Chapter 14
Measuring the Theoretical Quantities: What Could and Could Not Be Measured

14.1 Purpose of this chapter

This chapter tabulates which of the symbols introduced in Part I could actually be measured, and what values they take where they could. Each chapter of Part IV treats a separate domain, so the correspondence with the theoretical quantities is easily scattered. This chapter supplies the index.

Its main purpose is to make explicit what could not be measured. Fewer than half the quantities the framework defines could be connected to real data.

The stage reached differs by quantity (Remark 11.3). The list here is a record of where between stages 1 and 3 of Section 11.2 each one stalled.

14.2 The primitive sets made concrete

14.3 The observed range of the map $\mathrm{\Phi }$

Part II enumerated 28 types; those that could be connected to real data are limited to the following (Table 14.2).

That is five of the 28 types. Family 3 (renting assets), family 4 (complement recovery) and family 7 (payer separation) were not measured at all.

14.4 Cumulants and the credit position

14.4.1 $D$, $P$ and $\kappa$ in family 2

Substituting FY2024 values into the definitions of Chapter 2, in millions of yen:

$$\begin{array}{lll}\hfill P(\text{FY2024})& =28,150,350\text{(issued: settlement from customers)}& \hfill \text{(14.1)}\\ \hfill D(\text{FY2024})& =28,241,983\text{(redeemed: delivery by the firm)}& \hfill \text{(14.2)}\\ \hfill -{\kappa }_C(\text{FY2024})& =2,895,327\text{(unused balance: customers credit the firm)}& \hfill \text{(14.3)}\end{array}$$

Against (3.2), ${\kappa }_C=-2.90$ trillion yen. Being negative, customers extend credit to the issuer. Applying (5.1),

$$\mathrm{CCC}=\frac{W}{r}=\frac{2,895,327}{28,150,350}\times 365=37.5\text{days}.$$

14.4.2 $\mathrm{CCC}$ by medium

The same calculation by medium shows a difference of two orders of magnitude.

14.4.3 $\kappa$ in the corporate statistics

Applying (3.2) to the corporate sector as a whole, FY2024, all industries excluding finance and insurance.

$$\begin{array}{lll}\hfill {\kappa }_C& =\mathrm{DSO}=71\text{days (capital 1bn yen and over)},35\text{days (under 10m yen)}& \hfill \text{(14.4)}\\ \hfill {\kappa }_S& =-\mathrm{DPO}=-44\text{days (1bn and over)},-21\text{days (under 10m)}& \hfill \text{(14.5)}\\ \hfill \sum _i{\kappa }_i& =27\text{days (1bn and over)},14\text{days (under 10m)}& \hfill \text{(14.6)}\end{array}$$

The sign of $\kappa$ depends on both size and industry, and the industry effect dominates. In retail, ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{\kappa }_i=-16$ days for firms of 1bn yen and over: the sign reverses.

14.5 Measuring the growth constraint

The $g^{\star }=(m+d-I∕r)∕\mathrm{CCC}$ of Corollary 5.7 needs four quantities. The measurement proceeds in two stages.

$d$ and $I$ are not used on the long-run side because depreciation and fixed-asset balances were not obtained back to the 1990s. For $m$, the operating margin is used: the $m$ of Chapter 5 is what funds increases in working capital, so a figure net of selling and administrative expenses is closer to the definition.

14.5.1 The corporate sector over time ($m∕\mathrm{CCC}$)

The trend is shown in Figure 14.1.

14.5.2 $g^{\star }$ including depreciation and investment

Corollary 5.7 needs, besides $m$, the depreciation rate $d$ and investment $I$. As Remark 5.12 states, $I\approx dr$ is not assumed. $I$ is estimated as the change in fixed-asset balances plus depreciation. Land is excluded because it is not depreciated.

$$I\approx {\left(\text{tangible}+\text{intangible}-\text{land}\right)}_t-{\left(\text{tangible}+\text{intangible}-\text{land}\right)}_{t-1}+{\text{depreciation}}_t$$

(14.7)

Following Remark 14.5, judgments use the nine-year means.

In the single year, the under-10m class shows a high $34.7\%$, but with a standard deviation of 41.4 no meaning should be read into that value.

14.5.3 $g^{\star }$ by industry

The bottom two industries in Table 14.7 exhibit the divergence of Remark 5.13. The $\mathrm{CCC}$ of food services is 5 days, so a one-point movement in the numerator moves $g^{\star }$ by about 73 points. The standard deviation of 3334 is the consequence of that amplification: $g^{\star }$ is not functioning as an indicator.

14.5.4 The $m$ of small firms is decided by directors’ remuneration

Proposition 14.6 showed that $m$ is the constraint but not why $m$ is low. Decompose the cost structure.

