Land use dynamics

Discounting

Terminal Values

Initialization

Ages

Markov chains

Discounting

• t … time periods• l … length of time periods (years)• r … real interest rate• T … time horizon

0 0

1

(1 )

T T

t t tltt t

NPV V Vr

Discount factor

Terminal Values

• Model ends at period T but real life/ business is likely to continue thereafter

• Without consideration of life/business after T, the model would cease all investments as it approaches T

• To account for life/business after T, terminal conditions need to be specified which account for benefits outside the model horizon of investments inside the model horizon

Net Present Value in the last period

0

2

2

1

(1 )

1 1 1 1...

(1 ) (1 ) (1 ) (1 )

1 (1 ) 1 1 1...

(1 ) (1 ) (1 ) (1 ) (1 )

1 1 1 1(1 ) ...

(1 ) (1 ) (1 ) (1 )

T TT ltt

T T T TT T l T l T l

l

T T T TT l T l T l T l

lT T TT T T l T

NPV Vr

V V V Vr r r r

rV V V V

r r r r r

V r V Vr r r r

( )

1 1

(1 ) (1 )

Tl l

T TT l

V

V NPVr r

Net Present Value in the last period

0

1 1 1

(1 ) (1 ) (1 )

1 10

(1 ) (1 )

1 (1 ) 1 11

(1

(1 )

(1 ) 1 (

) (1 ) (

1 )

1 )

lim

T T T TT lt T lt

T kk

l

T T Tl l T

l

T T T Tl T

NPV V V NPVr r r

r r

rNPV NPV V

r

rNP

r

r

V Vr

r

V

Discount factors

Period 1% 2% 3% 4% 5% 6%0 1.000 1.000 1.000 1.000 1.000 1.0001 0.990 0.980 0.971 0.962 0.952 0.943

10 0.905 0.820 0.744 0.676 0.614 0.55820 0.820 0.673 0.554 0.456 0.377 0.31230 0.742 0.552 0.412 0.308 0.231 0.17440 0.672 0.453 0.307 0.208 0.142 0.09750 0.608 0.372 0.228 0.141 0.087 0.054

100 0.370 0.138 0.052 0.020 0.008 0.003100 + Infinity 7.618 1.464 0.379 0.111 0.035 0.012

Discount Rate

Initialization

• Represent past investments for activities which are still alive at the beginning of the model horizon

• Industry capacities created in the past

• Current forests planted in the past

• State of soil carbon

Represent Age

• If time is discrete, so should age be

• Width of age classes should correspond to length of time periods

• Last age class should represent all ages above the upper boundary on the second highest age class

Mathematical Representation

, ,1

1, 1 1,0 0,

0 0

0,1 0

t a t t aa

t a t at t a A

a at a Aa A t

H P X

X X

x x

Discrete – Time Markov Chains

• Many real-world systems contain uncertainty and evolve over time.

• Stochastic processes and Markov chains are probability models for such systems

• A discrete-time stochastic process is a sequence of random variables x0, x1, x2, … typically denoted {xt}.

State Occupancy Probability Vector

Let π be a row vector. Denote πi to be the ith element of the vector with n elements. If π is a state occupancy probability vector, then πi(t) is the probability that a DTMC has value i (or is in state i) at time-step t

( )1

1n

ii

t tp=

= "å

State Transition Probabilities

4.5.1.

2.6.2.

2.3.5.321

3

2

1ijPP

Transient Behavior of DTMC

π(t) = π(t-1)P

π(t-1)= π(t-2)P

π(t) = [π(t-2)P]P = π(t-2)P2

π(t-2)= π(t-3)P

π(t) = [π(t-3)P]P2 = π(t-3)P3

π(t) = π(0)Pt

Convergence t+1 = t P

State ... Year t Year t+1

1 ... 50 50

2 ... 75 75

3 ... 83 83

Empirical Example

• Soil carbon sequestration from land use will receive premium

• Continuous application of a certain tillage system leads to specific soil carbon equilibrium (after few decades)

• How to model optimal decision path?

Empirical Example

• Two tillage systems

• Annual decisions over multi-decade

horizon

• Limited land availability

• Carbon price

Time

Soil Carbon

Tillage Effect on Soil Carbon

Zero Tillage

Intensive Tillage

Land Use Decision Model

t … time indexr … region indexi … soil type indexu … tillage index

M Ct t,r,i,u t ,r,i,u t ,r,i,u

t ,r,i,u

max v v X

t ,r,i,u t ,r,iu

X LL … available landvM … market profitvC … carbon profit … discount factor

Soil Carbon Status Dynamics

t ,r,i,u,o r,i,u,o,o t 1,r,i,u,ou u,o t 1

r,i,o t 1

X X

t, r, i,o

t … time index r … region indexi … soil type index u … tillage indexo … soil carbon state

I II III IV V

Case (see Schneider 2007)

Transition Probabilities

C a s e I : I f u p u p u pr , i , u , oo s o a n d lo w lo w lo w

r , i , u , oo s o , th e n u p lo w lo w

r , i , u , o

r , i , u , o , or , i , u , o

o o s

w

C a s e I I : I f u p u p u pr , i , u , oo s o a n d lo w lo w lo w

r , i , u , oo s o , th e n u p u p lo w

r , i , u , o

r , i , u , o , or , i , u , o

o s o

w

C a s e I I I : I f u p u p u pr , i , u , oo s o a n d lo w lo w lo w

r , i , u , oo s o , th e n u p lo w

r , i , u , o , or , i , u , o

o o

w

C a s e IV : I f u p u p u pr , i , u , oo s o a n d lo w lo w lo w

r , i , u , oo s o , th e n r , i , u , o , o 1

C a s e V : I f u p u p lo wr , s , u , o , oo s o o r lo w lo w u p

r , s , u , o , oo s o , th e n r , s , u , o , o 0

Transition Probabilities