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On Optimal Forest Management: A Bifurcation Analysis

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The theory of optimal forest management is a key component of the economic theory of natural resources due to the fact that forests constitute a major renewable resource. It also constitutes one of the key examples of vintage capital theory, making it an important factor in understanding the general theory of intertemporal allocation. Swapan Dasgupta explores the theory of optimal forest management, focusing on the forester’s (optimal) policy function.

Image courtesy of interviewee. September 13, 2024

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This video discusses the development of optimum growth literature, beginning with Frank Ramsey's seminal, yet initially overlooked, paper on optimum savings. The discussion evolves from Ramsey to David Gale, who approached the problem in a discrete time setting. The main focus is on a framework for analyzing optimal growth models, emphasizing the importance of price support properties that correlate competitive prices with Pareto optimal allocations in static and dynamic settings.

The narrative highlights the difference between undiscounted and discounted utility functions. In undiscounted models, the behavior is more straightforward, often leading to a unique optimal stationary state, and is characterized by a turnpike theorem, indicating convergence towards this optimal state over time. Conversely, in discounted models, cyclical behaviors can emerge, indicating multi-period cycles rather than a singular optimal state, especially when discounting is not minimal.

A specific application to forest management is presented, citing Martin Faustman’s historical contributions. He emphasized the importance of considering future land use rather than merely immediate timber harvesting. Faustman’s findings suggest cyclical harvesting patterns, particularly with linear utility functions, while nonlinear utility functions exhibit convergence properties over time.

This video further discusses the research conducted by Henry Wan and others, exploring optimum stationary programs and their behavior under different discounting scenarios. Key findings reveal complex interactions in the dynamics of resource management, emphasizing history dependence with respect to initial stock conditions and presenting bifurcation phenomena within the policy function.

Finally, the text underlines that understanding the value function is crucial for grasping optimal paths. The analysis concludes that while the model offers insights into dynamic behavior, incorporating additional complexities could yield more intricate dynamics in optimal forest management practices.

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Here are five key points from the video discussing optimum growth literature and forest management optimization:

1. **Historical Foundation and Key Figures**: The discussion starts with the foundational work of Frank Ramsey on optimum savings, which laid the groundwork for growth literature. David Gale further developed these ideas in a discrete time setting, emphasizing the importance of price support properties in allocating resources effectively over time.

2. **Optimal Paths and Welfare Maximization**: The literature addresses how to determine time paths of capital that provide the highest social welfare, particularly through undiscounted models. The turnpike theorem is introduced, indicating that all optimal paths tend to converge towards an optimal stationary state over time, showcasing a long-term predictability in resource management.

3. **Role of Discounting in Optimization**: When introducing discounting to utilities, the behavior of the optimal paths becomes more complex, leading to possible cycles in optimal behavior rather than convergence to a single state. It was revealed that with lower discount rates, cyclical behaviors emerge, contrasting with undiscounted cases where steady states dominate.

4. **Dynamic Optimization in Forest Management**: The concept of dynamic optimization applied to forest management originates from Martin Faustman's work, which assessed sustainable forestry practices over time. Faustman's findings stressed the importance of considering future land use alongside immediate timber harvests, leading to cyclical cutting and replanting strategies based on tree maturity.

5. **Bifurcation and Policy Functions**: A study led by Henry Wan and others explored the dynamics of forest management using bifurcations in policy functions. They identified critical thresholds (termed the "Fausman's threshold") for mature trees, which influenced whether management strategies would lead to cyclical or stationary approaches. The behavior of optimal paths is sensitive to initial conditions, indicating complex long-term outcomes based on starting stock levels of trees.

These points collectively capture the essence of the interplay between theoretical models in economics and practical applications in forest management, highlighting the significance of historical convideo, optimization strategies, and the impact of discounting on long-term resource management decisions.

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Here are some key quotes from the video:

1. **On the genesis of optimal growth literature**:
- "This really started with, uh, uh, Frank Ramsey, uh, the mathematician, uh, with a paper on optimum savings."

2. **Seminal yet underappreciated work**:
- "But that paper, which was a seminal paper, but it was not really... it did not draw too much attention despite CA's, you know, pushing for... the paper as being interesting."

3. **Description of the problem**:
- "So what one is looking for is that starting with a given initial stock, uh, what kind of time path... can, uh, gave us the largest welfare, some of welfare."

4. **On the formulation of Gale and Ramsey**:
- "In Gale's formulation as well as Ramsey's, they did not use any discounting."

5. **Turnpike theorem**:
- "There is a stationary program, which if you start with that stock, it doesn't change the stock in the future... and they all sort of towards that, uh, optimal stationary state."

6. **Transition to discounted models**:
- "When we move from that... into a model where there is some discounting of the utilities taking place."

7. **Cyclical behavior discovery**:
- "It was gradually discovered that no, there are examples where one could provide a... cyclical behavior... instead of... a single unique optimal stationary state."

8. **Bifurcation and its consequences**:
- "This policy function has what is called a bifurcation, okay? Splitting up... into two branches."

9. **Description of the Faustman problem**:
- "I said before, it is a forest management problem... first really investigated properly by Martin Faustman... in the mid-19th century."

10. **On the complexity of the solutions**:
- "But generally speaking, the idea is still the same, that there is this cycle, et cetera."

11. **Characteristics of the policy function**:
- "If we get to know about the policy function in some detail, then one can have a much better handle over how the... paths are going to behave over time."

12. **Exploration of history dependence**:
- "It is... we call the history dependence of these two period cycles... it matters what the final state is going to be."

