Chapter 15
Empirics of families 2 and 5: prepayment and outcome-contingent forms

15.1 The propositions tested in this chapter

Among the types enumerated in Part II, families 2 (continuing access) and 5 (outcome- or state-contingent) are taken as subjects, and Remark 2.22 (conditions for divergence) together with the measurability condition of Chapter 4 is tested. The reason for choosing these two families is given in Section 11.5: each contains, singly, one of the two terms of the three-way decomposition of surplus that depend on the shape of $\mathrm{\Phi }$, and they sit on the diagonal of Figure 2.1. Following the policy of Part III, the implications were fixed before the investigation, and whether they were supported is recorded as it stands, including the parts that failed.

The two implications fixed in advance are as follows.

Registered implication (family 2): the lower the transaction frequency $\nu$ of a product, the larger the divergence between entitlement and exercise. Concretely, gift vouchers show a larger divergence than payment instruments used daily.

Registered implication (family 5): the harder the ex post verification of the outcome in an occupation, the stronger the refund scheme (a higher refund rate, or a longer covered period).

The conclusions first. The family-2 implication was supported as to level, but what could be measured was $\kappa$, not ${\varphi }_{\mathrm{cog}}$. The family-5 implication could not be tested: the necessary variable does not exist in the aggregate statistics.

15.2 Family 2: prepaid payment instruments

15.2.1 Data source and population

The 27th Survey of the Actual State of Issuing Businesses (FY2024 edition) of the Japan Payment Service Association was used. As set out in Section 12.9, it is a complete frame covering listed and unlisted issuers alike.

15.2.2 Measuring the residence period

The quantities of this section correspond to the notation of the theory as follows: issuance is cumulative settlement $P$, redemption is cumulative delivery $D$, and the unused balance is $-{\kappa }_C$ in equation (3.2). The FY2024 values are

$$P=28,150,350,D=28,241,983,{\kappa }_C=-2,895,327\text{(millions of yen)}$$

(15.1)

so ${\kappa }_C<0$: the customer extends credit to the issuer. Taking working capital as $W=-{\kappa }_C$ and sales velocity as $r=P$, equation (5.1) gives

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

which converted to days is called the residence period. The results by industry are given in Table 15.3.

The ordering is consistent with the registered implication. Transport (transit IC cards, daily use) and supermarkets (everyday shopping) are around 20 days, whereas accommodation vouchers, travel vouchers and department-store gift certificates run from 1,200 to 2,100 days.

As corroboration, the tabulation of reasons for purchase serves as a proxy for the transaction frequency $\nu$ of equation (6.4). For paper instruments, 41.5% are for the purchaser’s own use and 30.1% are gifts; for IC instruments, 95.5% are for own use and 0.0% gifts. In a gift the agent holding $\overline{\delta }$ and the agent performing $\delta$ are separated, so $\nu$ is structurally low.

15.2.3 Separating the alternative

There is a strong alternative. If the contraction of the paper-voucher market reduces issuance (a flow) while the unused balance (a stock) remains as an accumulation from the past, the ratio rises mechanically. This is precisely the period effect of Section 12.5.

A cross-section by industry cannot separate the two. But a panel by medium contains growing and contracting media at the same time, which makes identification possible.

The verdict splits in two.

(1)

The difference in levels cannot be explained by market contraction. Paper instruments lost only 6.0% of issuance over five years, yet already showed a residence of 775 days in FY2020 — about fifty times the 15 days of IC instruments. Since the difference exists at a starting point where contraction had not proceeded, the level derives from structure. Moreover the residence of the two growing media (IC and server) is almost immobile over five years: no mechanical rise in the ratio occurs.

(2)

Part of the variation can be explained by market contraction. Magnetic instruments lost 32.2% of issuance and their residence rose from 1,307 to 1,904 days. Paper instruments have also returned from 668 days in FY2022 to 845. Here the alternative bites.

The path of the levels is shown in Figure 15.1.

The separation is thus level from $\nu$, part of the variation from a period effect. Against the problem described as generally unidentified in Section 12.5, the simultaneous presence of a growing segment permitted partial identification. This is not a general solution but a piece of luck specific to this subject matter.

As an auxiliary observation, the share with no expiry date set is 71.2% for paper, 52.0% for magnetic, 37.9% for IC and 19.0% for server (physical stores), which apart from magnetic orders the same way as residence days. The direction of causation cannot be pinned down, however: is it that no expiry is set and so the balance sits, or that products whose balance sits — gifts, for instance — are the ones given no expiry? Probably the same $\nu$ determines both, but these data cannot settle it.

