Research on ENKR's Blog | Jing Hui PANG

Why the Hundredth Cake Costs Less: Marginal Cost, Scale, and Sunk Cost

Mon, 29 Jun 2026 00:53:31 +0800

i got this as a knowledge card on douyin/xiaohongshu, cannot recall which: marginal cost falls as you produce more, your personal assets keep accumulating, and sunk costs should stay out of big decisions. the intuition is lovely. some of the economics underneath it is slightly off, and one of the historical claims is just wrong. so i had my AI teach me the proper version and draw the curves. this note is that proper version. the personal reflection at the bottom is mine to write after.

the puzzle

you open a bakery. the first cake is brutal. you buy an oven, a set of moulds, you burn three batches getting the recipe right. all in, that first cake might cost you 500 dollars.

by the hundredth cake, the oven is paid for and your hands know the motion. one more cake costs you some flour, a couple of eggs, and a little electricity. call it 10 dollars.

so why does the hundredth cake cost a fraction of the first, and what exactly is the thing that fell? the honest answer is more interesting than the knowledge card lets on, and it comes apart into two different costs that people constantly mix up.

marginal cost is not quite what the card says

a quick key for the shorthand, so the formulas below read easily:

symbol	what it means
$Q$	quantity: how many units you make
$TC$	total cost of producing all of them
$MC$	marginal cost: the cost of one more unit
$ATC$	average total cost: total cost divided by units
$\Delta$	“change in” (so $\Delta Q$ is the change in quantity)

marginal cost is the cost of making one more unit. formally it is the change in total cost divided by the change in quantity, the slope of the total cost curve:

$$ MC = \frac{\Delta TC}{\Delta Q} $$

average total cost is the total cost spread over every unit you made:

$$ ATC = \frac{TC}{Q} $$

these are not the same number, and the bakery is the cleanest way to feel the difference. split the cost in two:

fixed cost, the oven and the moulds, say 500 dollars. you pay it once whether you bake one cake or a thousand.
variable cost, the flour and eggs and electricity, about 10 dollars per cake. you pay it again for every cake.

now look carefully. the marginal cost of the hundredth cake is not falling much at all. it is roughly 10 dollars, the same flour and eggs as the second cake. what collapses is the average cost, because the one off 500 dollars gets shared across more and more cakes:

$$ ATC = \frac{500}{Q} + 10 $$

at one cake that is 510 dollars. at ten cakes it is 60. at a hundred cakes it is 15, sliding towards the 10 dollar floor. here is the whole story in one picture:

Fixed cost of 500 dollars spread over more cakes. The marginal cost never really moved; the average just stopped being dominated by the oven.

so the knowledge card’s “marginal cost falls as you scale” is really, for the most part, average cost falling as a fixed investment gets amortised over more output. the marginal cost was low the whole time. that distinction is the entire game, and it is exactly the part the card skips.

why cost per unit falls: economies of scale

amortising a fixed cost is the simplest reason cost per unit drops, but it is not the only one. when bigger genuinely makes each unit cheaper to produce, economists call it economies of scale, and it comes from a few real places:

spreading indivisible costs. the oven, the brand, the one good recipe, the R&D. you buy them once and reuse them forever.
specialisation. at one cake a day you do everything. at a thousand, one person pipes, one person bakes, each gets fast at their slice. this is Adam Smith’s pin factory from 1776, still the cleanest example we have.
bulk and learning. flour gets cheaper by the tonne, and you get quicker every time you repeat the motion. more on that learning effect at the end, because it is the bridge to the personal version.

economies of scale are why a single chip fab or a single search engine can serve the planet. but the card quietly implies cost per unit falls forever. the textbook is more careful, and more honest.

the honest picture: marginal cost is U-shaped

in the standard short-run model, marginal cost does not fall forever. it falls, bottoms out, then rises. the reason is diminishing returns: with the kitchen size fixed, the fifth baker is crammed against the same two ovens as the first four, and each extra baker adds less than the one before. past some point, one more cake costs more than the last, not less.

so the real marginal cost curve is U-shaped, and it cuts the average total cost curve at exactly the lowest point of average cost. that crossing is not a coincidence: while making one more unit is cheaper than your running average, it drags the average down; once it costs more than the average, it pulls the average back up. they can only meet at the bottom.

