Global Public Goods

Abstract

Global public goods affect people all over the earth. Sandler stresses the aggregation technology of different global public goods; the mitigation of carbon dioxide is a summation public good while checking biological invasions is a weakest link public good. Simple game theory models of summation and weakest link public goods are considered. While climate change is widely perceived to be the major environmental issue for humanity, Perrings claims that biological invasions are one of the most important challenges. I consider the ingredients of a model where global change consists of climate change and linked biological invasions; the formal model is in the Appendix. A key issue in climate change is the discount rate, and Partha Dasgupta argues that if climate change is truly disastrous the discount rate could be negative, strengthening the case for acting to avoid the disaster. Chhatre and Agarwal investigated the following question: if we have an interest in sequestering carbon in the forests of developing countries, how are livelihoods likely to be affected; this involves some conceptual and empirical issues.

Appendix

In this appendix I develop a model for interlinked climate change and biological invasions. It combines the considerations considered separately in different sections of this chapter (Table A.1) .

Table A.1 Symbols and variables in the model

Symbol	Variable
R	Rich country
P	Poor country
U	Social utility
C	Aggregate consumption
ρ	Utility discount rate or rate of pure time preference
M	Emissions of greenhouse gases
Q	Gross or potential output
A	Abatement of greenhouse gases or exotic invasives
G	Stock of greenhouse gases
F	Growth of exotic invasives
S	Spread of exotic invasives
δ	Depreciation of man-made capital
K	Man-made capital
λ	Multiplier or co-state variable for stock of greenhouse gases
ω	Multiplier or co-state variable for stock of exotic invasives
μ	Multiplier or co-state variable for man-made capital
D	Damage
O	Outlay

I establish a conceptual framework using an optimal control model. Two countries, one rich (R) and another poor (P) confront the issue of exotic invasives (E) and greenhouse gases (G). Both countries get together to maximize utility (U)—a function of aggregate consumption (C)—over time; so they seek to maximize

$$ \int\limits_{0}^{\infty } {[U(C^{P} (t)} ) + U(C^{R} (t))]e^{ - \rho t} dt $$

(6.1)

subject to several dynamic constraints, relating to the capital stock (K), and G and E.

Output is determined solely by capital, K. Apart from consumption and depreciation (δ), some potential output is lost due to damages (D) because of greenhouse gases and exotic invasives. Some output is used as an outlay (O) to abate (A) greenhouse gases and exotic invasives. Suppressing time notation, we thus have

$$ {\text{dK}}^{\text{P}} /{\text{ dt }} = {\text{ Q}}^{\text{P}} \left( {{\text{K}}^{\text{P}} } \right) \, - {\text{ D}}^{\text{P}}_{\text{G}} \left( {\text{G}} \right) \, - {\text{ D}}^{\text{P}}_{\text{E}} \left( {{\text{E}}^{\text{P}} } \right) \, - {\text{ C}}^{\text{P}} - {\text{ O}}^{\text{P}}_{\text{E}} \left( {{\text{A}}^{\text{P}} {\text{E }}} \right) \, - {\text{ O}}^{\text{P}}_{\text{G}} \left( {{\text{A}}^{\text{P}}_{\text{G}} } \right) $$

(6.2)

where the superscript P stands for poor country and the subscript E and G stand for exotic invasives and greenhouse gases. The equation of motion for K^R, is symmetric to Eq. (6.2).

The stock of greenhouse gases increases with emissions (M) which are a function of output. Emissions are reduced by abatement (A) and there is some absorption (γG). Hence, dG/dt is assumed to be

$$ {\text{dG}}/{\text{dt }} = {\text{ M}}^{\text{P}} \left( {{\text{Q}}^{\text{P}} \left( {{\text{K}}^{\text{P}} } \right)} \right) \, - {\text{ A}}^{\text{P}}_{\text{G}} + {\text{ M}}^{\text{R}} \left( {{\text{Q}}^{\text{R}} \left( {{\text{K}}^{\text{R}} } \right)} \right) \, - {\text{ A}}^{\text{R}}_{\text{G}} - {{\upgamma}}\,{\text{G}} $$

(6.3)

