U.S. stocks cost 40.7 times their average earnings, close to the 1999 record

40.7 Shiller’s price-earnings ratio of the U.S. stock market in October 2026; its average since 1881 is 17.8

Is that a bubble, or are investors right to pay so much? The equation that answers it is the same one a pizza restaurant should use before it buys another oven.

Open with the number. It is the last observation of the notebook’s Fig. 3: Robert Shiller’s cyclically adjusted price-earnings ratio, the S&P 500 price index divided by the average real earnings of the previous ten years. In October 2026 it was 40.7. The record, 44.2, was set in December 1999, at the top of the dot-com boom. The average over all months since 1881 is 17.8.

Ask the room: who owns stocks, directly or through a pension fund? Would you buy at 40 times earnings? What would make that price reasonable?

The thread of the chapter: one tool, the arbitrage equation, prices a machine, a share and a house. We will come back to 40.7 in Act III with that tool in hand.

Today: three questions

• Why does investment swing so much? Its three parts, in the United States and in 35 countries, and which of them falls in a recession.
• How much capital should a firm own? The arbitrage equation, the user cost of capital and the corporate income tax.
• What are stocks and houses worth, and what can policy do? Present values, the P/E ratio, Tobin’s \(q\), house prices; then taxes, bubbles and housing.

These are the three acts of the deck and the three branches of the mind map on the title slide.

The headline result: every investment must earn what the bank pays. That one condition sets the user cost of capital, the capital stock firms want, the investment rate and the prices of stocks and houses. Because those prices rest on expectations of the future, investment is the most volatile part of spending.

Connections: Chapter 4 had the rule MPK = R; Chapter 5 took the investment rate as given; Chapter 10 showed that recessions are largely investment slumps. This chapter explains the investment rate and adds finance.

Investment swings

Act I

In macroeconomics, investment means new capital, not buying shares

“I put my savings into the stock market, so investment went up.”

Buying a share moves the ownership of existing capital from one saver to another. In the national accounts, investment is spending on new capital: equipment, buildings, software and research, new houses, and additions to inventories.

“Invest” has many meanings: in ideas, in schooling, in a stock. In the national accounts (Chapter 2) it means the accumulation of physical capital. Buying shares is saving; it becomes investment only if a firm uses the money to buy new capital.

The two meanings are not unrelated, and that is the point of the chapter: the same arbitrage argument prices a machine and a share, and Tobin’s q links the stock market to the purchase of machines.

The three parts of private investment, which we see in the next figure:

• nonresidential fixed investment: equipment and structures bought by businesses, plus intellectual property products (research and development, software, films and music);
• residential fixed investment: new housing;
• inventory investment: the change in the stock of goods produced but not yet sold.

What falls in a slump?

Which part of investment falls most when the economy goes into a recession?

From 2006 to 2009, U.S. private investment fell by about 6 points of GDP. Was it mostly business investment, housing or inventories? Write down your guess.

Collect guesses. Many students say business investment, because firms stop building factories in a recession. Others remember the housing bust. Ask for a rough split of the 6 points, not just a ranking.

Investment averages 17.8% of GDP and drops in every recession

Data: BEA, NIPA Tables 1.1.5 and 1.1.10, via FRED (GPDIA, PNFIA, PRFIA, CBIA, GDPA, A006RE1Q156NBEA, A011RE1Q156NBEA); NBER recessions (USREC). Chart inspired by Fig. 17.4 in Jones (2024), Macroeconomics, 6th ed.

How to read it: each line is a share of U.S. GDP in percent, from 1975 to 2025. Total private investment on top, then nonresidential fixed (business) investment, residential investment (housing) and inventory investment, which is small and crosses zero. Shaded bands are NBER recessions.

Do live:

1. Point at the total: about 20 percent of GDP at its peaks, near 13 percent in 2009.
2. Click “Nonresidential fixed” in the legend to hide it, and look at housing alone: the hill of the early 2000s and the cliff after 2006.
3. Look at inventories in each shaded band: below zero almost every time.
4. Move the slider to “quarterly”: the quarterly total fell to 12.7 percent of GDP in 2009Q3. That is the textbook’s “less than 13 percent”; the annual value is 13.3.

Data caveat: shares of nominal GDP, so they mix quantities and relative prices. Equipment, especially computers, got cheaper over time.

Housing explained more than half of the fall of 2006–09

• Of the 6.22-point fall in total investment from 2006 to 2009, housing accounted for 3.37 points, inventories for 1.54 and business investment for 1.31.
• Total investment went from 19.6% of GDP in 2006 to 13.3% in 2009, the lowest value since 1975.
• Residential investment peaked at 6.6% of GDP in 2005 and fell to 2.4% in 2011, less than half.

Recessions are largely investment slumps, and in 2006–09 above all a housing slump.

Answer to the prediction: housing, more than half of the fall. Business investment fell least in that episode.

Numbers: the notebook’s Fig. 4 “What to notice” and the recession table. Average of total investment, 1975 to 2025: 17.8 percent of GDP (the textbook says about 17 percent in recent decades).

Why so volatile? Investment is a decision about the future: a small change in expected returns or in the interest rate moves the capital stock firms and households want, and investment is the flow that gets them there. Act II makes that precise.

Every recession hits a different part of investment

Change from the year before to the worst year, points of GDP	Total	Business	Housing	Inventories
1979 to 1982: high interest rates	−3.16	+0.28	−2.31	−1.13
2000 to 2002: the dot-com bust	−2.22	−2.24	+0.37	−0.35
2006 to 2009: the housing bust	−6.22	−1.31	−3.37	−1.54
2019 to 2020: the pandemic	−0.47	−0.46	+0.47	−0.48

There is no single recession story: high rates hit housing, a tech bust hit business investment.

The notebook computes, for four recessions, how much each part changed from the year before the recession to its worst year, in percentage points of GDP. “Business” is nonresidential fixed investment.

Read the rows:

• Early 1980s: the Volcker disinflation pushed mortgage rates to record highs; housing fell 2.31 points while business investment even rose a little.
• 2000–02: business investment collapsed with the internet boom; housing grew.
• 2006–09: the largest fall, more than half of it housing.
• 2020: the recession lasted two months, so annual data hide most of it; housing grew as people moved and worked from home.

