7 A Catalogue of Φ: Seven Families, Twenty-Eight Types

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v0.28.9 — This text is under construction. The structure of the theory, the propositions, and the empirical conclusions may all change. Overview

Chapter 7
A Catalogue of $Φ$ : Seven Families, Twenty-Eight Types

7.1 How the catalogue is built

Within the space generated by the six degrees of freedom of Section 2.5, the points and regions that recur are organized into seven families and listed. For each type the sign of the credit position $κ$ , the character of settlement, and representative examples are given. The catalogue claims no exhaustiveness; it is a working classification used to select objects of study.

7.2 Family 1: one-off exchange

Forms in which one $δ$ corresponds to one $π$ .

No.	Type	$κ$ / character of settlement	Examples
1-1	immediate exchange	$\approx 0$ / simultaneous with $δ$	retail, restaurants, vending
1-2	one-off advance	$< 0$ / precedes $δ$	made-to-order, weddings, tickets, crowdfunding
1-3	one-off arrears	$> 0$ / follows $δ$	B2B trade credit, invoiced sales
1-4	instalments	$> 0$ (long) / one $δ$ split into many $π$	hire purchase, BNPL, mortgages

Table 7.1: Family 1: one-off exchange.

Only 1-4 is qualitatively different in structure. The delivery $δ$ occurs at a point in time while $π$ is spread over a long horizon; it commercializes the difference in discount factors of Section 4.2 on its own. Its surplus is weighted towards $ϕ_{barg}$ and $ϕ_{cog}$ .

7.3 Family 2: continuing access

Forms in which a right $\bar{δ}$ is sold and $δ$ is exercised against it. The principal home of cognitive surplus.

No.	Type	$κ$ / character of settlement	Examples
2-1	flat access	annual $< 0$ / monthly in arrears $> 0$ , $π = F$	SaaS, membership retail, facility use
2-2	usage	$\approx 0$ / $π = p δ$	electricity, cloud, telecoms
2-3	two-part tariff	mixed / $π = F + p δ$	mobile, dues plus goods, booking discounts
2-4	prepaid	$< 0$ / $π$ first, timing of $δ$ at the customer’s discretion	gift cards, ticket books, transit cards
2-5	options and warranties	$< 0$ / $π$ first, $δ$ only on exercise	extended warranties, cancellable bookings, credit lines

Table 7.2: Family 2: continuing access.

2-1 divides in two according to whether a capacity constraint is present (the proposition of Section 2.7). 2-4 and 2-5 are formally sales of options, leaving the operator short.

7.4 Family 3: renting an asset over time

No.	Type	$κ$ / character of settlement	Examples
3-1	lease and rental	the asset comes first / proportional to time	equipment leasing, property rental
3-2	shared asset	the asset comes first / tied to utilization	lodging, car sharing, coworking
3-3	transfer plus long collection	$> 0$ (large) / $π$ continues after title passes	vendor finance, equipment finance

Table 7.3: Family 3: renting an asset over time.

What characterizes this family is that the dominant term is not $κ$ but the $A$ of (3.8). In 3-2 a capacity constraint is explicitly present, so the utilization rate decides feasibility.

7.5 Family 4: recovery through a complement

Forms that split $δ$ and deliberately skew $π$ .

No.	Type	$κ$ / character of settlement	Examples
4-1	lock-in consumables	early $> 0$ / $δ_{1}$ below cost, recovered on $δ_{2}$	printers, coffee machines, games consoles
4-2	freemium	early $> 0$ / $π = 0$ for some, recovered from converts	storage, business chat, mobile games
4-3	hardware subsidy plus finance	$> 0$ / device discounted, recovered through recurring charges	mobile carriers, auto sales finance

Table 7.4: Family 4: recovery through a complement.

By definition this family deliberately creates $κ > 0$ early on. If the payback period on customer acquisition lengthens, the whole family becomes dependent on capital markets.

7.6 Family 5: outcome- and state-contingent

Forms in which $π$ is measurable with respect to $ω$ . The arguments of Chapter 4 bear directly here.

