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# Chapter 20: The Demand for Assets

*Part IV: The Investor Ecology — Financial Economics: Claims, Prices, and Holders*

***

## Opening Episode: One Question, Forty Years Apart

In 1986 Andrei Shleifer published a paper in the *Journal of Finance* whose title was a question that most of his readers thought had already been answered: "Do demand curves for stocks slope down?"

The answer everyone knew was no. A share of General Motors is not a consumption good. It is a bundle of exposures — to the market, to the business cycle, to the automobile industry — and every one of those exposures is available elsewhere, in other shares, in portfolios of other shares, in futures and options written on all of them. If an investor with a large order pushed GM's price above the value implied by that bundle, other investors would sell GM and buy the substitutes, and the price would come back. Assets have near-perfect substitutes; goods do not, and the residual demand curve facing any individual security should therefore be, to a first approximation, horizontal. Scholes had made the argument in 1972 with block-trade data, and the profession had taken his verdict as settled: what looks like price pressure is really information.

Shleifer's contribution was to find a demand shock with no information in it. Standard and Poor's states as policy that inclusion in the S\&P 500 is not a judgment about a company's prospects; it is a decision about index composition. Yet inclusion obliges every fund tracking the index to buy the stock, in a known amount, on a known date. Here was a purchase order arriving with no news attached. If demand curves are flat, the price should not move at all.

It moved. Shleifer found abnormal returns on the order of three percent around additions to the index over 1976-1983, and no reversal afterward. The result on its own would have been contestable — a committee that picks stocks might be picking on something. What made the paper hard to dismiss was a second finding. The size of the price response *grew over the sample period*, and it grew roughly in step with the assets managed against the index. A demand shock that was larger produced a price response that was larger. That is not what news looks like. It is what a demand curve looks like.

The reception was instructive. In the very next issue of the same journal, Harris and Gurel reported a comparable announcement effect and a reversal within about two weeks, and read the same phenomenon as temporary price pressure rather than as a permanent slope. The distinction mattered enormously, because a temporary effect is a liquidity cost — a fee paid to whoever supplies the shares in a hurry — while a permanent one says that the marginal holder's willingness to pay depends on how much he has been asked to hold. One is microstructure. The other is asset pricing. Chapter 17 §17.4 describes where the inclusion evidence itself ended up: a large effect in the early decades, an effect close to zero today, and an unresolved argument about whether the slope flattened or the shock shrank.

Now move forty years forward.

Xavier Gabaix and Ralph Koijen ask Shleifer's question of the entire stock market rather than of one stock, and report an estimate: a dollar invested in the aggregate equity market raises the aggregate value of that market by about five dollars. Not five cents. Five dollars. Their figure has been contested, revised, and attacked on identification grounds, and this chapter states the objections rather than burying them. But the order of magnitude has survived enough independent work to be taken seriously. If it is right, a pension fund shifting one percent of a portfolio from bonds into equities is not a portfolio decision with a negligible market footprint; it is a macroeconomic event. And the level of the stock market is not, even to a first approximation, a discounted stream of expected cash flows plus a risk premium. It is a market-clearing price, set by who is buying and how much they have.

Between the two dates sits a change in what kind of object the question is. In 1986 it was a heresy defended with an event study. Today it is a research program with a name — **demand-system asset pricing** — an estimation method, regulatory holdings data on both sides of the equation, and a set of contested magnitudes that specialists argue about. That is the arc this chapter traces, and it is the destination the whole book has been driving toward. Chapter 3 asked whose marginal utility appears in the pricing equation and refused to answer "the representative household's." Part IV has spent six chapters describing the holders. This chapter puts them in one system, lets prices clear it, and gives Chapter 3's question the only complete answer this book has.

***

## 20.1 From Stochastic Discount Factors to Demand Curves

Chapter 3 §3.7 posed the problem and named the difficulty. The pricing equation $$p = E\[mx]$$ is an implication of no arbitrage where trading is frictionless; what it does not do is tell you whose $$m$$ it is. Where shorting or replication fails it does less than that: no arbitrage then pins a band rather than a point, as Chapter 3 §3.3 argues and its Problem 6 computes, so markets with binding trading frictions generally deliver bounds, or a wedge between different holders' $$m$$'s, rather than one price. The Euler equation that produces $$m = \delta u'(c\_1)/u'(c\_0)$$ belongs to an investor who is *marginal* in the claim — free to buy or sell a little more at the going price. In a frictionless complete market everyone is marginal in everything, all the $$m$$'s coincide, and the question is uninteresting. In actual markets it is the whole problem.

Part IV has accumulated two partial answers.

**The wealthy household** (Chapter 5 §5.7, with the evidence in Chapter 14 §14.6). Equity risk is borne overwhelmingly by the top of the wealth distribution. Build the consumption series of equity *holders* rather than of the population, and it is more volatile and more strongly correlated with equity returns than aggregate consumption is, which shrinks the risk aversion needed to rationalize the equity premium without eliminating the puzzle. This answer works in the markets households actually trade.

**The constrained dealer** (Chapter 19 §19.5). For claims households do not hold at all — credit default swaps, over-the-counter index options, foreign sovereign debt, commodity futures — the marginal investor is a levered intermediary, and the relevant $$m$$ is the marginal value of its equity capital, $$m\_{t+1} = \delta\Lambda\_{t+1}/\Lambda\_t$$. He, Kelly and Manela show that a factor built from primary dealers' capital ratio carries a price of risk of similar magnitude across markets that share nothing except their intermediary. This answer works exactly where the first cannot apply.

Both answers have the same shape: pick the marginal investor, write down his first-order condition, let it price everything. Both are therefore vulnerable to the same objection — that the choice of marginal investor is an assumption imposed before the data are consulted, and different assumptions are defensible in different markets with no principle for choosing among them.

The demand-system move is to stop choosing.

Instead of hunting for the one investor whose $$m$$ prices the market, estimate a **demand curve for every holder**: how much of each claim each sector wants to hold, as a function of that claim's price and characteristics and of the holder's own wealth and constraints. Then impose the accounting identity that has been in this book since Chapter 2 — every share of every claim is held by somebody — and let it determine prices. The pricing equation is not discarded; it is inverted. Rather than deriving quantities from an assumed $$m$$, one estimates quantities and derives the price system that makes them add up.

The change is not merely technical. Under the representative-agent approach, holdings data are a curiosity: if everyone holds the market portfolio, who holds what tells you nothing you did not already know. Under the demand-system approach, holdings data are the dependent variable. This is why Chapter 2 §2.5 closed by promising that "when Chapter 20 estimates a demand system, the holdings in this table are, quite literally, its left-hand side." Table 2.5's cells — households' $25 trillion of directly held equity, insurers' and pensions' $5 trillion of corporate bonds, the rest of the world's $8.5 trillion of Treasuries — are, at higher frequency and finer disaggregation, the observations a demand system is fitted to. The master map is not background. It is the data.

Two things make this feasible now and not in 1986. The first is disclosure: institutions above a size threshold report their US equity positions quarterly on Form 13F, insurers file statutory schedules, funds file portfolio holdings, and the Financial Accounts aggregate the residual sectors. The second is that the holdings are concentrated enough to matter. When households directly held most of the equity market and traded it in odd lots, no register of holders could have been assembled. Chapter 2 §2.4's fifty-year reallocation from direct to intermediated ownership is what made the ownership of the market *observable*.

***

## 20.2 The Logic of Demand-System Asset Pricing

Koijen and Yogo's 2019 paper is the canonical statement. What follows is its logic, stripped of the estimation machinery, which is gated below.

### Characteristics-based demand

Index the assets by $$j$$ and the holders by $$h$$ — an institution, or a sector such as "insurers" or "the household residual." Let $$A\_h$$ denote holder $$h$$'s total assets under management and $$w\_{h,j}$$ the share of those assets it places in claim $$j$$, so that $$A\_h w\_{h,j}$$ is the dollar position and $$\sum\_j w\_{h,j} \le 1$$, the remainder sitting in an outside asset such as cash.