Ask: which part would you expect to fall first if the central bank raised interest rates sharply? Act II answers: the part with the longest life and the most borrowing, housing.

Housing usually turns down before a recession begins

• The residential share of GDP fell in the year before 11 of the 12 U.S. recessions since 1948; the exception was 2020. Total investment fell before only 8.
• Before December 2007 it fell by 1.4 points.
• But in early 2023 it fell from 4.8% to 4.0% of GDP, and no recession followed: a false alarm.

Housing is a warning light, not a forecast.

This is Practice Problem 4 of the book chapter, a test of Leamer (2007, NBER Working Paper 13428), who argued that residential investment gives the best early warning of U.S. recessions. Quarterly shares from NIPA Table 1.1.10; change from four quarters before the peak to the quarter of the peak.

Before July 1953 and March 2001 the share fell by only 0.1 point, too little to count; Leamer traces those recessions to the cut in defense spending after Korea and to the end of the internet boom.

Why housing first? Its user cost reacts most to interest rates (Act III). In 2022–23 rates jumped, housing fell, but the rest of private investment fell only from 14.2 to 13.9 percent of GDP, and the slump did not spread.

Unsold goods do not pile up in recessions: inventories shrink

“In a recession shops cannot sell their goods, so inventories pile up.”

In the data, inventory investment was negative in 1975, 1980, 1982, 1991, 2001, 2008, 2009 and 2020. Firms cut production by more than their sales fell and ran down their stocks: inventory investment is procyclical.

The myth has a grain of truth: in the first weeks of a downturn, unplanned inventories can build up. But over a year, firms respond by cutting production hard.

Three motives for holding inventories, in the textbook’s terms:

• production smoothing: produce steadily and let inventories absorb swings in sales. If this dominated, inventory investment would rise in recessions (countercyclical). It does not.
• the pipeline: parts and materials are part of production, so they grow and shrink with output.
• stockout avoidance: shops keep goods on the shelf so as not to lose customers; fewer customers, fewer goods on hand.

Davis and Kahn (2008, Journal of Economic Perspectives) document that better inventory management cut inventory-to-sales ratios, and argue, more controversially, that this helped calm the U.S. business cycle (the Great Moderation).

Beyond the United States

Were the housing boom and bust of the 2000s an American story?

Ask: which countries do you think had the largest housing booms? Students often name Spain or Ireland; some name China (not in the OECD data). Collect guesses before showing the menu.

Housing booms were far larger in Ireland and Greece than in the U.S.

Data: OECD, National Accounts at a Glance, investment (DSD_NAAG@DF_NAAG_III, DSD_NAAG@DF_NAAG_III_GFCF), and national accounts Table 1, GDP and changes in inventories (DSD_NAMAIN10@DF_TABLE1_EXPENDITURE), via the OECD Data Explorer. Chart inspired by Fig. 17.4 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the same three parts and total as the U.S. chart, built from OECD national accounts for 35 countries, from 1970 or 1995. Private investment is all fixed investment minus the government’s, plus inventories. The menu picks the country; the title reports the lowest and highest total.

Do live:

1. Start on the United States: the OECD numbers match the BEA’s within 0.4 points of GDP.
2. Choose Ireland: residential investment at 13.5 percent of GDP in 2006, twice the U.S. peak, then 2.0 percent in 2011. Then the jump of the total to 51.5 percent in 2019.
3. Choose Greece: housing at 11.0 percent in 2007; total investment down to 7.4 percent in 2017.
4. Choose Germany: no housing boom in the 2000s, and still a fall in 2009.

The vertical axis keeps the U.S. scale unless a country invests more than 25 percent of GDP.

The slump of 2009 was worldwide, with or without a housing boom

• Ireland’s residential investment reached 13.5% of GDP in 2006 and fell to 2.0% in 2011; Greece’s reached 11.0% in 2007.
• Germany had no housing boom, yet its total fell to 15.1% of GDP in 2009. Inventories fell by 1.1% of GDP, as in Japan (−1.0).
• Ireland’s jump to 51.5% of GDP in 2019 is accounting: multinationals moved their patents there, and that counts as Irish investment.

Read investment numbers with care: accounting can move them as much as new factories do.

Numbers: the notebook’s Fig. 4b “What to notice”.

Ireland is a data lesson for the policy students. Intellectual property products are part of investment, so when a multinational moves the legal ownership of its patents to an Irish subsidiary, Irish investment and GDP jump, with no new machines. Irish statisticians now publish a modified measure of national income for this reason. Chapter 2’s caution applies: always ask what a number measures.

Germany shows the second channel of 2009: a collapse in world trade and in orders for machinery, which hit business investment and inventories everywhere.

The user cost of capital

Act II

A firm can put its cash in the bank or into a machine

\[\underset{\text{deposit } p_k \text{ in the bank}}{R\cdot p_k} \;=\; \underset{\text{buy the machine, sell it after a year}}{MPK + \Delta p_k}\]

• The bank pays the interest rate \(R\) on the \(p_k\) dollars.
• The machine pays its marginal product, plus a capital gain (or loss) when it is sold.
• If one paid more, moving a dollar into it would raise profits. A profit-maximizing firm ends where both pay the same.

Two safe uses of the same money must pay the same return: the arbitrage equation.

The textbook’s example is a pizza restaurant deciding whether to buy another oven. It could keep the money in the bank instead.

Why the two sides end up equal: if the oven paid more, the firm buys another one. Diminishing returns lower the MPK of each extra oven, so buying continues until the oven pays no more than the bank. If the oven paid less, the firm owns fewer ovens, which raises the MPK.

Assumptions: no risk, so the two options are compared only by their size; diminishing returns to capital; profit maximization. The full derivation is in the book chapter’s Model overview, equations (1) and (2).

The user cost: interest, wear and tear, minus any price gain

\[MPK = \underset{\text{user cost of capital, } uc}{R + \bar{d} - \frac{\Delta p_k}{p_k}}\]

• \(R\): the interest given up. \(\bar{d}\): the share of the machine that wears out. \(-\Delta p_k/p_k\): a rise in its price, recovered when it is sold.
• Baseline: \(0.02 + 0.08 - 0 =\) 10%. Each unit of capital must produce 10 cents of output a year to be worth owning.
• Chapter 4’s rule, \(MPK = R\), is the case with no wear and a constant price.