15.2.4 What could not be tested

A long residence does not imply that the entitlement is ultimately never exercised. The only figure corresponding to ${\varphi }_{\mathrm{cog}}$ is ¥20,634 million recovered through expiry and the like — 0.71% of the unused balance and 0.073% of issuance. The association further notes that this amount is thought to include recoveries through refunds, so it is not a pure non-exercise figure.

The conclusion for family 2 is therefore confined to the following.

In a prepayment $\mathrm{\Phi }$, the lower the transaction frequency $\nu$ of a product, the larger $|\kappa |$, and the difference reaches two orders of magnitude. Whether this residence converts into cognitive surplus, however, cannot be judged from public data.

Chapter 5 ($\kappa$ and residence) is supported, while Section 2.7 (cognitive surplus) remains untested. The former is only a necessary condition for the latter.

15.2.5 The absence of per-firm data

The association states explicitly that officials of the Local Finance Bureaux and other administrative bodies are under a statutory duty of confidentiality and cannot pass matters reported by issuers to the association as a third party. The Financial Services Agency publishes a list of registered and notified issuers, but it does not include unused balances. In addition, issuers themselves have asked that the state of non-redemption (expiry and the like) also be disclosed — that is, the breakage rate is not currently tabulated.

As a result, no dispersion within an industry or within a medium can be observed at all. Only means are visible.

15.2.6 The nature of the remaining obstacle

${\varphi }_{\mathrm{cog}}$ cannot be measured in this domain because the breakage rate is not tabulated. Issuers themselves have requested of the association’s statistics that “the state of non-redemption (expiry and the like) also be disclosed”, so measurement is not being carried out within the frame of this statistic.

This looks like (i) absence of measurement. But the obstacle is more narrowly located. Of the three conditions of Proposition 14.13, what is missing in this domain is (2) realized use at the individual-record level and (3) an exogenous event that forces attention; (1) the contractual entitlement is observable as the expiry date by medium. That is, the method of measurement is established, and what is missing is the granularity of the data.

Under the classification of Section 13.2, then, ${\varphi }_{\mathrm{cog}}$ for family 2 belongs to (iv) access constraint. Unlike an ordinary access constraint, however, data of the required granularity exist only inside the issuer, and there is no route by which publication can be awaited.

That per-firm unused balances are confidential is secondary. Even if per-firm data were obtained, ${\varphi }_{\mathrm{cog}}$ could not be estimated so long as the reported items contain no record of individual usage.

15.3 Family 5: fee-charging employment placement

15.3.1 Data source and population

The Ministry of Health, Labour and Welfare’s “Tabulation of Employment Placement Business Reports, FY2024 (preliminary)” was used. It is a complete enumeration of all operators under the Employment Security Act and, like family 2, escapes listing bias.

15.3.2 The choice of contractual regime

The Employment Security Act provides two methods of charging fees: the capped scheme and the notified scheme. The capped scheme is a formula-based contract fixed at a set rate of the wage; the notified scheme is a contract the operator sets freely and files.

The capped scheme has all but disappeared.

15.3.3 Fees per placement by occupation

Fees mix in temporary and day work. Nationally, fees relating to regular placements account for ¥913.6 billion, about 93% of the whole, but the breakdown by occupation does not separate the two. Defining the degree of contamination as the number of temporary and day placements (person-days) per regular placement, the calculation is restricted to occupations where this is below one.

The ordering corresponds to seniority, and if the fee rate is roughly constant this merely reflects annual income. Without a denominator of annual income, the table has no power to identify the measurability hypothesis. Whether officers’ fees are high because outcomes are hard to observe or simply because incomes are high cannot be distinguished from these data.

15.3.4 Why the measurability hypothesis cannot be tested

Data on refund schemes do not exist in this tabulation. They appear only firm by firm on the Jinzai Service General Site. The measurability hypothesis is therefore untested.

There is a more serious problem.

15.3.5 Fees and turnover

Data source

On checking the office detail pages of the Jinzai Service General Site, the problem of unstructured data feared in Part IV did not arise, except for the content of refund schemes.

Moreover the detailed search allows occupation, realized fee rate and turnover rate to be specified as conditions, and displays all three together in the results. This removes the need to open offices one at a time and made the retrieval of the 307 firms of Section 15.3.5.0 possible. The occupations that can be specified as conditions, however, are limited to nine in medicine, care and childcare.

The source is favourable in three respects.

