Chapter 8
A Subspace Under Constraints: Alone, Without Assets, Without Credit

8.1 The premises imposed in this chapter

Chapter 7 enumerated the whole space of $\mathrm{\Phi }$. This chapter imposes constraints on it and derives a subspace: which $\mathrm{\Phi }$ are feasible under given conditions, derived from the equations of Part I.

The object is an IT service of a scale one person can run. The premises are as follows.

(1)

There is one operator, and elements requiring human coordination are kept to a minimum

(2)

No outside funding is raised (including crowdfunding)

(3)

Assets are few and the business starts with no commercial credit

This chapter is deduction, not evidence. The validity of the conclusions depends on the validity of the premises.

8.2 Translating the premises

8.2.1 The financial constraint

From premises (2) and (3), set $E(0)\approx 0$ and $A\approx 0$. For software, $\mathrm{Inv}\approx 0$. These are the conditions of Corollary 5.4, which requires $\mathrm{CCC}\le 0$, that is ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{\kappa }_i\le 0$. Written in gross form,

$$\sum _i\max <!--\; FUNCTION\; APPLICATION\; -->\{-{\kappa }_i,0\}\ge \sum _i\max <!--\; FUNCTION\; APPLICATION\; -->\{{\kappa }_i,0\}$$

(8.1)

Equation (8.1) states that the credit drawn must always exceed the credit extended. This is not an assumption but the special case of the fundamental inequality (5.3) with $E=A=0$. It is also the substitution of Proposition A.15 seen at the corner $E=0$. The assumption $A\approx 0$ agrees with data on capital expenditure by sole proprietors (Remark 14.11). Because there is no cushion of equity, the path constraint of Section 4.2 must hold at every instant.

8.2.2 The capacity constraint

Premise (1) puts a ceiling on total delivery.

$$\sum _i{\delta }_i(t)\le \mathrm{Cap},\mathrm{Cap}=\text{one person’s time}$$

(8.2)

$\mathrm{Cap}$ is the same quantity that governed the feasibility of memberships in Proposition 2.24. There it was the capacity of facilities on the supply side; here it is the operator’s disposable time. Its value differs by operator and is not disclosed, so it is not measured in this text.

8.2.3 Which one binds

Equations (8.1) and (8.2) were introduced separately but are imposed together. Converting both into upper bounds on the scale $r$ makes them comparable.

Effort to reduce $\mathrm{CCC}$ increases scale only in the range $\mathrm{CCC}>{\mathrm{CCC}}^{†}$. Below that, capacity binds and changing contractual terms does not increase $r$.

For a solo business with $E\approx 0$, ${\mathrm{CCC}}^{†}\approx 0$. In Part I, which treated firms, the financial constraint $M(t,\omega )\ge 0$ bound; here the binding constraint moves to capacity as soon as $\mathrm{CCC}$ is made negative. As customers increase, time runs out before cash does. That this chapter runs into a capacity constraint after choosing an advance-payment form is due to this move.

The design principles therefore come in two.

The region satisfying both is the feasible set (Figure 8.1).

8.3 What the cloud means structurally

A by-product is the following correspondence.

Before the cloud, procuring servers required capital and $A$ was large. After the cloud $A\approx 0$, and because usage billing is in arrears, ${\kappa }_S<0$ arises instead: credit extended by the infrastructure provider.

$$\text{Effect of the cloud:}A\to {\kappa }_S<0$$

(8.4)

The requirement moved from the physical layer to the credit layer, and its magnitude fell by orders of magnitude — from the purchase price of a server to one month’s usage fee. That one person can now start an IT business is explained by this.

8.4 Sieving the types

The 28 types of Part II are passed through the sieve of (8.1) and (8.2).

About ten of the 28 types remain.

8.5 The credit bootstrap problem

Equation (8.1) requires ${\kappa }_C<0$. But being paid in advance requires credit, and premise (3) says there is none. The contradiction is structural.

There are only three solutions.

(a)

Make the amount small. If the absolute amount paid in advance is negligible to the counterparty, no credit is needed. The consequence is that many small transactions are structurally required. This is also consistent with the path constraint of Section 4.2: dependence on one large customer reaches $T\mathrm{ins}$ on that customer’s single delay.