No.	Type	$κ$ / character of settlement	Examples
5-1	success fee	$> 0$ (large) / $π = h (outcome)$	recruitment, M&A advisory
5-2	revenue share	$> 0$ / $π =$ customer revenue $\times$ rate	franchising, app distribution, agencies
5-3	cost plus	intermediate / $π =$ cost $+$ margin	some construction, consulting, public procurement
5-4	underwriting	$< 0$ / $π$ taken first, $δ$ depends on $ω$	life and non-life insurance, guarantees
5-5	IP licensing	mixed / royalties, zero marginal cost of $δ$	semiconductor IP, character licensing, patents

Table 7.5: Family 5: outcome- and state-contingent.

5-1 and 5-2 transfer risk because effort is unobservable; 5-3 transfers it in the opposite direction because the outcome is uncertain (the remark in Chapter 4). Only 5-4 has a mirrored time structure within the family.

7.7 Family 6: intermediation

Forms that stand between two other parties’ transaction.

No.	Type	$κ$ / character of settlement	Examples
6-1	transaction fee	$\approx 0$ / volume $\times$ rate	securities, property brokerage, marketplaces
6-2	escrow	$< 0$ (other people’s funds) / held then remitted	payment processing, lodging platforms, travel
6-3	spread	$> 0$ (inventory) / the gap between bid and ask	wholesale, currency exchange, market making
6-4	charging for opportunity	$< 0$ / charged for listing or contact, not the transaction	job advertising, search advertising

Table 7.6: Family 6: intermediation.

The $κ$ of 6-2 is not the operator’s own money. It occupies the same place in (3.8) as other negative $κ$ but differs in that it drains rapidly when volume falls. The difference between 6-1 and 6-3 is whether inventory risk is held, which is again the transfer question of Chapter 4.

7.8 Family 7: separating beneficiary from payer

The operation of moving the index $i$ itself.

No.	Type	$κ$ / character of settlement	Examples
7-1	advertising	mixed / $π = 0$ for the beneficiary, a third party pays	broadcast, search, social
7-2	cross-subsidy	mixed / one side free, recovered from the other	single-sided charging in two-sided markets
7-3	public payment	$> 0$ / an insurer or government pays	healthcare, long-term care, education
7-4	donation and support	$< 0$ / no explicit consideration	support-based charging, tipping, non-profits

Table 7.7: Family 7: separating beneficiary from payer.

7-3 is special: because the payer sets the price, the design freedom of $Φ$ is absorbed into the institutional layer. A revision of the official price changes $κ$ and $m$ exogenously and at the same time.

7.9 Assigning an observation to a type

To use the catalogue empirically, a rule is needed for assigning an observed $Φ$ to a type. A rule decided after the fact allows the assignment to be chosen to fit the conclusion. The rule is fixed below.

The information needed for assignment is the degrees of freedom of Section 2.5, which coincide with the variables listed in Chapter 11.5 as those to be recorded for each $Φ$ .

The conditions defining the families are not mutually exclusive, so the order of the tests is fixed.

Order	Test	If it applies
1	Do beneficiary and payer differ (degree of freedom (3))?	family 7
2	Does it stand between two other parties’ transaction?	family 6
3	Is $π$ measurable with respect to $ω$ (degree of freedom (4))?	family 5
4	Is delivery split and $π$ deliberately skewed?	family 4
5	Does an asset come first, recovered in proportion to time or utilization?	family 3
6	Is there repetition (degree of freedom (6))?	family 2; otherwise family 1

Table 7.8: Order of the family tests. Applied from the top, the first family that applies is taken.

Once the family is fixed, the number within it is specified by degree of freedom (1) (timing), (2) (form of dependence) and (5) (separation of right from exercise).

Remark 7.1 (Some assignments depend on the order). There exist $Φ$ whose assignment changes if the order in Table 7.8 is changed.

An app-distribution fee applies a rate to the volume distributed, which fits 6-1 (transaction fee), and moves with the seller’s revenue, which fits 5-2 (revenue share). Under the order above, family 6 is tested first.

There is no deductive ground for the order. It follows a policy of testing the composition of parties before the form of settlement, and is a convention. Being a convention is precisely why it must be fixed in advance.

Remark 7.2 (Handling composite contracts). As Section 7.10.2 notes, actual contracts are often composites of several types. In that case $Φ$ is first decomposed into components and Table 7.8 applied to each.

The unit of decomposition is the range over which one correspondence between delivery and settlement closes. Selling a device and supplying consumables are separate correspondences, so they split into two components, assigned to 1-1 and 2-2 respectively and recorded as the composite 4-1.

