Multiple equilibria

[hdrake]

I came up with a simple example of multiple equilibria, which could be used to test the limits of optimization schemes (cc @danielkoll). All I had to do was decrease the exponent of the control costs (here for geoengineering and mitigation) to below p=1, which can be thought of as a crude parameterization for "economies of scale" or "learning rates". This is because marginal costs scale like p-1 and thus decrease with more deployment and so it makes more sense to put all of your eggs in one basket. In this example I tuned the parameters so that it happens that the two local minima are roughly similar, so that there are two roughly equivalent policy options.

Here is a minimal example to reproduce the plot.

# # A simple two-dimensional optimization problem

# ## Loading ClimateMARGO.jl
using ClimateMARGO # Julia implementation of the MARGO model
using PyPlot # A basic plotting package

# Loading submodules for convenience
using ClimateMARGO.Models
using ClimateMARGO.Utils
using ClimateMARGO.Diagnostics

# ## Loading the default MARGO configuration
params = deepcopy(ClimateMARGO.IO.included_configurations["default"])

# Modify parameters to make geoengeering and mitigation costs similar
params.economics.geoeng_cost *= 0.1;
params.economics.mitigate_cost *= 5;

# ### Instanciating the `ClimateModel`
m = ClimateModel(params)

# ## Brute-force parameter sweep method to map out objective function

# ### Parameter sweep
Ms = 0.:0.005:1.0;
Gs = 0.:0.005:1.0;

net_benefit = zeros(length(Gs), length(Ms)) .+ 0.; #

for (o, option) = enumerate(["adaptive_temp", "net_benefit"])
    for (i, M) = enumerate(Ms)
        for (j, G) = enumerate(Gs)
            m.controls.mitigate[t(m) .<= 2100] = zeros(size(t(m)))[t(m) .<= 2100] .+ M
            m.controls.geoeng[t(m) .>= 2150] = zeros(size(t(m)))[t(m) .>= 2150] .+ G
            ### KEY CHANGE: Significantly decrease exponent of both control costs, so that costs are now concave rather than convex functions
            net_benefit[j, i] = net_present_benefit(m, discounting=true, p=0.75, M=true, G=true)
        end
    end
end

# ### Visualizing the two-dimensional optimization problem

fig = figure(figsize=(8, 6))

subplot()
q = pcolor(Ms, Gs, net_benefit, cmap="Greys_r")
cbar = colorbar(label="Net present benefits, relative to baseline [trillion USD]", extend="both")
contour(Ms, Gs, net_benefit, colors="k", levels=20, linewidths=0.5, alpha=0.4)
grid(true, color="k", alpha=0.25)

xlabel("Emissions mitigation level [% reduction]")
xticks(0.:0.2:1.0, ["0%", "20%", "40%", "60%", "80%", "100%"])
ylabel("Geoengineering rate [% of RCP8.5]")
yticks(0.:0.2:1.0, ["0%", "20%", "40%", "60%", "80%", "100%"])
title("Cost-benefit analysis")
gcf()

[hdrake]

Note: this example requires making some small changes to the diagnostic functions to make cost exponents a keyword argument. See branch https://github.com/ClimateMARGO/ClimateMARGO.jl/tree/multiple-equilibria

[hdrake]

@CommonClimate, this is somewhat related to questions you had about exploring multiple policy equilibria. In this case I've gotten here by modifying the economic functions, but you could also introduce multiple equilibria through non-linear physics.

[CommonClimate]

Interesting. I'll take a look later this semester when I prepare that lab. Thanks for keeping that question in mind!