These quotes capture the essence of the discussion around optimal growth models, the influential works within the field, the implications of discounting, and the exploration of cyclical behavior in economic modeling—particularly in the convideo of forest management.

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In the optimum growth literature.

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This really started with, Frank Ramsey,

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the mathematician, with a paper on optimum savings.

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Uh, but that paper, which was a seminal paper,

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but it was not really, it did not

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draw too much attention despite CA's, you know, pushing for,

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the paper as being interesting.

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It did not draw too much attention at that time,

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and it, only attracted attention to begin

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with of mathematician.

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I also interested in economics as, David Gale

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in, in, in Berkeley, right?

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Uh, so Gale worked on this problem,

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but in a discreet time setting.

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And whatever I'm going to talk about is

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in a discreet time setting instead of continuous stem.

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They all talk about in a reduced form model,

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where basically what it all means is

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that the technology set is described

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by a transition from initial stock

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of capital to final stock.

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And there is a social welfare

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function somehow defined over this, each pair.

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So what one is looking for is that starting

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with a given initial stock, what kind of time path,

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which is technically feasible, which can,

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gave us the largest welfare, some of welfare,

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and in Gale's formulation as well as Ramsey's,

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they did not use any discounting.

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So it was undiscounted some,

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which presents some technical problems,

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but they caught around it with very clever.

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Uh, but

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after analyzing the way that these things,

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and then follow up lots of other papers, which came in once

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that set the ball rolling.

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And the main, approach,

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what I can see is that it involves what is,

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called the price support properties

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of the optimal program.

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And that is really correspond to the notion

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of competitive prices somehow, becoming associated

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with the Pareto optimal, allocation

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of resources in a static problem.

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So these are the ones that are doing it

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for the dynamic problems.

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So in other words, in every period,

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the initial terminal stock of capital,

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which is being chosen, that would be in some sense

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value maximizing at those prices, okay?

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The supporting prices,

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and, now it, it,

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these supporting prices.

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So in other words, is this maximization,

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short period maximization is basically something like Oilers

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variation problem, but in a discrete setting.

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And those equations are all inequalities depending upon,

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the situation

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and that they are called the oiler equations just

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after oiler who first talked about it.

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And, but those by itself, of course, do not pinpoint the

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optimal programs because in oiler situation,

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variation problem, those were finite horizon.

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So at the end of the planning horizon, there is some target

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that has to be met.

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And so that provides a, a thing

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to tie up the whole thing,

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and that determines then the optimal path.

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Whereas in this kind of problem,

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it is a infinite horizon problem.

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There is no terminal date, for obvious reasons

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that you can't really figure out what that would be.

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And, so it, it requires a further

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kind of a terminal condition,

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but a limiting condition on the value of the capital stocks,

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usually in terms of these prices

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and their limiting behavior.

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So under certain conditions, then that together with the,

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those price, price support is going to identify

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an optimal program.

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So this is then used

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in analyzing long-term behavior.

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In particular, it was found that it has the,

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quite remarkable property that

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all optimal parts are going to come closer to the

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optimal stationary state.

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There is a stationary program, which if you start with

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that stock, it doesn't change the stock in the future,

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the optimal program, and they all sort of

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towards that, optimal stationary state.

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So in other words, the long-term behavior of the thing,

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in some ways is very well defined, okay?

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Uh, and that is what is called typically

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the turnpike theorem.

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Yeah. Now, when we move from that

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into, a model where there is some discounting

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of the utilities taking place, okay?

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It's not just simply the straightforward, sum,

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but, we have a discount factor, which is used to

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like we do for, you know, finance models kind of thing,

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and that constant dis rate or discount factor,

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and that is added up. So this,

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Uh, in some ways, this problem is much easier to sort of,

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you know, deal with

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because there's no problem of that, divergency,

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et cetera, that kind of thing.

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But, it was, it was thought

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that maybe the long-term behavior would be something similar

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thought for some while,

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but then it was gradually discovered that no,

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there are examples where one could provide a,

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cyclical behavior, in fact,

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that was investigated quite a bit.

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So there could be optimum cycles, a two period

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or multi period cycles instead of, a single

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unique optimal stationary state.

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So there, of course, the, you know,

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convergence theorem is not going to take place,

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but, in what was then seen is that,

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when the discount factor is very close to one, so

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that the discounting is,

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there is almost no discounting taking place in that case,

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the model behaves in a similar fashion

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as the un discounted case,

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and there is a, turnpike theorem, which will be valid

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for stocks starting in the neighborhood of that,

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and further the results which are

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of interest in that direction.

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But the, in general, that is not going to be the case.

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So for smaller discount factors, there is,

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this would not be valid, okay?

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Now, in, in these things, all of them,

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this price support property together with the trans, the,

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so-called limiting condition is called the

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ality condition.

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Uh, that together is the one,

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the main state people had been looking at the optimum

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stationary program

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by looking at the price support properties and so forth.

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And then finally, you know, the, the,

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all the developments took, took place in that way.

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Now, let me come to the actual problem

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that we are discussing in the paper itself.

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I said before, it is a forest management problem where,

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which was first really investigated properly by

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Martin Faustman, and this was in Germany,

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and it was very long ago.

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I mean, it was in the mid 19th century,

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some somewhere thereabouts.

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And this faustman, what what he saw is

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that very often people really did not analyze the question

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as to what may be the best way of exploiting a forest.

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So give, let's, in other words, the forest,

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the trees are providing timber content.

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So making some assumptions,

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obviously about the timber content

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and how that, changes with, you know,

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different ages and so forth.

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Uh, the question is that what would be the best way

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I i, in order to make use of the forest in the sense

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of this optimal growth model,

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but his utility function is taken to be linear,

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whereas the utility function,

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generally speaking in all these models, is taken

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to be non-linear, okay?