Invest until the last unit of capital just earns its user cost.

Divide the arbitrage equation by p_k (measure capital in units that cost one dollar) and add depreciation as its own capital loss. Then solve for MPK.

The user cost is what it costs to use one more unit of capital for one year. It does not depend on how much capital the firm owns, so in a diagram it is a flat line.

The baseline values, R = 0.02 and d̄ = 0.08, are the textbook’s. The notebook adds Ā = 0.3 and L = 1, chosen so that the desired capital stock is exactly 1.

Experiment: the interest rate rises from 2% to 4%

The user cost is 10% and the firm wants a capital stock of \(K = 1\). The interest rate rises from 2% to 4%. What is the new user cost? Does desired capital fall by a fifth, by more, or by less? Write down your guess.

A 2-point rise in R raises the user cost from 10 to 12 percent, a 20 percent rise. Students often expect desired capital to fall by the same 20 percent. Ask them to commit to “a fifth, more or less” before the figure.

The firm stops where the falling MPK curve meets the user cost

Model figure: the marginal product of capital of \(Y = \bar{A}K^{1/3}L^{2/3}\) (\(\bar{A} = 0.3\), \(L = 1\)) against the user cost \(R + \bar{d} - \Delta p_k/p_k\) with \(\bar{d} = 0.08\). Adapted from Fig. 17.1 in Jones (2024), Macroeconomics, 6th ed.

How to read it: capital K across, the return per unit of capital up. The falling curve is the MPK; the flat line is the user cost. They cross at the desired capital stock. The arrow runs from a young firm’s capital, K0 = 0.5, to the desired stock. The slider sets the interest rate from 1 to 6 percent; the title reports the user cost, desired capital and the investment needed from K0.

Do live:

1. At R = 0.02: user cost 10 percent, desired capital 1.00, investment 0.54 (0.50 to grow and 0.04 to replace wear).
2. Move to 0.04: user cost 12 percent, desired capital 0.76, investment 0.30.
3. Move to 0.01: user cost 9 percent, desired capital 1.17. Point out how far the crossing moves along the flat tail of the curve.

The Colab version also has sliders for depreciation, the expected price change and productivity.

A 2-point rise in the interest rate cuts desired capital by 24%

• Answer: the user cost rises to 12% and desired capital falls from 1.00 to 0.76, more than a fifth. Investment from \(K_0 = 0.5\) falls from 0.54 to 0.30.
• With \(Y = \bar{A}K^{1/3}L^{2/3}\), desired capital is proportional to \(uc^{-3/2}\): a 1% higher user cost lowers it by about 1.5%.
• The MPK curve flattens out, so when the user cost is low, each point of it moves the crossing a long way.

Investment is very sensitive to interest rates: one channel through which monetary policy moves the economy.

Answer to the prediction: by more than a fifth, 24 percent. The figure’s titles: R = 0.02 → 1.00; R = 0.04 → 0.76.

The notebook’s elasticity check: a 1 percent higher user cost changes desired capital by −1.481 percent, close to the elasticity of −1.5 (the exact number is not −1.5 because 1 percent is not infinitesimal).

Chapters 11 and 12 use exactly this channel: the central bank moves the real interest rate, and investment, the most interest-sensitive part of spending, moves the IS curve.

Interest, wear and price changes move the line; productivity moves the curve

A rise in …	Effect on the user cost	Desired capital	What moves
interest rate \(R\)	+1 point per point	falls	user-cost line up
depreciation rate \(\bar{d}\)	+1 point per point	falls	user-cost line up
expected capital gain \(\Delta p_k/p_k\)	−1 point per point	rises	user-cost line down
productivity \(\bar{A}\)	none	rises	MPK curve up

Ask of every shock: does it move the flat line or the falling curve?

This is the table of the book chapter’s Model overview. The habit of comparative statics: name the line that moves, its direction, and read the new crossing.

A rise in labor L also shifts the MPK curve up: more workers per machine make each machine more productive.

Ask: a new technology makes every machine 10 percent more productive. Which line moves? (The curve, up; the firm wants more capital at the same user cost.)

Cheap computers carry a high user cost

“Computers keep getting cheaper, so they are cheap capital to own.”

A falling price is a capital loss on every machine the firm already owns. If its price falls 4% a year, the user cost rises from 10% to 14% and desired capital falls to 0.60. A building whose price rises 2% a year has a user cost of only 8%.

Numbers: the notebook’s user-cost check (a capital gain of 2 percent gives 8.0 percent; a loss of 4 percent gives 14.0 percent) and Fig. 1’s “What to notice” (desired capital 0.60 with Δp/p = −0.04).

The myth confuses the price of buying a unit with the cost of using it. A laptop bought today is worth much less in a year, partly because it wears out and partly because better ones are cheaper. Structures often gain value because the land under them gets scarcer.

So, for a given MPK curve, firms should hold fewer units of fast-falling capital and more of appreciating capital. The book chapter’s “Your turn” asks how much more capital a firm wants in buildings than in computers: the answer is (0.14/0.08)^(3/2).

An expected price rise pulls investment forward

A tariff will raise machine prices 3% this year	User cost	Desired capital	Investment
No tariff: the firm holds \(K = 1\)	10%	1.00	0.08
This year: the price rise is a capital gain	7%	1.71	0.79
Next year: prices stop rising	10%	1.00	none

An announced price rise or a temporary subsidy moves investment in time: a boom now, a bust later.

This is Practice Problem 5 of the book chapter. The firm of Fig. 1 holds its desired capital, K = 1, and normally invests only the 8 percent that wears out. A tariff announced for next year makes it expect the price of its machines to rise 3 percent this year. A machine bought now can be sold dearer later, so owning it costs 3 points less: the user cost falls from 10 to 7 percent, desired capital rises to (0.10/0.07)^(3/2) = 1.71, and investment jumps to 1.71 − 0.92 = 0.79, almost ten times normal.

Next year the firm owns more than it wants: the formula gives investment of −0.57, so it buys nothing and lets depreciation shrink its capital; that takes 7 years.

The real-world echo: in early 2025, Federal Reserve staff linked a surge in U.S. imports to purchases made ahead of expected tariff increases (FOMC minutes, May 2025), and U.S. equipment investment grew at an annual rate of 22.5 percent in the first quarter in the BEA’s first estimate. Real firms face installation costs, so the boom is smaller and spread out, but the direction is the same.