The modeling choice that makes the problem tractable is to write each holder's portfolio weight as a function of a short list of the asset's observable **characteristics** $$c\_j$$ — market equity, book-to-market, profitability, investment, dividend yield, credit rating, duration, index membership — together with the asset's price:

$$
w\_{h,j} = w\_h\big(p\_j, c\_j; \theta\_h\big).
$$

Two features of this specification deserve attention before anything is estimated.

First, the characteristics are the *same* for every holder; what differs is $$\theta\_h$$, the coefficient vector describing how much weight this holder puts on each one. A value manager loads on book-to-market. An index fund loads on index membership and market equity and on nothing else. An insurer loads on rating and duration, because the risk-based capital schedule of Chapter 16 §16.2 attaches a price to both. The system does not assume these differences; it estimates them, holder by holder.

Second, price enters as its own argument, which is what makes the object a demand curve rather than a portfolio. The coefficient on price is the holder's **demand elasticity** $$\zeta\_h$$: the percentage reduction in the quantity of claim $$j$$ this holder wants for a one percent increase in its price, holding characteristics fixed. Frictionless theory has a strong prediction about that coefficient, worth stating precisely because the empirical finding is a rejection of it. A mean-variance investor facing a stock whose price has risen with no change in expected cash flows now faces a lower expected return, and — because the stock has close substitutes — reallocates aggressively toward them. Reasonable calibrations of that logic imply elasticities well above five, and in the pure-substitution limit, arbitrarily large.

### Why demand slopes down: the book's inventory

The reason estimated elasticities are far smaller than five is not that investors have failed to notice the substitutes. It is that most of the money is not permitted to move. Part IV has assembled the inventory one chapter at a time; it is worth reading as a single list.

* **Mandate, statutory and self-imposed.** The 1940 Act's issuer, industry, illiquidity and borrowing limits (Box 20.1) bind a fund's holdings regardless of price — each a wall the demand curve runs into.
* **Regulatory capital.** An insurer moving from BBB to BB corporate bonds sees its required capital more than triple, and holding equities costs it seventy-five times the capital of holding AAA paper (Chapter 16 §16.2, Table 16.3). That schedule is a price attached to a characteristic, and it makes an insurer's demand a function of ratings rather than of expected returns.
* **Benchmarks and index tracking.** A fund evaluated against a benchmark holds the benchmark's names in something close to the benchmark's weights, because tracking error is the risk its employer measures (Chapter 16 §16.3). A pure index fund goes further: its weight in a stock is a mechanical function of index membership and float-adjusted capitalization, with a price coefficient of exactly zero (Chapter 17 §§17.4-17.6).
* **Redemption and delegation.** An arbitrageur whose capital belongs to investors who judge him on realized returns sells into a widening mispricing rather than buying, because the money leaves when the position moves against him (Chapter 15 §15.5). A private fund's investors cannot leave, but its policy-portfolio limits bind mechanically when public marks fall and the private share rises — the denominator effect of Chapter 18 §18.6.
* **Balance-sheet capacity.** A dealer's willingness to absorb inventory is a function of its equity, its leverage ratio, and the haircut its funding counterparties demand, all of which tighten together (Chapters 16 §16.5 and 19 §§19.3, 19.5).
* **Defaults and inattention.** The largest single pool of US equity demand arrives through retirement vehicles on payroll dates, is allocated by a glide path indexed to a birthday, and rebalances on a calendar (Chapters 14 §§14.2-14.3 and 17 §17.8). Its price coefficient is not merely small. It is, by design, zero.

Each is a reason for a holder to want a particular quantity of a claim more or less regardless of what the claim costs. Aggregate them and the market's demand curve has slope.

> **Box 20.1 — What a fund may hold, and what it discloses**
>
> The 1940 Investment Company Act limits what a registered fund may hold: a fund *electing* diversified status may not, within 75 percent of its assets, exceed 5 percent in one issuer or 10 percent of an issuer's voting stock; a 25 percent industry concentration is allowed but commits the fund to a disclosed policy changeable only by shareholder vote; an open-end fund may hold at most 15 percent of net assets in illiquid investments; and Section 18's 300 percent asset-coverage test holds borrowing near zero (Chapter 16 §16.3).
>
> Section 13(f) of the Securities Exchange Act runs the other way, making holdings visible. A manager with discretion over $100 million of listed US equities files Form 13F-HR within 45 days of quarter end, position by position, long positions only, aggregated across all its funds. An ownership share built from it is a floor, not a measurement; Appendix B §B.3 catalogues what else it misses.

### Market clearing

The last step turns a set of estimated demand curves into an asset pricing model. Shares outstanding are given in the short run. Write $$Q\_j$$ for the shares outstanding of claim $$j$$ and $$p\_j$$ for its price. Then for every claim,

$$
\sum\_h A\_h w\_h\big(p\_j, c\_j; \theta\_h\big) = p\_j Q\_j .
$$

The left side is the total dollar demand of every holder. The right side is the total dollar supply. There is one such equation per claim, and the system determines the vector of prices jointly, given the characteristics, the holders' estimated coefficients, and the distribution of wealth across holders.

Read the system and the book's title reads back. A price is what makes the holdings of a claim add up to its supply. Change who has the money — a wealth transfer from an elastic holder to an inelastic one — and prices change with no change in any cash flow. Change a mandate and prices change.

### What the estimates say

Koijen and Yogo fit this system to 13F holdings of US equity from 1980 onward and to the residual household sector, and the headline finding has held up across replications in other markets and asset classes: **institutional demand curves are steep**. Estimated elasticities cluster near one and often below it, against the frictionless benchmark above five. The order of magnitude of the gap, not its second decimal, is the finding.

![Figure 20.1: How elastic is demand](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-91115ca74b3c4083fea51af103498474c5a63267%2Ffig_20_01_how_elastic_is_demand.png?alt=media)

**Figure 20.1: How elastic is demand.** Three bands on a logarithmic scale, with the price multiplier each implies on the top axis. The first is the frictionless benchmark of Part II: a claim with close substitutes has a nearly flat demand curve, and an elasticity above five is the conservative end of what that logic delivers. The second is what the estimates find — clustering near one and often below it, an order of magnitude away. The third is the aggregate market, taken from the multiplier debate's own range of roughly two to eight and inverted, which puts the whole market's elasticity between about an eighth and a half. They are bands rather than points on purpose. The chapter states these magnitudes as orders of magnitude, and nothing in the argument turns on the second decimal place of any of them: what the figure has to show is the distance between the first band and the second, and that distance is a factor of ten. The dashed rule at an elasticity of one is where a dollar of flow moves a dollar of value. *Source: Ranges as stated in §§20.2-20.3, from Koijen and Yogo (2019) and the multiplier literature the next section reviews. No point estimates.*

A second finding matters as much and is less often quoted. Most of the variation in individual stock prices in the fitted system is attributed not to the characteristics but to **latent demand** — the holder-specific residual, the part of a portfolio weight the observed characteristics do not explain. That is a candid admission of how much the system does not know, and it is the mechanism by which one investor's idiosyncratic reallocation moves a price. In a flat-demand world, arbitrageurs absorb latent demand at unchanged prices. In a steep-demand world it is a price.

### ★ Identification: why you cannot regress holdings on prices

The obvious estimator is the wrong one, and understanding why is worth more than the formula.

Suppose you regress a holder's portfolio weight in a stock on the stock's price. The coefficient you recover is not a demand elasticity, because price is not exogenous. Prices are set by demand — that is the entire content of the market-clearing condition above. Any shock that raises a holder's desired position also raises the price, so the regression mixes movement *along* a demand curve with movement *of* it. If enthusiasm for a stock raises both the quantity held and the price, the fitted relation can even come out upward-sloping. This is the oldest problem in demand estimation, and it arrives here in an unusually severe form, because in asset markets the shifts and the price are contemporaneous by construction.