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And, non-linear

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and sort of decreasing marginal utilities,

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the usual kinda stuff in economics.

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And, now, which of course means that

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what in control theory,

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optimal control theory is very often referred to

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as a bang bang solution.

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So it is either this extreme

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or that extreme, that kind of a thing.

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And what fasman, he found

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that many people just simply took the problem with,

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the question, when to cut the forest, and that was it.

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They didn't consider what the land could be used in future,

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whereas faus wind did very explicitly as to what the value

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of the land may be in the future if it were

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to be used in the same kind of fashion.

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And so that set up his dynamic optimization problem

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for which he found that the solutions would be of that kind,

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that extreme behavior, you know, so the land, at some point,

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the mature trees would be all cut down, okay?

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Let's say that it just starts with zero put in,

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saplings grows to maturity within a certain period,

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and then it is all cut down, the timber is harvested

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and another replanting occurs.

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The optimal program would proceed in such

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a cyclical fashion.

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Also, a and, of course,

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this cyclical thing is going

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to be a little bit more complicated if, the forest

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that the land to begin with is not,

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only mature trees,

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but it has got trees of different, you know, lifespan

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that it has covered.

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So that obviously introduces some more,

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a little bit more complexity in the problem,

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but generally speaking, the idea is still the same,

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that there is this cycle, et cetera.

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But we have to keep in mind that the, the,

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the utility function that is being taken,

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that is really something like profit maximization

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by a competitive firm, that kind of thing.

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So it's a linear function.

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So what we, what then later happened,

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this problem was then not really very much followed up

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as we have seen some references in, people who are

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numerically trying to, get some,

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so numerically meaning approximate numerical calculation,

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simulated things which, using,

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dynamic programming methods,

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but not really the theoretical part

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investigated at any length, which was done by,

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Murray Kemp and Associates.

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And then on their suggestion, Henry won

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at Cornell and Top Metra at Cornell,

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they worked out this problem.

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And now that I've mentioned top Metro's name,

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I should simply say that, this paper

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as well as many of the joint work that I have done

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with Pon in the past, that this was very much influenced

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by his, you know, his expertise,

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and he was a friend, a colleague as well as a mentor.

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So when he passed away not too long ago,

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unfortunately at a, you know, reasonably young

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youthful age kind of thing,

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but, we really had a severe loss in,

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in that form.

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Uh, but I just wanted to mention

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that like in all the other papers, most of them that it was,

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he was the prime driver behind this,

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this idea in this paper.

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Okay? So I wish he were there, here to explain matters.

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He would've probably done a much better job than I'm doing,

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but I'll have to go on site.

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So, having said that, so he

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and Henry won, they worked on this problem

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and they found in the Undiscounted model,

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but they studied it in the same kind

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of a reduced form framework, okay?

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Basically the optimized the load model, it could be

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cast always, the special example could always be cast in

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that kind of a very general formulation.

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So they started it that,

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and using of course, the price support kind of properties,

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but there's special things, you know, in terms of

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optimal stationary programs

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and the price supports, et cetera.

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Uh, but using those, they found that,

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in the linear case, if the utility function is linear, then

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Mann's conclusion is valid.

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It is a cyclical process where its sort of bang, bang kind

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of a solution.

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But if it is a nonlinear utility function, a strictly

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concave utility function, so that

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marginal utilities are decreasing, then in that case

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that over time these, optimum parts are going

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to exhibit it on bike property, the way

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that theorem is usually referred to as the, convergence

15:05.045 --> 15:08.165
to the optimum stationary status, the turnpike theorem,

15:08.745 --> 15:11.645
it would exhibit that turnpike property, okay?

15:12.545 --> 15:14.685
Now, that was in the Undiscounted case,

15:14.825 --> 15:19.245
but in the discounted case, they found that, they

15:19.775 --> 15:21.605
tried the similar kind of methods,

15:21.735 --> 15:24.485
which is the price support properties, found

15:24.485 --> 15:27.205
that there is some optimum stationary program.

15:27.355 --> 15:29.205
They could characterize that well,

15:29.745 --> 15:34.005
but then beyond that, they could not make too much

15:34.005 --> 15:38.645
of a headway except to find some examples where

15:39.365 --> 15:41.285
I think that they would see immediately

15:41.285 --> 15:42.885
that there would be some problems

15:43.315 --> 15:47.365
with the long-term behavior, which is not going to be,

15:47.765 --> 15:51.125
convergent to the optimal stationary state, okay?

15:51.945 --> 15:53.245
But they left it at that,

15:53.545 --> 15:57.485
and this is where PON had the idea

15:58.115 --> 16:02.045
that instead of trying those conventional approaches

16:02.075 --> 16:03.685
with the price support property

16:03.705 --> 16:07.885
and so forth, that maybe we should give it a go, especially

16:08.625 --> 16:11.835
in a very stripped down,

16:12.665 --> 16:14.755
very basic framework of the model,

16:14.805 --> 16:17.955
which essentially was devised by Henry W

16:18.495 --> 16:21.815
and investigating that context

16:23.525 --> 16:27.275
using, Richard Mann's approach

16:27.455 --> 16:30.555
to optimal control problems, which is,

16:30.905 --> 16:32.795
that there is a value function.

16:33.295 --> 16:36.155
The value function is something which is

16:36.675 --> 16:38.115
summarizing the total value

16:38.115 --> 16:41.075
that can be generated from the initial stock.

