Cyclical versus non-cyclical harvesting policies in renewable resource economics

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(1)Cyclical versus non-cyclical harvesting policies in renewable resource economics Katrin Erdlenbruch, Alain Jean-Marie, Michel Moreaux, Mabel Tidball, . European Association of Environmental And Resource Economists. To cite this version: Katrin Erdlenbruch, Alain Jean-Marie, Michel Moreaux, Mabel Tidball, . European Association of Environmental And Resource Economists. Cyclical versus non-cyclical harvesting policies in renewable resource economics. 15. EAERE Annual conference, Jun 2007, Thessalonique, Greece. 18 p. �hal02822507�. HAL Id: hal-02822507 https://hal.inrae.fr/hal-02822507 Submitted on 6 Jun 2020. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(323) ' !5 ' ". ∂φ ∂φ ∂φ (t; x) = (t; x)/ (0; x) = F (φ(t; x))/F (x). ∂x ∂t ∂t τ (x, y) ! x y ! :" φ( ) !. 7++8. φ(τ (x, y); x) = y.. . 7++8 "! x5 ∂φ ∂φ ∂τ (x, y) (τ (x, y); x) + (τ (x, y); x) = 0, ∂x ∂t ∂x. ' ! 6 " %5 ∂φ ∂φ F (y) 1 1 ∂τ (x, y) = − (τ (x, y); x)/ (τ (x, y); x) = − =− . ∂x ∂x ∂t F (x) F (y) F (x). . 7++8 "! y 5 ∂φ ∂τ (x, y) (τ (x, y); x) = 1 ∂y ∂t. ∂τ 1 = . ∂y F (y). ⇒. ! τ (x, y) ! ! " '. 6 5 1 ∂τ =− ; ∂x F (x). !. τ (x, x) = 0.. τ (x, y) = A(y) − A(x). " ' !5 '. ∂τ 1 = ; ∂y F (y). 1 , A (x) = F (x) . A(x) = . y. τ (x, y) = x. x 0. 1 du F (u). 1 du. F (u). 7+,8. # )$ *. &< G(x, x¯, x0 ) = g(¯ x, x). e−rτ (x0 ,¯x) e−rτ (x,¯x)−τ (x,x0 ) = g(¯ x , x) 1 − e−rτ (x,¯x) 1 − e−rτ (x,¯x). x ≤ x0 ≤ x¯ ! ! ! '& τ (x, x¯) ! '5. = τ (x, x0 ) + τ (x0 , x¯). ln G = ln g − rτ − ln(1 − e−rτ ),. 3G. τ = τ (x, x¯).. ' !.

(324) ' 1 ∂G 1 ∂g ∂τ e−rτ = − r (x, x¯) , G ∂x g ∂x ∂x 1 − e−rτ. 1 ∂g ∂τ 1 ∂G e−rτ = − r (x, x¯) . G ∂ x¯ g ∂ x¯ ∂ x¯ 1 − e−rτ. )! ! '& τ " !5 e−rτ 1 ∂G 1 ∂g 1 = +r , G ∂x g ∂x F (x) 1 − e−rτ. 1 ∂g 1 1 1 ∂G = −r . G ∂ x¯ g ∂ x¯ F (¯ x) 1 − e−rτ. > " ! 7( 6 73B8 7338 385 1 e−rτ 1 ∂g = −r , g ∂x F (x) 1 − e−rτ. ! 6 &. 1 ∂g 1 e−rτ . =r g ∂ x¯ F (¯ x) 1 − e−rτ. 7+G8. F (¯ x) −rτ (x,¯x) ∂g ∂g =− e / . ∂x ∂ x¯ F (x). ! ! 6 ' "! 6 7J8( #

(325) . ' x0 < x¯∗ ( ' ! 3 ! % / 15 v(x) =. max. y∈[0,K],t≥0. e−rt [g(φ(t, x), y) + v(y)] ,. 7+F8. "! φ(t, x) ! '& ! &( % φ(t, x) = x¯ ' t = τ (x, x¯)( ) !" ! −rτ (x,¯ x ∗. e ) v(x) = G(x , x¯ , x) = g(¯ x ,x ) ∗ ,x∗ ) −rτ (x 1−e ∗. ∗. ∗. ∗. 1 6 7+F8 (( 15. v(x) = max ψ(x, x¯, y), x ¯,y∈[0,K]. ψ(x, x¯, y) := e−rτ (x,¯x) [g(¯ x, y) + v(y)] .. 7+28. " ! ! # ψ !' 7x∗ , x¯∗8 5 . ∗ g(¯ x∗ , x∗ ) e−rτ (y,¯x ) ∂ −rτ (x,¯ x) ∂g = 0. ψ(x, x¯, y) = e +r ∂y ∂y F (y) 1 − e−rτ (y,¯x∗ −re−rτ (x,¯x) ∂ ∂g ψ(x, x¯, y) = [g(¯ x, y) + v(y)] + e−rτ (x,¯x) =0 ∂ x¯ F (¯ x) ∂ x¯ . ∗ e−rτ (y,¯x ) −r ∗ ∗ g(¯ x, y) + g(¯ x ,x ) = 0. ⇔ F (¯ x) 1 − erτ (x∗ ,¯x∗ ). 7+C8. 7+J8. 6 7+C8 ' 7+J8 ' ! 1 ' ' /( & " ! v(x) = G(x∗ , x¯∗, x) 1 ! '& 6 3F.

(326) 5. max. y∈[0,K]. −rτ (x,¯x) [g(¯ x, y) + G(x∗ , x ¯∗ , y)] e. ∗ ∗ e−rτ (x ,¯x ) x ,x ) =e g(¯ x , x ) + g(¯ 1 − e−rτ (x∗ ,¯x∗ ) . 1 −rτ (x,¯ x∗ ) ∗ ∗ =e g(¯ x ,x ) = G(x∗ , x¯∗ , x) 1 − e−rτ (x∗ ,¯x∗ ) −rτ (x,¯ x∗ ). & !' 0 x > x¯. ∗. . ∗. ∗. ∗. ∗.

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(331) v(x) = g(x, x∗ ) + v(x).. v(x) =. ⎧ ⎨ G(x∗ , x¯∗ , x) ⎩. if x ≤ x¯∗. g(x, x∗ ) + G(x∗ , x¯∗ , x) if x > x¯∗. 32.

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