The instrument has to change the price a holder faces without being related to that holder's own reason for wanting the stock. Koijen and Yogo's proposal exploits a feature of institutional portfolios with no counterpart in the demand for groceries: institutions hold from restricted and observable menus. Define a holder's **investment universe** as the set of assets it has actually held in the recent past — a small mandate-determined subset of the market, stable over time and different for every institution. A stock's price then depends on the size of the institutions in whose universes it happens to sit: a stock eligible for many large investors' portfolios faces more demand, at given characteristics, than an otherwise identical stock eligible for few small ones.

That gives the instrument. For each holder and stock, construct the counterfactual price that would prevail if every *other* institution allocated its assets evenly across its own universe. The resulting variation is driven by other institutions' sizes and menus rather than by anything specific to this holder's view of this stock, which is what an instrument for price needs to be.

The exclusion restriction it requires should be stated plainly, because it is the hinge of the whole literature: universe membership and the size of the institutions holding the stock must be unrelated to the stock's latent demand. Whether that is true is contested, and §20.5 returns to it. The reader who skips this subsection keeps the point that matters: **the elasticity estimates are not regression coefficients on prices, and every disagreement about their magnitude is at bottom a disagreement about this instrument.**

***

## 20.3 Inelastic Markets

Demand-system estimation is about the cross-section — which stock, held by whom, at what price. The **inelastic markets hypothesis** asks the same question of the aggregate.

### The multiplier

Gabaix and Koijen define the object the aggregate question needs. Let $$V$$ be the total market value of equities and let $$F$$ be a **flow**: a dollar of demand entering the market, financed by a sale of bonds or cash, with no change in any firm's expected cash flows. The **price multiplier** is

$$
\mathcal{M} = \frac{dV}{dF},
$$

the dollars of aggregate market value created per dollar of flow. Frictionless theory says $$\mathcal{M}$$ is essentially zero: an inflow is absorbed by whoever sells, at a price that barely moves, because the aggregate supply of risk-bearing capacity is elastic. Gabaix and Koijen's estimate is about **five**, which they report as a central figure in a range that their own methods and the subsequent literature place somewhere between roughly two and eight depending on the identification strategy and the horizon. The estimate is contested. Its sign and order of magnitude are much less so.

The multiplier is the reciprocal of an elasticity. If aggregate demand for equities has elasticity $$\zeta$$ with respect to the market's value, and supply is fixed in the short run, then

$$
\mathcal{M} = \frac{1}{\zeta}.
$$

A multiplier of five is an aggregate elasticity of $$0.2$$. That number is small enough to be worth restating in words: a ten percent rise in the aggregate value of the stock market induces the market's holders, collectively, to want to hold two percent fewer shares. They mostly do not respond to price at all.

### The multiplier is an aggregation

The connection to §20.2 is the part of the argument most often lost, and it is the reason these two literatures are one chapter rather than two. The aggregate elasticity is not a separate parameter estimated from separate data. It is a wealth-weighted average of the micro elasticities of the holders in the market. Write $$s\_h$$ for holder $$h$$'s share of the market's value; then

$$
\zeta = \sum\_h s\_h \zeta\_h .
$$

Steep individual demand curves are what makes the aggregate curve steep, and the composition of ownership — which of Chapter 2's sectors holds what fraction — is what determines the weights. Chapter 14 §14.6's two household demand curves, Chapter 17 §17.8's mandated holders, Chapter 16's capital-constrained insurers, Chapter 19's balance-sheet-limited dealers: each is a row in that sum, and the fifty-year reallocation of Chapter 2 §2.4 is a fifty-year reweighting of it.

### A two-sector economy, fully worked

The mechanism is clean enough to see in closed form, and the arithmetic is worth doing once.

An economy has one risky claim in fixed supply of $$Q = 100$$ shares, trading initially at one dollar a share, so $$p = 1$$ and $$V = 100$$. There are two holders.

**Sector A is mandated.** It holds $$50$$ shares whatever the price: $$\zeta\_A = 0$$. Read it as the index complex, the target-date defaults, and the insurer whose allocation is set by a capital schedule.

**Sector B is active.** Its demand is $$q\_B(p) = 50p^{-\zeta\_B}$$, so it holds $$50$$ shares at $$p = 1$$ and less as the price rises. Take $$\zeta\_B = 0.4$$.

At $$p=1$$ the market clears: $$50 + 50 = 100$$. The aggregate elasticity is the value-weighted average, $$\zeta = 0.5 \times 0 + 0.5 \times 0.4 = 0.2$$, so $$\mathcal{M} = 5$$.

Now let sector A receive an inflow and raise its desired holding by two percent, to $$51$$ shares. At the initial price that is a flow of $$F = 1$$ dollar, one percent of the market. Market clearing requires

$$
51 + 50p^{-0.4} = 100 \quad\Longrightarrow\quad p^{-0.4} = 0.98 \quad\Longrightarrow\quad p = 0.98^{-2.5} = 1.0518.
$$

The market's value rises to $105.18. **One dollar of flow created** $$5.18$$ **of market value**, against the multiplier of exactly $$5$$ that the elasticity formula gives in the limit of small flows. Nobody's expected cash flows changed. Nobody learned anything. Sector B was simply induced, by a five percent higher price, to hand over one share. Figure 20.2 draws it, sector by sector and then in aggregate.

![Figure 20.2: The multiplier, worked](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-039a7764cacf4e3355b59ce7f5043a6ed871d4ea%2Ffig_20_02_multiplier_two_sector.png?alt=media)

**Figure 20.2: The multiplier, worked.** Section 20.3's two-sector economy drawn. Panel (a): sector A's vertical demand and sector B's sloped demand, share by share. Panel (b): aggregate demand, fixed supply, and the price move from a $1 flow, with the outcome labelled. Sector A's inflow raises its desired holding from 50 to 51 shares — a flow of one percent of the market, and no change in anyone's expected cash flows. Clearing then requires 51 + 50p^(−0.4) = 100, so p = 1.0518 and market value rises from $100 to $105.18: one dollar of flow created $5.18 of market value, against the multiplier of 5 that the elasticity formula gives in the limit of small flows. Both panels are drawn on the same ten-share horizontal scale and the same price axis, so the aggregate curve in panel (b) is visibly the horizontal sum of the two curves in panel (a); a chevron on the bottom frame marks a schedule that leaves the panel rather than ending there. *Source: Author's construction from the worked economy of §20.3.*

Three variations, each of which is a section of this book.

**Table 20.1: The multiplier under four ownership structures**

| Change                                                          | New aggregate $$\zeta$$ | Multiplier $$\mathcal{M}$$ | Where it lives         |
| --------------------------------------------------------------- | ----------------------- | -------------------------- | ---------------------- |
| Baseline: half the market mandated, active $$\zeta\_B = 0.4$$   | 0.20                    | 5.0                        | §20.3                  |
| Active sector as elastic as theory says ($$\zeta\_B = 4$$)      | 2.00                    | 0.5                        | Part II                |
| Mandated share rises to 70%                                     | 0.12                    | 8.3                        | Chs 14 §14.6, 17 §17.8 |
| Baseline plus issuers who supply shares with elasticity $$0.3$$ | 0.50                    | 2.0                        | Ch 23                  |

*Source: Author's calculation from the two-sector example in this section.*

The second row is Part II's world, the third is the composition argument of Part IV expressed as a number, and the fourth is the supply side, taken up next.