16:41.815 --> 16:45.755
And that is going to satisfy in a, a kind

16:45.755 --> 16:49.235
of a particular, what he calls optimality principle,

16:49.855 --> 16:53.835
and then it gives rise to the optimum policy function.

16:54.575 --> 16:56.635
Now, normally in these things,

16:56.705 --> 17:00.395
they have not really looked at the policy function

17:00.395 --> 17:03.955
because the policy function is going to give us

17:04.225 --> 17:06.955
that if you start with a certain initial stock,

17:07.505 --> 17:10.635
what should be the next initial stock

17:10.655 --> 17:14.435
or the terminal stock, that it should reach the optimum one,

17:14.855 --> 17:17.235
and it sort of unfolds that way.

17:17.295 --> 17:20.955
So if we, if we are, in other words, if we get

17:20.955 --> 17:24.595
to know about the policy function in some detail,

17:24.945 --> 17:29.635
then one can have a much better handle over how the,

17:30.125 --> 17:32.995
paths are going to behave over time, okay?

17:33.935 --> 17:37.905
And as it happens, he,

17:38.285 --> 17:41.845
we found that the,

17:44.185 --> 17:47.235
because of the very simple, simple structure

17:47.375 --> 17:48.835
of the problem, but it does

17:48.835 --> 17:50.725
Have the main elements of the problem

17:51.115 --> 17:55.005
because of the very simple structure of the problem,

17:55.275 --> 17:59.765
that we could find, get a very good handle on

18:00.235 --> 18:03.125
what the optimum policy function is going

18:03.125 --> 18:04.485
to look like, okay?

18:05.545 --> 18:09.485
And now this policy function has what is called a

18:10.115 --> 18:11.885
bifurcation, okay?

18:12.035 --> 18:15.805
Splitting up, into two branches, if you will.

18:16.345 --> 18:20.005
So in one of the branches, it is approaching a certain,

18:20.505 --> 18:21.725
value.

18:22.025 --> 18:24.565
And in the other branch it is, sort

18:24.565 --> 18:26.725
of following completely different.

18:27.035 --> 18:29.565
It's not like that, okay?

18:29.825 --> 18:33.835
And this, this value

18:33.865 --> 18:36.075
that it is approaching,

18:36.295 --> 18:40.355
it is something which we are calling the, fasman

18:40.995 --> 18:42.485
threshold, okay?

18:42.945 --> 18:45.325
So below that threshold,

18:45.945 --> 18:48.725
the behavior takes one kind of form.

18:49.695 --> 18:54.365
Above that threshold, what is happening is that if we start

18:54.435 --> 18:56.445
with any initial stock,

18:56.895 --> 19:00.485
which is there is a optimum stationary stock in this

19:00.485 --> 19:03.765
problem, and which is between this threshold

19:03.865 --> 19:05.525
and the optimum station stock,

19:05.795 --> 19:09.645
then they all follow two period cycles, okay?

19:09.865 --> 19:13.325
All of them. Now, that means that

19:14.145 --> 19:17.685
we can really divide, we can get a very clear picture of

19:17.685 --> 19:21.325
what is happening with the dynamics of this, problem,

19:22.105 --> 19:26.125
namely that there is a certain region, which we are calling

19:26.675 --> 19:31.365
this global attractor set the stocks,

19:31.415 --> 19:36.005
which are, you know, effectively stationary,

19:36.065 --> 19:39.125
but not really, because they are cycles starting with

19:39.125 --> 19:42.005
that stock, you get optimum cycles, okay?

19:42.945 --> 19:47.285
Uh, now if we call that the, set m

19:47.795 --> 19:51.605
then outside that on each side, basically the

19:52.285 --> 19:56.165
starting 0.1 can see, given the nature of the policy

19:56.685 --> 20:01.085
function, that the paths are going to, eventually enter,

20:01.705 --> 20:05.325
this region of aim in finite time.

20:06.035 --> 20:10.805
That means that in a manner of speaking, one could say,

20:10.805 --> 20:14.365
well, this is, looks something like the turn by theorem,

20:14.395 --> 20:18.485
because after all, there is this global attractor set,

20:18.545 --> 20:22.725
and all the parts are converging close to the set,

20:23.595 --> 20:27.295
not to a specific point, but close to the set itself.

20:28.635 --> 20:32.255
Uh, but however, having done, thought of it that way,

20:32.315 --> 20:34.935
it becomes clear that it's not quite

20:35.115 --> 20:36.535
as straightforward a set.

20:36.565 --> 20:39.055
Because what is happening is that

20:39.085 --> 20:43.575
with this particular transition dynamics, which by the way,

20:43.635 --> 20:47.845
it is obtained by using essentially the oiler fun,

20:48.125 --> 20:52.765
equations in the interior of the, of the production set

20:52.785 --> 20:57.205
or the transformation set, that it seen that,

20:57.745 --> 21:02.675
the starting from say, low levels of stock,

21:03.325 --> 21:07.275
where it would eventually up in a

21:08.195 --> 21:11.715
terminal state where it is going through two period cycles,

21:13.965 --> 21:18.655
another stock, which is pretty close by, okay, might make it

21:18.675 --> 21:19.815
as close as you wish,

21:20.595 --> 21:22.985
but it, it, it

21:23.715 --> 21:26.545
lands up also in a two period cycle,

21:26.965 --> 21:31.585
but it may be quite far apart, I guess, in dynamics,

21:32.045 --> 21:35.225
kind of thing that they're interested in.

21:35.615 --> 21:38.385
It's not like a chaotic path, which,

21:38.435 --> 21:40.545
those things have been well investigated.

21:40.895 --> 21:43.225
It's not like a chaotic path,

21:43.685 --> 21:48.385
but there is a certain degree of very extreme sensitivity

21:48.925 --> 21:50.505
to the initial conditions.

