Economic Growth and Environmental Sustainability: The Prospects for Green Growth

By Paul Ekins | Go to book overview

A4

APPENDIX

A4.1

Deriving the strong sustainability formula for the depletion of non-renewable resources

In a two-period sequence, where i refers to the relevant period, let:

 S0 = Stock of unextracted non-renewable resource at beginning of first period S1 = Stock of unextracted non-renewable resource at beginning of second period (excluding discoveries in first period) di = Depletion (extraction) during period Ri = Stock of renewable substitutes for the resource Ci = Stock of the non-renewable resource in use ei = Level of economic/environmental service delivered by one unit of the resource wi = Quantity of resource disposed of as waste during period ai = Multiplication factor to reflect increase in renewable substitute during period ESi = Total environmental service delivered by the resource ⇒ ES0 = e0(S0+R0+C0) S1 = S0−d0; R1=a0R0; C1=C0+d0−w0 ES1 = e1(S1+R1+C1) =e1(S0−d0+a0R0+C0+d0−w0) = e1(S0+a0R0+C0−w0) =e1(ES0/e0+(a0−1)R0−w0)

For sustainability, ES1≥ES0

⇒ e1(S0+a0R0+C0−w0)≥e0(S0+R0+C0)

⇒ w0≤(1−e0/e1)(S0+C0)+(a0−e0/e1)R0

A4.2

Deriving the formula for the depletion rate at minimum life expectancy
 Let: S = Resource reserves (stock) d = Depletion in a given time period, then L = Life expectancy =S/d Lmin = Agreed minimum life expectancy D = Discoveries

Subscripts 1,2 refer to time-periods, then

 S2 =( S1−d1)+D1 At Lmin: Lmin = S1/d1=S2/d2=[(S1−d1)+D1]/d2 ⇒ d2 = d1[(S1−d1)+D1]/S1=d1[1+(D1−d1)/S1]

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