Natural Resource Economics Academic year: Prof. Luca Salvatici Lesson 8: Forestry

Outline 1.Trasversality conditions 2.Forestry 3.Forest resources: the optimal single rotation 4.The optimal single rotation: Excel example

Transversality conditions We consider a variety of alternative ending conditions for continuous-time dynamic optimization problems: the state variable at the terminal time or the ending time are flexible 3 Natural Resource Economics - Luca Salvatici ( )

Transversality conditions: taxonomy STATEFixFlexible TIME Fix graph 1graph 2 Flexiblegraph 3steady-state 4 Natural Resource Economics - Luca Salvatici ( )

Vertical or Free-endpoint problems: graph 2 By vertical end point, we mean that T is fixed and X(T) can take on any value. This would be appropriate if you are managing an asset or set of assets over a fixed horizon and it doesn't matter what condition the assets are in when you reach T. When looked at from the perspective of the beginning of the planning horizon, the value that X takes on at T is free and, moreover, it has no effect on what happens in the future. So it is a fully free variable and we would maximize V over X(T). 5 Natural Resource Economics - Luca Salvatici ( )

Trasversality: optimal (final) stock Relevant for renewable resources Example: payment according to the level of stock at the end of the period Instead of X(T) = a  (T) = 0 Rationale: if the shadow price of the stock was positive, the level could not be optimal (by definition) B(X(T)) = salvage value If B(.) is positive, how does transversality condition change? 6 Natural Resource Economics - Luca Salvatici ( )

Horizontal terminal line or fixed- endpoint problem: graph 3 Economia delle risorse naturali - a.a In this case there is no fixed endpoint, but the ending state variables must have a given level. For example, you can keep an asset as long as you wish, but at the end of your use it must be in a certain state. Natural Resource Economics - Luca Salvatici ( )

Trasversality: optimal period of duration Relevant for non-renewable resources It adds both the contributions of the state variable to the objective function. i.e. the " direct" (  ) as well as the "indirect" one (value of the stock increase) 8 Natural Resource Economics - Luca Salvatici ( )

The current state of world forest resources Comprehensive assessments of the state of the world’s forest resources are published every five years by the Food and Agriculture Organisation (FAO) of the United Nations in its ‘Global Forest Resources Assessment’ series: One can partition the earth’s total land area into four categories: 1.forest land with a high density of tree cover; 2.other wooded land, that is extensively wooded, but where the density or extensiveness of trees is insufficient to warrant description by the word ‘forest’; 3.other land with tree cover, which consists of land that has substantial wooded coverage, but where the density of that coverage is fairly low or the trees are relatively small; 4.other land (that does not have forest or tree coverage); this includes agricultural land, meadows and pastures, built-up areas, barren land, and land incapable of supporting large trees or trees at anything other than very low density. The earth’s total land area is a little larger than 13 thousand million hectares. Of this, approximately 40%) consists of forest, other wooded land, or other land with tree cover. Natural Resource Economics - Luca Salvatici ( ) 9

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Characteristics of forest resources 1.While fisheries typically provide a single service, forests are multi-functional. 2.Woodlands are capital assets that are intrinsically productive. 3.Trees typically exhibit very long lags between the date at which they are planted and the date at which they attain biological maturity. 4.Unlike fisheries, tree harvesting does not involve a regular cut of the incremental growth. Forests, or parts of forests, are usually felled in their entirety. 5.Plantation forestry is intrinsically more controllable than commercial marine fishing. Tree populations do not migrate spatially, and population growth dynamics are simpler, with less interdependence among species and less dependence on relatively subtle changes in environmental conditions. 6.Trees occupy potentially valuable land. The land taken up in forestry often has an opportunity cost. 7.The growth in volume or mass of a single stand of timber, planted at one point in time, resembles that illustrated for fish populations Natural Resource Economics - Luca Salvatici ( ) 17

Definitions Even-aged forest: a forest consisting of trees of the same species and age Q = Q(t) => volume of merchantable timber Average annual yield: Q(T)/T Maximum Annual Increment (MAI): d[(Q(T)/T]/dT = 0 MAI = Q(T)/T = Q’(T) Q’(T)/T – Q(T)/T 2 = 0 => TQ’(T) – Q = 0 Example: forestry1.xls (downloadable from the website) Is it the efficient solution? Natural Resource Economics - Luca Salvatici ( ) 18

Commercial plantation forestry An economist derives the criterion for an efficient forest management and felling programme by trying to answer the following question: What harvest programme is required in order that the present value of the profits from the stand of timber is maximised? The particular aspect of this question that has most preoccupied forestry economists is the appropriate time after planting at which the forest should be felled. As always in economic analysis, the answer one gets to any question depends on what model is being used. We begin with one of the most simple forest models, the single-rotation commercial forest mdel. Natural Resource Economics - Luca Salvatici ( ) 19