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Land use dynamics

Land use dynamics. Discounting Terminal Values Initialization Ages Markov chains. Discounting. Discount factor. t … time periods l … length of time periods (years) r … real interest rate T … time horizon. Terminal Values.

audrey-holman
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Land use dynamics

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  1. Land use dynamics Discounting Terminal Values Initialization Ages Markov chains

  2. Discounting Discount factor • t … time periods • l… length of time periods (years) • r … real interest rate • T … time horizon

  3. 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

  4. Net Present Value in the last period

  5. Net Present Value in the last period

  6. Discount factors

  7. 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

  8. 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

  9. Mathematical Representation

  10. 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}.

  11. State Occupancy Probability Vector Let π be a row vector. Denote πito 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

  12. State Transition Probabilities

  13. 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

  14. Convergence t+1 = t P

  15. 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?

  16. Empirical Example • Two tillage systems • Annual decisions over multi-decade horizon • Limited land availability • Carbon price

  17. Tillage Effect on Soil Carbon Zero Tillage Soil Carbon Intensive Tillage Time

  18. Land Use Decision Model t … time index r … region index i … soil type index u … tillage index L … available land vM … market profit vC … carbon profit  … discount factor

  19. Soil Carbon Status Dynamics t … time index r … region index i … soil type index u … tillage index o … soil carbon state

  20. Transition Probabilities I II III IV V Case (see Schneider 2007)

  21. Transition Probabilities

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