Bottom-Up Analysis
Pielke’s analysis rely on the Kaya identity: \[ F = P \times g \times e \times f, \] where
- F is energy-related CO2 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 GDP2),
- and f is the carbon intensity of the energy supply (here, measured in millions of metric tons of CO2 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.
Recent Trends in Population and GDP
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()
#> `geom_smooth()` using formula 'y ~ x'
Trend in population for U.S. from 1990–2019
plot_kaya(kaya, "g", log_scale = TRUE, start_year = 1990, trend_line = TRUE, points = FALSE) +
theme_bw()
#> `geom_smooth()` using formula 'y ~ x'
Trend in per-capita GDP for U.S. from 1990–2019
plot_kaya(kaya, "e", log_scale = TRUE, start_year = 1990, trend_line = TRUE, points = FALSE) +
theme_bw()
#> `geom_smooth()` using formula 'y ~ x'
Trend in primary-energy intensity of the U.S. economy from 1990–2019
plot_kaya(kaya, "f", log_scale = TRUE, start_year = 1990, trend_line = TRUE, points = FALSE) +
theme_bw()
#> `geom_smooth()` using formula 'y ~ x'
Trend in the carbon-dioxide intensity of the U.S. energy supply from 1990–2019
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()
#> `geom_smooth()` using formula 'y ~ x'
Trend in the carbon-dioxide intensity of the U.S. energy supply from 2005–2019
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.
Historical Trends
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"))| Variable | Rate |
|---|---|
| P | 0.94% |
| g | 1.48% |
| e | -2.05% |
| f | -1.08% |
Implied Decarbonization Rates
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)| year | Target F | Implied Rate |
|---|---|---|
| 2025 | 4346 | -1.51% |
| 2050 | 1175 | -3.58% |
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")| Year | implied F | historical G | implied ef | historical ef |
|---|---|---|---|---|
| 2025 | -1.51% | 2.43% | -3.93% | -3.12% |
| 2050 | -3.58% | 2.43% | -6.00% | -3.12% |
Results
To meet the goals for 2025 would require increasing the rate of reducing ef from -3.12% per year to -3.93% per year: 1.3 times faster.
To meet the goals for 2050 would require increasing the rate of reducing ef from -3.12% per year to -6.00% per year: 1.9 times faster.