# Rt dt s o is the exhaustion condition so it equals to

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T= when there is no more resources. 0 Rt .dt = s o is the exhaustion condition (so it equals to surface under the curve). Because of the Hotelling rule u ' ( R ) c ' ( R ) is decreasing when youhaveless lesscapacity ¿ extract resources. It means R is also decreasing in time. Example No extraction cost C(r) = 0 Quadratic utility function U ( R ) = AR R 2 2 U ' ( R ) = A R

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U ' ( 0 ) = A The optimality condition: U ' ( R ) = λt = A Rt = λt Rt = λ t Hotelling rule: ° λt λt = ρ Rt = ρλ t = ρ ( A Rt ) Rt = ρ ( Rt At ) Solution : Rt = A (1- e ρ ( T t ) ) You must use the exhaustion condition, because resources are scarce. The goal is to compute T to know the date where the resources are exhaustible, when you don’t have resources anymore. The results we get at the end are: - T is an increasing function of S 0 . It the resources are very abundant, it will take longer to exhaust it. Rt 2 1 Surfaceunder the curve = s o T T. t If ρ becomes bigger, what happens with extraction? It is difficult to say but we can first look at the scarcity rent λ with this figure:
What happens with the scarcity rent λ when ρ increases? We don’t know the initial ρ but we can see the curve either way is steeper. Case 1 : λ 0 is higher and T is smaller. This case is impossible because the Surfaceunder the curve is the stock. So, it is not possible that they are the same stock. Case 2 : when ρ increases then λ 0 is lower and you exhaust the resources more quickly. If you have a big preference for the present , then you extract more and your resources will exhaust quicker.
• Fall '19
• Ecological economics, Renewable resource

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