21:50.925 --> 21:54.665
So if the initial conditions change slightly, you don't know

21:54.665 --> 21:58.225
what would happen to the path later on down the road

21:58.225 --> 22:02.985
that it would make, stay far apart from where the

22:04.575 --> 22:06.195
al path is meeting up.

22:07.815 --> 22:12.265
So that is turns out to be one feature

22:12.445 --> 22:14.865
of the thing which we found to be very interesting,

22:14.965 --> 22:18.625
and, which is quite unlike anything that we have seen.

22:19.045 --> 22:21.785
It is, in other words, what,

22:22.245 --> 22:26.745
we call the history dependence of these two period cycles,

22:27.295 --> 22:30.545
depending upon what is the initial stock that history,

22:31.365 --> 22:34.385
it matters what the final state is going to be,

22:34.485 --> 22:36.185
and it might be far apart.

22:36.455 --> 22:41.145
Okay? Now, aside from that, then the other thing is

22:41.175 --> 22:46.125
that this, cutoff point, this point where the

22:46.755 --> 22:51.005
bifurcation of the policy function is taking place, the

22:51.005 --> 22:53.285
what we are referring to as X delta.

22:53.625 --> 22:58.565
So X delta is standing for that stock of mature fees,

22:59.375 --> 23:03.685
which, starting, which will be the smallest,

23:04.075 --> 23:06.845
such stock at, from which the,

23:06.905 --> 23:08.925
we have these cycles, okay?

23:09.195 --> 23:12.165
Between that and the optimum stationary state.

23:13.385 --> 23:14.925
Now one time, then

23:14.925 --> 23:19.645
because of the, in this case, again,

23:19.655 --> 23:21.445
given the structure of the problem,

23:22.145 --> 23:26.765
one can get a pretty good handle on the, value,

23:27.255 --> 23:28.965
value function in this case.

23:29.765 --> 23:34.165
Normally, it's not all together that easy to obtain,

23:34.475 --> 23:36.005
idea about the value function.

23:36.145 --> 23:39.005
It, it can, you might be able to show

23:39.005 --> 23:41.605
that it has smoothness properties, et cetera,

23:41.755 --> 23:43.565
concavity properties and whatnot.

23:44.225 --> 23:47.205
But anything coming close to

23:47.275 --> 23:51.525
what the solution may be like of the optimality principle,

23:51.595 --> 23:53.885
it's not altogether very clear at all.

23:54.475 --> 23:58.805
Whereas in this case, one can use the value function.

23:58.905 --> 24:02.325
You don't have a simple, a break solution.

24:02.595 --> 24:03.605
It's not possible

24:03.785 --> 24:06.965
unless we know actually the welfare function.

24:07.705 --> 24:11.405
But one can obtain, on the basis

24:11.545 --> 24:15.685
of this bifurcation, this, xl, at the x delta value,

24:16.425 --> 24:20.605
we can obtain a equation which involves the marginal utility

24:21.065 --> 24:25.175
at the two endpoints of the cycle, okay?

24:25.755 --> 24:28.495
Uh, x delta and one minus x delta.

24:28.755 --> 24:32.615
That's one being the, we were just,

24:32.685 --> 24:35.935
normalizing the unit of measurement so that,

24:35.935 --> 24:39.415
the mature trees in the, is all of them,

24:39.635 --> 24:43.295
it forms one unit for a particular piece of land.

24:44.075 --> 24:48.375
So what we found is that then in that case,

24:48.915 --> 24:53.495
one can obtain this very, clear looking equation,

24:53.945 --> 24:56.095
which involves those marginal utilities

24:56.395 --> 25:01.295
and which involves the discount factor delta, as well

25:01.295 --> 25:05.455
as the other technological parameters like the,

25:05.555 --> 25:09.655
growth fact, the growth rate of the productivity

25:10.155 --> 25:12.415
of the timber content of the trees.

25:13.025 --> 25:16.495
Those things enter into the equation in a very clear cut

25:16.555 --> 25:21.175
way, and one can then analyze that in order to see

25:21.435 --> 25:25.015
how that is changing as the parameters

25:25.075 --> 25:26.295
of the problem changes.

25:27.195 --> 25:32.135
Now, some of them are, straightforward in the sense

25:32.135 --> 25:33.735
that they are very clearly kind

25:33.735 --> 25:37.015
of monotone changes one direction or the other.

25:37.195 --> 25:40.975
In the case of the discount factor, it is different,

25:41.075 --> 25:44.415
in other words, for the discount factor, that x delta value,

25:44.435 --> 25:48.815
it follows a certain way where it reaches a certain

25:49.365 --> 25:52.815
maximum value, and then on either side it falls off.

25:54.205 --> 25:56.065
And when it falls off,

25:56.095 --> 25:58.945
that is when the discount factor in one end,

25:59.285 --> 26:02.705
it is going closer and closer to the value of one.

26:03.115 --> 26:04.545
There is no discounting.

26:05.135 --> 26:09.385
Well, in that case, it does approach exactly the,

26:09.965 --> 26:12.945
the, you know, the turnpike,

26:13.325 --> 26:15.465
the optimum stationary state, okay?

26:15.765 --> 26:19.345
And it would be following really the fon kind of a cycle.

26:20.325 --> 26:23.425
The other side, it is this is

26:23.425 --> 26:27.465
because in order to have a optimum stationary state,

26:27.805 --> 26:29.985
it must satisfy certain condition,

26:31.465 --> 26:33.025
a technological condition.

