Chapter 19
Where This Stands: What Was Established and What Remains Open

19.1 What was established and what was not

19.1.1 Derived as theory

19.1.2 Judged empirically

As a descriptive instrument the framework worked. As a predictive one it did not. These are different properties, and only the former is claimed here.

19.2 Fixing the implications in advance

The implications derived from the theory were fixed in writing before the observations, so that support and rejection could not be chosen afterwards. Twenty-two were registered: six supported, eleven rejected, five untestable. The wording and the verdicts are listed in Appendix A.4.1.

The number of rejections is not a failure of the method. Without the fixing, an after-the-fact account could have been given for any fact observed. Appendix A.4.1 records four places where a post-hoc interpretation was nearly adopted.

19.3 Open problems

The open problems are organized by the six categories of Section 13.2 (Section 13.1). Because the remedy differs by category, the allocation of effort depends on the distinction.

19.3.1 (i) No measurement exists

The quantity is not measured anywhere. Secondary data will not fill the gap; an original survey is required.

(1)

Is directors’ remuneration a cost or an appropriation of profit? The cause of the low $m$ in the small classes is directors’ remuneration (Proposition 14.8). As a fraction of sales it is fifty times that of large firms, and adding it back to operating profit removes the ordering by size.

The remaining question is whether that remuneration is a return to labour or a discretionary appropriation. That the remuneration ratio does not move when profit does, and that the operating margin of unincorporated businesses rises in a downturn (Remark 14.10), suggest discretion; but the labour-return account cannot be rejected. Deciding requires directors’ hours worked, and directors are not employees under the Labour Standards Act, so no official statistics on their hours exist (Remark 14.12).

An alternative route would identify from changes in the corporate tax rate or the thresholds for reduced rates as exogenous shocks, but cutting a sample at a threshold requires microdata.

19.3.2 (ii) Ill-posed hypothesis

The theory does not fix the sign. More data does not help; the theory must be refined.

(2)

Testing the measurability hypothesis. The formula for $b^{\star }$ itself cannot be tested because the required quantities are not available (Section 15.3.5.0).

A test on the price side was carried out. Across 307 firms, fee rates achieved and separation rates were uncorrelated (Section 15.3.5.0). Separation rates differ 5.5-fold across occupational averages while fee rates differ only 1.16-fold, and prices cluster on the customary 20–30%.

The obstacle is therefore not only (ii). It belongs to (vi) of Section 13.1 — the explanatory variable does not vary over the range the theory assumes — and is resolved neither by refining the theory nor by adding data.

(3)

The objective under R2: are $H$ and $\beta$ enough? A route was shown that explains the change through $H$ (time horizon) and $\beta$ (discount factor) without changing the form of the objective (Section 8.6.4). Whether that suffices is untested.

Testing requires observing contract choices among older people, and no distribution of contract-form choice by age is available. The supply side’s price design is consistent with the alternative, but it is not consumers’ choice.

19.3.3 (iii) Not identified

The data exist but confounding cannot be separated. Better design or exogenous variation is required.

(4)

Interpreting the lengthening of receivables. The long-run rise in $\mathrm{CCC}$ was decomposed in Section 16.8. Changes in industrial composition are excluded (the composition effect is 17%, and 2% since 2010), and the rise in the 2000s is explained by shorter DPO as trade bills disappeared.

What remains is the lengthening of receivables in the 2010s ($+6.62$ days). Whether this is a genuine lengthening of collection or a reclassification accompanying the shift to electronically recorded monetary claims cannot be separated. Separating them requires balances of such claims, which the corporate statistics do not survey (Remark 16.5).

(5)

What drives the variation in $\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->W$. The contemporaneous negative correlation was separated in Section 16.9, giving a denominator-effect contribution of about 17%. That the denominator effect is not dominant could be shown.

An estimate allowing for lag structure was attempted but nothing was detectable in annual data (Remark 16.6). The $R^2$ of the elasticity estimate is only 0.081, so most of the annual variation in working capital is driven by something other than sales. Year-end inventory adjustment, timing shifts on large transactions, and changes in accounting policy plausibly contaminate it, but none is separated.