Small firms have high gross margins: 46.3% against 23.6% for large firms. Their SG&A ratio, however, is 44.8% against 15.8%. As Remark 2.17 notes, part of that difference may be $C_{\mathrm{admin}}$: at a scale where invoicing and chasing are done by hand, the operating cost of contracts is booked in SG&A. This text does not separate it. The low $m$ is therefore not due to a lack of pricing power; the drop occurs at the SG&A stage.

Moving personnel costs back into cost of sales narrows the gap in gross margin from 22.7 points to 7.7 (15.9% against 23.6%). Two-thirds of the gap was a difference of accounting classification. Even after the adjustment, however, the small classes remain higher.

Decomposing personnel costs into directors and employees identifies the cause.

14.5.5 Comparison with unincorporated businesses

Proposition 14.8 rests on the corporate statistics. Whether the same structure appears in a different statistic, with different size classes, and in a sample including entities without corporate personality, is checked here.

The Basic Survey on Small and Medium Enterprises was used: a sample of about 110,000 firms in which four corporate size classes by headcount and unincorporated businesses appear in the same table. Unincorporated businesses have no line item for directors’ remuneration, so the proprietor’s return to labour remains in operating profit.

Registered implication: the operating margin of unincorporated businesses is close to the corporate small classes’ figure “after adding remuneration back”. The two are substantively the same economic agent and differ only in accounting classification.

The implication is supported. The operating margin of unincorporated businesses is about five times the corporate figure in all six years, with a standard deviation of 1.3. This is not a single-year peculiarity. The corporate figure after adding remuneration back in Section 14.5.4 was 9.5%. The 16.9% of unincorporated businesses exceeds that, but is far closer to it than to the 1.5–3.0% before adding back.

The personnel-cost ratio explains the difference.

Unincorporated businesses have the lowest personnel ratio, because the proprietor’s labour is not booked as a cost. What appears in personnel costs as directors’ remuneration in a company remains in operating profit for an unincorporated business.

14.5.6 By industry (FY2024)

Where the numerator $m+d-I∕r$ is negative, $g^{\star }<0$, and by Proposition 5.5 free cash flow is negative even at zero growth.

A long $\mathrm{CCC}$ and a small $g^{\star }$ do not coincide (Table 14.13). Leasing has the longest $\mathrm{CCC}$ at 210 days but a high $m=6.6\%$, so on $m$ alone it sits at the same level as petroleum and coal products ($\mathrm{CCC}=20$ days, $m=0.6\%$). $\mathrm{CCC}$ alone cannot decide the growth constraint. The $g^{\star }$ including $d$ and $I$ is given in Section 14.5.

14.6 Right and exercise, and cognitive surplus

The ${\varphi }_{\mathit{𝑐𝑜𝑔}}=p\cdot 𝔼[\overline{\delta }-\delta ]$ of (2.15) is a measurable quantity, and measurements exist. Four are given below. What this text obtained itself, however, is only an upper bound; the value is unknown.

14.6.1 ${\varphi }_{\mathit{𝑐𝑜𝑔}}$ is measurable

This text’s measurement: family 2 recoveries on expiry.

$$\begin{array}{lll}\hfill p\cdot 𝔼[\overline{\delta }-\delta ]& \le 20,634\text{million yen}& \hfill \text{(14.8)}\\ \hfill \frac{20,634}{2,895,327}& =0.71\%\text{(of unused balances)}& \hfill \text{(14.9)}\\ \hfill \frac{20,634}{28,150,350}& =0.073\%\text{(of issuance)}& \hfill \text{(14.10)}\end{array}$$

The inequality holds because this figure also contains recoveries through refunds. What was obtained is an upper bound, not ${\varphi }_{\mathit{𝑐𝑜𝑔}}$ itself. The three examples below all give the value itself.

Gym memberships. [6] matched the contract choices and daily attendance records of 7,752 members across three US gyms, measuring the $\overline{\delta }-\delta$ of (2.15) directly. Members predicted 9.50 visits a month and made 4.17. Members on plans above $70 a month attended on average only 4.3 times, paying more than $17 a visit where a ten-visit pass would cost $10. The loss over a membership averaged $600 against total payments of about $1,400, so ${\varphi }_{\mathit{𝑐𝑜𝑔}}$ is about 43% of the amount paid.

Subscription renewals. [9] covers a wider range. Using comprehensive data from a payment-card network, it exploits as an exogenous shock the fact that card renewal forces an active renewal decision. Across ten subscriptions, inattention raises sellers’ revenue by 89% on average (estimates ranging from 14% to over 200%). The attention parameter $\lambda$ is estimated at 0.18 on average, ranging from 0.04 to 0.50 across services.