A corporate tax divides the user cost by \(1-\tau\)

\[R = (1-\tau)\,MPK - \bar{d} + \Delta p_k/p_k \quad\Longrightarrow\quad MPK = \frac{R + \bar{d} - \Delta p_k/p_k}{1-\tau}\]

• The government takes a share \(\tau\) of what capital produces, but shares none of its costs.
• With \(\tau = 0.35\) the user cost rises from 10% to 15.4%.
• The tax magnifies everything else: each point added to \(R\) now adds 1.54 points to the user cost.

The textbook’s version leaves depreciation and capital gains untaxed. Real tax codes let firms deduct depreciation, often faster than machines actually wear out (bonus depreciation, expensing), which lowers the effective tax on investment. That is why the user cost depends on the whole tax code, not only the headline rate.

The notebook checks the arbitrage: with the tax, the bank pays R = 0.02 and capital pays (1 − τ)MPK − d̄ = 0.02 at the desired stock.

In logs, ln uc = ln(R + d̄ − Δp/p) − ln(1 − τ): a 1 percent fall in the after-tax share 1 − τ raises the user cost by 1 percent.

Experiment: introduce a 35% corporate tax

The economy has no corporate tax. A tax of 35% raises the user cost from 10% to 15.4%. Does the capital stock firms want fall by 35%, by less, or by more? Write down your guess.

Most students answer “by 35 percent” or “by less, because the tax only raises the user cost by half”. Ask for a number. The answer combines the two steps: the tax raises the user cost by the factor 1/0.65, and capital reacts to the user cost with an elasticity of −1.5.

The tax lifts the user-cost line, and firms move up their MPK curve

Model figure: the marginal product of capital (\(\bar{A} = 0.3\), \(L = 1\)) against the user cost \((R + \bar{d})/(1-\tau)\) with \(R = 0.02\), \(\bar{d} = 0.08\). Adapted from Fig. 17.2 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the same MPK curve as before. The lower flat line is the user cost with no tax, 10 percent; the upper one is the user cost with the tax on the slider. The firm starts at K = 1 (K initial) and moves to the new crossing (K desired). The title reports both user costs and the change in desired capital.

Do live:

1. Move the slider to 0.35: user cost 15.4 percent, desired capital 0.52, a fall of 48 percent.
2. Move to 0.21, today’s U.S. federal rate: 12.7 percent and 0.70.
3. Move to 0.50: 20 percent and 0.35, a fall of 65 percent.

In Colab, the widget has a button “Introduce the tax!” and also lets you change R.

A 35% tax cuts desired capital by almost half

• Answer: desired capital falls from 1.00 to 0.52, by 48%, more than the tax rate.
• Desired capital is proportional to \(uc^{-3/2}\), so the tax cuts it by the factor \(0.65^{3/2} \approx 0.52\).
• Getting there takes time: with no investment at all, depreciation needs 8 years to bring capital from 1.00 down to 0.52.

Because returns diminish slowly, a small rise in the required return removes a lot of capital.

Answer to the prediction: by more, 48 percent.

Numbers: Fig. 2’s title, the notebook’s check (factor 0.5240 = 0.65^(3/2)) and the while loop that counts the years of zero investment (8 years).

This is the textbook’s thought experiment of a country that moves from no tax to 35 percent. In practice, firms do not shed capital in one year; they stop replacing machines, and the capital stock drifts down. That is also why a tax cut raises investment for many years, not at once.

With diminishing returns, desired capital is proportional to \(uc^{-3/2}\)

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

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

• Baseline: \(\bar{A}/(3\,uc) = 0.3/0.3 = 1\), so \(K^{\text{desired}} = 1\) and \(Y/K = 0.3\): capital worth about 3.3 years of output.
• The capital-output ratio depends only on the user cost, not on \(\bar{A}\) or \(L\).

Step by step (Model overview, equations 4 to 6): the MPK of the Cobb-Douglas function is (1/3)ĀK^(−2/3)L(2/3), which can be written as (1/3)(Y/K) or as (1/3)Ā(L/K)^(2/3). Set the second form equal to uc, raise both sides to the power 3/2, and invert.

The textbook prints the production function as ĀK^(1/3)L(1/2) in this section; with constant returns to scale the labor exponent must be 2/3, which is what its MPK formula uses. A small teaching moment: check exponents against the formula that uses them.

The notebook verifies: desired capital 1.000, MPK there 0.1000 = user cost; Y/K = 0.3000 = 3 × uc.

The investment rate falls in proportion to the user cost

\[K_{t+1} = I_t + (1-\bar{d})K_t \quad\Longrightarrow\quad \frac{I_t}{Y_t} = \frac{g_K + \bar{d}}{3\,uc}\]

• With \(g_K = 0.02\) and \(\bar{d} = 0.08\): \(0.10/0.30 =\) 33.3% with no tax, 21.7% with the 35% tax.
• The numerator is the investment needed per unit of capital; \(3\,uc\) turns it into a share of output.
• This is a theory of the Solow model’s \(\bar{s}\): taxes, interest rates and depreciation move it through the user cost.

With the growth models and the Euler equation, this closes a long-run model of the economy.

Derivation (Model overview, equation 7): write the accumulation equation as a change, ΔK/K = I/K − d̄; split I/K into (I/Y)(Y/K); replace Y/K by 3uc and solve.

g_K = 2 percent is an illustrative long-run growth rate of capital. The notebook checks the formula against a simulation of the accumulation equation: 0.3333 and 0.2167 both ways. The tax lowers the investment rate by the factor 1 − τ = 0.65.

The textbook’s closing argument: the growth models (Chapters 5 and 6) set g_K; the Euler equation (Chapter 16) sets the long-run interest rate and so the user cost; MPK = uc sets the capital-output ratio; this equation sets the investment rate, and so the consumption share of GDP. These are the key endogenous variables of a closed economy in the long run.

Productivity raises capital, but not the investment rate

A 1% rise in …	\(K^{\text{desired}}\)	\(I/Y\)
user cost \(uc\) (from \(R\), \(\bar{d}\), a capital loss or \(\tau\))	−1.5%	−1%
productivity \(\bar{A}\)	+1.5%	0
labor \(L\)	+1%	0
the after-tax share \(1-\tau\)	+1.5%	+1%

More productive firms want more capital, but output rises in proportion, so the share invested stays the same.