![Figure 20.3: The multiplier under four ownership structures](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-02f16f4960b46814d6d55435d372bbb498c6cc70%2Ffig_20_03_multiplier_four_structures.png?alt=media)

**Figure 20.3: The multiplier under four ownership structures.** Table 20.1 as four bars, with the aggregate elasticity that produces each printed above it. The dashed rule at one is the line separating a market in which a dollar of flow moves a dollar of value from one in which it moves more. Three readings. The distance between the first bar and the second is the whole of the dispute between Parts II and IV, and it is a factor of ten: the same economy with the same claims and the same information moves five dollars per dollar of flow or half a dollar, depending on nothing but how elastic the active sector is. The distance between the first bar and the third is the composition argument of Part IV with a number attached — raising the mandated share from a half to seven-tenths, which is roughly what the last four decades did, raises the multiplier from five to eight. And the fourth bar is the one that says the argument is not only about holders: a supply elasticity of 0.3, modest by the standards of any goods market, would cut the multiplier from five to two on its own. Section 20.3 explains why the short-run supply response is smaller than that, and Chapter 23 §23.7 takes up the issuer's side properly. The statement to carry forward is that **the multiplier is one over a wealth-weighted average of the holders' elasticities, and the constraints catalogued in §20.2 are what keeps that average small.**

### The supply side, and a forward pointer

Nothing in the argument requires the number of shares to be fixed. Firms issue equity and retire it, and if issuance responded strongly to price, the multiplier would collapse: an inflow would be met by new shares rather than by a higher price. The fourth row of the table above prices this out. A supply elasticity of $$0.3$$ — modest by the standards of any goods market — cuts the multiplier from five to two.

![Figure 20.6: Flows and returns, and why the regression lies](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-4bd39c6e077cfda7180ec69da63b7531695b6e4d%2Ffig_20_06_flows_and_returns.png?alt=media)

**Figure 20.6: Flows and returns, and why the regression lies.** The chapter's own data exercise Part A, run, and then refused. Panel (a) regresses the market's monthly excess return on net flows into domestic equity mutual funds and ETFs, scaled by the prior month's market value so both sides are in percent of market value and the slope reads directly as a multiplier. The slope is 21. That is three times the top of this section's range and an order of magnitude above its centre, and it is not a discovery — it is what the naive regression gives, and the reason to distrust it is next to it. Panel (b) is the same correlation at leads and lags. The only positive value is contemporaneous. At every lead and every lag it is zero or negative, *including* the one Chapter 16 §16.3's flow-performance relation predicts should be positive, so the pattern discriminates between nothing. It is what flows moving prices inside the month looks like; it is also what returns pulling flows inside the month looks like; it is also what both responding to a third thing looks like. Two further reasons the 21 is not the multiplier. The flow measure covers fund vehicles only — a fraction of Chapter 2's Table 2.5 equity row — while the return is the whole market's, so the denominator is far too small. And the ICI series reclassifies funds between categories; one month in the sample carries a reclassification of about 300 billion dollars and is dropped, which is stated here because dropping it silently would be the same sin at a smaller scale. *Source: ICI Fact Book cumulative-flow exhibit, differenced to monthly; Kenneth French's market excess return; Financial Accounts market value via FRED. Author's calculations.*

Two facts make the short-run supply response smaller than that. Net equity issuance by US nonfinancial corporations has been *negative* for most of the past four decades — persistently so since the mid-1980s, as Chapter 12 §12.6 documents: buybacks and cash-financed mergers retire more equity than offerings create, so the corporate sector has been a net purchaser of its own shares rather than a net supplier. And the timing is wrong for a stabilizer — firms issue after prices have risen, which is closer to the market timing of Chapter 23 §23.4 than to an elastic supply curve.

The theory of the supply side is Part V's, and Chapter 23 §23.7 is where it lands: Greenwood and Hanson's gap-filling result — that firms adjust the maturity and type of the securities they issue to fill gaps left by other issuers and by holder demand — is the supply-side counterpart of this chapter's demand system.

### Reconciling with Chapter 7

A reader arriving from Part II should be told directly what this chapter does and does not overturn, because the natural misreading is large and this book does not hold it.

Inelastic markets do **not** mean that information is irrelevant to prices. Chapter 7's evidence stands: earnings surprises move prices within seconds, event studies detect the announcement and not the leak, and the return series is close to unforecastable at short horizons. A theory denying this would be denying the best-measured facts in the field.

What inelastic markets say is something narrower and more corrosive. The mechanism that gets information into prices is *trade by informed investors*, and trade by informed investors is executed by holders with balance sheets. Chapter 7 §7.2's Grossman-Stiglitz equilibrium already establishes that over the intermediate range of information costs in its Table 7.1 the informed fraction is interior and finite — and that at neither corner of that table is the price fully revealing. Chapter 15 §15.5 establishes that its capital contracts exactly when it is most needed. Chapter 16 §16.5 states the general case. Chapter 19 shows that the constrained sector's shadow value of capital is itself a pricing factor. Put those together and the conclusion is that **information is incorporated into prices at a rate and to a degree set by the capital available to the traders who do the incorporating** — which is finite, mandated, benchmarked, levered, and redeemable.

That is Part IV in one sentence, and it is why this chapter is a completion of Part II rather than a refutation of it. The efficient-markets benchmark describes what prices would be if arbitrage capital were unlimited. The demand system describes what prices are when it is not, and it nests the benchmark: let every holder's elasticity go to infinity and the multiplier goes to zero and latent demand stops mattering. That limit is the flat-demand benchmark, and it is all the limit delivers by itself. A demand system contains no payoff and no SDF equation, so what a flat curve pins down is the *price impact* of a flow, not the level the price is flat at; the level is set by the demand intercepts. Getting back to the discounted cash flows of Chapter 3 takes the further assumption that those intercepts embed $$E\[mx]$$ — that somebody in the clearing condition is pricing off fundamentals and is unconstrained enough to be marginal.

***

## 20.4 The Ecology Assembled

This is the section the part was written for. Every chapter of Part IV has described a holder and the constraint that binds it. Each is a row.

**Table 20.2: The Investor Ecology as a Demand System**

| Holder                             | Chapter         | Binding constraint                                  | Demand behavior                                                              | Price elasticity                                        |
| ---------------------------------- | --------------- | --------------------------------------------------- | ---------------------------------------------------------------------------- | ------------------------------------------------------- |
| Households, default-driven         | 14 §§14.2-14.3  | Inertia, plan defaults, glide path                  | Contributions on payroll dates; allocation by birthday; calendar rebalancing | Near zero                                               |
| Households, wealthy direct holders | 14 §14.6        | Taxes, realization decisions                        | Discretionary, levered, tax-sensitive reallocation                           | Positive, the household sector's only elastic component |
| Households, behavioral             | 15              | Reference points, extrapolation, attention          | Belief-driven; sells winners, holds losers; chases returns                   | Slope of the wrong sign in some states                  |
| Pensions and insurers              | 16 §16.2        | Liability duration, risk-based capital              | Derived demand: buys duration and rating buckets, not expected returns       | Low; discontinuous at rating boundaries                 |
| Active funds                       | 16 §16.3        | 1940 Act limits, benchmark, career risk             | Tracks a benchmark with tilts; forced seller on redemptions                  | Low                                                     |
| Index funds and ETFs               | 17 §§17.4-17.6  | Index membership, float weights                     | Mechanical: buys what the committee adds, in the published weight            | Exactly zero by construction                            |
| Private funds                      | 18 §§18.4, 18.6 | Lockups, commitments, policy-portfolio limits       | Patient until the denominator effect binds, then a forced seller             | Low, and state-dependent                                |
| Banks and dealers                  | 19 §§19.3, 19.5 | Equity capital, leverage ratio, haircuts            | Absorbs inventory while capacity permits, withdraws when it does not         | Positive but capacity-limited                           |
| Rest of world                      | 2 §2.3, Box 2.2 | Reserve policy, hedging costs, global index weights | Responds to variables no domestic pricing model contains                     | Low                                                     |

*Source: Author's construction from the chapters cited.*

### What breaks the representative agent

Line the rows up and the standard defense of aggregation fails.