19.3.4 (iv) Access constraint

It exists but cannot be obtained. A route must be opened, or the attempt abandoned.

(6)

Measuring ${\varphi }_{\mathit{𝑐𝑜𝑔}}$. Of the three conditions of Proposition 14.13, only (1), the contractual right, was reached here. (2), individual realized use, and (3), an exogenous event forcing attention, are missing.

A detour was constructed through ${\delta }^{\ast }=F∕p$, but it applies only to binary menus and the proxy is not uniquely determined for a stepped tariff (Remark 14.14).

For family 2-4 (prepaid), [13] shows that about 40% of prepaid value goes unused, so the quantity this text could not measure for family 2 has already been measured. The remaining gaps are an estimate for family 2-5 (options) and a cross-industry estimate of the share of ${\varphi }_{\mathit{𝑐𝑜𝑔}}$ within total $\varphi$.

(7)

An indicator of how readily surplus is taken. No indicator corresponding to the horizontal axis of the two in Chapter 16 could be constructed. The reason is structural.

An intermediary is an entity that spans industries, not an attribute of one (Remark 16.1). It cannot in principle be captured by industry aggregates. Attempting a cross-section on the intermediary side fails because rates decided by individual screening are not disclosed. That a rate is negotiated presupposes that nobody outside the transaction knows it. Collecting only published rates biases the sample towards thin intermediaries operating from a price list.

An implication was registered for the risk component (Section 17.5.1), but the data needed to estimate $\rho$ lie inside intermediaries, and industries with high $\rho$ are screened out, so it cannot be tested.

(8)

Curvature of the age effect. By Proposition A.7 the levels of the three coefficients are not identified, but the second difference — the curvature — is (Remark A.8).

Pursuing this requires a sample carrying firm age. The corporate statistics used here have three axes — industry, size and year — and do not include the year of founding. As Remark A.10 states, an age effect is latent in the $\mathrm{CCC}$ time series, and there is no means of separating it.

19.3.5 Matters of the framework’s descriptive range

Items that do not fit the taxonomy of obstacles and concern the setting of the object itself.

(9)

The simplification of the capacity constraint. As Remark 2.23 states, $\mathrm{Cap}$ was treated as a constant. Section 16.11 records an observation with which this does not agree, but with $n=1$ the theory was not amended (Remark 16.10). Whether the same limit appears in other families is unconfirmed.

(10)

Social optimality of pooling. This text is written from the standpoint of an individual operator and does not treat the effect of the choice of $\mathrm{\Phi }$ on the system (Remark 2.12). Families 5-4 (underwriting) and 6-2 (escrow) make pooling their business, and pooling that is individually optimal may be excessive for the system.

(11)

The credit position of sole proprietors. No published statistic measures the $\kappa$ of sole proprietors. The Survey of Unincorporated Enterprises dropped its assets and liabilities items in FY2019, and the Basic Survey on Small and Medium Enterprises has a questionnaire for sole proprietors but does not tabulate assets and liabilities (Section 15.7).

As a result, the effect of the Freelance Act (in force November 2024) setting an upper bound on $\tau$ cannot be tested. Against the three forms of the institutional layer organized in Section 17.8, this is a fourth form that regulates time directly, and it is of high theoretical interest as a case in which law imposes a ceiling on the ${\tau }_i$ of Section 3.1.4.

(12)

Forms in which claims circulate. The $\mathrm{\Phi }:\mathrm{\Delta }\to \mathrm{\Pi }$ of Chapter 2 presumes that delivery and settlement correspond between the same counterparty. When settlement is resold to a third party, as with token issuance, the party receiving delivery changes after the fact and cannot be expressed in a setting with $N$ fixed (Section 7.10.3). An extension of the domain is required.

(13)

Observation through protocol records. The tests were carried out. They were carried out as reported in Chapter 18. This was the first test of the deductions of Part II, and only $\mathrm{\Phi }$2 was supported.