26:33.615 --> 26:36.785
That is to say the discounted value of the growth

26:36.845 --> 26:39.705
of productivity must be enough

26:40.095 --> 26:44.565
because sufficiently large, that larger than one, so

26:44.565 --> 26:48.565
that there is a incentive to move away from the,

26:48.995 --> 26:51.765
zero stock to grow, okay?

26:52.035 --> 26:54.845
Otherwise, all, all paths are going to be

26:55.545 --> 26:56.965
all optimal paths are going

26:56.965 --> 26:58.925
to be very trivial looking in the sense

26:58.925 --> 27:01.365
that they all generate down to zero.

27:01.755 --> 27:05.045
There's not much to be, you know, investigated over there.

27:05.705 --> 27:07.525
So in order for this thing,

27:07.555 --> 27:09.885
that condition has to be satisfied.

27:09.945 --> 27:11.965
So it has a certain lower bound,

27:12.265 --> 27:17.205
but as it approaches that, then once again, the,

27:17.635 --> 27:21.205
optimum path, the amplitude of the cycles

27:21.945 --> 27:23.005
are getting reduced

27:23.185 --> 27:27.605
and finally becoming that of zero, you know,

27:27.605 --> 27:30.125
converging on the optimal stationary state.

27:30.505 --> 27:34.485
So in that sense, over there, in those extreme cases,

27:34.625 --> 27:37.845
it is following the kind of turnpike property.

27:38.135 --> 27:42.085
There are some other comparisons in the, the,

27:42.185 --> 27:46.285
the competitive statics of it, which there's not much point,

27:46.345 --> 27:48.565
but those are two interesting features

27:48.595 --> 27:50.085
that emerge over there.

27:50.705 --> 27:53.805
And, yeah, we, we just thought that it,

27:53.985 --> 27:55.565
it contributes something

27:55.565 --> 27:57.245
because as I said,

27:57.445 --> 28:01.885
that the model is the simplest possible in some sense,

28:01.945 --> 28:05.085
if you just complicate it even a little bit

28:05.585 --> 28:09.725
by putting in some additional features about the maturity,

28:09.985 --> 28:12.525
threes, different ages, varieties

28:12.525 --> 28:16.685
and so on, so forth, who knows what kind of a further

28:17.595 --> 28:19.725
complicated behavior that can produce.

"Foundations and Developments in Optimum Growth Literature: From Ramsey to Dynamic Resource Allocation"

00:00:00 - In the optimum growth literature. This really started with, Frank Ramsey, the mathematician, with a paper on optimum savings. but that paper, which was a seminal paper, but it was not really, it did not draw too much attention despite CA's, you know, pushing for, the paper as being interesting. It did not draw too much attention at that time, and it, only attracted attention to begin with of mathematician. I also interested in economics as, David Gale
00:00:47 - so Gale worked on this problem, but in a discreet time setting. And whatever I'm going to talk about is in a discreet time setting instead of continuous stem. They all talk about in a reduced form model, where basically what it all means is that the technology set is described by a transition from initial stock of capital to final stock. And there is a social welfare function somehow defined over this, each pair. So what one is looking for is that starting
00:01:29 - which is technically feasible, which can, gave us the largest welfare, some of welfare, and in Gale's formulation as well as Ramsey's, they did not use any discounting. So it was undiscounted some, which presents some technical problems, but they caught around it with very clever. but after analyzing the way that these things, and then follow up lots of other papers, which came in once that set the ball rolling.
00:02:09 - what I can see is that it involves what is, called the price support properties of the optimal program. And that is really correspond to the notion of competitive prices somehow, becoming associated with the Pareto optimal, allocation of resources in a static problem. So these are the ones that are doing it for the dynamic problems. So in other words, in every period, the initial terminal stock of capital, which is being chosen, that would be in some sense
00:02:53 - The supporting prices, and, now it, it, these supporting prices. So in other words, is this maximization, short period maximization is basically something like Oilers variation problem, but in a discrete setting. And those equations are all inequalities depending upon, the situation and that they are called the oiler equations just after oiler who first talked about it. And, but those by itself, of course, do not pinpoint the
00:03:36 - variation problem, those were finite horizon. So at the end of the planning horizon, there is some target that has to be met. And so that provides a, a thing to tie up the whole thing, and that determines then the optimal path. Whereas in this kind of problem, it is a infinite horizon problem. There is no terminal date, for obvious reasons that you can't really figure out what that would be. And, so it, it requires a further

"Exploring Long-Term Behaviors and the Turnpike Theorem in Optimal Growth Models"

00:04:15 - but a limiting condition on the value of the capital stocks, usually in terms of these prices and their limiting behavior. So under certain conditions, then that together with the, those price, price support is going to identify an optimal program. So this is then used in analyzing long-term behavior. In particular, it was found that it has the, quite remarkable property that all optimal parts are going to come closer to the
00:04:59 - There is a stationary program, which if you start with that stock, it doesn't change the stock in the future, the optimal program, and they all sort of towards that, optimal stationary state. So in other words, the long-term behavior of the thing, in some ways is very well defined, okay? and that is what is called typically the turnpike theorem. Yeah. Now, when we move from that into, a model where there is some discounting of the utilities taking place, okay?
00:05:41 - but, we have a discount factor, which is used to like we do for, you know, finance models kind of thing, and that constant dis rate or discount factor, and that is added up. So this, in some ways, this problem is much easier to sort of, you know, deal with because there's no problem of that, divergency, et cetera, that kind of thing. But, it was, it was thought that maybe the long-term behavior would be something similar thought for some while, but then it was gradually discovered that no,
00:06:24 - cyclical behavior, in fact, that was investigated quite a bit. So there could be optimum cycles, a two period or multi period cycles instead of, a single unique optimal stationary state. So there, of course, the, you know, convergence theorem is not going to take place, but, in what was then seen is that, when the discount factor is very close to one, so that the discounting is,
00:07:05 - the model behaves in a similar fashion as the un discounted case, and there is a, turnpike theorem, which will be valid for stocks starting in the neighborhood of that, and further the results which are of interest in that direction. But the, in general, that is not going to be the case. So for smaller discount factors, there is, this would not be valid, okay? Now, in, in these things, all of them,
00:07:45 - so-called limiting condition is called the ality condition. that together is the one, the main state people had been looking at the optimum stationary program by looking at the price support properties and so forth. And then finally, you know, the, the, all the developments took, took place in that way. Now, let me come to the actual problem that we are discussing in the paper itself. I said before, it is a forest management problem where,