First, regular and temporary or day work are separated. The contamination that made the aggregates for physicians and nurses unusable in Part IV does not arise at the level of the individual firm.

Second, fees are published as rates (percentages). The problem stated in Part IV — that without a denominator of annual income there is no identifying power — does not arise, since the figures are published already divided, and no external income data need be procured. In one office, for instance, the FY2025 realized rates were published by occupation as 40.1% for software development, 37.4% for designers and 37.7% for sales.

Third, there is a column for “separation not ascertained”. This is a figure showing whether the operator can observe what becomes of those placed, and is very nearly a direct observation — not a proxy — of the breadth of $\mathcal{𝒢}$ (verifiable shared information) of Chapter 4. It may answer the problem recorded in Part IV as “the greatest weakness of the design is the failure to construct a proxy on the B1 side”. It is self-reported, however, and the possibility that operators who do not follow up enter zero cannot be excluded.

The individual pages of certified operators also carry a field for “MHLW Jinzai Service General Site URL / licence number”, confirming that a join key exists from the list of certifications to the office detail pages.

The extent to which the regressor is prescribed by regulation

The regressor to be used in the test is the strength $b$ of the refund scheme, concretely the refund rate and the covered period. Of these, the covered period is subject to a regulatory floor in some fields.

Under the certification scheme for medicine, care and childcare, certification from FY2024 requires that a refund on early separation cover the case in which the jobseeker separates within at least six months of placement.

For certified operators in this field alone, the floor on the covered period is a product of regulation rather than of market choice. Keeping certified operators in the sample while using the covered period as a regressor measures compliance with regulation, not the choice of a contract. The reach of this point is limited, however (Remark 15.4).

This belongs to (iii) non-identification, but it is at the same time an opportunity for identification. Because the regulation bears on one field only, a three-group comparison can be constructed.

B against C is the hypothesis test proper. A against B answers a different question: how far did the regulation move the market? If most of group B also offers six months, the regulation is non-binding and merely ratifies what the market had already chosen. If group B centres on one to three months, the regulation is binding and has imposed a level away from the market equilibrium.

A against B is distinct from the measurability hypothesis, but it is a direct measurement of how far the institutional layer can rewrite $\mathrm{\Phi }$, and thus a test of the claim of Section 6.3.

The two sources record at different granularities

There is a more practical but serious problem. The refund scheme and the fee rate must be taken from different sources, and the two record at different granularities.

In the list of certified operators, one firm is registered as separate records for the three fields of medicine, care and childcare, and the certification number differs by field. The licence number on the Jinzai Service General Site, by contrast, denotes a particular office.

For a firm with several offices it is therefore unclear whether the licence number tied to the certification covers the firm’s performance as a whole. One large operator is certified for nine occupations in care and six in medicine, but there is no guarantee that office-level fee rates by occupation cover that granularity (the office checked in Section 15.3.5.0 had only five occupations).

This error correlates with size: the larger the operator, the greater the mismatch, and certified operators are disproportionately large. It is a systematic bias.

The three-way split of occupational classifications

Three classifications must be joined, and no correspondence table exists.

The certification scheme notes that “nursing occupations” includes nurses, assistant nurses, public health nurses and midwives, but the others are unclear. The system of numeric codes does not match the ordering of occupations in the Employment Placement Business Report. Building a correspondence table is within the range of what labour can solve.

Constraints on an analysis restricted to certified operators

The list of certified operators runs to about 100 entries. Since certification is by firm $\times$ field and one firm may appear in several fields, the number of actual firms is estimated at 60 to 70.

An analysis using certification as a regressor cannot be carried out at this scale. Only part of the 307-firm sample of Section 15.3.5.0 is certified, and no power for a between-group test is obtained. As stated in Section 12.9, the completeness of a frame does not guarantee the size of a sample.

This is, however, a constraint only on analyses using certification as a variable. The relation between the realized fee rate and the turnover rate can be measured for all operators irrespective of certification, and was in fact measured across 307 firms.

The sign is not uniquely determined

What family 5 sets out to test is the claim that the measurability of Chapter 4 determines the contractual form. Put naively, it becomes “the harder the ex post verification of the outcome in an occupation, the stronger the refund scheme”, but this form ignores a force running the other way.

Two forces follow from the optimal strength $b^{\star }\propto 1∕(1+r{\sigma }^{2}k)$ of Example 4.5.

• B1 (numerator, the incentive side): the more the operator’s effort bears on retention in an occupation, the more valuable it is to load $\pi$ onto $y$. The refund becomes stronger.
• B2 (denominator, the risk side): the more separation is determined by factors outside the operator’s control, the more loading it onto $y$ merely transfers risk. The refund becomes weaker.

