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+(a01)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]

-113-

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