Breakage on prepaid balances. [13] treats family 2-4 (prepaid) directly. Analysing transaction data on over four million people for a scheme adding a bonus to prepaid restaurant balances, it records that about 40% of prepaid value goes unused. Operators earn a median of $5.50 of breakage profit per dollar of bonus, and the profit from breakage exceeds the revenue from increased spending.

This text’s family 2 measurement stopped at an upper bound because (2) and (3) were missing. Aggregate statistics contain no individual records and there is no exogenous shock. In the taxonomy of Section 13.2 this is not (i), no measurement exists, but (iv), access constraint: the method of measurement is established and what is needed is reaching the data.

14.6.2 A detour through the menu

Condition (2) of Proposition 14.13, individual realized use, is not available in published statistics. But a substitution of definitions can sometimes provide a detour.

When a flat rate $F$ and a per-use price $p$ coexist on the same menu, the break-even number of uses can be computed.

$${\delta }^{\ast }=F∕p$$

(14.11)

Even where $\overline{\delta }$ is “unlimited” and unmeasurable, (14.11) follows from the menu alone. Using it as a proxy for $\overline{\delta }$, all that is needed is the industry-average number of uses $\delta$.

Since (14.11) is equivalent to $p{\delta }^{\ast }=F$, substituting into (2.15) gives

$${\varphi }_{\mathrm{cog}}\approx p𝔼[{\delta }^{\ast }-\delta ]=F-p𝔼[\delta ]$$

(14.12)

Under this proxy, cognitive surplus is the difference between the flat fee and the per-use equivalent. The registered implication ${\delta }^{\ast }-\delta >0$ becomes the same claim as (14.12) being positive, that is, the flat fee exceeding the per-use equivalent of average use.

Registered implication: for flat-rate members, ${\delta }^{\ast }-\delta >0$; the average member does not reach break-even.

Result. From the Survey on Selected Service Industries, $\delta$ is 89 visits a year per individual member, that is 7.42 a month. ${\delta }^{\ast }$ was computed from the published prices of one large full-service operator (2026, tax included).

The implication is rejected. The sign is negative in all four categories (Table 14.14): for the average member the flat contract is rational.

The effective unit price — total dues divided by total visits — points the same way, falling 29% over 25 years from 1,147 yen in 2000 to 815 yen in 2023. Against a per-visit market rate of 2,530–3,630 yen, members are using the facility more cheaply than paying per visit.

The reason for purchase by medium serves as a proxy for the transaction frequency $\nu$.

The higher the gift share, the lower $\nu$ and the longer $\mathrm{CCC}$. The $\lambda (\nu )$ of (6.4) itself was not measured.

14.7 The information structure $\mathcal{𝒢}$ made concrete

The $\mathcal{𝒢}$ of Chapter 4 — verifiable shared information — is observable in family 5 as a concrete set of fields. The items published on the Employment Service Information Site are what $\mathcal{𝒢}$ consists of.

Measured values of $y$ at the two establishments that could be confirmed:

$$\frac{y}{\text{open-ended placements}}=\frac{124}{4,348}=2.85\%,\frac{3}{189}=1.59\%.$$

Testing the $b^{\star }=1∕(1+\gamma {\sigma }^{2}k)$ of Example 4.5 requires both $b$ and ${\sigma }^2$ as continuous quantities, but refund rates and periods are not in a uniform format and the uncontrollable part cannot be extracted from ${\sigma }^2$.

A test on the price side was possible, however. Fee rates achieved by occupation were obtained as a continuous quantity for 307 firms and correlated with separation rates (Section 15.3.5.0).

14.8 The three-way decomposition of surplus

The three components of (6.1) can be separated only in the case of Chapter 17.

This is the only case in which two of the three components could be separated numerically for a single transaction. No procedure for separating the components exists in the other domains.

The two firms are intermediaries operating at uniform rates published in a price list, so the risk component ${\varphi }_{\mathit{𝑟𝑖𝑠𝑘}}$ was negligible and separation was possible. Where an intermediary sets rates by individual screening, a third term is added and the separation fails (Remark 17.2).

14.9 The terms of the launch cost

For each term of (8.11), the price of externalization was measured, as the difference in effective rate on a $50 transaction.

$T_{\mathit{𝑎𝑐𝑞}}$ exceeds the sum of all the other items. $T_{\mathit{𝑑𝑒𝑣}}$ itself (the development period) and the enumeration of $\widehat{\mathrm{\Omega }}$ cannot be externalized and therefore have no price.

14.10 Quantities relating to selection and identification

14.11 Quantities that could not be measured

The following are defined by the framework but could not be connected to real data (Table 14.20).

${\varphi }_{\mathit{𝑐𝑜𝑔}}$ is measurable in principle (Proposition 14.13); that this text could not reach it is a matter of data granularity. The rest belong to (i), no measurement exists, or are unobservable in principle.