7.10 Limits of the catalogue

The twenty-eight types above are not exhaustive. Of three omissions considered, two can be described with existing types.

7.10.1 The $Φ$ of labour: no family is added

An employment contract is also a map from delivery to settlement, but the direction is reversed: the worker supplies $δ$ and the firm pays $π$ . Because the sign convention for $X$ in Chapter 2 makes inflows to the firm positive, all twenty-eight types presume the firm is the side supplying $δ$ .

But seen from the worker’s side, the worker is an operator supplying $δ$ . The structure is the same as the solo business of Chapter 8, and the catalogue applies unchanged.

Contract form	Type	$κ_{L}$
Monthly salary in arrears	repeated 1-3, one-off arrears	$> 0$ (the worker extends credit)
Daily or weekly pay	1-1, immediate exchange	$\approx 0$
Advances and signing payments	1-2, one-off advance	$< 0$
Bonuses	5-1, success fee	$> 0$
Equity compensation	5-2, revenue share	$> 0$ (extreme)
Retirement payments	1-4, instalments	$> 0$ (very long)

Table 7.9: Contract forms seen from the worker’s side and the corresponding types.

No new family is needed. A worker holds a composite contract combining families 1 and 5.

Remark 7.3 (Not a special case but an object under different constraints). Table 7.9 shows that the catalogue applies once the viewpoint is reversed, but a worker cannot be regarded as a special case of a solo business. The constraints differ in both $Φ$ and $ϕ$ .

Constraints on $Φ$ . Five of the six degrees of freedom of Section 2.5 are constrained by law. The Labour Standards Act requires wages to be paid in currency, directly, in full, at least once a month, and on a fixed date; timing is fixed to arrears and the paying party to the employer. State dependence is also permitted only to a limited degree, so tying wages to performance is restricted. It is not that there is no choice, but that the choice set is small.

Constraints on $ϕ$ . Defining $ϕ_{prod}$ for a worker would require a marginal cost, and none is determined. For $ϕ_{cog}$ no counterpart is even clear. The three-way decomposition of Chapter 6 does not apply to workers as it stands.

Furthermore the insolvency time of Section 4.2 is absorbing, whereas a worker does not cease to exist upon being unable to pay. This text does not treat workers as an independent object.

Two points are worth recording. First, because monthly pay in arrears is the norm, $κ_{L} > 0$ dominates. Payroll lags are usually 15–45 days, so workers extend credit to firms on the same order as trade payables. That $κ_{L}$ belongs in the third term of (3.8) should be made explicit.

Second, an employment contract is also a trade in which the stability of $κ$ is bought at the price of assigning credit to the organization (Section 8.7). In the language of the catalogue:

The worker avoids 5-1 (success fee, high variance) and chooses 1-3 (one-off arrears, low variance), giving up the accumulation of public credit in exchange.

This has the same form as Proposition 4.10: whoever chooses stability discounts a future claim and gives it away.

7.10.2 Composite contracts: combinations of existing types

Actual B2B contracts are often composites of several types. This is not a limit of the catalogue but the correct way to use it. By Proposition 3.9, $κ$ is additive over parallel $Φ$ , so a description that lists several types is meaningful at the level of $κ$ . But, as Remark 3.10 states, the decomposition is not unique. Several sets of types give the same $κ$ , so the types are a generating set for $Φ$ , not a basis. The assignment procedure is fixed in Remark 7.2.

7.10.3 Claims that circulate: outside the framework

Forms in which $π$ itself circulates as a claim on future $δ$ — token issuance, for instance — still cannot be handled.

The map $Φ : Δ \to Π$ of (2.8) presumes that delivery and settlement correspond between the same counterparty. When a claim is resold to a third party, the party that receives delivery changes after the fact. This is a problem of the domain and cannot be expressed in the setting of Chapter 2, which fixes $N$ . An extension of the framework is required.

7.11 A real firm is a bundle of $Φ$

A large firm normally runs $Φ$ from several families in parallel. Because the sign of $κ$ differs by family, computing $CCC$ at the level of the firm yields a mixed average of terms with opposite signs. By Corollary 5.19 that average is weighted by the rate of sales, not a simple average of the families’ $CCC$ . To restate the conclusion of Section 12.10.3: the unit of observation should be $Φ$ , not the firm.

Chapter 7 A Catalogue of Φ: Seven Families, Twenty-Eight Types