0:
x =
dx
dt
= f ( x, u , t ), x( t 0 ) = x0
• For example: non-renewable resource stock x, extraction share u
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x = −u ⋅ x, x(0) = 1000
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Differential equations (ctd.)
• Solving a differential equation:
x = f ( x, u , t ), x(0) = x0
• Numerical approximation: “Start from the initial position x(0) and the initial strategy u(0).
Determine the speed f(x,u,0). Continue with this speed for a marginal instance ε. You then
arrive at the position x(ε)= x(0)+ ε ⋅ f(x,u,0). Now continue with the strategy u(ε), and so on
and so forth.”
• Algebraically: “Find a function x(t) in time with the following property: If x(t) is plugged into
f (together with u(t)), this yields exactly the derivative of x(t).”
• Note: explicit solutions are usually difficult or even impossible to find, but important
properties can frequently be derived
There are various sources that help you to learn ODEs. In this class, you only need the very
basics. For example:
https://www.youtube.com/watch?v=f8xtjVGnnXo
https://www.khanacademy.org/math/ap-calculus-bc/bc-differential-equations-new/bc7-5/v/eulers-method
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Differential equations: example
Exponential growth
• A physical capital stock x(t) at time t produces µ·x(t) units of goods that can
be invested to increase the capital stock. The initial capital stock is x(0)=x0.
− How does the capital stock develop over time if its product is completely reinvested?
• Differential equation:
x = µ x
• The solution is a function x(t) with the following properties:
x(0) = x0 , and
for all t ≥ 0 :
dx ( t )
dt
= µ x(t )
• Exercise: show that the exponential function is the solution:
x (t ) = x0e µt
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Intertemporal decisions: first
thoughts
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Thought experiment: cake eating problem
• How to invest in and make use of a capital stock over time?
• Cake eating problem: simplest version
− Robinson Crusoe has saved 1000 pieces of zwieback from the ship, and stays on the island for two
years: how much should he eat in the first, and how much in the second year?
− Think of extensions: 40 years instead of two; uncertain time on the island; …
• One way to solve the problem: optimal plan (utilitarian)
− Cakes in year one (R1), and in year two (R2), maximize total utility W(R1,R2) subject to R1+R2=1000
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
− Lagrangean yields 𝑑𝑑𝑅𝑅
=𝑑𝑑𝑅𝑅
1
2
 Equalization of marginal utility over both periods
(vgl. Ströbele, 1987, Rohstoffökonomik, Verlag Franz Vahlen, p. 15-18)
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Intertemporal efficiency and optimality
• Recall standard Pareto efficiency…
• Dynamic Pareto efficiency
− You cannot extract more of a resource at one point in time without sacrificing extraction at another
time
• Static optimality / allocative efficiency
− A welfare function is maximized (at each time)
− Marginal net benefits of all goods and services equalized (at each time)
• Intertemporal optimality / dynamic efficiency
− A welfare function that aggregated benefits over time is maximized
− Marginal net benefit of a resource equalized over time
Perman et al. Ch. 11.1, Ch. 14.5.2
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Discounting: how to aggregate
benefits over time
Pe r m a n e t a l . C h . 3 . 5
Complementary reading:
R . P. M c A fe e , T. L e w i s , D. D a l e ( 2 0 1 6 ) I n t ro d u c t i o n t o E c o n o m i c A n a l y s i s , M i c ro s o f t / D u ke
U n i v e rs i t y / M u h l e n b e rg C o l l e g e ,
h tt p s : / / w w w. ke l l o g g . n o r t h w e st e r n . e d u / fa c u l t y / d a l e / i e av 2 1 . p d f [ 2 2 . 4 . 2 0 ] , C h . 1 1 . 1
C o r e Te a m ( 2 0 1 7 ) T h e Ec o n o my, O n l i n e E b o o k , h tt p : / / w w w. c o re - e c o n . o r g / , U n i t s
10.3, 10.4.
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Comparing present and future utility:
Discounting and net present value
• Usually: present utility of a given quantity larger than future
(expected) utility of the same quantity, but valued today
W = U ( R0 ) + 1+1ρ U ( R1 ) +
( ) U ( R ) + ... +
+ ( ) U (R )
+
− Time preference / patience
− Uncertainty about future conditions (risks, death, doomsday, technological
progress)
− Economic growth
− Current value(s): U(R0), U(R1), …, U(RT); (Net) present value: W
• To consider time preference for present consumption: derive net
present value W from stream of current value utility U(·), using a
discount rate ρ
1. (discrete time)
2. (continuous time)
• For private investments: consider a stream of profits π(t) and a
market interest rate i
1. (discrete time)
2. (continuous time)
T
W =∑
t =0
T
1 2
1+ ρ
2
1 T
1+ ρ
T
( ) U (R )
1 t
1+ ρ
t
W = ∫ U ( R(t ))e − ρt dt
W = ∑ (1+1 i ) tπ (t )
t =0
T
0
T
W = ∫ π (t ) e −it dt
0
Discount admits to compare intertemporal decisions
Example
• Suppose a firm can choose between two investment options, which both cost the
same amount
• Both investments promise a certain stream of profits over 4 years (see table)
• If profits are put to a bank account, the firm earns an interest rate of 10% per year.
