Chapter 17 Model Apps

Investment

Explore the models of Chapter 17 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 17: Investment

See also: Chapter 17 Data Apps, the chapter’s data without code.

How firms choose their capital stock

Fig. 1: How much should a firm invest?

Move the slider to change the interest rate; watch the flat user-cost line shift and the desired capital stock slide along the MPK curve.

\[ MPK = \frac{1}{3}\,\bar{A}\left(\frac{L}{K}\right)^{2/3} \]

\[ MPK = R + \bar{d} - \frac{\Delta p_k}{p_k} = uc \]

\[ K^{\text{desired}} = L\left(\frac{\bar{A}}{3\,uc}\right)^{3/2} \]

where \(uc\) is the user cost of capital, \(R\) the interest rate, \(\bar{d} = 0.08\) the depreciation rate, \(\Delta p_k/p_k\) the capital gain (zero here) and \(\bar{A}\) productivity.

Takeaway. The firm adds capital as long as one more unit earns more than it costs, so it stops where the falling MPK curve meets the flat user-cost line. A higher interest rate raises the user cost point for point and moves the desired capital stock to the left, and a lower one moves it to the right; because the curve flattens out, each point of interest moves capital more when the user cost is low.

Source. Adapted from Fig. 17.1 in Jones (2024), Macroeconomics, 6th ed.

Fig. 2: An increase in the corporate income tax

Move the slider to set the corporate tax rate; the dashed line is the old user cost, and the title reads the fall in desired capital.

\[ MPK = \frac{R + \bar{d} - \Delta p_k/p_k}{1-\tau} \]

\[ K^{\text{desired}} = L\left(\frac{\bar{A}}{3\,uc}\right)^{3/2} \]

where \(\tau\) is the corporate tax rate and \(uc\) the user cost with the tax, the right side of the first equation; without the tax it is 0.10.

Takeaway. The government takes a share of what each unit of capital earns but shares none of its costs, so the tax divides the user cost by \(1-\tau\) and shifts the user-cost line up. The firm moves up its MPK curve to a smaller capital stock, which falls by a larger percentage than the tax rate itself; at a zero tax rate the two lines coincide and nothing changes.

Source. Adapted from Fig. 17.2 in Jones (2024), Macroeconomics, 6th ed.

The stock market

Fig. 3: The model’s price-earnings ratio against the U.S. stock market

Move the slider to change expected dividend growth in the model; compare the teal model line with the data and the dashed long-run average.

\[ p_s = \frac{\text{dividend}}{R - g} \]

\[ \frac{p_s}{\text{earnings}} = \frac{\text{dividend}/\text{earnings}}{R - g} \]

where \(p_s\) is the share price, \(R = 0.05\) the interest rate, \(g\) the growth rate of dividends, and the payout ratio \(\text{dividend}/\text{earnings}\) is 0.5.

Takeaway. A share is worth the present value of its dividends, so the model’s P/E ratio is the payout ratio divided by the gap \(R - g\): faster dividend growth raises it, slowly at first and then steeply. Growth of about 2 percent puts the line near the long-run average, but only growth close to the 5 percent interest rate justifies peaks like those of 1929, 1999 and the mid-2020s, which is why a bubble is hard to tell from a change in expected growth.

Source. Data: Robert J. Shiller, Online Data, cyclically adjusted price-earnings ratio (CAPE) of the S&P composite index (file ie_data.xls), formerly at econ.yale.edu. The model line is our own, from equation (10) of the chapter. Prepared file: data/ch17/fig17-3.csv (see the data notes). Chart: inspired by Fig. 17.3 in Jones (2024), Macroeconomics, 6th ed.