"Historical Perspectives on Optimal Forest Management: The Legacy of Martin Faustman"

00:08:30 - Martin Faustman, and this was in Germany, and it was very long ago. I mean, it was in the mid 19th century, some somewhere thereabouts. And this faustman, what what he saw is that very often people really did not analyze the question as to what may be the best way of exploiting a forest. So give, let's, in other words, the forest, the trees are providing timber content. So making some assumptions, obviously about the timber content
00:09:08 - different ages and so forth. the question is that what would be the best way I i, in order to make use of the forest in the sense of this optimal growth model, but his utility function is taken to be linear, whereas the utility function, generally speaking in all these models, is taken to be non-linear, okay? And, non-linear and sort of decreasing marginal utilities, the usual kinda stuff in economics. And, now, which of course means that
00:09:49 - optimal control theory is very often referred to as a bang bang solution. So it is either this extreme or that extreme, that kind of a thing. And what fasman, he found that many people just simply took the problem with, the question, when to cut the forest, and that was it. They didn't consider what the land could be used in future, whereas faus wind did very explicitly as to what the value of the land may be in the future if it were to be used in the same kind of fashion. And so that set up his dynamic optimization problem for which he found that the solutions would be of that kind,
00:10:35 - the mature trees would be all cut down, okay? Let's say that it just starts with zero put in, saplings grows to maturity within a certain period, and then it is all cut down, the timber is harvested and another replanting occurs. The optimal program would proceed in such a cyclical fashion. Also, a and, of course, this cyclical thing is going to be a little bit more complicated if, the forest that the land to begin with is not,
00:11:13 - but it has got trees of different, you know, lifespan that it has covered. So that obviously introduces some more, a little bit more complexity in the problem, but generally speaking, the idea is still the same, that there is this cycle, et cetera. But we have to keep in mind that the, the, the utility function that is being taken, that is really something like profit maximization by a competitive firm, that kind of thing. So it's a linear function. So what we, what then later happened, this problem was then not really very much followed up
00:12:00 - numerically trying to, get some, so numerically meaning approximate numerical calculation, simulated things which, using, dynamic programming methods, but not really the theoretical part investigated at any length, which was done by, Murray Kemp and Associates. And then on their suggestion, Henry won at Cornell and Top Metra at Cornell, they worked out this problem.

"Tribute to a Mentor: Reflections on Collaborative Research and Optimal Control Models"

00:12:39 - I should simply say that, this paper as well as many of the joint work that I have done with Pon in the past, that this was very much influenced by his, you know, his expertise, and he was a friend, a colleague as well as a mentor. So when he passed away not too long ago, unfortunately at a, you know, reasonably young youthful age kind of thing, but, we really had a severe loss in,
00:13:17 - but I just wanted to mention that like in all the other papers, most of them that it was, he was the prime driver behind this, this idea in this paper. Okay? So I wish he were there, here to explain matters. He would've probably done a much better job than I'm doing, but I'll have to go on site. So, having said that, so he and Henry won, they worked on this problem and they found in the Undiscounted model, but they studied it in the same kind of a reduced form framework, okay?
00:14:01 - cast always, the special example could always be cast in that kind of a very general formulation. So they started it that, and using of course, the price support kind of properties, but there's special things, you know, in terms of optimal stationary programs and the price supports, et cetera. but using those, they found that, in the linear case, if the utility function is linear, then Mann's conclusion is valid. It is a cyclical process where its sort of bang, bang kind
00:14:41 - But if it is a nonlinear utility function, a strictly concave utility function, so that marginal utilities are decreasing, then in that case that over time these, optimum parts are going to exhibit it on bike property, the way that theorem is usually referred to as the, convergence to the optimum stationary status, the turnpike theorem, it would exhibit that turnpike property, okay? Now, that was in the Undiscounted case, but in the discounted case, they found that, they
00:15:21 - which is the price support properties, found that there is some optimum stationary program. They could characterize that well, but then beyond that, they could not make too much of a headway except to find some examples where I think that they would see immediately that there would be some problems with the long-term behavior, which is not going to be, convergent to the optimal stationary state, okay? But they left it at that, and this is where PON had the idea that instead of trying those conventional approaches
00:16:03 - and so forth, that maybe we should give it a go, especially in a very stripped down, very basic framework of the model, which essentially was devised by Henry W and investigating that context using, Richard Mann's approach to optimal control problems, which is, that there is a value function. The value function is something which is summarizing the total value that can be generated from the initial stock. And that is going to satisfy in a, a kind

"Exploring the Bifurcation of the Optimum Policy Function in Dynamic Resource Management"