Take logs of the desired-capital and investment-rate equations: the coefficients are elasticities.

Depreciation has two effects on I/Y: it raises the replacement investment in the numerator and the user cost in the denominator. The net effect has the sign of R − Δp/p − g_K. In the baseline R = g_K = 0.02 with no capital gain, so the two cancel exactly: the notebook prints I/Y = 0.3333 for d̄ = 0.05, 0.08 and 0.12. The investment rate equals capital’s exponent, 1/3.

Ask: why might a poor country with low productivity still have a high investment rate? (Its user cost or its growth rate, not its productivity, sets the rate.)

Corporate tax rates differ widely; user costs differ much less

Jurisdiction, as printed in the textbook (2022)	Tax rate	User cost
Brazil	34.0%	12.0%
Argentina	30.0%	11.3%
United States	21.0%	10.0%
Germany	15.8%	9.4%
Ireland	12.5%	9.0%
Switzerland	8.5%	8.6%
Cayman Islands	0.0%	7.9%

A 34-point spread in tax rates raises the user cost by only about half.

How the table is built: assume a user cost of 10 percent in the United States, where the rate is 21 percent. The user cost before tax is then 10 × (1 − 0.21) = 7.9 percent everywhere, and each country’s user cost is 7.9/(1 − τ). It leaves out everything else in the tax codes, such as depreciation allowances and investment credits.

The book chapter rebuilds this table from OECD Corporate Tax Statistics for every year since 2000, with a year menu. Three 2022 rates differ from the textbook, a lesson in checking data:

• Chile: the textbook’s 10 percent is the small-business rate; the general rate is 27 percent (user cost 10.8).
• South Africa: the textbook uses the 27 percent rate of 2023; in 2022 it was 28.
• Argentina: since 2021 profits are taxed in brackets of 25, 30 and 35 percent; the OECD reports the top one.

The U.S. federal rate was 35 percent until 2017 (the textbook says 34, the rate on middle incomes) and 21 percent from 2018.

Stocks, houses and policy

Act III

A share is worth the present value of its dividends

\[R = \frac{\text{dividend}}{p_s} + \frac{\Delta p_s}{p_s} \quad\Longrightarrow\quad p_s = \frac{\text{dividend}}{R - g}\]

• Same arbitrage: the bank or a share. The dividend plays the role of the MPK; the price change is the capital gain.
• If the dividend grows at \(g\), the price grows at \(g\) too, so the capital gain is \(g\).
• A dividend of $3 with \(R = 5\%\): worth $60 with no growth, $100 if it grows 2% a year.

A growing dividend is discounted at \(R - g\), not at \(R\).

The return on a share is its dividend yield plus its percentage capital gain. If a share paid more than the bank, savers would buy it, its price would rise today, and both terms would fall until the gap closed.

The formula is the present discounted value of all future dividends: D/(1+R) + D(1+g)/(1+R)^2 + … = D/(R − g), the sum of a geometric series (book Appendix C.12). The notebook checks it numerically: the sum of discounted dividends is 100.0000, the same as equation (9).

The big assumption: no risk. Real shares are risky, and much of finance is about how risk lowers prices (a risk premium raises the effective R). Keep that in mind for the bubble debate.

The P/E ratio rests on a thin gap between \(R\) and \(g\)

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

• With a payout ratio of one half, \(R = 5\%\) and \(g = 2\%\): \(0.5/0.03 \approx\) 16.7, close to the long-run average.
• Cutting \(R - g\) from 3 to 2 points, a third, raises the price by half.
• A P/E ratio of 40 needs a gap of only 1.25 points.

Small changes in expected growth or interest rates can justify very large changes in stock prices.

Divide the price formula by earnings, the firm’s accounting profits per share. The share of earnings paid out as dividends is the payout ratio.

In logs, the price has an elasticity of −1 with respect to the gap R − g. When the gap is small, a small change in it is a large percentage change. That is the key to reading Fig. 3.

Experiment: is today’s stock market a bubble?

In October 2026 the P/E ratio was 40.7. With a payout ratio of one half and \(R = 5\%\), how fast would investors need to expect dividends to grow for the model to give a ratio of about 40? Write down your guess.

Let students solve 0.5/(0.05 − g) = 40 if they like: g = 0.05 − 0.0125 = 3.75 percent. Ask the second question too: is 3.75 percent real growth of dividends forever plausible? Real GDP per person has grown about 2 percent a year over the long run (Chapter 3).

The P/E ratio swings far from its average of 17.8

Data: Robert J. Shiller, Online Data, cyclically adjusted price-earnings ratio (CAPE) of the S&P composite index, shillerdata.com; the model line is our own. Chart inspired by Fig. 17.3 in Jones (2024), Macroeconomics, 6th ed.

How to read it: Shiller’s ten-year P/E ratio, monthly since 1881, as a filled area; the dashed line is its average over all months, 17.8. The teal line is the model’s ratio, 0.5/(R − g), with R = 5 percent. The slider sets expected dividend growth g from 0 to 3.75 percent; the title shows the latest ratio and the model’s.

Do live:

1. At g = 0.02: the model line sits at 16.7, near the average.
2. Move g up step by step: 20, 25, 33.3, then 40.0 at g = 3.75 percent. The line rises slowly at first and then very fast.
3. Point at the peaks: September 1929, December 1999, and today.

Data caveat: the ratio uses ten-year average earnings to smooth the cycle; Shiller’s data are the only free long series. Other countries’ ratios are sold commercially.

Today’s prices need dividends to grow almost as fast as the interest rate

• Answer: 3.75% gives 40.0. Read the other way, a ratio of 40.7 implies \(R - g =\) 1.23 points; the long-run average, 17.8, implies 2.81.
• The ratio peaked at 32.6 in September 1929 and at its record, 44.2, in December 1999; it was below 7 in 1920 and 1982.
• The textbook’s average of 17.4 runs to March 2023; with data to October 2026 it is 17.8.

A high P/E ratio is either a bubble or optimism about growth and rates, and the data can hardly tell the two apart.

Answer to the prediction: 3.75 percent.