(b)

Borrow credit. If a payment platform provides escrow and a refund guarantee, the customer trusts the platform rather than the operator. Performing 6-2 oneself is prohibited by regulation, but riding on someone else’s is possible. A $\kappa >0$ arises to the extent of the fee and the payout lag.

(c)

Complete the delivery first. Sell a finished product. The question of what supports $M(t)$ during development then remains.

A consequence follows from (c). Writing $T_{\mathit{𝑑𝑒𝑣}}$ for the development period, $T_{\mathit{𝑎𝑐𝑞}}$ for the acquisition period, and $c_{\mathit{𝑙𝑖𝑣}}$ for the rate of living and infrastructure costs,

$$E_{\mathit{𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑}}=c_{\mathit{𝑙𝑖𝑣}}(T_{\mathit{𝑑𝑒𝑣}}+T_{\mathit{𝑎𝑐𝑞}})$$

(8.5)

Under premise (3) this cannot be met, so a route is needed for injecting $E$ from outside. Salaried income is that route. That is, starting while holding another job is not a matter of attitude but a requirement of the constraint. Designing $\kappa$ is a question only after this valley has been crossed.

8.5.1 Cash flow of the surviving types

Annually billed SaaS is the typical case. Both ${\kappa }_C<0$ (annual prepayment) and ${\kappa }_S<0$ (cloud billed in arrears) are negative, so $W<0$ and hence $\mathrm{CCC}<0$, and the $g^{\star }$ of (5.6) does not exist. Growth generates cash.

Even when financially unconstrained, however, (8.2) remains. In a design where the support load is proportional to the number of customers, the capacity constraint breaks first. In practice this, and not funding, is what stops a solo business.

The design guidance in summary:

(1)

Be paid in advance (${\kappa }_C<0$)

(2)

Keep amounts small and numbers large (no credit needed, and it lowers variance under the path constraint)

(3)

Choose a form whose delivery is not proportional to the number of customers (capacity)

(4)

Avoid $\mathrm{\Phi }$ that the institutional layer prohibits

(5)

Secure from outside the $E$ needed to cross the valley

Item (3) is the hardest, and failure at (3) despite satisfying (1) and (2) is likely common.

8.6 Relaxing a premise: those who hold assets

Relax the asset part of premise (3) and set $E(0)>0$. The case in mind is people who hold accumulated salary assets after retirement.

8.6.1 The social background

The average age at founding rose from 39.7 in 1979 to 49.7 in 2012. The share of founders aged 60 and over rose from 19.3% in 1992 to 35.0% in 2012 for men, and from 7.2% to 20.3% for women. Teikoku Databank’s 2024 survey of newly incorporated companies records an all-time high average age of 48.4 for company representatives. The premise is realistic.

8.6.2 The constraint does not loosen; it changes type

The effect of $E(0)>0$ is not simply a loosening of the financial constraint. What is decisive is whether $E$ is a flow or a stock.

$$\begin{array}{lll}\hfill \text{Salaried:}& E(t)=E(0)+(w-c_{\mathit{𝑙𝑖𝑣}})t(\text{increasing if }w>c_{\mathit{𝑙𝑖𝑣}})& \hfill \text{(8.6)}\\ \hfill \text{Retired:}& E(t)=E(0)-c_{\mathit{𝑙𝑖𝑣}}t(\text{monotonically decreasing})& \hfill \text{(8.7)}\end{array}$$

For the salaried, the $M\ge 0$ of (4.3) is a floor: on approaching it one can pull back, and time is an ally. For the retired it is a deadline. With no revenue the insolvency time is bounded above deterministically by

$$T\mathrm{ins}\le \frac{E(0)}{c_{\mathit{𝑙𝑖𝑣}}}$$

(8.8)

and arrives regardless of the choice of $\mathrm{\Phi }$.

Proposition A.11 supplies this.

The advice “keep amounts small and numbers large to reduce variance” derived in Section 8.5(a) was a conclusion under a floor constraint. It can reverse when the type of constraint changes.