Eiswerth and Johnson (2002) had modeled an alien invasive species following its establishment using a logistic growth function. Following them, we think of an aggregate stock of invasives (E) whose growth is density-dependent and also increased by the stock of greenhouse gases. There may also be some spread (S) from the other country, and growth of invasives may be abated. Hence, we assume that

$$ {\text{dE}}^{\text{R}} /{\text{dt }} = {\text{ F}}^{\text{R}} \left( {{\text{E}}^{\text{R}} ,{\text{G}}} \right) \, + {\text{ S}}^{\text{R}} \left( {{\text{E}}^{\text{P}} } \right) \, - {\text{ A}}^{\text{R}}_{\text{E}} $$

(6.4)

The current value Hamiltonian is:

$$ {\text{H = U}}^{\text{R}} \left( {{\text{C}}^{\text{R}} } \right){\text{ + U}}^{\text{P}} \left( {{\text{C}}^{\text{P}} } \right){ + }\lambda {\text{ dG/dt + }}\omega^{\text{R}} {\text{dE}}^{\text{R}} / {\text{dt + }}\omega^{\text{P}} {\text{dE}}^{\text{P}} / {\text{dt + }}\mu^{\text{R}} {\text{dK}}^{\text{R}} / {\text{dt + }}\mu^{\text{R}} {\text{dK}}^{\text{P}} / {\text{dt}} $$

where λ, ω and μ are co-state variables. The current value Hamiltonian can be interpreted as a performance indicator, balancing the flow of consumption benefits with the value of increases in man-made capital, the value of the increases in stock of bad greenhouse gases, and the value of exotic invasives.

Our control variables are consumption and the abatement of greenhouse gases and exotic invasives (three controls for each country).

Assuming an interior solution, there are three static efficiency conditions for the rich country, and symmetric conditions for the poor country. These are as follows:

$$ \partial U/\partial C^{R} = \mu^{R} $$

(6.5)

$$ \mu^{R} *\partial O_{G}^{R} /\partial A_{G}^{R} = - \lambda $$

(6.6)

$$ \mu^{R} *\partial O_{E}^{R} /\partial A_{E}^{R} = - \omega^{R} $$

(6.7)

In Eq. (6.5) we see that the marginal value of consumption from output needs to be balanced against the marginal value of an addition to the stock of man-made capital. Equations (6.6) and (6.7) are marginal cost should be equal to marginal damage conditions. The stocks will change, so will the marginal values and so will the shadow prices. We have dynamic efficiency or portfolio balance conditions as follows.

$$ \begin{aligned} d\mu^{R} /dt &= \rho \mu^{R} - \mu^{R} *\partial Q^{R} /\partial K^{R} + \delta \mu^{R} \\ &\quad - \lambda *\partial M^{R} /\partial Q^{R} *\partial Q^{R} /\partial K^{R} \\ \end{aligned} $$

(6.8)

$$ \begin{aligned} d\lambda /dt &= \rho \lambda + \mu^{P} *\partial D^{P} /\partial G + \mu^{R} *\partial D^{R} /\partial G \\ &\quad + \gamma \lambda - \omega^{P} *\partial F^{P} /\partial G - \omega^{R} *\partial F^{R} /\partial G \\ \end{aligned} $$

(6.9)

$$ \begin{aligned} d\omega^{R} /dt &= \rho \omega^{R} + \mu^{R} *\partial D^{R} /\partial E^{R} - \omega^{P} *\partial S^{P} /\partial E^{R} \\ &\quad - \omega^{R} *\partial F^{R} /\partial E^{R} \\ \end{aligned} $$

(6.10)

Equation (6.10) shows that the optimal rate of growth of the shadow price of the exotic invasive stock in the rich country depends on the instantaneous marginal damage caused by the exotic invasive species in the rich country, the marginal value of the spread of the invasive in the poor country due to the rich country exotic invasive stock, and the marginal value of the change in growth due to a stock size change. Equation (6.10) has a symmetrical counterpart for the poor country. Equation (6.10) and the equation of motion for the exotic invasive reflect the fact that exotic invasives are weakest-link public goods. Similarly Eq. (6.9) and the equation of motion for G reflect the summation public good nature of abatement of greenhouse gases. In Eq. (6.9) the evolution of the shadow price of G depends on marginal damages in both the rich and poor country plus the marginal values of increased exotic invasive growth in the two countries.