A very simple version of the normative model described in the text involves a simple economic growth

Question:

A very simple version of the normative model described in the text involves a simple economic growth process converging to a steady-state, where values do not change over time. ${ }^{14}$

A simple model of the climate would involve a greenhouse gas accumulation equation and a temperature change equation: $d M / d t=\beta E(t)-\delta M(t)$ and $d T / d t=\alpha[\lambda M(t)-$ $T(t)]$ where $M$ is the stock of greenhouse gases (in billions of tons), $E(t)$ is the emission level (in billions of tons per year) and $T(t)$ is the global temperature increase (in ${ }^{\circ} \mathrm{C}$ over preindustrial levels), with values of $\beta, \delta, a$, and $\lambda$ of $0.50,0.005,0.02$, and 0.003 , respectively. Note that in 2005 world GDP was about $\$ 60$ trillion and approximately 8 billion tons of carbon were emitted (in the form of carbon dioxide equivalent).

a. Using the simple climate model, what is the relationship between the steady-state emission level $\left(E^{*}\right)$ and the steady-state temperature $\left(T^{*}\right)$ ? Recall that steady-state means that neither $M$ nor $T$ are changing.

b. Convert Figure 2.2 into a plot of the damage as a function of temperature change, assuming steady-state global GDP of US $\$ 100$ trillion per year. Generate a plot of the marginal damage (\$ per degree temperature increase) as a function of the temperature rise.

c. Use the results from parts (a) and (b) to generate a plot showing the marginal damage ( $\$$ per ton of carbon emissions) as a function of the fraction of emissions controlled (assume steady-state uncontrolled emissions of 12 billions tons of carbon).

d. What level of emission control balances the marginal cost of control with the marginal damage?

Figure 2.2

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