Numbers: the notebook’s Fig. 3 key numbers. The textbook dates the dot-com peak “in 2001”; in the data it is December 1999, and by 2001 the ratio had already fallen to between 28 and 37.

The two readings of the late 1990s, then and now: a bubble (Shiller’s “irrational exuberance”) or a “new economy” of faster growth that justified higher prices. In 2026 the same debate is about artificial intelligence. The model does not decide it; it tells us what you must believe to justify the price.

If prices already reflect what is known, only news moves them

• A market is informationally efficient if prices fully reflect all available information. Expected news is already in the price.
• Only surprises move prices, and good and bad surprises are equally likely: the price follows a random walk.
• Evidence: most actively managed funds fail to beat a simple index fund for long. But prices swing more than dividends seem to justify, and bubbles recur.

Markets are close to efficient: hard to beat, yet still prone to bubbles.

Example: if everyone expects a firm to report record profits next week, the price already includes it; when the report matches expectations, the price does not move.

The evidence. Fama (1970, Journal of Finance) set out the efficient-markets hypothesis. Malkiel (2003, Journal of Economic Perspectives), cited by the textbook, reports that more than three-quarters of comparable actively managed funds underperformed the S&P 500 in the decade to 2002, and that past winners did not keep winning. Shiller (1981, American Economic Review) showed that stock prices move far more than the later dividends would justify, “excess volatility”. Fama and Shiller shared the 2013 Nobel Prize with Hansen: the profession still argues about how to read these facts.

Why bubbles can survive: betting against one is risky, because a mispriced market can keep rising for longer than a short seller can afford to wait.

Tobin’s \(q\) compares a firm’s market value with the cost of its capital

\[q = \frac{V}{p_k K} = \frac{MPK - \bar{d}}{R} \qquad\qquad q > 1 \iff MPK > uc\]

Capital of the firm	\(MPK\)	\(q\)	What it should do
\(K = 0.5\)	0.159	3.94	invest
\(K = 1\) (desired)	0.100	1.00	hold
\(K = 1.2\)	0.089	0.43	shrink

When the market values a dollar of new capital at more than a dollar, building more creates value.

James Tobin (Nobel Prize 1981) compared the stock market value of a firm, V, the present value of its future profits, with the replacement cost of its capital.

Derivation (Model overview, equations 11 and 12): a firm that owns only capital, with p_k = 1 and no taxes, earns MPK·K, spends d̄K to replace wear, and pays the rest as a constant dividend. With g = 0 its value is (MPK − d̄)K/R. Divide by K.

Numbers: the notebook’s q check, with R = 0.02 and d̄ = 0.08. At K = 0.5, each dollar of capital adds almost four dollars of market value; the firm could issue shares, buy machines and be worth more. At K = 1.2 it is worth less than its machines; its owners would gain by selling some.

In the data \(q\) is often well above 1, and cash matters too

• Firms own brands, patents and know-how that the stock market values but \(K\) leaves out.
• Installing capital is slow and costly, so firms close a gap in \(q\) gradually.
• Studies of many firms find that tax reforms move investment through \(q\) and the user cost, as the theory says; but cash flow and access to loans also matter.
• Through \(q\), a higher interest rate also cuts investment: at \(R = 3\%\) the firm’s \(q\) falls to 0.667 and its desired capital to 0.867.

Evidence, presented as mixed (as the textbook does). Cummins, Hassett and Hubbard (1994, Brookings Papers on Economic Activity) use tax reforms as natural experiments and find investment responds to q and the user cost once measurement error is reduced. Fazzari, Hubbard and Petersen (1988, Brookings Papers on Economic Activity) find that investment of firms likely to face financing constraints moves with their cash flow, which the frictionless model leaves out; that finding was itself disputed (Kaplan and Zingales 1997, Quarterly Journal of Economics).

The last bullet is part (d) of Practice Problem 6: the firm of Fig. 1 holds K = 1 at R = 2 percent; if R rises to 3 percent, q = 0.667 and desired capital 0.867. It stops investing until depreciation has cut its capital by about 13 percent, about two years.

A house is priced like any capital good, adjusted for the mortgage

\[P^{\text{house}} = \frac{\text{rent}}{\tau\bar{x}R + (1-\tau)R + \bar{d} - \Delta P^{\text{house}}/P^{\text{house}}}\]

• The denominator is the user cost of a house: interest on the down payment and on the loan, depreciation, minus the expected price gain.
• \(\bar{x}\) is the down payment as a share of the price (leverage is \(1/\bar{x}\)); \(\tau\) is the tax rate at which mortgage interest is deducted.
• A rent of $6,000 a year, \(R = 4\%\), \(\bar{d} = 2\%\), \(\tau = 0.3\), \(\bar{x} = 0.2\) and 2% expected growth: a price of $197,368.

The arbitrage: an investor can keep her down payment in the bank, or buy a condominium with a mortgage, rent it out for a year and sell it. The house pays the rent, minus depreciation, plus the capital gain, minus the after-tax interest on the loan. Equate, divide by the price, and solve (Model overview, equation 13).

The notebook checks it: the bank and the house both pay 1,578.95 dollars on the down payment at that price.

With no tax, τ = 0, the denominator is just R + d̄ − ΔP/P, the user cost of any capital good: housing is not special, only leveraged and tax-favored.

Expected price growth moves house prices far more than cheap credit

Change from the baseline house	Price change
the down payment falls from 20% to 10% (more leverage)	+4.1%
expected price growth rises from 2% to 3% a year	+49.0%

Expected gains can feed on themselves: the arithmetic of a housing bubble.

Why so different: a smaller down payment lowers the denominator by τR per unit of x̄, a small number (0.3 × 0.04 = 0.012), so a 10-point cut lowers it by only 0.0012. One more point of expected growth lowers it by a full 0.01, about eight times as much. And the denominator is small (0.0304), so a 0.01 cut is a third of it.

If rising prices make people expect further rises, the price can climb on its own. As with stocks, it is hard to tell such a bubble from a genuine change in rents, interest rates or credit.

Ask: which policy would you use to cool a housing boom, given this table: tighter down-payment rules or something that changes expectations? (The policy slide returns to this.)

Do corporate tax cuts raise investment, and who gains?

In 2017 the U.S. cut its federal corporate tax rate from 35% to 21%. Across the 33 jurisdictions of the tax table, the average rate fell from 30.7% in 2000 to 23.8% in 2025. Who gains, and at what cost?

