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()
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.