What remains is per-party quantities. What is observable is per identifier, and because one party may hold several identifiers the per-party ${\kappa }_i$ cannot be recovered (Remark 10.5). The obstacle is not the absence of aggregation but its impossibility, and it differs from both forms added in Section 13.1. The tests revealed a second form of this divergence: the concentration of identifiers pointing to one implementation (Remark 18.2).

What could be verified, moreover, is the correctness of a deduction, not whether that deduction applies to real solo businesses (Remark 10.14).

This is the same place as item (12): a constraint originating in the setting of the index.

(14)

A quantity expressing a degree of unmannedness. “How unmanned is it?” cannot be expressed as a ratio. Measured on the delivery side, the valuation map $v$ measures only value; measured on the response side, no measure is defined on $\mathrm{\Omega }\setminus \widehat{\mathrm{\Omega }}$ (Remark 9.9). The obstacle is the absence of a measure, not of a unit, and sharpening the definition does not remove it.

(15)

Separating ${\varphi }_{\mathit{𝑝𝑟𝑜𝑑}}$ from ${\varphi }_{\mathit{𝑏𝑎𝑟𝑔}}$. Definition 6.2 requires the price $p^{\ast }$ that would obtain without market power, which is a counterfactual and is not observed (Remark 6.3). The numerical separation achieved in Chapter 17 rests on the exceptional availability of a within-platform comparison differing only in function. For a general object, an additional assumption identifying $p^{\ast }$ is required.

(16)

The degree to which the third party is removed. Chapter 10 treats only the cases where the protocol bears recording and enforcement of settlement entirely and not at all, not the middle (Remark 10.4). The composition of the party writing external facts into the state varies, and most existing arrangements are in the middle.

19.3.6 Items with no obstacle

The following fall under none of the obstacles of Section 13.2. Either the route of acquisition is established or the work can be done with data already obtained. They have simply not been started.

(17)

Decomposing ${\lambda }_{\mathit{𝑛𝑜𝑣𝑒𝑙}}$ into components. Section 18.3 measured the response rates: 0.048 a year for modification and 0.030 for cessation, so the $1∕{\lambda }_{\mathit{𝑛𝑜𝑣𝑒𝑙}}$ of Proposition 9.2 is shorter than 20 years.

What remains is the decomposition into components. What was measured is the total of responses, without separating whether they originate in vulnerabilities, dependencies, institutions or demand. Separating them requires reading the content of each modification, which is possible only where verified sources exist, so the sample is biased.

(18)

Correcting the scope of inventory. Chapter 16 uses only “finished goods and merchandise” and omits work in progress and raw materials, so $\mathrm{DIO}$ is understated.

This was carried out for 62 industries (Section 16.10). The median $\mathrm{CCC}$ moves from 28.0 to 43.9 days, but the correlation of $\mathrm{\Delta }\mathrm{CCC}$ is 0.882 and the denominator effect moves only from 0.160 to 0.172. The chapter’s conclusions stand.

What remains is the cross-section with size classes. “Inventory (period end)” is available for 62 industries only for all sizes together, and by size class for only 10 industries. A cross-section split by both industry and size exceeds e-Stat’s 100,000-cell limit. It is possible by splitting the retrieval, but what would be obtained is information linking the two cross-sections of Remark 16.8, which would not move the chapter’s conclusions.

19.3.7 Priorities

Assessed on feasibility, breadth of consequence and cost, separating what can be started from what effort will not advance.

Items in categories (i) and (ii) do not advance with effort.

Among them, (11), the $\kappa$ of sole proprietors, differs in character. The obstacle is neither absence of measurement nor non-identification but a choice in aggregation: responses that may already have been collected are not published. As Section 15.7 notes, the Basic Survey on Small and Medium Enterprises has a questionnaire for sole proprietors. A request to the statistical authority for the tabulation, or an application to use anonymized microdata, is a possible route.

Item (13), protocol records, is one where quantities this text abandoned as requiring per-firm data are in fact published in the “inside” partition of Chapter 10. But because Remark 10.5 blocks aggregation to parties, what can be measured is confined to the distribution per identifier.