The representative-agent argument is not that everyone is identical. It is that heterogeneity *averages out*: if households differ idiosyncratically around a common mean, complete markets let them insure the differences away, and what remains to be priced is the aggregate. Under that reading, heterogeneity is noise and the mean investor is a real object that the aggregate data measure.

Table 20.2 is not noise around a mean. The rows differ systematically, along the dimension that determines prices, in ways that do not cancel. An index fund's elasticity is zero as a matter of contract; a dealer's is positive but shrinks precisely when the index fund's flows are largest; an insurer's is discontinuous at the investment-grade boundary; a target-date fund's is zero and its flow is a function of the wage bill. Average these and the resulting "mean investor" has no counterpart in the world — a fictional holder with an average mandate, an average leverage ratio, and an average redemption term, none of which anyone faces. Worse, the average is not even stable, because the weights $$s\_h$$ have been moving in one direction for fifty years.

The demand system's response is to stop averaging. Estimate the rows, weight them by wealth, and clear the market. The identity of the marginal investor then becomes an output rather than an input: whichever holder's demand curve is the steep one in a given market, in a given state, is the one whose constraint appears in the price.

Figures 20.4 and 20.5 are that argument in two forms: the construction, and the census.

![Figure 20.4: One claim, many holders](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-b70227f2a3fb1b870fa780adcb5c2d58e7ad7cc1%2Ffig_20_04_one_claim_many_holders.png?alt=media)

**Figure 20.4: One claim, many holders.** Panel (a) is five of Table 20.2's rows as demand curves for one claim, each drawn relative to its own holding at a price of one so that the slopes can be compared. An index fund's is vertical as a matter of contract; a default-driven household's is nearly so, for a different reason; a dealer's has slope until its balance sheet fills. Panel (b) sums them horizontally against a fixed supply and applies a flow of one percent of the market. The price rises 5.2 percent, which is a multiplier of five on a wealth-weighted elasticity of 0.20. The point of drawing the two panels together is that the steepness of the aggregate curve is not a property anybody chose. It is an accounting consequence of who holds the claim, and it would be different tomorrow with the same investors, the same beliefs and the same cash flows if the weights were different. The shares and elasticities are constructed, to an aggregate that reproduces §20.3's.

![Figure 20.5: The ecology as a demand system](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-f51711fc8c10f1b9d3245079dfc8e7a19f57c156%2Ffig_20_05_ecology_as_demand_system.png?alt=media)

**Figure 20.5: The ecology as a demand system.** Table 20.2 read along the one dimension that sets a price, with each row's binding constraint beside it. Two things become hard to unsee. The rows do not scatter idiosyncratically around a mean — they are ordered, and the ordering tracks the *kind* of constraint: a mandate at the bottom, a liability in the middle, a balance sheet above it, and discretion at the top, with the behavioral household the one row whose slope can take the wrong sign. Averaging over a distribution shaped like that produces a holder with an average mandate, an average leverage ratio and an average redemption term, which is not an approximation to anybody. And the arrow at the bottom is the scale of the thing: Part II's benchmark assumes an elasticity of four or more, which is twice as far off this axis as the axis is long. The elasticities attached to the table's verbal grades are illustrative; the ordering and the grades are the table's. Chapter 19 §19.5 anticipated this. The intermediary discount factor is what the system produces when the dealer's curve is the binding one; the wealthy-household answer of Chapter 5 §5.7 is what it produces when hers is. Neither is wrong. Each is a special case, and the system tells you which case you are in.

### Who bought when prices fell

The payoff of thinking in these terms shows up most clearly in an episode where the sectors moved in opposite directions at once.

In February and March of 2020 the US equity market fell by roughly a third in five weeks and the Treasury market became disorderly in the way Chapter 19's opening episode describes. A representative-agent account of that month has one investor, so it has nothing to say about who was on which side. The sectoral data say a great deal, and work drawing on the Financial Accounts, the New York Fed's primary dealer statistics, and brokerage records has converged on a consistent picture.

**Institutions sold, or were sold for.** Mutual funds and ETFs met large redemptions, which are forced sales under any mandate; levered funds unwound positions into rising haircuts; dealers, having absorbed inventory in the first week, reached the limits of their balance sheets and stopped. Foreign official holders sold Treasuries in size to raise dollars. Every one of these is a row in Table 20.2 with its constraint binding.

**Households bought.** The default-driven flow continued, because it is a function of payroll rather than of price, and the discretionary retail flow rose sharply — new brokerage accounts, direct equity purchases, and, over the year as a whole, a household sector that was a net buyer of equities into the drawdown while the institutional sectors were net sellers.

**And the central bank bought.** The Federal Reserve's purchases of Treasuries and agency mortgage-backed securities over the following months were the largest and fastest in its history. They were finite and policy-determined — an announced program, not a standing commitment to absorb any quantity at an unchanged price — but they were very large and, over the range of the intervention, close to insensitive to price, which is what makes them legible here. Chapter 9 treats the same episode from the claim's side and Chapter 19 from the dealer's; the demand-system reading is that a buyer whose purchases were set by policy rather than by a balance-sheet constraint entered the clearing condition, at a moment when the private holders' curves had gone vertical, and the price cleared.

![Figure 20.7: Who bought when prices fell](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-e3daec912fe2ef548d99cd5157248d7deb3c464e%2Ffig_20_07_who_bought_when_prices_fell.png?alt=media)

**Figure 20.7: Who bought when prices fell.** Quarterly net purchases by holder sector, 2019 through 2021, with 2020:Q1 and Q2 shaded; this is the chapter's own data exercise Part B, drawn. Buyers stack up from zero and sellers down, so the two halves of a bar are never netted into one number. Panel (a) is corporate equities. Across the two shaded quarters the household sector bought 287 billion dollars of equity while the pension and fund sectors sold; the household line is the default-driven flow and the new brokerage accounts together, and it did not pause. Panel (b) is Treasuries, and it contains the row §20.4 is about. In 2020:Q1 the Federal Reserve bought 1,019 billion dollars of Treasury securities while every private domestic sector together sold 244 and the rest of the world sold 256 — the first quarter in the window in which the central bank's purchases exceeded the private sectors' combined net sales, and the demand curve that entered the clearing condition when the private curves had gone vertical. Two cautions the section states and the figure inherits. The Z.1's household row is a residual containing hedge funds and personal trusts, so panel (a)'s blue band is not only households. And a quarterly flow nets purchases against sales inside the quarter, which is why the fund sector's equity selling looks modest here: the March redemption spike and the April recovery are one number, and nothing in this figure can date an event within a quarter. *Source: Financial Accounts of the United States (Z.1), tables F.223 and F.210, unadjusted quarterly transactions through the FRED mirror. Author's calculations.*

No framework starting from a single marginal investor can state those three facts together. Stated together they are a demand system with one row shocked, one row constrained, and one row whose quantity was set by policy rather than by price.

***

## 20.5 What Is Established, and What Is Not

This chapter teaches a live research literature, and honest teaching requires marking the line, as the book does with the common-ownership evidence of Chapter 17 §17.7 and the intermediary-factor tension of Chapter 19 §19.5.

**Table 20.3: The state of the evidence**

| Claim                                                                      | Status                                                                    |
| -------------------------------------------------------------------------- | ------------------------------------------------------------------------- |
| Demand curves for individual stocks slope down                             | **Established.** Multiple methods, four decades, several markets          |
| Estimated institutional elasticities are far below frictionless benchmarks | **Established** as a qualitative gap; the magnitudes are method-dependent |
| Holder identity and mandate affect prices                                  | **Established**; it is the finding Part IV is built from                  |
| The aggregate multiplier is positive and economically large                | **Widely accepted**, on several independent approaches                    |
| The multiplier is approximately five                                       | **Contested.** A central estimate in a range, not a constant of nature    |
| The investment-universe instrument satisfies its exclusion restriction     | **Contested**, and the most active dispute in the field                   |
| Rising passive ownership has mechanically lowered market elasticity        | **Contested**, with a serious theoretical objection                       |

*Source: Author's assessment of the literature discussed in §§20.2-20.3.*

Three of the contested rows deserve a sentence more than the table gives them.

