1. Natural Resource Economics Academic year: 2015-2016 Prof. Luca Salvatici luca.salvatici@uniroma3.it Lesson 24: Optimal (harvesting) effort

2. Outline Dynamic vs. Static solution Dynamic models using E as control variable “Optimal” extinction «Micro-foundations» of the rent dissipation

3. Gordon-Schaefer model: dynamic version Maximize rent using the catch as control variable: 3 Natural Resource Economics - a.a.2015/16

4. Singular solution Differentiating the maximum principle: (1) Co-state equation: Then we eliminate the costate variable from (1) 4 Natural Resource Economics - a.a.2015/16

5. Singular solution and arbitrage condition From the equation for the singular stock: Since the singular solution is a steady-state: (P - AC)*f(x) = R it’s a perpetuity: what is its present value? [(P - AC)f]/  Interpretation: an optimal solution implies that the instantaneous profit (P – AC) is equal to the present value of the change in the sustainable rent 5 Natural Resource Economics - a.a.2015/16

6. 6 Economia delle risorse naturali a.a. 2008/09 Dynamic vs. Static solution 6 Natural Resource Economics - a.a.2015/16

7. Maximizing the Present Value of Resource Rent in a Gordon-Schaefer Model  The classical Gordon–Schaefer model presents equilibrium revenue (TR) and cost (TC), including opportunity costs of labor and capital, in a fishery where the fish population growth follows a logistic function.  Unit price of harvest and unit cost of fishing effort are assumed to be constants.  In this case, the open access solution without restrictions (OA) is found when TR=TC and no rent (abnormal profit, P=TR-TC) is obtained. Abnormal profit (here resource rent) is maximized when TR'(X)=TC'(X) (maximum economic yield, MEY).  Discounted future flow of equilibrium rent is maximized when P'(X)/d=p, where p is the unit rent of harvest and d is the discount rate. This situation is referred to as the optimal solution (OPT), maximizing the present value of all future resource rent.  The open access solution and MEY equilibriums are found to be special cases of the optimal solution, when the discount rate is infinite or null, respectively. Natural Resource Economics - a.a.2015/16

8. 8 Control variable: E Problem structure: Are we going to have bang-bang solutions? Natural Resource Economics - a.a.2015/16 8

9. Singular solution I  From the maximum principle (1)  From the costate equation (2) Natural Resource Economics - a.a.2015/16 9

10. Singular solution II Using (1) e (2) substituting out  : Natural Resource Economics - a.a.2015/16 10

11. 11 Bang-bang solutions Natural Resource Economics - a.a.2015/16

12. 12 Optimal «extinction»: costs depending on the stock  TC(x) =>  Extinction only with critical depensation: from the property rights distribution point of view, when is it more likely? Natural Resource Economics - a.a.2015/16 12

13. 13 Optimal «extinction»: costs independent from the stock  AC(y) = c, if P > c what is going to be x*(T) with free access?  Single owner with pure compensation: Given that 2bx>0, what is going to happen if a<  ? Natural Resource Economics - a.a.2015/16 13

14. Optimal «extinction»: depensation 14 Natural Resource Economics - a.a.2015/16 14

15. 15 Rent dissipation: single owner (  = 0)  Steady-state (singular solution of the optimal control) ==>  Static solution = dynamic solution ==> 15 Natural Resource Economics - a.a.2015/16

16. 16 Rent dissipation: two owners (  = 0)  Steady-state:  Solution firm 1:  Solution firm 2: Natural Resource Economics - a.a.2015/16 16

17. Economia delle risorse naturali a.a.2007/0817 Nash equilibrium Natural Resource Economics - a.a.2015/16 17

18. 18 Rent dissipation: n firms (r = 0)  In N:  n firms: steady-state  Optimal effort Natural Resource Economics - a.a.2015/16 18

19. 19 Rent dissipation: synoptic table  nEnE  1   2(2/3)  (4/3)  ......................  10(2/11)  (20/11)  .........................  infinite  0  2  19 Natural Resource Economics - a.a.2015/16