Chapter 5 Model Apps

The Solow Growth Model

Explore the models of Chapter 5 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.

← Back to Chapter 5: The Solow Growth Model

See also: Chapter 5 Data Apps, the chapter’s data without code.

The Solow diagram and the steady state

Fig. 1: The Solow diagram

Move the slider to change the investment rate; watch where the two curves cross and how tall the green net-investment bar is.

\[ \Delta k_{t+1} = \bar{s}\bar{A}k_t^{1/3} - \bar{d}k_t \]

\[ k^* = \left(\frac{\bar{s}\bar{A}}{\bar{d}}\right)^{3/2} \]

where \(k\) is capital per person, \(\bar{s}\) the investment rate, \(\bar{d}\) the depreciation rate and \(\bar{A}\) productivity.

Takeaway. Capital per person grows while investment is above depreciation and shrinks while it is below, so the economy always moves toward the steady state \(k^*\) where the curves cross. A higher investment rate lifts the investment curve and moves \(k^*\) to the right; a lower one moves it to the left.

Source. Adapted from Fig. 5.1 in Jones (2024), Macroeconomics, 6th ed.

Fig. 2: The Solow diagram with output

Move the slider to change the investment rate; the black bar is steady-state consumption, and the title reads its size.

\[ y^* = \bar{A}^{3/2}\left(\frac{\bar{s}}{\bar{d}}\right)^{1/2} \]

\[ c^* = (1 - \bar{s})\,y^* \]

where \(y^*\) is steady-state output per person and \(c^*\) steady-state consumption per person.

Takeaway. Every unit of output is either invested or consumed, so consumption is the gap between the output curve and the investment curve. A higher investment rate always raises steady-state output, but it leaves a smaller share to consume: consumption rises up to an investment rate of about one third and falls beyond it.

Source. Adapted from Fig. 5.2 in Jones (2024), Macroeconomics, 6th ed.

Does the model fit the data?

Fig. 3: The Solow model’s prediction for capital across countries

Move the slider to change the depreciation rate in the model line; hover over a point to name the country.

\[ \frac{K^*}{Y^*} = \frac{\bar{s}}{\bar{d}} \]

where \(K/Y\) is the capital-output ratio, \(\bar{s}\) the investment rate and \(\bar{d}\) the depreciation rate.

Takeaway. In the steady state investment just covers depreciation, so the model predicts that countries that invest more hold more capital relative to output, along a straight line with slope \(1/\bar{d}\). The points do slope upward, as predicted. A higher depreciation rate flattens the line, and rates of about 7 to 9 percent run it through the middle of the cloud.

Sources. Data: Penn World Table 11.0, Groningen Growth and Development Centre (DOI 10.34894/FABVLR), variables csh_i (investment share) and rgdpo (real GDP); cite as Feenstra, Inklaar and Timmer (2015), “The Next Generation of the Penn World Table”, American Economic Review 105(10). Capital is our own perpetual-inventory estimate from PWT investment, with 5 percent depreciation. The straight line is the Solow model’s prediction, \(K/Y = s/d\). Prepared file: data/ch05/fig5-3.csv (see the data notes). Chart: inspired by Fig. 5.3 in Jones (2024), Macroeconomics, 6th ed.

Experiments: more investment, faster depreciation

Fig. 4: An increase in the investment rate

Move the slider to set the new investment rate; the dashed curve is the old one, and the arrow shows where capital moves.

\[ k^* = \left(\frac{\bar{s}\bar{A}}{\bar{d}}\right)^{3/2} \]

\[ \frac{k^{**}}{k^*} = \left(\frac{\bar{s}'}{\bar{s}}\right)^{3/2} \]

where \(\bar{s} = 0.4\) is the old investment rate, \(\bar{s}'\) the new one, and \(k^{**}\) the new steady state.

Takeaway. Only the investment curve moves; the depreciation line stays put. A new rate above 0.4 rotates the curve up, so at the old \(k^*\) investment exceeds depreciation and capital builds up to a higher \(k^{**}\). A rate below 0.4 runs the experiment in reverse, and at 0.4 nothing changes.

Source. Adapted from Fig. 5.4 in Jones (2024), Macroeconomics, 6th ed.

Fig. 5: Output after an increase in the investment rate

Move the slider to set the new investment rate from 2030, and switch between the ratio and linear scale with the buttons.

\[ \Delta k_{t+1} = \bar{s}'y_t - \bar{d}k_t \]

\[ \frac{y^{**}}{y^*} = \left(\frac{\bar{s}'}{\bar{s}}\right)^{1/2} \]

where \(\bar{s} = 0.4\) is the old investment rate and \(\bar{s}'\) the rate from 2030 on.

