Definition 2.1 (Primitive sets). Four sets are taken as primitive.

The parties are grouped into customers $C$, suppliers $S$, capital providers $K$, labour $L$, and third parties $R$ who assume risk. These are subsets of $N$ and cover $N\setminus \{0\}$. An individual party is written as an element of $N$, as in $i\in C$. When a group symbol is used directly as an index it denotes the sum over the parties in that group; for instance ${\kappa }_C={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i\in C}{\kappa }_i$. $R$ is where degree of freedom (3) moves the paying party, which is the transfer operation of Section 2.6.

Regulation and commercial custom are not counterparties to cash and are therefore not in $N$. They constrain the range from which $\mathrm{\Phi }$ may be chosen and are treated as the institutional layer in Section 6.3.

Definition 2.2 (Business model). Using $\mathrm{\Delta }$ and $\mathrm{\Pi }$ from (2.7), a business model is defined as the map

Remark 2.3 ($\mathrm{\Phi }$ and $\varphi$ are different objects). Capital $\mathrm{\Phi }$ denotes the map (2.8) itself, that is, the structure. Lower-case $\varphi$ denotes the size of the surplus that arises in one period under that structure; it is decomposed into three sources in (6.1) of Chapter 6. $\mathrm{\Phi }$ fixes the shape of the correspondence between delivery and settlement; $\varphi$ measures the share the firm retains under that shape. No general relation between the two is given here. How the choice of $\mathrm{\Phi }$ moves each term of $\varphi$ is discussed term by term in Chapter 6.

Definition 2.4 (Cash flow). Given a delivery $\delta \in \mathrm{\Delta }$, the map $\mathrm{\Phi }$ returns as its image the tuple of settlements $\pi =\mathrm{\Phi }(\delta )\in \mathrm{\Pi }$. An element of $\mathrm{\Pi }$ is a tuple of functions indexed by party, so its $i$-th component ${\pi }_i$ is a function on $T\times \mathrm{\Omega }$. Write

Then $X(i,t,\omega )$ is the net movement of cash between the firm and party $i$ at time $t$ in state $\omega$. Inflows to the firm are positive. Payments to suppliers or to labour appear as ${\pi }_i<0$.

Remark 2.5 (Every quantity here is a level). Every quantity introduced after (2.9) — including ${\kappa }_i$, $\mathrm{CCC}$ and $\varphi$ — is a level. Each is defined as a value at a time $t$, and no notation for change is provided.

This is not a presentational choice but a limitation. As Part IV confirms repeatedly, the levels of these quantities differ by two orders of magnitude across industries and product forms, and cross-sectional comparison cannot be interpreted. What can be read is change over time within one object.

The text has no theoretical apparatus for change and deals with it case by case in the empirical work (Section 19.4.3).

Example 2.6 (The three indices made concrete). Consider a service priced at 1,000 yen a month and supplied to 100 customers. If customer $i\in C$ pays at the end of each month, then

which does not depend on $\omega$. If cancellation is possible, $\omega$ contains the cancellation time and $X$ does depend on $\omega$.

With a single supplier, $S=\{s\}$, and a server cost of 30,000 yen a month, $X(s,t,\omega )=-30,000$. Written with the group index, ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i\in S}X(i,t,\omega )=-30,000$.

Remark 2.7 (The typeface of $F$, and the use of $c$). The $F$ in (2.10) is the fixed part of a tariff and is a different quantity from the $\sigma$-algebra $\mathcal{F}$ on the state space. The two are distinguished by typeface.

Throughout this text $c$ denotes marginal cost and is always written with its argument, $c(\delta )$ (Definition 2.14). A flat fee is written $F$.

Remark 2.8 (Three groups). (1), (3) and (4) move the indices of (2.9): they change where $X$ is placed for the same $\delta$. (2) fixes the form of the map itself (Remark 2.7), and (5) and (6) are properties of the domain $\mathrm{\Delta }$.

The three risk operations of Section 2.6 are the operations that move (1), (3) and (4).

Proposition 2.9 (Variance reduction by pooling). For identically distributed risks $X_1,\dots ,X_n$ with variance ${\sigma }^2$ and pairwise correlation $\rho$, provided $\rho$ does not depend on $n$,

Remark 2.10 (Independence is a finite resource). The limit in (2.11) is $\rho {\sigma }^2$, which is non-zero whenever $\rho >0$. Pooling consumes independence, and independence is a finite resource. Underwriting limits on earthquake insurance, the sensitivity of platforms to the business cycle, and systemic risk among financial institutions all come from the same term of this equation.