 Which option is the best for the firm?
• Exercise: determine and compare the present value
of both options
year profit (option 1)
(See supplied excel sheet for a way to solve the task)
0
100
80
1
100
100
2
100
120
3
100
140
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profit (option 2)
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Dynamic optimization: determining
optimal intertemporal decisions
about dynamic systems
Perman et al. Ch 14.5 and Appendix 14.1
Compementary reading:
Ströbele (1987) Rohstoffökonomik, Vahlen, p. 15-21, 175-177
Dynamic optimization (or: dynamic control)
• Problem:
Find a control path in time 𝑢𝑢(�), so that
𝑇𝑇
max 𝑊𝑊 𝑢𝑢(�) ≔ ∫0 𝐼𝐼 𝑥𝑥 𝑡𝑡 , 𝑢𝑢(𝑡𝑡) 𝑒𝑒 −𝜌𝜌𝑡𝑡 𝑑𝑑𝑑𝑑
𝑢𝑢(�)
subject to
𝑥𝑥̇ = 𝑓𝑓 𝑥𝑥 𝑡𝑡 , 𝑢𝑢(𝑡𝑡) ,
𝑥𝑥 0 = 𝑥𝑥 0
for all 𝑡𝑡 ∈ 0, 𝑇𝑇 : 𝑢𝑢 𝑡𝑡 ∈ 𝑈𝑈
(and further technical assumptions)
• Cake Eating Problem?
• Many applications, e.g.
𝑥𝑥(𝑡𝑡): state of system at time 𝑡𝑡
𝑢𝑢(𝑡𝑡): strategy/control chosen at 𝑡𝑡 from admissible set 𝑈𝑈
𝐼𝐼(𝑥𝑥, 𝑢𝑢): current value at state 𝑥𝑥 if strategy 𝑢𝑢 is chosen
𝜌𝜌: discount rate
𝑊𝑊: present value if control path 𝑢𝑢(�) is chosen
− Engineering: mechanics with minimal energy costs
− Physics: movement of bodies in mechanical systems
− Military: rockets reaching their target in minimal time
− Economics: optimal growth, optimal resource exploitation, dynamic game theory
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Cake eating problem, revisited
• Robinson Crusoe can dispose of 1000 packs of zwieback
• He assumes an annual 10% probability to be rescued
• How much packs of zwieback shall he consume per year?
− Further assumption: isoelastic current value of zwieback
• Formulation as dynamic optimization problem
∞
max W := ∫ U (C (t ) ) e − µt dt ,
C ( ⋅)
0
s.t.
S = −C , S (0) = 1000, C ≥ 0,
with U = 1−1a C 1− a , µ = 10%
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Dynamic optimization: „cooking recipe“
• Introduce a dynamic shadow price λ(t) and Hamiltonian H (cf. Lagrange)
• (1) For every point in time: maximize H with respect to u(t)
• (2,3) Differential equations for x and λ
• (4) transversality condition
(These conditions are necessary but not always sufficient)
Ideas:
•
•
•
Reduce the intertemporal problem to problems (1) that
can be solved for each time t independently
The role of intertemporal trade-offs
it taken over by λ
The shadow price λ „integrates back“ all future value
of the state variable (see 3, 4).
Klaus Eisenack
H ( x, u , λ , t ) = I ( x, u , t ) + λ ⋅ f ( x, u , t )
(1) for all t ∈ [0, T ] : max H (x(t ), u (t ), λ (t ), t )
u (t )
(in the simple case : dudH( t ) = 0)
(2) x = dH
dλ = f ( x, u , t )
(3) − λ + ρλ = dH
dx
(4) lim λxe − ρt = 0
t →T
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Let us solve the cake eating problem (solution)
• “Cooking recipe”
H = U (C ) − λC ,
dH
 = −C , S (0) = S ,
S
=
0
,
0
dC
− µt
µλ − λ = dH
λ
t
S
t
= 0.
,
lim
(
)
(
)
e
dS
t →∞
• Leads to the solution:
λ (t ) = λ0 e µt , lim S (t ) = 0,
t →∞
µ
C (t ) = a S 0 e
− µa t
.
• All cake is eaten, with decreasing rate
• Exponentially increasing shadow price = constant marginal value of resource (in present value)
• Concrete solution with µ=0.10, a=½: C(t)=200 e-0.2t
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Core reading and guiding questions
• Perman et al. Ch 3.5, 11.1, 14.5 and Appendix 14.1
• Compare difference and differential equations: what pros and cons do you see for
the two approaches?
• Show that 𝑥𝑥 𝑡𝑡 = 2 � 𝑒𝑒 −3�𝑡𝑡 + 1 is a solution of the ODE 𝑥𝑥̇ = −3𝑥𝑥 + 3.
• Dynamic Pareto efficiency and intertemporal optimality: does one of these criteria
imply the other one? Which criterion is more strict?
• Solve the exercise to compare two decision alternatives by determining their net
present value
• Dynamic optimization:
− Solve the supplementary training tasks on Moodle
− Describe another problem of your choice as a dynamic optimization problem (you don’t need to
solve it)
− Compare dynamic optimization with the Lagrange approach: what is similar, what different?
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