What the model predicts

• Desired capital: 0.52 at a 35% tax, 0.70 at 21%.
• Revenue peaks only at \(\tau = 2/3\): below it, cuts lose revenue.
• At 35%, workers lose $1.37 of wages per $1 collected.

What the evidence adds

• After 2017, firms with bigger cuts invested more.
• Workers bear about half the tax in German data.
• Firms also move profits to low-tax places.

Model numbers: Fig. 2’s slider (0.52 at τ = 0.35, 0.70 at τ = 0.21) and Practice Problems 3 and 7 of the book chapter. In the model the interest rate is set on world markets, so owners of capital still earn R after tax; they simply invest less, and with less capital per worker, wages fall. Revenue τ·MPK·K = (τ/3)Y peaks at τ* = 2/3, far above any actual rate, so cuts do not pay for themselves. Twenty-five of the 33 jurisdictions cut their rate after 2000; the median cut raised desired capital by a factor of 1.12 in the model.

The evidence, contested in its size:

• Chodorow-Reich, Smith, Zidar and Zwick (2024, NBER Working Paper 32180) compare U.S. firms by how much the 2017 Tax Cuts and Jobs Act lowered their tax costs. Domestic investment rose markedly more in firms with larger cuts; in their model the long-run gain in domestic capital is several percent and the revenue cost is large, with little offset from growth.
• Zwick and Mahon (2017, American Economic Review) find that bonus depreciation, a cut in the user cost, raised investment strongly, especially in small firms.
• Fuest, Peichl and Siegloch (2018, American Economic Review) use thousands of local business-tax changes in Germany and find that workers bear about half of the tax.
• Profit shifting: Ireland’s 12.5 percent rate attracted reported profits. Under the OECD/G20 minimum tax of 15 percent (agreed in 2021), the model says a large multinational’s desired capital in Ireland falls only 4.3 percent (Practice Problem 1): the minimum tax matters more for where profits are reported than for where factories are built.

Should the central bank lean against a stock-market boom?

The P/E ratio is 40.7, more than twice its average. Should the central bank raise interest rates to cool stock prices before they fall on their own?

What the model predicts

• At \(R = 4\%\), \(g = 2.75\%\): a P/E of 40. One more point of \(R\): 22.2, a fall of 44.4%.
• At the average P/E, the same rise: a fall of 26.3%.

What it leaves out

• Nobody observes \(R - g\), so the dose is a guess.
• Bubbles are hard to recognize in real time.
• Higher rates also cost output and jobs.

Model numbers: Practice Problem 6 of the book chapter. The price is proportional to 1/(R − g), so a market priced on a thin gap is very sensitive to interest rates: one point is 80 percent of a gap of 1.25 points, but only 36 percent of a gap of 2.81 points.

The debate, which is unsettled:

• Bernanke and Gertler (2001, American Economic Review Papers and Proceedings) argue that central banks should respond to asset prices only through their effect on expected inflation and output, not target them.
• Svensson (2017, Journal of Monetary Economics) estimates the costs and benefits of “leaning against the wind” with higher rates and finds that, for plausible numbers, the costs in lost output and employment exceed the benefits from a lower risk of a crisis.
• Others, at the Bank for International Settlements for example, argue that booms fed by credit are dangerous enough to justify leaning. Many central banks now prefer targeted tools (bank capital, loan limits) to the interest rate.

Ask the policy students: what would you need to know, and when, to time such a move? Recall from Chapter 10 that the stock market fell by half in 2008–09.

What moves house prices: rates, credit or expectations?

Real interest rates rose about 2.6 points from 2021 to 2023, yet real U.S. house prices rose 4.6%. Should policy target credit, interest rates or the supply of homes?

What the model predicts

• 2.6 more points of \(R\): prices fall 39.4%.
• To hold prices, expected growth must rise 2 points.
• Smaller down payments: only +4.1%.

What it leaves out

• Owners with cheap mortgages stopped selling.
• Cheap credit explains only part of the 2000s boom.
• Expectations are hard to observe or steer.

Model numbers: Practice Problem 2 of the book chapter (the baseline house: the denominator rises from 0.0304 to 0.0502, so the price falls from 197,368 to about 119,617 dollars) and the leverage table. The data: 10-year inflation-indexed Treasury yield (FRED DFII10), −0.91 percent in 2021 and 1.68 in 2023; real Case-Shiller index (CSUSHPISA over CPIAUCSL), 4.6 percent higher.

The evidence:

• Batzer, Coste, Doerner and Seiler (2024, Federal Housing Finance Agency Working Paper 24-03) estimate that owners holding low fixed-rate mortgages stayed put: the “lock-in” prevented about 1.33 million home sales between mid-2022 and the end of 2023, and the smaller supply raised prices by 5.7 percent, more than the 3.3 percent by which higher rates lowered them directly. The arbitrage model assumes anyone can buy or sell at no cost.
• Glaeser, Gottlieb and Gyourko (2013, in Housing and the Financial Crisis, University of Chicago Press) find that lower real interest rates explain only about a fifth of the rise in U.S. house prices from 1996 to 2006, and that looser down-payment and approval standards do not explain the bulk either.
• Case, Shiller and Thompson (2012, Brookings Papers on Economic Activity) survey buyers and find very high expected long-run price growth in boom markets: the expectations channel of the model.
• Mian and Sufi (2009, Quarterly Journal of Economics) argue, by contrast, that the expansion of mortgage credit to subprime ZIP codes drove the boom and the defaults; the debate is open.

For the policy students: loan-to-value limits are popular macroprudential tools. The model says they barely move prices directly, but they do reduce the leverage that turns a price fall into a banking crisis (Chapter 10).

“Firms do not compute user costs”

A skeptic says: “No manager calculates \(R + \bar{d} - \Delta p_k/p_k\). Firms invest when they have cash and feel confident, and stock prices follow moods. The arbitrage equation is a fairy tale.”

Reply: the equation describes where competition pushes firms, not how managers think. When tax reforms changed the user cost, investment moved as predicted. Cash and moods matter too, and the model says what they must change: expected gains, the gap \(R - g\), or access to funds.

