# Weak versus Strong Sustainability

## Exploring the Limits of Two Opposing Paradigms, Third Edition

## Eric Neumayer

# Appendix 3: The Hotelling Rule and the Ramsey Rule in a More Complex Model

## Eric Neumayer

## Extract

The derivation of OS in Section 5.1.1, p. 127, depends on an already quite complex dynamic optimisation model where both the Hotelling and Ramsey rule are no longer as simple as introduced in Appendix 2. Here it is shown how the rules must be amended and how to interpret the amerrlments. The static and dynamic first-order conditions of interest here are (A3.i.l) (A3.L2) (A3.i.5) (A3.iLl) (A3.iL2) Plugging (A3.L2) into (A3.iL2) and rearranging gives (A3.1) Plugging (A3.i.5) into (A3.iLl) and rearranging leads to 202 Appendix 3 203 -+ FK(l-iAY)=P A Using (A3.2) and (A3.I) leads to )1. (A3.2) (A3.3) (modified Hotelling rule) This is the modified Hotelling rule for non-renewable resources. How to interpret this result? Let us start at the left-hand side of the equation. This represents the rate of change of a modified Hotelling rent in including not only price minus marginal cost but also subtracting the marginal pollution caused by resource extraction valued at marginal abatement cost. This modification is due to resource extraction causing pollution which was not included in the simpler model of Appendix 2. As concerns the right-hand side of the equation, the first term looks familiar to the interest rate. It is corrected by the expression in brackets, however. Since iAY is positive, the first tenn on the right-hand side is smaller than the uncorrected interest rate. The interest rate is corrected because it is taken into account that the accumulation of manmade capital to produce output causes pollution which lowers the...

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