knitr::opts_chunk$set( echo = TRUE, collapse = TRUE, comment = "#>" )

The United States's Nationally Determined Contribution (NDC) to the 2015 Paris climate accord committed to reducing greenhouse gas emissions to 26--28% below 2005 levels by 2025 and with a longer-term target of reducing emissions to 80% below 2005 levels by 2050 [@us:indc.cover.note:2015].^[The Trump administration has announced its intention to withdraw from the Paris accord. The analysis in this vignette does not address that development and simply presents the implications of the pledged contribution.] This vignette will compare what these goals imply for rates of improving energy efficiency and transitioning from fossil fuels to clean energy sources. The methods will follow Roger Pielke, Jr.'s approach for both bottom-up and top-down analysis [@pielke:tcf:2010; @pielke:uk.decarbonization:2009; @pielke:japan.decarbonization:2009; @pielke:australia.decarbonization:2011].

Pielke's analysis rely on the Kaya identity: $$ F = P \times g \times e \times f, $$ where

*F*is energy-related CO~2~ emissions (here, measured in billions of metric tons),*P*is the population in billions,*g*is per-capita GDP (here, measured in thousands of dollars),*e*is the energy intensity of the economy (here, measured in quads of primary energy consumed per trillion dollars of GDP^[A quad is short for a quadrillion British Thermal Units.]),- and
*f*is the carbon intensity of the energy supply (here, measured in millions of metric tons of CO~2~ per quad).

Pielke begins his bottom-up analysis by examining projections of future population and per-capita GDP. In this vignette, I develop those projections not from demographic and economic models, but simply by extrapolating from recent trends.

Begin by loading historical values of the Kaya identity parameters for the United States:

suppressPackageStartupMessages({ library(magrittr) library(dplyr) library(stringr) library(tidyr) library(purrr) library(broom) library(knitr) library(scales) library(kayadata) }) kaya <- get_kaya_data("United States")

Let's start by looking at trends in *P*, *g*, *e*, and *f* starting in 1990.
I plot these with a log-scale on the y-axis because a constant growth rate
produces exponential growth, so it should look linear on a semi-log plot.

plot_kaya(kaya, "P", log_scale = TRUE, start_year = 1990, trend_line = TRUE, points = FALSE) + theme_bw()

plot_kaya(kaya, "g", log_scale = TRUE, start_year = 1990, trend_line = TRUE, points = FALSE) + theme_bw()

plot_kaya(kaya, "e", log_scale = TRUE, start_year = 1990, trend_line = TRUE, points = FALSE) + theme_bw()

plot_kaya(kaya, "f", log_scale = TRUE, start_year = 1990, trend_line = TRUE, points = FALSE) + theme_bw()

All of these trends look reasonable, except for *f*. Let's try *f* again, but
fitting the trend only since 2005:

plot_kaya(kaya, "f", log_scale = TRUE, start_year = 2005, trend_line = TRUE, points = FALSE) + theme_bw()

That looks better. The abrupt changes in trend at 1990 and again at 2005 illustrate the difficulties of predicting future values by extrapolating from past trends, and indicates that the extrapolations we will use here should be taken with several grains of salt.

Now let's calculate the historical trends:

vars <- c("P", "g", "e", "f") historical_trends <- map_dbl(vars, ~kaya %>% gather(key = variable, value = value, -region, -year) %>% filter(variable == .x, year >= ifelse(.x == "f", 2005, 1990)) %>% lm(log(value) ~ year, data = .) %>% tidy() %>% filter(term == "year") %$% estimate ) %>% set_names(vars) tibble(Variable = names(historical_trends), Rate = map_chr(historical_trends, ~percent(.x, 0.01))) %>% kable(align = c("c", "r"))

Next, calculate the implied rate of change of *F* under the policy. This is
not an extrapolation from history, but a pure implication of the policy goals:
From the Kaya identity, $F = G \times e \times f$, so the rates of change are
$r_F = r_G + r_e + r_f = r_G + r_{ef}$.

ref_year <- 2005 target_years <- c(2025, 2050) target_reduction <- c(0.26, 0.80) F_ref <- kaya %>% filter(year == ref_year) %$% F F_target <- tibble(year = target_years, F = F_ref * (1 - target_reduction)) %>% mutate(implied_rate = log(F / F_ref) / (year - ref_year)) F_target %>% mutate(implied_rate = map_chr(implied_rate, ~percent(.x, 0.01))) %>% rename("Target F" = F, "Implied Rate" = implied_rate) %>% kable(align = c("crr"), digits = 0)