8.6.3 The objective bifurcates

There is a more important bifurcation.

Under R2 the business need not cover $c_{\mathit{𝑙𝑖𝑣}}$.

The bifurcation is also observed across age groups. Analyses of the labour economy by the Ministry of Health, Labour and Welfare show that economic reasons for working among those aged 55–64 have declined over the long run while purpose and social participation have risen.

8.6.4 The change in the objective can be expressed through parameters

The observed change in motives can be written either as a change in the form of the objective or as a change in its parameters. The former is the stronger claim.

Socioemotional selectivity theory ([3]) takes the latter. It holds that as time horizon contracts, goals shift from acquiring knowledge to emotional meaning, and that what drives the shift is not age itself but perceived remaining time. Experimentally inducing a limited time horizon in younger people reportedly produces preference patterns close to those of older people.

Translated into the formulation of Section 4.2, what changes is $H$ and $\beta$.

In the limit of both, the objective approaches an evaluation of the present state itself rather than of accumulated future $\varphi$. That is what the theory means by prioritizing present-oriented emotional goals over future-oriented ones.

This formulation requires no rewriting of the objective. The framework of Section 4.2 is used unchanged.

It has also been noted that founders in their fifties frequently cite securing income, while those in their sixties more often seek connection with society.

8.6.5 Under R2 the feasible region is reversed

The upper left of Figure 8.1 was assessed as “capped”, but under R2 being capped is not a defect.

Equation (8.2) binds only where expansion of scale is required. If $H$ is short and $\beta$ small, accumulated future $\varphi$ carries less weight in the objective and stopping at a handful of customers causes no difficulty (Section 8.6.4). A delivery that takes personal effort in fact contributes directly to present emotional goals.

Within one and the same framework, the optimal quadrant reverses. The guidance “choose a replicable delivery” was correct only when $\varphi$ is being maximized.

8.6.6 Family 3 revives, and a reconciliation with data

In the sieve, family 3 (renting assets over time) dropped out entirely for requiring $A$. Introducing $E(0)>0$ should revive it.

By field of business at founding by age in the 2014 White Paper on Small and Medium Enterprises, among those aged 60 and over the leading fields are services 39.1%, real estate and goods rental 12.2%, and scientific research and professional and technical services 11.9%, while real estate and goods rental is only 0.2% among those under 30.

A sixtyfold difference. The deduction that the presence of $E(0)$ decides the availability of family 3 agrees with the distribution by age. This is an after-the-fact reconciliation and shows no causation, but because the sieve was stated in advance it has value as a check.

8.6.7 The structural role of pensions

The following correspondence holds.

Both remove the binding constraint on a solo business. Pensions generate the very option that R2 represents. This is an instance of the institutional layer widening the feasible set of $\mathrm{\Phi }$, the opposite of the removal in the previous section.

8.6.8 Restating the path constraint

The constraint of Section 4.2 has been written $M(t)\ge 0$, which is inappropriate for this group.

$$M(t,\omega )\ge M_{\mathit{𝑓𝑙𝑜𝑜𝑟}},M_{\mathit{𝑓𝑙𝑜𝑜𝑟}}\gg 0$$

(8.9)

$E(0)$ cannot be re-earned through labour. Even for an identical loss, the asymmetry of utility differs from that of the working-age population. In practice the design is to carve out in advance a business budget $B$ that may be spent and leave $E(0)-B$ untouched. The R2 trajectory of Figure 8.2 halts at a lower bound where this discipline operates.

Note that support services and information aimed at this group face an incentive to have $E$ spent beyond $B$. Fixing $M_{\mathit{𝑓𝑙𝑜𝑜𝑟}}$ in advance is not a modelling device but a practical defence.

8.7 The formation of credit

8.7.1 Why credit attributable to an individual is zero

Independently of $E$, consider whether a retiree can bring credit with them. The conclusion is $S_{\mathit{𝑖𝑛𝑑}}\approx 0$, and this is not a rule of thumb but a consequence of the measurability condition of Chapter 4.