Two items have no obstacle, and both can be started with published data or an established acquisition procedure. Item (18) has been done for 62 industries and what remains is the cross-section with size classes (Section 16.10). The kind of work required differs from that of the items in Table 19.4.

19.4 Known weaknesses of the framework

The weaknesses revealed through verification fall into three types. None is an error in an individual proposition; all originate in the construction of the framework.

19.4.1 (A) Excessive assumptions of independence

Several propositions treat as independent quantities that are in fact determined jointly.

Each proposition states its independence assumption explicitly, but no consequence is derived for the case where the assumption is dropped. The proofs are formally correct; the weakness is that their range of application is narrow.

Four of the implications fixed in advance failed because of this weakness.

B and F have the same form: a variable acting on both sides was handled on one side only.

19.4.2 (B) Excessive assumptions of exogeneity

Institutions, $\mathrm{\Phi }$, $v$, $\mathrm{Cap}$ and $\mathcal{𝒢}$ are all treated as exogenous.

(1)

Subjectivity of the valuation map $v$. The $v$ that measures delivery in money was taken as given, but it is itself subjective, and the definition of ${\varphi }_{\mathrm{cog}}$ depends on it.

(2)

Endogeneity of $\mathrm{\Phi }$, and switching cost. $\mathrm{\Phi }$ was treated as a firm’s choice variable, but it is in fact an equilibrium of bargaining between parties and cannot be chosen unilaterally. In addition the switching cost is set to zero (Remark 2.18). Where that cost is of the same order as the gain in surplus, waiting has value and a question of timing arises — when to fix $\mathrm{\Phi }$. This text does not treat it.

(3)

Exogeneity of the institutional layer. Institutions were treated only as an exogenous constraint. In fact they change endogenously in response to firm behaviour. The correspondence between antitrust litigation and rate reductions in Chapter 17 is an example.

(4)

Simplification of $\mathrm{Cap}$. It was treated as a constant, but it is properly a function of time and is also a design variable through restrictions on hours of use (Remark 2.23).

(5)

Endogeneity of $\mathcal{𝒢}$. The information structure was taken as given, but an operator can extend it through follow-up surveys (Remark 4.3).

19.4.3 (C) Level and change are not separated

This is the weakness most easily missed. (A) and (B) can be recognized as “the assumption is too strong”, whereas (C) has no corresponding concept in the theory and is hard to notice as an absence.

As Remark 2.5 states, every quantity here is defined as a level. Yet in the empirical work, four times over, cross-sectional comparison of levels could not be interpreted and only change could be discussed.

The theory has only levels while the empirical work can speak only of change.

Part of this mismatch was resolved by introducing the lag ${\tau }_i$ in Section 3.1.4. Proposition 5.16 gives $\mathrm{CCC}={\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{\tau }_i+{\tau }_{\mathrm{inv}}$, from which it is deduced that the level of $\mathrm{CCC}$ is fixed by $\mathrm{\Phi }$ (Corollary 5.17) and that cross-sectional comparison is meaningless (Corollary 5.18). A tool for splitting $\mathrm{\Delta }\mathrm{CCC}$ into change in contracts and change in operations was also obtained (Corollary 5.20).

That level and change are separate constraints was likewise settled at the boundary of the fundamental inequality of Proposition 5.2: $M\ge 0$ bounds the level and $\dot{M}\ge 0$ bounds change (Remark 5.8). The two are independent, yet their sensitivity to ${\tau }_i$ carries the same coefficient $-1∕\mathrm{CCC}$ (Proposition 5.22, Table 5.3).

What remains is constructing a comparable quantity. The candidate is the deviation from the contractual ${\tau }_i^{\mathrm{contract}}$, but contractual terms do not exist in published statistics (Remark 5.23). The theoretical problem is solved and the measurement problem has taken its place.

19.4.4 The three types, and mid-level theory

For (A) and (C), the existing theory that ought to be connected can be identified.

These were referred to but not connected systematically. Borrowing the informativeness principle in Chapter 4 while failing to check its contested status when framing an implication in Section 15.3.5.0 happened because the same theory was handled at different depths when citing it and when framing an implication.

For (C), no theory to connect to has been identified.