Chapter 6 Model Apps

Growth and Ideas

Explore the models of Chapter 6 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 6: Growth and Ideas

See also: Chapter 6 Data Apps, the chapter’s data without code.

Why ideas are different

Fig. 1: A fixed cost leads to increasing returns

Move the slider to change the fixed research cost; compare the dashed rays in panel (b), and read the output gain in the title.

\[ Y = \frac{X - \bar{F}}{c} \]

\[ \frac{Y}{X} = \frac{1}{c}\left(1 - \frac{\bar{F}}{X}\right) \]

where \(X\) is spending, \(Y\) the number of doses, \(\bar{F}\) the fixed cost of research and \(c = 10\) dollars the cost of each dose.

Takeaway. Once the research cost is paid, every dose costs the same, so with a fixed cost the output per dollar rises with the scale of production: doubling spending more than doubles output, and the larger the fixed cost, the bigger that gain. With no fixed cost, panel (b) is panel (a), with constant returns: the increasing returns come from the fixed cost alone.

Source. Adapted from Fig. 6.1 in Jones (2024), Macroeconomics, 6th ed.

The Romer model: growth from ideas

Fig. 2: The balanced growth path in the Romer model

Move the slider to change the importance of ideas \(\gamma\) and watch the line’s slope; switch between the ratio and linear scale with the buttons.

\[ y_t^* = (1-\bar{\ell})\left(\frac{\bar{z}\bar{\ell}N_t}{\bar{n}}\right)^{\gamma} \]

\[ g_y^* = \gamma\bar{n} \]

where \(y_t^*\) is output per person on the balanced growth path, \(\bar{z}\) research productivity, \(\bar{\ell}\) the research share, \(N_t\) the population and \(\bar{n} = 0.05\) its growth rate.

Takeaway. More people means more researchers and more ideas, and because ideas are nonrival every new one raises everyone’s income, so output per person grows at the constant rate \(\gamma\bar{n}\). A larger \(\gamma\) makes that growth faster and the straight line on the ratio scale steeper; on the linear scale the same path curves upward, because constant percentage growth adds more each year.

Source. Adapted from Fig. 6.2 in Jones (2024), Macroeconomics, 6th ed.

Fig. 3: A rise in research productivity

Move the slider to set research productivity from 2050; the dashed lines are the old and new balanced growth paths; the title reads growth rates.

\[ y_t^* = (1-\bar{\ell})\left(\frac{\bar{z}\bar{\ell}N_t}{\bar{n}}\right)^{\gamma} \]

\[ g_{At} = \bar{n}\,\frac{A_t^*}{A_t} \]

where \(\bar{z}\) changes from 1 to \(\bar{z}'\) in 2050, \(A_t\) is the stock of ideas, \(A_t^*\) its value on the new balanced growth path, \(\gamma = 0.6\) and \(\bar{n} = 0.05\).

Takeaway. A change in research productivity leaves long-run growth unchanged and only moves the balanced growth path, a level effect. Output per person does not jump in 2050, because the stock of ideas is what past research left behind: above 1, the economy then grows fast, and more and more slowly as it nears the higher path. Below 1 growth slows for a while and output settles on a lower path, and at 1 nothing changes.

Source. Adapted from Fig. 6.3 in Jones (2024), Macroeconomics, 6th ed.

Fig. 4: A rise in the research share

Move the slider to set the research share from 2050; watch output per person drop on impact and then climb toward the new dashed path.

\[ y_t = A_t^{\gamma}(1-\bar{\ell}) \]

\[ y_t^* = (1-\bar{\ell})\,\bar{\ell}^{\gamma}\left(\frac{\bar{z}N_t}{\bar{n}}\right)^{\gamma} \]

where the research share rises from \(\bar{\ell} = 0.03\) to \(\bar{\ell}'\) in 2050, \(A_t\) is the stock of ideas and \(\gamma = 0.6\).

Takeaway. On impact the stock of ideas is unchanged but fewer people make goods, so output per person drops, and then it grows fast toward a higher balanced growth path while long-run growth stays the same. The share enters that path twice, as more ideas but fewer workers, so the level effect is hump shaped: it is largest at a share of \(\gamma/(1+\gamma) = 0.375\) and shrinks beyond it, while the drop on impact keeps deepening.

Source. Adapted from Fig. 6.4 in Jones (2024), Macroeconomics, 6th ed.

Fixed costs in practice: software

Fig. 7: The word processor production function

Move the slider to change the price per copy; the black dot is the break-even point, and the title reads how many copies it takes.

\[ Y = X - 100 \quad \text{for } X > 100 \]

\[ pY = 100 + Y \quad\Longrightarrow\quad Y = \frac{100}{p - 1} \]

where \(X\) is spending and \(Y\) the number of copies, both in millions, and \(p\) the price per copy in dollars; each copy costs 1 dollar to distribute.

Takeaway. The fixed cost of creating the program makes the production function flat at zero and then rise one for one, so output rises more than in proportion to spending: increasing returns. At any price above the 1-dollar marginal cost the firm breaks even once its sales cover the fixed cost, and the lower the price, the more copies that takes, so a low price works only in a huge market.

Source. Adapted from Fig. 6.7 in Jones (2024), Macroeconomics, 6th ed., which illustrates a worked exercise.

Combining Solow and Romer

Fig. 8: Output after a rise in the investment rate

Move the slider to set the investment rate from 2050; watch growth in the title and the gap between the two dashed paths.

\[ g_y^* = \tfrac{3}{2}\gamma\bar{n} \]

\[ y_t^* = \left(\frac{\bar{s}}{g_Y^*+\bar{d}}\right)^{1/2}(A_t^*)^{3\gamma/2}\,(1-\bar{\ell}) \]

where the investment rate rises from \(\bar{s} = 0.05\) to \(\bar{s}'\) in 2050, \(g_Y^*\) is the growth rate of total output, \(\gamma = 2/3\), \(\bar{n} = 0.03\) and \(\bar{d} = 0.02\).

Takeaway. Ideas set long-run growth and capital amplifies it, while the investment rate sets only how high the balanced growth path lies. A higher rate does not move output in 2050, because capital takes time to build, but growth jumps and then slows back toward the old rate as output climbs to a higher path, a level effect as in the Solow model. At 0.05 nothing changes.

Source. Adapted from Fig. 6.8 in Jones (2024), Macroeconomics, 6th ed.