Remark 6.1 (Where the return to bearing risk belongs). A party that makes a business of the transfer and pooling of Section 2.6 — an insurer or a payment processor, say — earns a return for bearing risk. Where does it belong in (6.1)?

The part by which variance is actually reduced through pooling is ${\varphi }_{\mathrm{prod}}$. By Proposition 2.9, raising $n$ removes the ${\sigma }^2∕n$ term; this is a technical advantage and reduces nobody else’s share.

The part corresponding to the limit $\rho {\sigma }^2$ of (2.11), by contrast, cannot be removed. The return for bearing it is a transfer and satisfies ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{\varphi }_i=0$; it belongs to ${\varphi }_{\mathrm{barg}}$.

In practice the two are not separated and appear as a single rate. Where convenient this text writes ${\varphi }_{\mathrm{risk}}$ (Section 17.5.1), which is not a new component but the undivided sum of ${\varphi }_{\mathrm{prod}}$ and ${\varphi }_{\mathrm{barg}}$.

Definition 6.2 (Production surplus and bargaining surplus). Using the marginal cost $c(\delta )$ of Definition 2.14 and the price $p^{\ast }$ that would obtain without market power, define

where $p$ is the actual price. The character of $p^{\ast }$ is discussed in Remark 6.3.

Remark 6.3 ($p^{\ast }$ is a counterfactual). Definition 6.2 requires $p^{\ast }$, which is not observed. It is a counterfactual price posited for a state without market power, and has the same structure as the standard procedure for measuring market power in industrial organization.

Separating ${\varphi }_{\mathrm{prod}}$ from ${\varphi }_{\mathrm{barg}}$ therefore requires an additional assumption identifying $p^{\ast }$.

One assumption suffices: that a control is observable which is identical except for market power. Given such a control, its price can be read as $p^{\ast }$ and the difference attributed to ${\varphi }_{\mathrm{barg}}$. Chapter 17 satisfies this in the form of two rates on the same platform that differ only in function. Where the assumption fails, this text goes no further than the sum ${\varphi }_{\mathrm{prod}}+{\varphi }_{\mathrm{barg}}$.

Proposition 6.4 (Implication of the decay). The solution of (6.4) is ${\varphi }_{\mathrm{cog}}(t)={\varphi }_{\mathrm{cog}}(0)e^{-\lambda (\nu )t}$, with half-life $\ln <!--\; FUNCTION\; APPLICATION\; -->2∕\lambda (\nu )$. The higher the transaction frequency, the faster cognitive surplus disappears.

Example 6.5 (Frequency and half-life). Suppose $\lambda$ is proportional to the number of transactions a year, $\lambda =0.02\nu$.

6.2 Surplus and cash

$\varphi$ affects cash only through (3.10), and the three components do not affect it in the same way.

Cognitive surplus generates no cash at the moment of recognition. The cash arrived when the prepayment was received, and its effect there appeared not as ${\varphi }_{\mathrm{cog}}$ but as ${\kappa }_C<0$. The ${\varphi }_{\mathrm{cog}}$ of (2.15) and the undelivered balance of Chapter 3 are two cross-sections of the same transaction.

Substituting (6.5) into Corollary 5.3, for a business with no source of surplus other than cognitive surplus,

Equation (6.6) shows that the transaction frequency fixes the upper bound on the scale a business can reach. Equation (6.4) does not stop at a discussion of half-lives; it connects to the constraint of Chapter 5. The coefficient values have no empirical basis and only illustrate the structure of the ratio. What follows is an ordering: goods bought daily retain almost no cognitive surplus, while contracts on an annual cycle retain it for a long time.

Remark 6.9 (The institutional layer does not act uniformly). “The institutional layer imposes constraints” covers at least three distinct forms.

The three erode differently. The first two make a particular $\mathrm{\Phi }$ unavailable; the third leaves $\mathrm{\Phi }$ in place and lowers the price. The dominant form in recent years is the third; forms that regulate price directly are in decline (Section 17.8).

Saying that something is “eroded by the institutional layer” therefore says nothing about how fast or how surely. Chapter 17 treats a case in which fees arising from market power can be separated from fees arising from function, but what is observed there is a single reduction of the third form, not continuous decay.