Remark 2.11 (Endogeneity of $\rho$). Proposition 2.9 treats $\rho$ as an exogenous constant. The assumption is hard to sustain.

The mechanism that makes $\rho$ endogenous is not the exhaustion of poolable risks but the fact that the act of pooling itself creates correlation. The example of [30] is clear: if two parties each take half of two risks and diversify perfectly, each party’s individual probability of failure falls, but their portfolios become perfectly correlated. The nature of the risk is unchanged, yet correlation between the parties appears.

Hence $\rho$ is a function of the number of poolers and the overlap of their holdings, and the limit $\rho {\sigma }^2$ in (2.11) can rise as pooling proceeds. The proof is formally correct, but its range of application is confined to cases with few poolers and little overlap. This dependence is not treated here (Section 19.4.1).

Remark 2.12 (Social optimality of pooling). The mechanism of Remark 2.11 implies that the individual and the collective optimum can diverge. [19] derives conditions under which the diversification optimal for an intermediary is not socially optimal and restricting risk sharing is preferable.

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. Family 5-4 (underwriting) and family 6-2 (escrow) in Part II make pooling their business and are the types where this point bears directly.

Example 2.13 (How fast correlation bites). With $\sigma =100$ and $\rho =0$, at $n=100$ the standard deviation falls to $100∕\sqrt{100}=10$. With $\rho =0.3$, however,

Even as $n\to \infty$ the standard deviation does not fall below $\sqrt{0.3}\times 100\approx 54.8$. At a correlation of $0.3$ — a modest value — the benefit of pooling is more than halved.

Definition 2.14 (Splitting the cost). Total cost is split in two.

For the first term, write

Remark 2.15 (No shape is assumed for $c(\delta )$). Neither monotonicity nor convexity is assumed for $c(\delta )$. Cost structures differ greatly across industries, and imposing one shape would narrow the range of application.

The price is that the optimal quantity delivered $\delta$ cannot be fixed at an interior point. The capacity constraint (2.17) supplies an upper bound, but the point short of it at which cost balances price does not follow from the assumptions made here. Where the level of $\delta$ matters later — the allocation in Chapter 8, for instance — it is treated as a corner of the constraint set.

Remark 2.16 (Advances cost something too). ${\tau }_C<0$ (an advance) requires no collection cost, but it does create custody of the funds held, refund handling, and regulatory compliance. The deposit obligation under the Payment Services Act, treated in Section 8.5, is an example. $C_{\mathrm{admin}}>0$ whatever the sign of $\tau$.

Remark 2.17 (Conditions under which $C_{\mathrm{admin}}$ is ignored). From here on $C_{\mathrm{admin}}$ is not written explicitly. Two conditions make it negligible.

(1)

Operations are systematized. Once the machinery is built, the marginal cost per transaction of measurement, rights management and ongoing handling approaches zero.

(2)

No intermediary is used. Intermediation fees are proportional to transaction value and do not vanish with systematization. Chapter 17 measures this cost and maps it onto the kinds of $C_{\mathrm{admin}}$ in Remark 17.1.

The range in which the conditions hold is limited. They hold for the solo business of Chapter 8 and for family 2 in Part IV, but not for the small firms treated in Chapter 16. Where invoicing and chasing are done by hand, $C_{\mathrm{admin}}$ cannot be ignored.

The price of the omission should be stated. Because $C_{\mathrm{admin}}$ is dropped, this text cannot treat the route by which the choice of $\mathrm{\Phi }$ feeds back into $\varphi$ through cost. Equation (2.12) is not used again; only $C_{\mathrm{prod}}$ appears. A relation such as “lengthening ${\tau }_i$ raises collection cost” cannot be written within this framework.

Remark 2.18 (Switching cost and the value of waiting). Treating $\mathrm{\Phi }$ as a choice variable amounts to setting the switching cost to zero. The cost of changing from one $\mathrm{\Phi }$ to another belongs to the setup term of $C_{\mathrm{admin}}$ and is not generally zero.

