Finance According to Ecclesiastes, Wisdom and Leviticus

A sequel to “Finance According to Solomon”. Three books this time, three registers: Qoheleth the sceptic, who observes that money does not satisfy and that disaster does not announce itself; the Wisdom of Solomon, written in Greek in Alexandria, which stages the financial reasoning of the ungodly in order to refute it; and Leviticus, a legal code which — almost by accident — states the first known discounting formula and the first known actuarial table. Same method as before: the verse, the equation, the simulation. Six claims, six tests.

I.The thirst that water does not quench

“He that loveth silver shall not be satisfied with silver; nor he that loveth abundance with increase: this is also vanity.”

Ecclesiastes 5:10 (KJV)

The verse describes a mechanism, not a vice: satisfaction adapts to the level of spending, then demands more. Modern psychology calls it hedonic adaptation; let us model it. Consumption grows at rate $ g $: $ C(t) = C_0 e^{gt} $. The inner reference $ R $ — what one is used to — chases consumption with a lag $ 1/\lambda $: $ \dot R = \lambda (C - R) $. Satisfaction is logarithmic in the ratio, not the level:

\[ S(t) = \ln\frac{C(t)}{R(t)} \;\xrightarrow[t \to \infty]{}\; S_\infty = \ln\!\left(1 + \frac{g}{\lambda}\right). \]

The result is brutal: in the long run, satisfaction does not depend on the level of spending, only on the ratio between its growth $ g $ and the adaptation speed $ \lambda $. Spending ten times more, at zero growth, leaves you exactly as satisfied as before — that is the verse's “shall not be satisfied”, together with its reason: to keep $ S $ above zero, spending must grow forever, exponentially. Being satisfied by money is not hard; it is structurally impossible as soon as $ \lambda > 0 $.

Spending explodes, satisfaction plateaus — 40 years

Spending growth g: 5% Adaptation speed λ: 50%/yr

Observation. At g = 5% and λ = 50%/yr, spending is multiplied by 7.4 over forty years, yet satisfaction plateaus at $ \ln(1.1) \approx 0.095 $ — less than a tenth of the effect of an instant doubling, and that doubling itself evaporates within $ 1/\lambda = 2 $ years. The only slider that durably raises the plateau is $ \lambda $ — that is, the slowness of getting used to things, which is not a financial variable.

II.A portion to seven, and also to eight

“Cast thy bread upon the waters: for thou shalt find it after many days. Give a portion to seven, and also to eight; for thou knowest not what evil shall be upon the earth.”

Ecclesiastes 11:1-2 — see also 11:6 (KJV)

The previous article treated diversification through variance — the law $ \sigma\sqrt{\rho + (1-\rho)/n} $. Qoheleth is after something else: not smoothing fluctuations, but surviving the evil — the event that destroys a portion entirely. Suppose each portion, each year, is annihilated with probability $ p $ — shipwreck, bankruptcy, expropriation — independently of the others. Over a horizon $ T $, one portion survives with probability $ (1-p)^T $, and the probability of losing everything is

\[ P_{\text{ruin}}(n) = \left(1 - (1-p)^T\right)^{n}. \]

Expected wealth does not depend on $ n $: dividing gains you nothing on average. The entire benefit lives in the tail — ruin decays exponentially with the number of portions. And the exponential has an elbow: the first few divisions do almost all the work, the later ones almost none. The verse does not say “divide forever”; it says seven, and also eight — then stops. The reason to stop is in the formula: past the elbow, each extra portion costs more in fees and oversight than it removes in ruin.

Probability of losing everything vs number of portions (log scale)

Probability of disaster per portion per year p: 5% Horizon T: 20 yrs

Observation. At p = 5% and T = 20 years, a single portion dies in 64% of cases; eight portions all die together in 3% of cases — a factor of 20 for seven divisions, then almost nothing from 8 to 20. Verse 11:6 (“In the morning sow thy seed, and in the evening withhold not thine hand”) adds the temporal dimension: diversify the sowing dates as well as the plots. The hidden condition is independence — eight portions on the same ship are one portion.