The naive formulation sees only B1. In many occupations the two covary in the same direction, so the sign is not uniquely determined.

A test from the price side

The formula for $b^{\star }$ cannot itself be tested, since the required quantities are not available.

${\sigma }^2$ is not something aggregation suffices for. What is needed is the variance of the part the operator’s effort cannot control. The variance of turnover by occupation mixes in the operator’s screening and its skill at supporting retention: controllable and uncontrollable variation are not separated.

Instead, the response on the price side is examined. If the measurability hypothesis is correct, then in occupations where the outcome is more uncertain the operator should be forced to set a lower fee.

Registered implication: the realized fee rate by occupation is negatively correlated with the turnover rate of that occupation.

Alternative: the fee rate is determined by the income level of the occupation and by commercial custom, and is unrelated to turnover.

Result of the test

Using the detailed search of the Jinzai Service General Site, occupation, realized fee rate and turnover rate were retrieved together. The sample is Tokyo, licence numbers 13-Yu, four occupations.

The coverage of the sample is recorded in Remark 15.7, and a secondary observation in Remark 15.8.

The implication is not supported. The regressor does not move. This is case (vi) of Section 13.2 (Remark 13.1). In all four occupations the correlation is insignificant ($|t|\le 1.55$) and the sign is not consistent. Pooling the 307 firms gives $r=-0.095$, $t=-1.67$.

Controlling for size does not change this. The partial correlation controlling for the log of open-ended placements is $-0.084$ ($t=-1.19$), and on residuals after removing occupation means, $-0.055$ ($t=-0.78$).

The asymmetry across occupations

The contrast across occupations matters more.

Although turnover differs by a factor of 5.5, the fee rate differs by only 1.16. The correlation at the level of occupations is $r=-0.102$, essentially zero.

Fee rates also concentrate on a few values. Operators at exactly 20%, 25% or 30% make up 40 to 64% of the whole, and 90% fall within 15–35%. Concentration is highest (64%) for physicians, where retention is most stable.

A candidate proxy for B1: size and turnover

Table 15.13 records that no proxy for the informativeness of effort — the B1 side — could be constructed. The observation of Remark 15.8 is a candidate.

Mechanism. If growth in size thins the input per placement, then turnover rises only in occupations where the operator’s effort bears on retention. Where effort does not bear, turnover is unchanged even as input thins. The occupation-level “correlation of size with turnover” therefore gives an ordering of the magnitude of B1.

The observed ordering is as follows (Remark 15.8).

Registration of the implications. Following the procedure of Section 11.5, the following are fixed before testing.

Registered implication (1): the occupation-level “correlation of size with turnover” preserves the ordering of Table 15.18 in samples from other prefectures — that is, positive for institutional care and childcare workers, near zero for nurses.

Alternative: the correlation is an artefact of circumstances specific to Tokyo, in particular the concentration there of the head offices of nationwide operators, and does not appear in other prefectures.

Registered implication (2): the stronger the correlation of size with turnover in an occupation, the stronger its refund scheme, because in occupations with large B1 it is more valuable to load $\pi$ onto $y$.

Alternative: the strength of the refund corresponds to the level of turnover; that is, B2 (the risk side) dominates, and occupations with higher turnover have weaker refunds.

If (2) holds, a route is obtained for separating B1 from B2 in the measurability hypothesis, whose sign Section 15.3.5.0 found indeterminate. (1) is a precondition, confirming that the correlation is not an artefact of the sample.

Matters fixed in advance.

(1) is tested in Section 15.3.5.0 and (2) in Section 15.3.5.0.

The ordering of occupations by the correlation of size with turnover

The detailed search of the Jinzai Service General Site was run with the prefecture set to the whole country for the four occupations. Since the sample of Section 15.3.5.0 was confined to Tokyo, the part excluding Tokyo is out of sample.

The rows extracted agree with the search results. The licence-number letters are three: Yu (fee-charging) 3,221, Mu (free placement office) 200, and Toku 7.

The total of 1,402 after consolidation is a count of firm–occupation pairs, not of firms. Distinct firms number 1,130, with 226 appearing in more than one occupation. The duplication does not affect the correlations, which are taken by occupation, but the figure must not be cited as a number of firms.