For the bottom-up analysis of decarbonization, I will use $r_{ef}$, the implied rate of decarbonization of the economy, conditional on future economic growth following the historical trend. This is expressed in the equation $r_{ef} = r_F - r_G$, where $r_F$ is the rate of emissions-reduction implied by the policy (see above) and $r_G$ is the historical growth rate of GDP:

implied_decarb_rates <- F_target %>% transmute(year, impl_F = implied_rate, hist_G = historical_trends['P'] + historical_trends['g'], hist_ef = historical_trends['e'] + historical_trends['f'], impl_ef = impl_F - hist_G) implied_decarb_rates %>% mutate_at(vars(starts_with("hist_"), starts_with("impl_")), list(~map_chr(., ~percent(.x, 0.01)))) %>% select(Year = year, "implied F" = impl_F, "historical G" = hist_G, "implied ef" = impl_ef, "historical ef" = hist_ef ) %>% kable(align="rrrrr")

To meet the goals for `r target_years[1]`

would require increasing the rate
of reducing *ef* from
```
r (100 * filter(implied_decarb_rates, year == target_years[1])$hist_ef) %>%
formatC(format = "f", digits = 2)
```

% per year
to
```
r (100 * filter(implied_decarb_rates, year == target_years[1])$impl_ef) %>%
formatC(format = "f", digits = 2)
```

% per year:
```
r filter(implied_decarb_rates, year == target_years[1]) %>%
transmute(ratio = impl_ef / hist_ef) %$% ratio %>%
formatC(format = "f", digits = 1)
```

times faster.

To meet the goals for `r target_years[2]`

would require increasing the rate
of reducing *ef* from
```
r (100 * filter(implied_decarb_rates, year == target_years[2])$hist_ef) %>%
formatC(format = "f", digits = 2)
```

% per year
to
```
r (100 * filter(implied_decarb_rates, year == target_years[2])$impl_ef) %>%
formatC(format = "f", digits = 2)
```

% per year:
```
r filter(implied_decarb_rates, year == target_years[2]) %>%
transmute(ratio = impl_ef / hist_ef) %$% ratio %>%
formatC(format = "f", digits = 1)
```

times faster.

The top-down analysis is very similar to the bottom-up analysis, but instead of looking at the elements of the Kaya identity individually, we use predictions from macroeconomic integrated assessment models that consider interactions between population, GDP, and energy use to predict future energy demand:

top_down_trends <- get_top_down_trends("United States") top_down_trends %>% select(P, G, E) %>% mutate_all(list(~map_chr(., ~percent(.x, 0.01)))) %>% rename("P trend" = P, "G trend" = G, "E trend" = E) %>% kable(align="rrr")

In the bottom-up analysis, we calculated the implied rate of decarbonizing the economy by comparing the rate of emissions reduction implied by the policy ($r_F$) to the predicted rate of change of GDP ($r_G$). Here, in the top-down analysis, we calculate the implied rate of decarbonizing the energy supply ($r_f$) by comparing the rate of emissions-reduction implied by policy ($r_F$) to the predicted rate of growth of energy demand ($r_E$): $F = E \times f$, so $r_F = r_E + r_f$, which we rearrange to find that $r_f = r_F - r_E$.

implied_decarb_rates_top_down <- F_target %>% transmute(year, impl_F = implied_rate, top_down_E = top_down_trends$E, hist_f = historical_trends['f'], impl_f = impl_F - top_down_E) implied_decarb_rates_top_down %>% mutate_at(vars(starts_with("hist_"), starts_with("impl_"), starts_with("top_down")), list(~map_chr(., ~percent(.x, 0.01)))) %>% select(Year = year, "implied F" = impl_F, "top-down E" = top_down_E, "implied f" = impl_f, "historical f" = hist_f ) %>% kable(align="rrrrr")

To meet the goals for `r target_years[1]`

would require increasing the rate
of reducing *f* from
```
r (100 * filter(implied_decarb_rates_top_down,
year == target_years[1])$hist_f) %>%
formatC(format = "f", digits = 2)
```

% per year
to
```
r (100 * filter(implied_decarb_rates_top_down,
year == target_years[1])$impl_f) %>%
formatC(format = "f", digits = 2)
```

% per year:
```
r filter(implied_decarb_rates_top_down,
year == target_years[1]) %>%
transmute(ratio = impl_f / hist_f) %$% ratio %>%
formatC(format = "f", digits = 1)
```

times faster.

To meet the goals for `r target_years[2]`

would require increasing the rate
of reducing *f* from
```
r (100 * filter(implied_decarb_rates_top_down,
year == target_years[2])$hist_f) %>%
formatC(format = "f", digits = 2)
```

% per year
to
```
r (100 * filter(implied_decarb_rates_top_down,
year == target_years[2])$impl_f) %>%
formatC(format = "f", digits = 2)
```

% per year:
```
r filter(implied_decarb_rates_top_down,
year == target_years[2]) %>%
transmute(ratio = impl_f / hist_f) %$% ratio %>%
formatC(format = "f", digits = 1)
```

times faster.

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