III.The discount rate of the ungodly

“For the ungodly said, reasoning with themselves, but not aright, Our life is short and tedious […] Come on therefore, let us enjoy the good things that are present: and let us speedily use the creatures like as in youth.”

Wisdom of Solomon 2:1,6 (KJV)

The book of Wisdom does something rare: it stages a complete economic argument — premise, inference, decision — in order to refute it. The ungodly posit a short horizon and a terminal value of zero, then correctly deduce immediate consumption. Life-cycle theory gives the exact formula: an agent who believes they have $ n $ years left consumes each year the annuity fraction of their capital,

\[ c = W \cdot \frac{r}{1 - (1+r)^{-n}}, \]

which explodes as $ n $ shrinks — for small $ n $ it tends to $ W/n $: everything, now. The error of the ungodly is therefore not in the optimisation, which is flawless; it is in the estimation of the parameter $ n $. The text says exactly this: “reasoning with themselves, but not aright” — and concludes in 2:21, “such things they did imagine, and were deceived”. An agent who plans over 15 years and lives 60 spends three quarters of their existence in the state their own equation had prepared for them: zero capital.

Two optimal consumption plans — but only one correct horizon (actual life: 60 years)

Horizon believed by the ungodly T_b: 15 yrs Real return r: 3%

Observation. Slide T_b toward 60: the two trajectories merge — the optimisation was the same, only the parameter differed. This may be the chapter's most modern lesson: the worst financial errors are not calculation errors, they are horizon errors, and they produce decisions that are locally beyond reproach.

IV.The asset that begets all the others

“I preferred her before sceptres and thrones, and esteemed riches nothing in comparison of her. […] All good things together came to me with her, and innumerable riches in her hands.”

Wisdom of Solomon 7:8,11 (KJV)

Read these two verses as a portfolio trade: Solomon exchanged wealth (a stock) for wisdom (a capacity), and the stock came back to him multiplied. That is the definition of human capital. Compare the two uses of a sum $ W_0 $: invest it at return $ r $, or spend it on training that raises income from $ Y $ to $ (1+\beta)Y $ forever. With a savings rate $ s $, the financial route starts at $ W_0 $ and saves $ sY $; the wisdom route starts at zero and saves $ sY(1+\beta) $. The second overtakes the first if and only if

\[ \frac{s\,Y\,\beta}{r} \;>\; W_0 \qquad\Longleftrightarrow\qquad \beta \;>\; \beta^\* = \frac{r\,W_0}{s\,Y}. \]

The left-hand side is the perpetuity value of the extra saving: training is not an expense, it is the purchase of an annuity. Once $ \beta $ exceeds the threshold, the overtaking is inevitable and the gap then grows exponentially — “all good things together came to me with her” is the exact description of a flow that feeds every future year, against a capital that only compounds upon itself.

$30,000 invested vs $30,000 spent on oneself — wealth over 40 years

Income boost β: 35% Financial return r: 6%

Observation. At the default parameters (income $50k, savings rate 15%), the threshold is β* = 24% — but just above it, victory is only asymptotic: at β = 25%, the crossover lands near year 54; at β = 35%, near year 19 — and the financial return $ r $ works against training at the threshold, but for it after the crossover, since the extra savings compound at the same rate. The verse has the causal order right: first the capacity, then the goods.

V.The jubilee, or the first discounting

“According to the number of years after the jubile thou shalt buy of thy neighbour, and according unto the number of years of the fruits he shall sell unto thee: According to the multitude of years thou shalt increase the price thereof, and according to the fewness of years thou shalt diminish the price of it: for according to the number of the years of the fruits doth he sell unto thee.”