**The instrument.** The exclusion restriction requires that an institution's investment universe, and the size of the institutions in whose universes a stock sits, be unrelated to that stock's latent demand. Critics point out that menus are chosen, not assigned: a manager whose strategy is drifting toward a group of stocks adds them to its universe, so universe membership carries information about demand. Because an elasticity estimate is a ratio with this instrument in the denominator, an instrument that is weak or contaminated moves the estimates a great deal, and reassessments under alternative specifications report substantially different numbers.

**The strategic response.** The most substantive theoretical objection is due to Haddad, Huebner and Loualiche, and it targets the inference most often drawn from these estimates. Suppose a growing share of the market becomes inelastic, as Chapters 14 and 17 document. The naive conclusion — the arithmetic of the two-sector example above — is that aggregate elasticity falls in proportion. But the remaining active investors are not passive in the face of that change. Facing less competition, each has a stronger incentive to trade aggressively against mispricing, and their own elasticities rise. Whether the aggregate elasticity falls, and by how much, is then a question about the competitive response of a shrinking active sector rather than a mechanical consequence of the passive share. The point does not overturn the finding that markets are inelastic; it undercuts the projection that they are becoming steadily more so.

**The magnitude.** A multiplier of five was estimated on one market, over one sample, using a particular decomposition of flows into exogenous and endogenous components. Later work using different flow measures, horizons, and asset classes has produced numbers on both sides of it, and a textbook reporting five as a settled parameter would misrepresent the field. What the field does agree on is that the frictionless answer — a multiplier indistinguishable from zero — is rejected, decisively and everywhere it has been tested.

One further limitation is structural rather than statistical. A demand system needs observable holdings and observable prices, and Chapter 18 §18.6's private funds have neither at the frequency required: a holder whose position is marked quarterly by the manager who holds it does not have an estimable demand curve. The fastest-growing part of the ecology is the part the method sees worst.

***

## 20.6 Whose $$m$$? — and the Rest of the Book

Chapter 3 §3.7 asked whose marginal utility appears in $$p = E\[mx]$$ and promised that Part IV would answer. The answer is now available and can be stated in one line.

**Everyone's, weighted by wealth and by constraint — and the weights are measurable.**

Each clause is doing work. *Everyone's*, because the demand system does not select a marginal investor; every holder's first-order condition, constraint multipliers included, enters the clearing condition. *Weighted by wealth*, because a holder's influence on the price is its share of the market, which is why Chapter 14's distributional data are a pricing input rather than a distributional footnote. *Weighted by constraint*, because a large holder with a vertical demand curve contributes size to the market without contributing any price discipline to it, which is what the index complex, the defaults, and the capital-charged insurer do. And *measurable*, because 13F filings, statutory schedules, and the Financial Accounts make the weights data rather than assumptions. That last clause separates this answer from a philosophical position: it converts "who is the marginal investor?" from a modeling choice into an estimate with a standard error.

What the rest of the book does with it:

**Parts II and III supplied the objects.** The demand system needs characteristics, and the characteristics are what those parts constructed: betas and factor loadings (Chapters 4 and 6), duration and convexity (Chapter 9), ratings and default probabilities (Chapter 10), liquidity (Chapter 11), prepayment risk (Chapter 13). A demand curve is written over characteristics that somebody had to define and measure. Nothing in Part IV replaces that work; all of it consumes it.

**Part V shows the issuers responding.** If holders' demand for a security type is steep and shifts, the price of that type moves, and firms that can manufacture the security will manufacture more of it. That is the content of Chapter 23 §23.7 — gap-filling, safe-asset supply, and credit-supply effects on leverage — and Chapter 25 §25.6 generalizes it: every financing door a firm can walk through exists because some constrained balance sheet in this chapter's ecology wants the claim behind it. That is why Part V is titled "Firms as Issuers of Claims." Chapter 12 §12.9 has already run the experiment in the other direction, with index inclusion as the demand shock.

**Chapter 27 closes the loop.** If asset prices are set by an ecology of constrained holders rather than by a forecast of the economy's cash flows, then their informational content for the real economy is weaker than the standard reading assumes, and the channel from financial conditions to investment runs partly through who holds what. That is the synthesis chapter's problem: §27.4 runs this chapter's multiplier through the transmission channels, and §27.5 sets out the four readings a price move can carry once the holders are in the picture. This chapter is what both have to work with.

Part IV ends here. It began in Chapter 14 with a household who never chose a fund, and it ends with a system in which that household's payroll date is a determinant of the level of the stock market. The claims were mapped in Chapter 2, the prices derived in Parts II and III, and the holders have now been assembled into the mechanism that connects them.

***

## Elsewhere in the Series

* **The macroeconomic implications of an inelastic asset market** — *Institutionalist Macroeconomics*, Chapter 25, for what a flow-driven asset market does to wealth distribution and aggregate demand; this chapter supplies the pricing mechanism and stops at the sector boundary.
* **Issuance as a response to holder demand** — this book, Part V, and Chapter 23 §23.7 and Chapter 25 §25.6 in particular.

***

## Summary

1. **Shleifer's 1986 question was a heresy and is now a research program.** Index additions carry no information and yet moved prices, by an amount that grew with the assets tracking the index. Harris and Gurel read the same effect as temporary price pressure, and the distinction between a liquidity cost and a demand slope organized the next three decades.
2. **The demand-system move is to stop choosing a marginal investor.** Rather than assuming whose $$m$$ prices the market, estimate every sector's demand for every claim and let market clearing determine prices. Chapter 2's master holdings table is the left-hand side of that estimation, exactly as §2.5 promised.
3. **Demand is written over characteristics and price.** Each holder's portfolio weight is a function of an asset's observable characteristics and its price; holders differ in the coefficients, not in the specification. The coefficient on price is the demand elasticity.
4. **Estimated elasticities are far below the frictionless benchmark.** Institutional elasticities cluster near or below one against a theoretical benchmark above five. The reason is the inventory Part IV assembled: 1940 Act limits, risk-based capital, benchmarks, index rules, redemption terms, balance-sheet capacity, and plan defaults.
5. **Identification is the whole difficulty.** Prices are endogenous to demand, so a regression of holdings on prices recovers nothing. The investment-universe instrument uses the fact that institutions hold from restricted, observable menus, and its exclusion restriction is the most contested object in the literature.
6. **The aggregate multiplier is one over a wealth-weighted average of holders' elasticities.** In the two-sector example, a market half of which is mandated and half of which has elasticity $$0.4$$ has aggregate elasticity $$0.2$$ and a multiplier of $$5$$: a dollar of flow creates about five dollars of market value. Gabaix and Koijen estimate roughly that figure for the US equity market, in a contested range.
7. **Flows are not a sideshow.** If the multiplier is materially above zero, a portfolio reallocation with no information in it is a first-order determinant of prices, and the corporate sector's net issuance — negative for most of four decades (Chapter 12 §12.6), and timed to price rather than against it — does not undo it.
8. **Inelastic markets do not mean information is irrelevant.** They mean the arbitrage that transmits information into prices is executed by capital-constrained holders. Efficiency is the limiting case of the demand system as elasticities go to infinity.
9. **Heterogeneity is not noise around a mean.** The holders of Table 20.2 differ systematically along the dimension that sets prices, and the differences do not cancel. The mean investor has no counterpart in the world, and the weights defining him have been moving in one direction for fifty years.
10. **March 2020 is the demonstration.** Institutions sold under binding constraints, households bought on payroll dates and in brokerage accounts, and the central bank added a very large policy-driven demand that was insensitive to price over the range of its purchases. No single-marginal-investor framework can state those three facts together.
11. **Chapter 3's question has an answer.** Whose $$m$$? Everyone's, weighted by wealth and constraint — and the weights are measurable.

