# The economics of climate change

Economists have been studying climate change for at least two decades, starting probably with Nordhaus’s 1992 paper. The main aim of economists is not to predict temperature increases per se but to make the connection between climate change and economic activity clear.

Specifically, economists assume that there is a trade-off between economic activity and climate change. Building models based on this assumption can help us answer more practical questions about global warming, the most important of which is what policies to combat global warming are optimal.

The trade-off between climate change and economic activity is simply that climate change is detrimental to economic activity, but economic activity is the main cause of climate change. Thus we need to strike a balance between the two. I.e. too strict emissions controls can depress economic activity, but too little control can do the same indirectly via climate change.

The model I’d like to introduce here is called the DICE model (Dynamic Integrated model of Climate and the Economy; based on Nordhaus’s 2011 paper). The model is a simple neoclassical growth model with the addition of climate-emissions contraints that make the trade-off described above part of the model. Therefore, we are talking about an optimization problem where the policy-maker can choose the emissions control rate (i.e. to what extent greenhouse gases (GHGs) should be regulated).

Therefore agents maximize their discounted sum of utilities

$\max \sum_{t=1}^{\infty} U[c(t), L(t)] (1 + \rho)^{-t}$

subject to the usual economic constraints (for instance that output is divided into consumption and investment, or the law of motion of capital) and a set of climate-emissions constraints that connect economic activity and climate change.

Output is given by

$Q(t) = \frac{1-\Lambda(t)}{1 + \Omega(t)} A_t K(t)^{\gamma}L(t)^{1-\gamma},$

where the lambda and omega functions represent the effect of climate change on output. They obviously scale output down. We’ll return to their exact functional form later.

Now let us look at the climate-emissions constraints. First, emissions are a function of output but they are scaled down by the emissions control rate (mu),

$E(t) = (1-\mu(t)) \sigma_t Q_t,$

where sigma is the uncontrolled emissions rate, that is the amount we emit if there are no regulations. Mu is a control variable of policy-makers. It represents policies that decrease emissions.

After GHGs are emitted they accumulate in different layers of the “ecosystem” (not sure if this is the right word). The model divides the ecosystem into the atmosphere (AT), the upper oceans and the biosphere (UP), and the deep oceans (LO). We emit GHGs into the atmosphere, thus the amount of GHGs in the atmosphere are

$M_{AT}(t) = E_(t) + \phi_{11} M_{AT}(t-1) + \phi_{21} M_{UP}(t-1),$

where the phi’s are the transfer rates of GHGs from one layer to another. Phi_11 in particular is the atmospheric retention rate, or the amount of GHGs that the atmosphere retains. Similar equations (without E(t) of course) exist for the other two layers:

$M_{UP}(t) = \phi_{12} M_{AT}(t-1) + \phi_{22} M_{UP}(t-1) + \phi_{32} M_{LO} (t-1)$

$M_{LO}(t) = \phi_{23} M_{UP}(t-1) + \phi_{33} M_{LO} (t-1).$

The next equation is one for radiative forcing. This is a climate science term. The point is accumulation of GHGs increases radiative forcing, which in the end is responsible for temperature changes. Based on empirical evidence from the climate science literature, radiative forcing takes the form

$F(t) = \eta \log_2 \left( \frac{M_{AT}(t)}{M_{AT}(1750)} \right) + F_{EX}(t),$

where F_EX is exogenous forcing, while the first term is forcing due to CO2. Forcing due to CO2 is considerably larger, so the exogeneity of other GHGs is not problematic.

High radiative forcing increases the temperature of the atmosphere, and then just like in case of GHG accumulation (M), these effects propagate down through the upper ocean/biosphere to the deep ocean. We have thus the following equations for temperature changes

$T_{AT}(t) = T_{AT}(t-1) + \xi_1 [F(t) - \xi_2 T_{AT}(t-1) - \xi_3 (T_{AT}(t-1) - T_{LO}(t-1))]$

$T_{LO}(t) = T_{LO}(t-1) + \xi_4 [T_{AT}(t-1) - T_{LO}(t-1)].$

So in both cases we have the temperature from the previous period appearing, and then the atmosphere is warmed by radiative forcing but it loses temperature to the lower layers too. The deep ocean on the other hand only receives temperature from the upper layers.

We’ve gone from economic activity-induced emissions to temperature changes. Now let us link the temperature back to economic activity. Recall that the production function is

$Q(t) = \frac{1-\Lambda(t)}{1 + \Omega(t)} A_t K(t)^{\gamma}L(t)^{1-\gamma},$

then let us define

$\Omega(t) = \psi_1 T_{AT}(t) + \psi_2 [T_{AT}(t)]^2$

$\Lambda(t) = \Psi(t) \theta_1 \mu_t^{\theta_2}.$

Omega can then be interpreted as the damage that temperature rise causes to economic activity. And lambda can be interpreted as the cost of fighting climate change (as it is increasing in the emissions control rate, mu). Note that both omega and lambda are inversely related to output.

Our primary control variable is then the emissions control rate, mu. Mu decreases emissions (see the equation for E(t)), but it also has a cost (see lambda above). The policy-maker therefore must find an optimal level of mu that maximizes welfare (i.e. the sum of discounted utilites).

After calibrating the model, all kinds of scenarios can be simulated to check the effects of various policies. The optimal policy, that is the solution to the optimization problem above, is therefore one that maximizes welfare, and not one that maximizes economic activity or minimizes climate change.

To get an idea of what results we can get. Assume we have the following policy scenarios:

1. Baseline: no climate change policies.
2. Optimal: the solution to the optimization problem above; maximizes economic welfare.
3. Temperature-limited: solution to the optimization problem above with the constraint that temperatures cannot rise 2°C above the 20th century average.
4. Copenhagen Accord: High-income countries implement deep emissions cuts, developing countries follow suit in 20-50 years. Implemented using national emissions caps and emissions trading within countries.
5. Copenhagen Accord (only rich): same as (4) but developing countries start participatng only in 2100.

Some of the results of the model for these scenarios are:

In case of policy (4), the total costs (including carbon taxes and permit purchases) by region over time are as follows:

From the point of view of economic welfare, we have the following results:

This shows the present value of utility (i.e. economic welfare) under each scenario. Clearly, the policy closest to the optimal one would be the Copenhagen Accord with developing countries participating in 20-50 years (4). The worst policies are inaction (1) and blindly liming temperature change to less than 2 degrees (3). The former would let emissions soar, the latter would limit economic activity too much.

Just to show how sensitive these results are to calibration though, take a look at the ranking from Nordhaus’s original paper from 1992:

Here the policy alternatives are slightly different, but one can clearly see that (ignoring geoengineering because it’s a special case) the policy closest to the optimal one is the no controls policy, i.e. (1) from above. One of the main differences is that in the 1992 paper temperature in the no controls policy only rises by around 3.5 degrees C by 2105, while in the 2011 paper the increase reaches 6 degrees C.

Of course ideally, the calibration has gotten better since 1992 and the new figures are more accurate.

So what does this tell us? Well, if we are to trust the model, then in order to maximize our welfare, we should implement some policies resembling the Copenhagen Accord and we should get developing countries to participate in the next 2-5 decades while also setting up a worldwide emissions permit market. Dream on.