Take the objection seriously. In a survey of more than 500 chief financial officers, most said their investment plans would hardly respond to a cut in interest rates, and only a little more to a rise; many pointed to their own cash holdings (Sharpe and Suarez 2021, Management Science). Financing constraints are real (Fazzari, Hubbard and Petersen 1988). Keynes’s “animal spirits” are a classic account of investment driven by confidence.

The reply rests on natural experiments: Cummins, Hassett and Hubbard (1994) with tax reforms; Zwick and Mahon (2017) with bonus depreciation; Chodorow-Reich, Smith, Zidar and Zwick (2024) with the 2017 tax law. In each, a change in the tax part of the user cost moved investment in the predicted direction. The user cost goes back to Hall and Jorgenson (1967, American Economic Review), who used it to study exactly these tax effects.

A firm that ignored the arithmetic would be outbid by one that did not: arbitrage is a statement about the market, not about each manager. And the moods matter precisely because they enter the equation, as expected price growth for houses or as g for stocks.

What to remember from this chapter

• Investment averages 17.8% of U.S. GDP but swings hard: from 19.6% in 2006 to 13.3% in 2009. Housing and inventories swing most.
• The arbitrage equation: firms invest until \(MPK = uc = R + \bar{d} - \Delta p_k/p_k\). A corporate tax divides the user cost by \(1-\tau\).
• Desired capital is proportional to \(uc^{-3/2}\), so a 35% tax cuts it almost in half; the investment rate is \((g_K + \bar{d})/(3\,uc)\).
• A share is worth \(\text{dividend}/(R-g)\), so P/E ratios hinge on a thin gap and bubbles are hard to spot. \(q > 1\) says invest.
• House prices respond far more to expected price growth than to cheap credit.

Ask each student to answer the opening question in one sentence: when is a machine, a share or a house worth buying? A good answer: when it earns at least what the bank pays, after wear and tear and the expected change in its price; and because those expectations change, investment swings.

For the policy students, three habits: ask how a policy changes the user cost, not only the tax rate; ask what growth and interest rates a price implies before calling it a bubble; and remember that announced or temporary changes move investment in time.

Walk the three branches of the mind map in about a minute each: investment swings (cyan), the user cost of capital (lavender), stocks, houses and policy (green). Click the image to zoom in.

One arbitrage equation prices machines, shares and houses, and expectations make investment swing

Next: Chapter 18, The Government and the Macroeconomy

This chapter explained the investment rate that the Solow model took as given, and linked it to taxes, interest rates and the stock market. Chapter 18 turns to the government: how it spends, taxes and borrows, and what its budget constraint implies for debt and deficits, including the corporate taxes we met here.

Sources: data and models

• Fig. 1 and Fig. 2, desired capital and the corporate tax. Model figures, adapted from Figs. 17.1 and 17.2 in Jones (2024); the baseline \(\bar{A} = 0.3\), \(L = 1\) and the derivations of desired capital, the investment rate, Tobin’s \(q\) and the house price are the book chapter’s own, following Sections 17.2 to 17.4 of Jones (2024).
• Tax table. OECD, Corporate Tax Statistics, statutory corporate income tax rate (DSD_TAX_CIT@DF_CIT), via the OECD Data Explorer; textbook values from Table 17.1 in Jones (2024).
• Fig. 3, the P/E ratio. Robert J. Shiller, Online Data, CAPE of the S&P composite index (ie_data.xls), shillerdata.com. Inspired by Fig. 17.3 in Jones (2024).
• Fig. 4, U.S. investment. BEA, NIPA Tables 1.1.5 and 1.1.10, via FRED (GPDIA, PNFIA, PRFIA, CBIA, GDPA, A006RE1Q156NBEA, A011RE1Q156NBEA); NBER business cycle dates (USREC). Inspired by Fig. 17.4 in Jones (2024).
• Fig. 4b, 35 countries. OECD, National Accounts at a Glance (DSD_NAAG@DF_NAAG_III, DSD_NAAG@DF_NAAG_III_GFCF) and national accounts Table 1 (DSD_NAMAIN10@DF_TABLE1_EXPENDITURE), via the OECD Data Explorer. Inspired by Fig. 17.4 in Jones (2024).
• Textbook. Jones, C. I. (2024). Macroeconomics, 6th ed. W. W. Norton.
• Book chapter, with the data, code and every figure. Data Science and Macroeconomic Models, Chapter 17.

Sources: research cited

• The user cost, taxes and investment. Hall, R. and D. Jorgenson (1967), American Economic Review 57(3); Cummins, J., K. Hassett and R. G. Hubbard (1994), Brookings Papers on Economic Activity 1994(2); Zwick, E. and J. Mahon (2017), American Economic Review 107(1); Chodorow-Reich, G., M. Smith, O. Zidar and E. Zwick (2024), NBER Working Paper 32180; Fuest, C., A. Peichl and S. Siegloch (2018), American Economic Review 108(2).
• Finance, cash flow and inventories. Fazzari, S., R. G. Hubbard and B. Petersen (1988), Brookings Papers on Economic Activity 1988(1); Kaplan, S. and L. Zingales (1997), Quarterly Journal of Economics 112(1); Sharpe, S. and G. Suarez (2021), Management Science 67(2); Davis, S. and J. Kahn (2008), Journal of Economic Perspectives 22(4); Leamer, E. (2007), NBER Working Paper 13428.
• Stock prices and bubbles. Fama, E. (1970), Journal of Finance 25(2); Shiller, R. (1981), American Economic Review 71(3); Malkiel, B. (2003), Journal of Economic Perspectives 17(1); Bernanke, B. and M. Gertler (2001), American Economic Review 91(2); Svensson, L. (2017), Journal of Monetary Economics 90.
• Housing. Glaeser, E., J. Gottlieb and J. Gyourko (2013), in E. Glaeser and T. Sinai (eds.), Housing and the Financial Crisis, University of Chicago Press; Case, K., R. Shiller and A. Thompson (2012), Brookings Papers on Economic Activity Fall 2012; Mian, A. and A. Sufi (2009), Quarterly Journal of Economics 124(4); Batzer, R., J. Coste, W. Doerner and M. Seiler (2024), FHFA Working Paper 24-03.
• Tariffs and investment, 2025. FOMC minutes, May 2025; BEA, Survey of Current Business, May 2025; OECD/G20 statement on the two-pillar solution, October 2021.