***

## Key Terms

* **Demand system**: A set of estimated demand functions, one per holder, over the full set of claims, closed by market clearing; the asset-pricing analogue of a consumer demand system
* **Characteristics-based demand**: The specification in which a holder's portfolio weight in an asset is a function of the asset's observable characteristics and its price, with holder-specific coefficients
* **Demand elasticity (**$$\zeta$$**)**: The percentage reduction in the quantity of a claim a holder wishes to hold in response to a one percent increase in its price, holding characteristics fixed
* **Latent demand**: The holder-specific residual in a fitted demand system — the part of a portfolio weight the observed characteristics do not explain; the channel through which idiosyncratic reallocation moves prices
* **Investment universe**: The set of assets a holder has actually held in the recent past; the restricted menu whose variation supplies the instrument for price
* **Price multiplier (**$$\mathcal{M}$$**)**: The dollars of aggregate market value created per dollar of flow, $$\mathcal{M} = dV/dF$$; the reciprocal of the aggregate demand elasticity
* **Inelastic markets hypothesis**: The proposition that the aggregate demand curve for equities is steep, so that flows unaccompanied by information move the level of the market substantially
* **Market clearing**: The condition that the total dollar demand of all holders for a claim equals its total dollar supply; the equation that turns estimated demand curves into prices
* **Flow**: A dollar of demand entering or leaving a market, financed by a transaction in another asset and unaccompanied by news about cash flows

***

## Readings

### Required

* Koijen, R. S. J. and M. Yogo (2019). "A Demand System Approach to Asset Pricing." *Journal of Political Economy* 127(4): 1475-1515. *The canonical statement. Read §§I-III for the specification and market clearing; the instrument is the paper's hardest and most important section, and §20.2's starred subsection is a guide to it, not a substitute.*
* Gabaix, X. and R. S. J. Koijen (2021). "In Search of the Origins of Financial Fluctuations: The Inelastic Markets Hypothesis." NBER Working Paper 28967; also CEPR Discussion Paper 16290 and SSRN 3686935, subsequently revised. *Still a working paper, and cite it as one: as of August 2026 it appears under working papers on Koijen's own research page and carries no journal publication. It has circulated widely — the Q-Group Treynor Prize and the AQR Insight Award — and been debated intensively, so cite the version you read, and check the state of the debate over the multiplier before quoting a number from it.*

### Recommended

* Shleifer, A. (1986). "Do Demand Curves for Stocks Slope Down?" *Journal of Finance* 41(3): 579-590. *Short, and the origin of everything in this chapter. The finding that the effect grew with index-fund assets is the one to notice.*
* Harris, L. and E. Gurel (1986). "Price and Volume Effects Associated with Changes in the S\&P 500 List: New Evidence for the Existence of Price Pressures." *Journal of Finance* 41(4): 815-829. *The same experiment read as temporary price pressure. Reading the two papers together is the cleanest introduction to what is at stake in the word "permanent."*
* Haddad, V., P. Huebner and E. Loualiche (2025). "How Competitive Is the Stock Market? Theory, Evidence from Portfolios, and Implications for the Rise of Passive Investing." *American Economic Review* 115(3): 975-1018. *The strategic-response objection of §20.5: active investors' elasticities are endogenous to how much competition they face, so a rising passive share need not lower aggregate elasticity proportionally.*

***

## Discussion Questions

1. **What does an index inclusion reveal?** Suppose the aggregate multiplier is five and a stock added to a major index sees mechanical demand equal to 5 percent of its shares outstanding, with an announcement return of 5 percent. Compute the implied elasticity of demand for that individual stock, and explain why it is so much *higher* than the aggregate elasticity of 0.2. Then answer the question in the title: with a multiplier that size, is the inclusion return telling you about the company, about the index, or about the market's ownership structure — and what would you have to observe to distinguish them?
2. **Refutation or completion?** Part II derives prices from preferences and no arbitrage. This chapter derives them from estimated holdings and a clearing condition. Make the strongest case that demand-system asset pricing refutes Part II, then the strongest case that it completes it. In your answer, say what would have to be true empirically for each case to win, and state which limiting case of the demand system delivers Part II's results exactly.
3. **The zero-elasticity holder.** An index fund's price coefficient is zero by construction. Does that make index funds destabilizing? Work through the two channels separately — the fund's own trading in response to price, and the flow it receives from its investors — and then state the condition on the *other* holders under which a large zero-elasticity sector makes prices more volatile, and the condition under which it makes them less so. Compare your answer with Chapter 17 §17.6's fifth point.
4. **Whose constraint is in the price?** Choose one claim from Chapter 2's Table 2.5 — agency MBS, municipal bonds, or long-dated Treasuries. Using Table 20.2, name the holder you believe is marginal in that claim and the constraint that binds it. What evidence would confirm your choice, and what would falsify it? Note that "marginal" may differ across states of the world for the same claim.
5. **The instrument, restated as a research design.** The investment-universe instrument requires that a stock's presence on institutions' menus be unrelated to unobserved demand for it. Propose a different source of exogenous variation in the price a holder faces — a regulatory change, an index-rule change, a merger of two managers, a mandate reclassification. State what your design identifies, what it does not, and whether it recovers an elasticity for one holder or for the market.

***

## Problems

**Problem 1 — Two-sector clearing.** A market has $$Q = 200$$ shares outstanding, trading at $$p = 1$$ dollar. Sector A is mandated and holds $$120$$ shares at any price. Sector B has demand $$q\_B(p) = 80p^{-\zeta\_B}$$.

(a) Verify the market clears at $$p = 1$$ and compute the aggregate demand elasticity for $$\zeta\_B = 0.5$$. (b) Sector A receives an inflow and raises its desired holding to $$126$$ shares. Solve for the new price and the new aggregate market value. What multiplier does the flow realize? (c) Repeat (b) with $$\zeta\_B = 5$$. State in one sentence what the comparison shows about which sector's elasticity the multiplier is sensitive to. *Answers:* (a) $$\zeta = 0.2$$; (b) $$p^{-0.5} = 0.925$$, $$p = 1.1687$$, $$V = 233.75$$, a multiplier of about $$5.6$$ on a flow of 6 dollars; (c) $$p = 1.0157$$, $$V = 203.14$$, a multiplier of about $$0.52$$.

**Problem 2 — Composition and the multiplier.** Using the baseline of §20.3 — aggregate elasticity $$\zeta = s\_A\zeta\_A + s\_B\zeta\_B$$ with $$\zeta\_A = 0$$ and $$\zeta\_B = 0.4$$ — compute the multiplier when the mandated share $$s\_A$$ is 30, 50, and 70 percent. Now suppose that as $$s\_A$$ rises, the surviving active sector's elasticity rises with it: $$\zeta\_B = 0.4/(1-s\_A)$$. Recompute. Explain in two sentences which of these two calculations is the naive projection criticized in §20.5 and which is the strategic-response alternative, and what the second one implies about the aggregate elasticity.

**Problem 3 — Supply response.** Take the baseline two-sector economy with $$\zeta = 0.2$$ and add issuers who supply shares with elasticity $$\zeta\_S$$ with respect to price. The multiplier becomes $$\mathcal{M} = 1/(\zeta + \zeta\_S)$$. (a) What supply elasticity would be required to cut the multiplier from 5 to 1? (b) US net equity issuance has been negative for most of the past four decades, and issuance rises after prices rise rather than before. Explain why each of those two facts, separately, argues against treating the observed correlation between issuance and prices as an estimate of $$\zeta\_S$$. (c) Which chapter of Part V takes up the estimation problem you have just described?