Takeaway. After 2030 output moves fastest at first and then more and more slowly, until it settles at the new steady state \(y^{**}\). A permanent change in the investment rate changes the level of output per person for good, but not its long-run growth rate, which returns to zero. A rate below 0.4 sends output down along the same kind of path, and at 0.4 nothing changes.

Source. Adapted from Fig. 5.5 in Jones (2024), Macroeconomics, 6th ed.

Fig. 6: A rise in the depreciation rate

Move the slider to set the new depreciation rate; the dashed line is the old one, and the arrow shows where capital moves.

\[ k^* = \left(\frac{\bar{s}\bar{A}}{\bar{d}}\right)^{3/2} \]

\[ \frac{k^{**}}{k^*} = \left(\frac{\bar{d}}{\bar{d}'}\right)^{3/2} \]

where \(\bar{d} = 0.10\) is the old depreciation rate, \(\bar{d}'\) the new one, and \(k^{**}\) the new steady state.

Takeaway. This time only the depreciation line moves. A new rate above 0.10 makes it steeper, so at the old \(k^*\) depreciation exceeds investment and capital falls to a lower \(k^{**}\). A rate below 0.10 does the opposite, and at 0.10 nothing changes.

Source. Adapted from Fig. 5.6 in Jones (2024), Macroeconomics, 6th ed.

Fig. 7: Output after an increase in the depreciation rate

Move the slider to set the new depreciation rate from 2030, and switch between the ratio and linear scale with the buttons.

\[ \Delta k_{t+1} = \bar{s}y_t - \bar{d}'k_t \]

\[ g_{k,t} = \bar{d}'\left[\left(\frac{k^{**}}{k_t}\right)^{2/3} - 1\right] \]

where \(g_{k,t}\) is the growth rate of capital per person and \(k^{**}\) the new steady state.

Takeaway. This is the principle of transition dynamics: the farther the economy is from its steady state, in percent, the faster it moves toward it. With a higher depreciation rate output shrinks fastest right after 2030 and then more slowly as it nears \(y^{**}\); with a lower one it grows toward a higher \(y^{**}\) in the same way, and at 0.10 nothing changes.

Source. Adapted from Fig. 5.7 in Jones (2024), Macroeconomics, 6th ed.

Population growth

Fig. 11: The Solow diagram with population growth

Move the slider to change population growth; the dashed gray line is depreciation alone, without population growth.

\[ \Delta k_{t+1} = \bar{s}y_t - (\bar{n} + \bar{d})\,k_t \]

\[ k^* = \left(\frac{\bar{s}\bar{A}}{\bar{n} + \bar{d}}\right)^{3/2} \]

where \(\bar{n}\) is the growth rate of the population.

Takeaway. New workers must be equipped with capital too, so population growth works like extra depreciation: a higher \(\bar{n}\) rotates the \((\bar{n} + \bar{d})k\) line up and lowers \(k^*\) and \(y^*\). Output per person still settles at a constant level, while total GDP keeps growing at the rate \(\bar{n}\).

Source. Adapted from Fig. 5.11 in Jones (2024), Macroeconomics, 6th ed.

A disaster: the earthquake

Fig. 12: The Solow diagram after an earthquake

Move the slider to change the share of capital destroyed; the red arrow shows the jump, and the title reads the fall in output.

\[ k_0 = (1 - \text{share})\,k^* \]

\[ \frac{y_0}{y^*} = (1 - \text{share})^{1/3} \]

where share is the fraction of the capital stock the earthquake destroys.

Takeaway. An earthquake shifts no curve: the economy just jumps left along the capital axis. Output falls by less than capital, because with diminishing returns the lost units were the least productive ones. Below \(k^*\) investment exceeds depreciation, so capital grows back to exactly the same steady state.

Source. Adapted from Fig. 5.12 in Jones (2024), Macroeconomics, 6th ed.

Fig. 13: Output per person after an earthquake

Move the slider to change the share of capital destroyed, and switch between the ratio and linear scale with the buttons.

\[ \frac{y_{2030}}{y^*} = (1 - \text{share})^{1/3} \]

\[ g_{k,t} = \bar{d}\left[\left(\frac{k^*}{k_t}\right)^{2/3} - 1\right] \]

where share is the fraction of capital destroyed in 2030 and \(g_{k,t}\) the growth rate of capital per person.

Takeaway. Output drops at once in 2030 and then grows back to the same \(y^*\), fastest right after the shock, when the economy is farthest below its steady state. A bigger disaster starts the recovery with faster growth, so it takes longer to recover from, but less than proportionally longer.

Source. Adapted from Fig. 5.13 in Jones (2024), Macroeconomics, 6th ed.