Comparison with the reference values. The reference values of Table 15.18 were computed on a composition excluding amount-displayed fees. On the Tokyo portion under the same composition, all four occupations agree to within 0.02 in correlation and 2 in sample size. Under the registered rule (including amount-displayed fees), however, institutional care gives $+0.079$, which does not agree with the reference value of $+0.431$ (Remark 15.12).

Results. Both the sample excluding Tokyo and the nationwide sample including it are computed. The former is the pre-registered out-of-sample test; the latter is the best estimate of the ordering.

Verdict. The facts are stated separately.

(1)

The ordering agrees with the reference values in only one of the four combinations. There are four combinations of sample (excluding Tokyo / nationwide) and exclusion rule (including / excluding amount-displayed fees), and the reference ordering is reproduced only under “excluding Tokyo” with “including amount-displayed fees”. In the other three the top two swap.

(2)

The ordering is not reproduced nationally. The best estimate of the ordering is the nationwide sample including Tokyo, where childcare workers exceed institutional care under both rules.

(3)

The qualitative division depends on neither sample nor rule. That institutional care and childcare workers are positive and nurses near zero holds in all four combinations. Nationally both institutional care and childcare are significant ($|t|>2.8$) and nurses have $|t|<0.7$.

The verdict is partial support. What is supported is (3): the qualitative division of positive, positive and near-zero alone. The ordering itself is not reproduced.

As recorded in Remark 15.12, the registered rule did not match the composition of the reference values, so it cannot be said that the one combination agreeing in (1) was chosen in advance.

The strength of the refund

How strength is measured was fixed first. At registration (Table 15.19) the regressor was written only as “the refund scheme is strong (a higher refund rate, or a longer covered period)”, with no measurement laid down. Deciding after reading the documents would be fitting after the fact, so the following was fixed before reading them.

Let $t$ be the number of days to separation and $r(t)$ the refund rate at that point, and take as strength

$$\text{strength}=\frac{1}{180}{\int \; <!--\; nolimits\; -->}_0^{180}r(t)dt.$$

(15.2)

The window is 180 days because the site’s turnover rate is defined over six months from placement, aligning the observation windows of regressor and regressand. Where a rate is written as a range the lower bound is taken, and where intervals overlap the higher rate is taken. The details are recorded in data/raw/jinzai/henreikin/README.md.

Reading the documents. The 907 PDFs referenced from the refund field of the office detail pages were read. The 132 carrying no text layer were put through OCR, and places OCR could not settle were checked by eye. No document was illegible.

Results. The 854 with a computable strength are tabulated by occupation. The treatment of the 53 missing values is as in Remark 15.14.

Implication (2) is rejected. The ordering by B1 is institutional care, childcare workers, physicians, nurses; the ordering by strength is physicians, institutional care, nurses, childcare workers. The rank correlation is $\rho =0.000$.

The alternative is not supported either. The correlation of strength with turnover is $r=-0.037$ (1,080 observations) and with size (the log of open-ended placements) $r=-0.079$ (571 observations), both near zero. Neither B1 (thinning of input) nor B2 (the risk side).

The difference across occupations appears only for physicians, and is not explained here (Remark 15.15). The difference is also visible only once strength is measured; the presence or absence of a scheme cannot detect it (Remark 15.16).

15.3.6 Assignment of obstacles

The principal obstacle for family 2 is (iv) access constraint: data of the required granularity cannot be reached. For family 5 it is (ii) ill-posedness of the hypothesis as regards the formula for $b^{\star }$, and (vi) non-operation of the adjustment as regards the test on the price side.

For family 5 the data source proved favourable once the route was checked (Section 15.3.5.0). Indeed the test from the price side was carried out on 307 firms (Section 15.3.5.0). Verification nevertheless fails to advance because, for the formula for $b^{\star }$, the proposition does not determine the sign uniquely, and on the price side the regressor does not vary within the range the theory posits.

This distinction has a practical implication. Where the obstacle is (iii), (iv) or (v), investing labour produces progress; where it is (i), (ii) or (vi), the same labour produces nothing. Unless which one it is is determined first, labour is misallocated.

15.4 Comparison of the two families and its methodological implications

Family 5 is in a good position if per-firm data can be reached, but at present, with only the aggregate layer examined, it yields a weaker conclusion than family 2. The completeness of a frame and the richness of its variables are independent, and the prospects of an investigation cannot be assessed without checking both.

One further structure common to both families was observed.

To reach ${\varphi }_{\mathrm{cog}}$ in family 2 one must descend to the annual securities reports of listed firms that disclose breakage income separately; but that means returning to the 0.1% sample of Section 12.9, and the disclosure-selection bias of Section 12.3 revives. The constraint that structure is visible in the complete frame while surplus is visible only in a selected sample holds at least in these two domains.

