Proposition 4.1 (Implementability). A necessary condition for a settlement schedule $\pi$ to be implementable as a contract is that $\pi$ be $\mathcal{𝒢}$-measurable.

Remark 4.2 (What $\mathcal{𝒢}$ bounds is degree of freedom (4)). Proposition 4.1 bears directly on degree of freedom (4), state dependence, in Section 2.5. $\pi$ may be made to depend on $\omega$ only within $\mathcal{𝒢}$; distinctions among $\omega$ that lie outside $\mathcal{𝒢}$ cannot be written into a contract.

So $\mathcal{𝒢}$ bounds the codomain of $\mathrm{\Phi }$, not its domain. Any delivery $\delta$ may be chosen; what is constrained is how $\pi$ may be made to move with it.

The constraint appears in the assignment order of Part II. The family decided by degree of freedom (4) is family 5 (outcome- and state-contingent), and that family cannot be implemented unless the $y$ in $\pi =h(y)$ is $\mathcal{𝒢}$-measurable. The impossibility result for family 7 in Chapter 10 (Proposition 10.6) has the same shape.

Remark 4.3 (Endogeneity of $\mathcal{𝒢}$). This chapter treats $\mathcal{𝒢}$ as given. In practice an operator can invest in $\mathcal{𝒢}$: follow-up surveys make the outcome $y$ observable and widen the range of verifiable information.

The staffing-service disclosures treated in Section 15.3.5.0 include a field for the number of placements whose separation could not be ascertained, which shows that the breadth of $\mathcal{𝒢}$ differs by operator and is itself a choice.

Proposition 4.1 should therefore be read as a short-run condition with $\mathcal{𝒢}$ held fixed. In the long run $\mathcal{𝒢}$ becomes part of $\mathrm{\Phi }$.

Remark 4.4 (A distinction of direction). Risk transfer comes in two kinds that run in opposite directions. Success fees and royalties leave risk with the agent because effort is unobservable. Cost plus returns risk to the principal because effort is observable but the outcome is uncertain. Grouping both under “risk sharing” erases the difference.

Example 4.5 (Optimal strength of outcome contingency). Consider the linear contract $\pi =\alpha +by$. Writing $\gamma$ for the agent’s absolute risk aversion and ${\sigma }^2$ for the noise variance of the outcome ($r$ is reserved for the rate of sales in Chapter 5, hence $\gamma$ here), the optimal outcome-contingency coefficient takes the form

$$b^{\star }=\frac{1}{1+\gamma {\sigma }^{2}k}$$

(4.1)

where $k$ is a constant relating to the marginal cost of effort ([18]). The larger ${\sigma }^2$, the smaller $b^{\star }$: the noisier the outcome, the weaker the outcome contingency should be.

As a numerical illustration, with $\mathit{𝛾𝑘}=1$: $b^{\star }=0.67$ at ${\sigma }^2=0.5$, $b^{\star }=0.33$ at ${\sigma }^2=2$, and $b^{\star }=0.10$ at ${\sigma }^2=9$.

Definition 4.6 (The optimization problem). With period profit ${\varphi }_t$ and discount factor $\beta$,

$$\begin{array}{lll}\hfill \underset{\mathrm{\Phi }}{\max <!--\; FUNCTION\; APPLICATION\; -->}& 𝔼\left[\sum _{t\in T}{\beta }^t{\varphi }_t\right]& \hfill \text{(4.2)}\\ \hfill \text{s.t.}& M(t,\omega )\ge 0\forall <!--\; FUNCTION\; APPLICATION\; -->t\in T,\forall <!--\; FUNCTION\; APPLICATION\; -->\omega \in \mathrm{\Omega }& \hfill \text{(4.3)}\end{array}$$

Proposition 4.7 (The path constraint does not commute with expectation). $𝔼[M(t)]\ge 0$ does not imply (4.3). Moreover the insolvency time $T\mathrm{ins}=\inf <!--\; FUNCTION\; APPLICATION\; -->\{t:M(t)<0\}$ is absorbing, and no ${\varphi }_t$ for $t>T\mathrm{ins}$ is realized. Because ${\tau }_i$ denotes the settlement lag, the insolvency time is written $T\mathrm{ins}$.

Corollary 4.8 (Insolvency while profitable). Even if ${\varphi }_t>0$ for every $t$, (4.3) can be violated. Insolvency while profitable is thus a failure of the constraint (4.3), not of the objective (4.2).

Example 4.9 (Insolvency while profitable, numerically). A business with margin $m=10\%$ grows from 100 million yen of annual revenue to 200 million. Its profit is 20 million yen.

If credit to customers runs at 20% of revenue and inventory at 5%, the sum of the credit positions of (3.2) and inventory is 25 million yen at 100 million of revenue and 50 million at 200 million. The increase of 25 million yen leaves in cash.

Against a profit of 20 million yen, 5 million is short. The books show a profit and the cash is short, and (4.3) can be violated.

Proposition 4.10 (Gain from rearranging the settlement schedule). If ${\beta }_C<{\beta }_0$ (the customer discounts the future more heavily than the firm), there exists a rearrangement of the timing of $\pi$ alone, leaving the delivery $\delta$ unchanged, that improves both parties’ subjective valuations simultaneously.

Example 4.11 (The gain from instalments). Suppose the customer’s annualized discount rate is 20% (${\beta }_C=0.833$) and the firm’s cost of funds is 5% (${\beta }_0=0.952$). For an item with a cash price of 100,000 yen, a price $p^{\prime }$ payable in one year improves both parties for $105,000<p^{\prime }<120,000$. Taking $p^{\prime }=112,000$: the firm receives $112,000\times 0.952=106,624$ yen in present value, a gain of 6,624 yen; the customer values it subjectively at $112,000\times 0.833=93,296$ yen, a gain of 6,704 yen.

Remark 4.12 (Two readings). ${\beta }_C<{\beta }_0$ can have two different causes. For a customer facing a liquidity constraint, present funds are objectively more valuable. Alternatively the cause may be cognitive, as with hyperbolic discounting ([21]). Meeting the first and exploiting the second are almost indistinguishable from outside, yet their durability (Chapter 6) is opposite. This framework cannot settle the distinction.