Credit is the ground on which a counterparty accepts ${\kappa }_C<0$, that is, a verifiable record of performance, and it must be in $\mathcal{𝒢}$. But an individual’s contribution inside an organization is by definition not measurable from outside the organization. This is no accident: it is precisely because it is not measurable that a principal–agent problem exists inside the organization.

$$\begin{array}{c}\text{the organization is observed as the performing party}\\ \Rightarrow \text{the record of performance attaches to the organization}\\ \Rightarrow \text{the individual’s contribution is outside }\mathcal{𝒢}\end{array}$$

(8.10)

An organization is a device that absorbs measurability from outside, and losing that on departure is the natural consequence of having enjoyed it.

8.7.2 Separating three quantities

What becomes discontinuous on leaving employment is settled once the quantities are separated.

While employed, an individual borrows the organization’s credit rather than holding it. What falls on departure is the return of the borrowed part; the held part was zero to begin with and does not change. It looks discontinuous, but what is lost was never one’s own.

8.7.3 Two discontinuities, and a lag between them

Not everything falls on departure. During employment, non-compete and confidentiality obligations restrict publication, and leaving removes them: the freedom to convert is discontinuous upwards. But non-compete obligations run for a period and in most cases do not lift immediately.

The shaded region of Figure 8.3 is the structural problem. The non-compete period coincides with the period in which the stock decays fastest, and conversion is impossible during it. The higher the decay rate $\lambda$ in a field, the greater the loss.

The only remedy for this lag is to produce results under one’s own name while still employed. Structurally, remedies after leaving arrive too late.

8.7.4 Capital and credit are substitutes

Restating (8.1): when $E\approx 0$, one has no choice but to earn $\sum <!--\; FUNCTION\; APPLICATION\; -->\max <!--\; FUNCTION\; APPLICATION\; -->\{-{\kappa }_i,0\}$, and credit is the only means of funding. If $E>0$, the inequality holds even without that term.

This is the substitution of Proposition A.15 seen at this chapter’s corner $E\approx 0$. Hence a retiree’s advantage is capital, not credit. In the dimension of credit they start from zero, like a young founder without assets.

8.7.5 Starting with the sign reversed

The guidance of Section 8.5 required ${\kappa }_C<0$, which cannot be executed initially under the premise of zero credit. The following trajectory is taken instead.

The role of capital is to buy the period in which credit is built. Rewriting (8.5),

$$E_{\mathit{𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑}}=c_{\mathit{𝑙𝑖𝑣}}(T_{\mathit{𝑑𝑒𝑣}}+T_{\mathit{𝑎𝑐𝑞}}+T_{\mathit{𝑐𝑟𝑒𝑑𝑖𝑡}})$$

(8.11)

adds the term $T_{\mathit{𝑐𝑟𝑒𝑑𝑖𝑡}}$. Without assets this term cannot be covered, which is why credit had to be borrowed from a platform as in Section 8.5(b). With capital it can be built without borrowing. That is the concrete content of the substitution.

8.8 The trade-off between contract work and publishing

8.8.1 The conflict is not peculiar to individuals; the inability to parallelize is

Contract work is the efficient way to obtain assets; publishing results is what is needed to obtain credit efficiently. Organizations face the same conflict: development contractors and consulting firms alike find it hard to accumulate a public record because of confidentiality.

The difference lies in the form of the capacity constraint.

$$\begin{array}{lll}\hfill \text{Organization:}& \sum \delta \le \mathrm{Cap},\mathrm{Cap}\text{proportional to headcount}& \hfill \text{(8.12)}\\ \hfill \text{Individual:}& {\delta }_{\text{contract}}+{\delta }_{\text{public}}\le \mathrm{Cap},\mathrm{Cap}\text{fixed}& \hfill \text{(8.13)}\end{array}$$

An organization can run both in parallel and cross-subsidize. It performs family 7-2 (cross-subsidy) not between external customers but between internal activities. An individual has no such means. What is peculiar to the individual is not the conflict but the inability to parallelize.

8.8.2 Two kinds of credit

“Contract work generates no credit” is inaccurate. There are two kinds of credit.