Leviticus 25:15-16 (KJV)

Every fifty years, the jubilee returns each plot of land to its original family; in the meantime, a sale is only a lease. The text draws the pricing rule from this: one does not buy the land, one buys its remaining harvests — “according to the number of the years of the fruits doth he sell unto thee”. With an annual harvest $ y $ and $ n $ years before the jubilee, the legal price is $ P = y \cdot n $: the undiscounted sum of future cash flows. Modern finance discounts:

\[ P_{\text{DCF}}(n) = \sum_{k=1}^{n} \frac{y}{(1+r)^k} = y\,\frac{1 - (1+r)^{-n}}{r} \;\xrightarrow[r \to 0]{}\; y \cdot n. \]

The rule of Leviticus is thus the annuity formula taken at $ r = 0 $ — and this is not a naïve approximation, it is an internal consistency: twenty verses further on, the same chapter forbids interest between brothers (25:36-37, “take thou no usury of him, or increase”). In an economy where $ r = 0 $ is the law, the correct discounting is exactly linear. Chapter 25 thus contains, in a single block, the first known discounted-cash-flow valuation — and the discount rate consistent with its own credit law.

Land price in harvests: rule of Leviticus (linear) vs discounted annuity

Years before the jubilee n: 30 Discount rate r: 6%

Observation. Slide r toward zero: the blue curve straightens and comes to hug the yellow line of the verse. At r = 6%, the gap at 30 years is already more than half — all the distant value evaporates. That is the deep link between the two halves of the chapter: to forbid interest is to decree that the future is worth as much as the present.

VI.The first actuarial table

“And thy estimation shall be of the male from twenty years old even unto sixty years old […] fifty shekels of silver […] from five years old even unto twenty years old […] twenty shekels […] from a month old even unto five years old […] five shekels of silver […] from sixty years old and above; if it be a male, then thy estimation shall be fifteen shekels.”

Leviticus 27:3-7 (KJV)

Chapter 27 fixes the redemption tariff for vows: a person “devoted” may be redeemed at a price set by law, graduated by age — 5 shekels under five, 20 from five to twenty, 50 from twenty to sixty, 15 beyond (female scale: 3, 10, 30, 10). It is an administrative schedule; it is also, structurally, an actuarial table: the value of a person of age $ a $ as the present value of their remaining production,

\[ V(a) = q(a)\sum_{t \ge a}^{} \frac{w(t)}{(1+r)^{\,t-a}}, \]

where $ w(t) $ is productivity by age (full from 20 to 60, reduced afterwards) and $ q(a) $ the probability of surviving to productive age — far from negligible in a world where half of all children never reached adulthood. Two parameters, $ r $ and $ q $, and the schedule almost falls out by itself: the child is worth little not out of contempt, but because their production is distant (discounted) and uncertain (mortality); the sixty-year-old falls below the adolescent because only a few years of flow remain. The schedule encodes an implicit discount rate — a high one, as befits a risky agrarian economy.

Tariff of Leviticus (shekels, males) vs present value of remaining production

Discount rate r: 12% Infant survival to age 20 q: 50%

Observation. At r ≈ 12% and q ≈ 50%, the model reproduces the four brackets to within a shekel or two — including the elder/adolescent inversion (15 versus 20), which is not intuitive at all and which discounting produces by itself. Lower r toward 2%: the child becomes almost as valuable as the adult and the biblical schedule stops fitting. The tariff of Moses implies a double-digit discount rate — three thousand years before the term existed.

Synthesis

Three books, three registers, six mathematical objects. Qoheleth states two theorems of randomness: satisfaction plateaus at $ \ln(1+g/\lambda) $ whatever the level of spending (5:10), and ruin decays as $ q^n $ — hence the reasonable stop at seven or eight portions (11:2). Wisdom supplies the two theorems of capital: a horizon error makes immediate consumption optimal (2:1-9), and a capacity that raises the flow beats a stock that compounds, as soon as $ \beta > rW_0/sY $ (7:8-11). Leviticus, finally, gives the two theorems of valuation: the price of land is the annuity of its harvests — linear precisely because interest is forbidden there (25:15-16) — and the tariff of persons is a present value of remaining production, mortality included (27:3-7). The sceptic, the philosopher and the legislator had neither stochastic calculus nor mortality tables; they had watched the same trajectories we do.

Iron sharpeneth iron (Proverbs 27:17): comments, objections and counter-examples welcome. Verses quoted from the King James Version (public domain), whose Apocrypha include the Wisdom of Solomon, a deuterocanonical book. Parameters are illustrative; nothing here is financial advice.