***

## Selected Solutions

*Solutions to Problems 1 and 2 follow. Solutions to the remainder are in the instructor materials.*

**Problem 1.**

(a) At $$p = 1$$ sector B holds $$80 \times 1^{-\zeta\_B} = 80$$ shares whatever $$\zeta\_B$$ is, and sector A holds 120, so demand is 200 and the market clears against the 200 shares outstanding at an aggregate value of 200 dollars. The value weights are $$s\_A = 0.6$$ and $$s\_B = 0.4$$, so with $$\zeta\_B = 0.5$$ the aggregate elasticity is $$\zeta = 0.6 \times 0 + 0.4 \times 0.5 = 0.20$$ and the multiplier is $$\mathcal{M} = 1/\zeta = 5$$. Note that this reaches §20.3's baseline from a different pair of numbers: a more elastic active sector holding a smaller share of the market. The multiplier is a property of the product, not of either factor.

(b) Sector A's desired holding rises to 126, so clearing requires $$80p^{-0.5} = 200 - 126 = 74$$, hence $$p^{-0.5} = 74/80 = 0.925$$ and $$p = 0.925^{-2} = 1.1687$$. Aggregate value is $$200 \times 1.1687 = 233.75$$ dollars, a rise of 33.75. The flow is the six additional shares bought at the initial price, 6 dollars, or 3 percent of the market, so the realized multiplier is $$33.75/6 = 5.62$$. It exceeds the limiting value of 5 because the flow is not infinitesimal: $$\mathcal{M} = 1/\zeta$$ is the derivative $$dV/dF$$ evaluated at the initial point, and the demand curve is convex in price, so a flow this large travels up a steepening schedule. Sector B was induced to give up one and a half shares of the six by a 17 percent higher price. Nobody's expected cash flows changed.

(c) With $$\zeta\_B = 5$$ the same clearing condition reads $$p^{-5} = 0.925$$, so $$p = 0.925^{-1/5} = 1.0157$$ and aggregate value is 203.14 dollars — a realized multiplier of $$3.14/6 = 0.52$$, against the limiting $$1/(0.4 \times 5) = 0.5$$. **The multiplier is a statement about the elastic minority, not about the inelastic majority.** Sector A's zero contributes nothing to the wealth-weighted average at any share; the whole of the aggregate elasticity comes from sector B, and the mandated sector enters only through the weight it leaves B to carry. A tenfold rise in the active sector's elasticity cut the price response by a factor of about eleven, with the ownership structure held fixed.

**Problem 2.**

The naive calculation holds the active sector's elasticity at 0.4 and lets the mandated share thin the only elastic weight in the average. The strategic-response calculation lets each surviving active dollar trade more aggressively in exact proportion to the competition that has left.

**Table 20.4: The multiplier under the two projections in Problem 2**

| Mandated share $$s\_A$$ | Naive $$\zeta$$ | Naive $$\mathcal{M}$$ | Strategic $$\zeta\_B$$ | Strategic $$\zeta$$ | Strategic $$\mathcal{M}$$ |
| ----------------------- | --------------- | --------------------- | ---------------------- | ------------------- | ------------------------- |
| 30 percent              | 0.28            | 3.6                   | 0.571                  | 0.40                | 2.5                       |
| 50 percent              | 0.20            | 5.0                   | 0.800                  | 0.40                | 2.5                       |
| 70 percent              | 0.12            | 8.3                   | 1.333                  | 0.40                | 2.5                       |

*Source: Author's calculation on the parameters stated in the problem.*

The first three columns are the naive projection §20.5 criticizes: the multiplier rises from 3.6 to 8.3 — by a factor of 2.3 — as the mandated share goes from 30 to 70 percent, purely as arithmetic. The last three are the alternative. Under $$\zeta\_B = 0.4/(1-s\_A)$$ the two effects cancel exactly, $$\zeta = (1-s\_A) \times 0.4/(1-s\_A) = 0.4$$ at every mandated share, and the multiplier is a constant 2.5. The levels are not comparable across the two halves of the table — the strategic specification is normalized to give $$\zeta\_B = 0.4$$ in a market with no mandated sector, so it sits above the naive one everywhere — and the comparison to make is between the *slopes*.

Exact cancellation is an artifact of the functional form rather than a result; the honest reading is that the truth lies between the two halves, and where it lies is an empirical question about the competitive response of a shrinking active sector rather than an accounting one. What the exercise establishes is the distinction §20.5 draws. The strategic-response objection does not touch the finding that markets are inelastic: even the right-hand column has an aggregate elasticity of 0.4 against a frictionless benchmark above five, and a multiplier of 2.5 is an enormous number. What it undercuts is the *projection* — the claim that a rising passive share is mechanically making markets more inelastic over time. The left-hand column is that forecast; the right-hand column says the forecast has no mechanism behind it until somebody measures how the surviving active investors respond.

***

## Data Exercise: Flows, Returns, and Why the Regression Lies

**Part A — A naive flow-return relation (free data).** The Investment Company Institute publishes estimated long-term mutual fund and ETF flows — weekly and monthly, by broad category, free from its website — and Ken French's data library supplies the market return.

1. Build a monthly series of net flows into domestic equity mutual funds and ETFs, over the longest common sample, and a matching series of market excess returns. Scale flows by the prior month's aggregate equity market value so that the units are comparable across decades.
2. Regress the market's monthly excess return on contemporaneous scaled flows. Report the coefficient, its standard error, and the implied multiplier (a coefficient of $$\hat b$$ on a flow scaled by market value implies a multiplier of $$\hat b$$ directly, since both sides are in percent of market value). Compare it with the figure in §20.3.
3. Now write down what would have to be true for that coefficient to be the multiplier. In particular: does a mutual fund inflow in a month cause the return, or does the return cause the inflow? Chapter 16 §16.3's flow-performance relationship says the second channel is real and large. Which direction does it bias your estimate, and by how much can you bound the bias?
4. Re-run the regression with flows lagged one month and with returns lagged one month, and use the pattern of coefficients to argue for one direction of causation over the other. Explain why this argument, though standard, does not settle the question.
5. Finally, a scope check. ICI flows cover fund vehicles only. What fraction of Chapter 2 Table 2.5's equity row do they represent, and what does that omission do to your interpretation of the estimate?

**Part B — Who bought in 2020 (free data).** The Financial Accounts (Z.1) and the Distributional Financial Accounts are free from the Federal Reserve Board.

1. From the Z.1 corporate equity table, extract net purchases of corporate equity by household sector, mutual funds and ETFs, private and public pension funds, insurers, and the rest of the world, quarterly for 2019 through 2021.
2. Plot them on one axis and identify which sectors were net buyers and which were net sellers in 2020 Q1 and Q2. Compare your reading with §20.4's account and note any sector that does not fit it.
3. Repeat for Treasury securities, adding the monetary authority. Date the quarter in which the Federal Reserve's purchases exceeded the combined net sales of all private domestic sectors.
4. State two measurement caveats before drawing conclusions: the household sector in the Z.1 is a residual that contains hedge funds and personal trusts, and quarterly flows net purchases against sales within the quarter, so they cannot date anything within it.

**Part C ★ (if you have WRDS or the patience for EDGAR).** Using 13F filings, build a panel of institutional holdings of the S\&P 500 for two adjacent quarters.

1. For each stock, compute the share of shares outstanding held by 13F filers, and split it between filers you classify as index-tracking and all others, using fund names and reported turnover.
2. Regress the quarterly change in each institution's portfolio weight on the quarterly return, stock by stock, and interpret the coefficient. It is not an elasticity. Explain precisely which of §20.2's two problems — endogenous prices, or omitted characteristics — is doing the most damage in your specification.
3. Construct a crude version of the investment-universe instrument: for each stock, the total assets of institutions that held it in at least one of the previous four quarters but do not hold it this quarter. Re-estimate. Report how far the estimate moves, and state honestly whether your instrument plausibly satisfies the exclusion restriction of §20.5.