Contract work produces referral credit as a by-product at no extra cost, and its effect on $\kappa <0$ is in fact more direct than that of public credit.

8.8.3 Whether the conflict arises depends on the quadrant aimed at

By the conclusion of Section 8.6, the optimum for a retiree under R2 was the upper left. For that group, no conflict between contract work and publishing arises. The conflict becomes a structural problem for those who must maximize $\varphi$, that is, those aiming at the upper right.

8.8.4 Partial conversion and the optimal allocation

Even when aiming at the upper right, the substitution is not one for one. Writing $\mathrm{Cap}\mathrm{cl}$ for the part of the $\mathrm{Cap}$ of (8.2) devoted to contract work and $\mathrm{Cap}-\mathrm{Cap}\mathrm{cl}$ for the part devoted to publishing,

$$\text{accumulation of public credit}=\gamma (\mathrm{Cap}-\mathrm{Cap}\mathrm{cl})+𝜃\mathrm{Cap}\mathrm{cl},0<𝜃<\gamma$$

(8.14)

The second term is the leakage. Of the problems solved on a client engagement, the part that generalizes can be published. The engagement itself cannot be written about, but the general technical problem encountered in it can. And because the thinking is already done, the marginal cost of writing is low.

Equation (8.14) is linear in $\mathrm{Cap}\mathrm{cl}$ and, since $0<𝜃<\gamma$, strictly decreasing. On its own this makes the optimal allocation $\mathrm{Cap}\mathrm{cl}=0$, that is, no contract work at all. The allocation is settled at an interior value only because contract work brings in cash.

The optimum is not an interior point of the objective but the edge of the constraint. If $E(0)>0$ the constraint loosens to $w\mathrm{Cap}\mathrm{cl}\ge c_{\mathit{𝑙𝑖𝑣}}-E(0)∕T$ and $\mathrm{Cap}{\mathrm{cl}}^{\ast }$ falls. The statement in Section 8.6 that capital buys the period in which credit is built can be written as a fall in the right-hand side of (8.15).

Separately from the allocation, there is room to raise $𝜃$. The criterion for choosing engagements moves from the fee to $𝜃$. At equal rates, choose the engagement containing a generalizable problem. The wall of confidentiality is cleared at the level of abstraction, not at the level of the engagement.

8.9 Employment contracts reread

From this chapter’s argument, an employment contract can be written as follows.

$$\begin{array}{c}\text{the individual supplies }\delta \to \text{credit attaches to the organization}\\ \to \text{the individual receives stability of }{\kappa }_L\end{array}$$

(8.16)

An employment contract is a trade in which stability of $\kappa$ is bought at the price of placing the attribution of credit with the organization. In exchange for a certain monthly receipt, the credit asset accumulated does not become one’s own.

This has the same form as the argument about discount factors in Section 4.2, and the same structure as the choice between families 5-1 and 1-3 discussed in Chapter 7.

The terms of attribution are a negotiable variable. Securing what may be produced under one’s own name while employed — public talks, contributions to public repositories, writing — belongs to the same dimension as negotiating pay. In many cases it is granted at no cost.

8.10 Limits of this chapter

(1)

This is deduction, not evidence. Apart from the reconciliation of family 3’s revival with the distribution by age, the conclusions of this chapter are deductions from premises. If the premises are wrong, so are the conclusions.

(2)

$S_{\mathit{𝑖𝑛𝑑}}\approx 0$ depends on the field. In fields that are small enough that participants know one another as individuals — academic disciplines, particular technical communities, small professions — an individual’s performance is observed without passing through an organization. There the separation of the borrowed from the held part is already advanced during employment, and the discontinuity on departure is small. The size of $S_{\mathit{𝑖𝑛𝑑}}$ depends not only on individual effort but on the measurability structure of the field.

(3)

The objective under R2 is not identified. It was shown that the change can be written through $H$ and $\beta$, but not verified that this is sufficient. Strictly, therefore, no claim of optimality under R2 has been made.

(4)

The decay rate $\lambda$ of the convertible stock is not measured. How serious the lag of Figure 8.3 is in practice depends on $\lambda$, and there is no basis here for its value.