15.5 The structure of the data source

15.5.1 The detailed search

The site’s detailed search allows occupation, realized fee rate and turnover rate to be given as search conditions, and displays them in the result listing when they are given. The samples of Sections 15.3.5.0 and 15.3.5.0 come from this route.

The occupations specifiable as search conditions are limited to nine in medicine, care and childcare. The same retrieval is impossible in other fields such as information and communications or sales. The restriction of this work to four occupations follows from this constraint.

The detail page of each office carries more than the listing: placements and separations by fiscal year, all occupations the office has filed, and a reference to the refund-scheme filing. The refund data of Section 15.3.5.0 came from there.

15.5.2 Outcomes of the three routes

The outcomes of the three routes listed in the investigation plan (Part III) are recorded.

Measuring the publication rate. In the search results for the four occupations, about three quarters of operators display fees as rates and about a quarter as amounts. Publication itself is carried out by nearly all operators — good progress for the first year of the requirement.

Describing turnover by field. Achieved. As in Section 15.3.5.0, turnover rates for the four occupations were obtained at the level of the operator. The occupation means are 2.3% for physicians, 5.8% for childcare workers, 6.4% for institutional care and 12.8% for nurses — a spread of a factor of 5.5.

Measuring how binding the regulation is. Achieved. The strength of refunds was measured for 907 documents and implication (2) tested in Section 15.3.5.0. As to fee rates, the standard deviation among certified operators is 2.2, smaller than the 7.0 of the uncertified (Section 15.3.5.0). Since the certification conditions say nothing about fee rates, this is an indirect effect.

15.5.3 The mismatch of recording granularity

The mismatch between firm and office granularity noted in Section 15.3.5.0 appears in stark form in the national sample of Section 15.3.5.0 (Remark 15.11). Rows of the search results denote offices, but placements, turnover rates and fees are values at the level of the firm, replicated across all of that firm’s offices.

15.6 What was reached and what was abandoned

15.6.1 Subjects abandoned

(1)

Measuring ${\varphi }_{\mathrm{cog}}$ for family 2 with the data of this work. Lacking (2) and (3) of Proposition 14.13, it cannot be estimated without reaching data carrying individual usage records and an exogenous shock. Quantities of the same kind have been measured by others (Section 14.6.1); what is abandoned is measurement from published statistics, not measurement as such.

(2)

Testing the formula for $b^{\star }$ in family 5. Obtaining $b$ as a continuous quantity, and extracting from ${\sigma }^2$ only the uncontrollable part, are both impossible ((ii) ill-posedness of the hypothesis).

(3)

Continuing the test from the price side. Section 15.3.5.0 established the absence of correlation. Since the regressor concentrates on customary values and does not vary, enlarging the sample will not change the result ((vi) non-operation of the adjustment).

15.6.2 Subjects possible under conditions

(5)

Extension to other prefectures. The sample of Section 15.3.5.0 is confined to Tokyo (Remark 15.7). Since the detailed search takes a prefecture, reproduction elsewhere is possible by the same procedure.

(6)

Testing the relation between size and turnover. Two implications were registered in Section 15.3.5.0; the test requires samples from other prefectures.

(7)

Collecting refund rates and periods. Since the published format differs by operator, this is a matter of a person reading them one at a time. Restricted to the 60–70 certified operators it is of workable scale.

15.6.3 What families 2 and 5 reached

For family 2, the part of the registered implication concerning levels was supported.

For family 5, the formula for $b^{\star }$ was not reached, but the test on the price side yielded a descriptive result. In placement for medicine, care and childcare, although occupation means of turnover differ by a factor of 5.5, fee rates differ by only 1.16 and concentrate on the customary values of 20–30%.

Beyond identifying the structural defect of the hypothesis in Section 15.3.5.0, showing in Section 15.3.5.0 that the adjustment the theory posits does not operate in reality is a result that could not have been reached by adding data.

15.7 A quantity that is not tabulated: the credit position of sole proprietors

In attempting to measure $\kappa$ for sole proprietors, two statistics were consulted.

The first is (i) absence of measurement, but differs from the usual case in that it was measured in the past. Data through FY2018 exist, so the past can be analysed.

The second is more peculiar. Since a questionnaire exists, responses are very likely collected; but they do not appear in the published tables. This is neither (i) nor (iv) but a fifth obstacle: the choice of what to tabulate.