In the theory of irreversible investment ([8]), a decision is irreversible not because it cannot physically be undone but because undoing it is expensive and locks in a particular path. Writing $C_{\mathrm{sw}}$ for the switching cost and $\mathrm{\Delta }\varphi$ for the gain in surplus from changing $\mathrm{\Phi }$:

In the third case waiting has value: it can be optimal not to fix $\mathrm{\Phi }$ until the uncertainty resolves. This implication is not treated here. For the solo business of Chapter 8, where the switching cost is relatively large, the point may not be negligible.

Remark 2.19 (Where $C$ appears). The quantities of Definition 2.14 are used repeatedly. ${\varphi }_{\mathrm{prod}}$ is defined as an advantage in $c(\delta )$ (Chapter 6); a near-zero marginal cost is the condition that relaxes the capacity constraint (Chapter 8); and the fact that both $C$ and $\mathrm{\Phi }$ can be specified in advance is the condition for replicability (Chapter 9). The physical layer refers to these properties of $C$ (Section 6.3).

Example 2.20 (Identical $C_{\mathrm{prod}}$, different $\mathrm{\Phi }$). The same business software can be supplied in three ways.

The technology is identical and so is the marginal cost. Only $\mathrm{\Phi }$ differs, yet the sign of $\kappa$ reverses and the working capital required changes.

Definition 2.21 (Right and exercise). Let $\overline{\delta }$ be the ceiling the contract grants the customer and $\delta$ the use actually realized. Write the divergence as $\overline{\delta }-\delta \ge 0$ and define the gross margin arising from it as

Remark 2.22 (Which contractual forms admit divergence). Equation (2.15) can be positive only when the contract states $\overline{\delta }$ explicitly.

Under usage pricing $\pi (t)=p\delta (t)$, the customer’s total payment is $p\int \delta$. No use means no payment, so the contract need not state a ceiling $\overline{\delta }$ and the difference $\overline{\delta }-\delta$ is undefined.

Under a flat fee, where $\pi$ does not depend on $\delta$, the consideration is fixed irrespective of use, so the contract states the extent of the right $\overline{\delta }$. Here $\delta <\overline{\delta }$ can be realized. Under a two-part tariff $\pi =F+p\delta$, divergence arises only over the range corresponding to the fixed part $F$.

This is a consequence of Definition 2.21, not an independent claim. It presumes that $\overline{\delta }$ is given exogenously. When the customer chooses $\overline{\delta }$ from a menu, the two are determined jointly and the meaning of the difference changes (Section 19.4.1).

Remark 2.23 (The general form of the capacity constraint, and the simplification used here). Capacity on the supply side covers the occupancy of a facility, the number of machines, the disposable time of staff, and so on. Since ${\delta }_i$ is a function on $T\times \mathrm{\Omega }$ by the definitions of Chapter 2, the general form of the capacity constraint is evaluated at each instant.

Capacity itself depends on time because opening hours, equipment run-times and staffing all vary over time.

From here on $\mathrm{Cap}$ is treated as a constant, that is, (2.16) is used in the form integrated over the whole period,

where ${\delta }_j$ denotes a total.

The simplification is justified when demand is distributed over time in roughly the same way across members and capacity does not vary by time of day. When neither holds, only the peak instant binds and (2.17) is stronger than necessary. Chapter 16 treats an observation with which this simplification does not agree.

Proposition 2.24 (Membership under a capacity constraint). Suppose the capacity constraint (2.17) is present and the number of members $n$ and average use $𝔼[\delta ]$ are determined independently (the number of members is written $n$ because $m$ is reserved for the margin). Then a flat membership ${\pi }_j=F$ is feasible only if

that is, low utilization is a condition of feasibility.

Example 2.25 (Two kinds of membership). A facility with room for 100 has 1,000 members and charges 8,000 yen a month. If $\overline{\delta }$ is “use it every day”, $n\overline{\delta }$ far exceeds capacity. At an actual utilization of 10%, $\mathit{𝑛𝔼}[\delta ]=100$ and the constraint is just satisfied. If utilization rises to 30%, the business fails physically.

By contrast, where the fee grants a right to purchase discounts there is no capacity constraint, and higher utilization is favourable because it enlarges the purchasing volume. Outwardly both belong to family 2-1, but the shape of the feasible region differs.