Chapter 3 Model Apps

An Overview of Long-Run Economic Growth

Explore the models of Chapter 3 without any code. Every chart below is live: move its slider to change one part of the model and watch the curves respond, and hover over a line to read its values.

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See also: Chapter 3 Data Apps, the chapter’s data without code.

The mathematics of growth

Fig. 3: Population over time

Move the slider to change the growth rate; watch how sharply the curve bends upward, and read the population after 100 years in the title.

\[ L_{t+1} = L_t\,(1+\bar{n}) \]

\[ L_t = L_0\,(1+\bar{n})^t \]

where \(L_t\) is the population in year \(t\), \(L_0 = 6\) billion, and \(\bar{n}\) is the constant growth rate.

Takeaway. Each year the population is multiplied by the same growth factor \(1+\bar{n}\), so the yearly increase is a fixed share of an ever larger base and the curve bends upward. A higher growth rate makes it bend more sharply, and small differences in the rate become large differences in the population after a century; at a zero rate the line stays flat.

Source. Adapted from Fig. 3.3 in Jones (2024), Macroeconomics, 6th ed.

Fig. 4: Population over time, revisited

Move the slider to change the growth rate; watch the slope of the straight line in panel (b), and read the doubling time in the title.

\[ \ln L_t = \ln L_0 + t\,\ln(1+\bar{n}) \]

\[ t_{\text{double}} = \frac{\ln 2}{\ln(1+\bar{n})} \approx \frac{70}{100\,\bar{n}} \]

where \(L_t\) is the population in year \(t\), \(L_0 = 6\) billion, and \(100\,\bar{n}\) is the growth rate in percent.

Takeaway. On a ratio scale equal vertical steps are equal percentage changes, so constant growth draws a straight line whose slope is the growth rate, with the doublings evenly spaced. A higher growth rate makes the line steeper and the doubling time shorter, and a lower one does the opposite. At every rate shown, the Rule of 70 is within half a year of the exact doubling time.

Source. Adapted from Fig. 3.4 in Jones (2024), Macroeconomics, 6th ed.

Fig. 5: Constant growth against U.S. per capita GDP on a ratio scale

Move the slider to change the growth rate of the dashed constant-growth line; watch whether it can follow the data at both ends.

\[ y_t = y_0\,(1+\bar{g})^t \]

\[ \ln y_t = \ln y_0 + t\,\ln(1+\bar{g}) \]

where \(y_t\) is GDP per person in year \(t\) and \(\bar{g}\) the growth rate of the dashed line, whose level \(y_0\) is fitted to the data.

Takeaway. On a ratio scale constant growth draws a straight line whose slope is the growth rate. U.S. GDP per person has stayed close to such a line at about 2 percent a year for more than 150 years, and the fit is closest near that rate. A slower line runs above the data in the early years and below them at the end, and a faster one does the opposite.

Sources. Data: U.S. Bureau of Economic Analysis via FRED: real GDP (GDPCA), population (B230RC0A052NBEA) and GDP deflator (A191RD3A086NBEA), from 1929; before 1929 and for every other country: Maddison Project Database 2023, GDP per capita (gdppc) and population, Groningen Growth and Development Centre (Bolt and van Zanden 2024, doi:10.34894/INZBF2). The dashed constant-growth line is our own least-squares fit. Prepared files: data/ch03/fig3-2.csv and data/ch03/maddison_world.csv (see the data notes). Chart: inspired by Fig. 3.5 in Jones (2024), Macroeconomics, 6th ed.