00:16:49 - and then it gives rise to the optimum policy function. Now, normally in these things, they have not really looked at the policy function because the policy function is going to give us that if you start with a certain initial stock, what should be the next initial stock or the terminal stock, that it should reach the optimum one, and it sort of unfolds that way. So if we, if we are, in other words, if we get to know about the policy function in some detail, then one can have a much better handle over how the,
00:17:33 - And as it happens, he, we found that the, because of the very simple, simple structure of the problem, but it does Have the main elements of the problem because of the very simple structure of the problem, that we could find, get a very good handle on what the optimum policy function is going to look like, okay? And now this policy function has what is called a bifurcation, okay?
00:18:16 - So in one of the branches, it is approaching a certain, value. And in the other branch it is, sort of following completely different. It's not like that, okay? And this, this value that it is approaching, it is something which we are calling the, fasman threshold, okay? So below that threshold, the behavior takes one kind of form. Above that threshold, what is happening is that if we start
00:18:56 - which is there is a optimum stationary stock in this problem, and which is between this threshold and the optimum station stock, then they all follow two period cycles, okay? All of them. Now, that means that we can really divide, we can get a very clear picture of what is happening with the dynamics of this, problem, namely that there is a certain region, which we are calling this global attractor set the stocks, which are, you know, effectively stationary,
00:19:39 - that stock, you get optimum cycles, okay? now if we call that the, set m then outside that on each side, basically the starting 0.1 can see, given the nature of the policy function, that the paths are going to, eventually enter, this region of aim in finite time. That means that in a manner of speaking, one could say, well, this is, looks something like the turn by theorem, because after all, there is this global attractor set,
00:20:23 - not to a specific point, but close to the set itself. but however, having done, thought of it that way, it becomes clear that it's not quite as straightforward a set. Because what is happening is that with this particular transition dynamics, which by the way, it is obtained by using essentially the oiler fun, equations in the interior of the, of the production set or the transformation set, that it seen that, the starting from say, low levels of stock,

"Exploring History Dependence and Bifurcation in Two-Period Cycles of Optimal Resource Management"

00:21:08 - terminal state where it is going through two period cycles, another stock, which is pretty close by, okay, might make it as close as you wish, but it, it, it lands up also in a two period cycle, but it may be quite far apart, I guess, in dynamics, kind of thing that they're interested in. It's not like a chaotic path, which, those things have been well investigated. It's not like a chaotic path,
00:21:48 - to the initial conditions. So if the initial conditions change slightly, you don't know what would happen to the path later on down the road that it would make, stay far apart from where the al path is meeting up. So that is turns out to be one feature of the thing which we found to be very interesting, and, which is quite unlike anything that we have seen. It is, in other words, what, we call the history dependence of these two period cycles,
00:22:31 - it matters what the final state is going to be, and it might be far apart. Okay? Now, aside from that, then the other thing is that this, cutoff point, this point where the bifurcation of the policy function is taking place, the what we are referring to as X delta. So X delta is standing for that stock of mature fees, which, starting, which will be the smallest, such stock at, from which the, we have these cycles, okay?
00:23:13 - Now one time, then because of the, in this case, again, given the structure of the problem, one can get a pretty good handle on the, value, value function in this case. Normally, it's not all together that easy to obtain, idea about the value function. It, it can, you might be able to show that it has smoothness properties, et cetera, concavity properties and whatnot. But anything coming close to what the solution may be like of the optimality principle,
00:23:54 - Whereas in this case, one can use the value function. You don't have a simple, a break solution. It's not possible unless we know actually the welfare function. But one can obtain, on the basis of this bifurcation, this, xl, at the x delta value, we can obtain a equation which involves the marginal utility at the two endpoints of the cycle, okay? x delta and one minus x delta. That's one being the, we were just,
00:24:35 - the mature trees in the, is all of them, it forms one unit for a particular piece of land. So what we found is that then in that case, one can obtain this very, clear looking equation, which involves those marginal utilities and which involves the discount factor delta, as well as the other technological parameters like the, growth fact, the growth rate of the productivity of the timber content of the trees.

"Analyzing the Impact of Discount Factors on Optimal Growth and Stationary States in Forest Management Models"

00:25:16 - way, and one can then analyze that in order to see how that is changing as the parameters of the problem changes. Now, some of them are, straightforward in the sense that they are very clearly kind of monotone changes one direction or the other. In the case of the discount factor, it is different, in other words, for the discount factor, that x delta value, it follows a certain way where it reaches a certain maximum value, and then on either side it falls off.
00:25:56 - that is when the discount factor in one end, it is going closer and closer to the value of one. There is no discounting. Well, in that case, it does approach exactly the, the, you know, the turnpike, the optimum stationary state, okay? And it would be following really the fon kind of a cycle. The other side, it is this is because in order to have a optimum stationary state, it must satisfy certain condition, a technological condition. That is to say the discounted value of the growth
00:26:40 - because sufficiently large, that larger than one, so that there is a incentive to move away from the, zero stock to grow, okay? Otherwise, all, all paths are going to be all optimal paths are going to be very trivial looking in the sense that they all generate down to zero. There's not much to be, you know, investigated over there. So in order for this thing, that condition has to be satisfied. So it has a certain lower bound, but as it approaches that, then once again, the,
00:27:21 - are getting reduced and finally becoming that of zero, you know, converging on the optimal stationary state. So in that sense, over there, in those extreme cases, it is following the kind of turnpike property. There are some other comparisons in the, the, the competitive statics of it, which there's not much point, but those are two interesting features that emerge over there. And, yeah, we, we just thought that it, it contributes something because as I said, that the model is the simplest possible in some sense,
00:28:17 - by putting in some additional features about the maturity, threes, different ages, varieties and so on, so forth, who knows what kind of a further complicated behavior that can produce.

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