Diminishing returns make each extra unit eventually cost more, not less. Marginal cost cuts average cost at exactly the bottom of the average cost curve.

this is the picture the card never shows you, and it matters: “scale always wins” is false in the short run. real firms have an efficient size, and pushing past it makes the next unit more expensive, not less.

the limit case: zero marginal cost

now the one place the card is most right. for some goods the marginal cost is not just low, it is almost nothing.

think about software, a song, a pdf, this blog post. the first copy is enormously expensive: years of work, the whole idea, the late nights. this is the first-copy cost. every copy after that costs essentially zero. there is no flour, no eggs, no oven time. just a near free bit of bandwidth.

economists Carl Shapiro and Hal Varian called this the defining feature of information goods in Information Rules (1998): “information is costly to produce but cheap to reproduce.” when marginal cost sits at roughly zero, average cost is just the first-copy cost divided by however many copies you ship, and it falls towards zero without limit:

The first copy costs a fortune; every copy after is almost free, so average cost falls towards zero without limit. This is the cost shape of software, music, and writing.

this is the kernel of truth the card is reaching for when it says knowledge and software can be copied at almost no cost. Jeremy Rifkin built a whole popular book on it, The Zero Marginal Cost Society (2014), arguing this drives the price of information goods towards free. treat that grand conclusion with some caution, the economics profession does, but the underlying mechanic is real and it is the dominant cost structure of the digital economy.

who actually discovered this, because the card gets it wrong

the card credits “Chandler and Stone, first proposed in the 1940s.” that is not where any of this comes from.

the idea of thinking at the margin, one more unit at a time, is the Marginal Revolution of the 1870s: William Stanley Jevons, Carl Menger, and Léon Walras, independently, around 1871 to 1874. Alfred Marshall then drew the supply and demand scissors and the cost curves you saw above in his Principles of Economics (1890). that is the real lineage, and it is roughly fifty years before the date on the card.
Richard Stone was a real and important economist, a Nobel laureate in 1984, but for building the system of national income accounting, GDP and the national accounts, in the 1940s. nothing to do with originating marginal cost.
Alfred Chandler was a great business historian. his Scale and Scope (1990) is genuinely about economies of scale and scope, so the card is not pulling his name from nowhere, but he was documenting how big firms grew in the late nineteenth and twentieth centuries, not proposing marginal cost in the 1940s.

so: nice instinct, wrong plaque. the concept is Victorian, not wartime. worth fixing, because a research note that repeats a confident wrong citation is worse than one that says nothing.

sunk cost: the other half of the same idea

the card pairs all this with a second rule: keep sunk costs out of big decisions. this is not a separate topic. it is the exact same marginal thinking, pointed forwards.

a sunk cost is money, time, or effort you have already spent and cannot get back no matter what you choose next. the 80 dollar ticket to a film that turns out to be terrible. the two years already poured into a project that is clearly going nowhere. the half of a book you have read and are not enjoying.

the rational rule is almost insultingly simple: compare the marginal benefit of continuing against the marginal cost of continuing, looking only forwards. the 80 dollars is gone in both worlds, stay or leave, so it should carry exactly zero weight. the only live question is whether the next two hours are better spent enduring a bad film or doing literally anything else.

we are famously bad at this. Hal Arkes and Catherine Blumer demonstrated it cleanly in 1985 (“The Psychology of Sunk Cost”): people who had paid for a ski trip were more willing to go on it in bad conditions than people who got the same trip free, even though the money was already spent either way. the spending changed nothing about the future and everything about the choice. that is the sunk cost fallacy, also called escalation of commitment, and naming it is half the cure.

notice why it sits next to marginal cost so naturally. both say the same thing: decide at the margin, and look forward. what you already sank is a fixed cost of the past. it belongs in the history books, not the decision.

a person is a firm with fixed costs

here is where the card makes its leap, from a cost curve to a way of seeing your own life, and this part is better grounded than it first looks.

learning a skill has the bakery’s exact shape. the first unit is brutal: the first month of a language, the first real program, every step uphill. that is your fixed cost, your oven. but what you build does not get consumed when you use it. knowledge is non-rival: teaching it, or using it, does not use it up, and unlike the oven it barely depreciates. so every skill you acquire is a fixed asset that keeps paying out, and the marginal cost of your next unit of competence keeps dropping.

this is not just a metaphor, it is two real pieces of economics:

human capital. Gary Becker (Nobel, 1992) made the case in Human Capital (1964) that skills and knowledge are a form of capital you invest in, with real returns, exactly like a machine.
the learning curve. Theodore Wright measured in 1936 that each time cumulative aircraft production doubled, the labour cost per plane fell by a roughly constant percentage. the Boston Consulting Group generalised it into the experience curve. unit cost is a power law in cumulative experience, which looks like this:

Wright's law, 1936. Every time your cumulative experience doubles, the cost of producing one more unit falls by a roughly constant percentage. Skill compounds.

so the card’s line, “your assets keep accumulating,” is just human capital compounding. you do not see today’s two hours of study as a cost that vanishes. you see it as one more unit produced, which lowers the cost of every future unit and adds to a stock that does not wear out. the flywheel the card is pointing at is real: keep producing, watch your marginal cost fall and your accumulated asset rise, and refuse to let what you already sank decide what you do next.

that is the economics. the personal version is below, and that part is mine.

a personal note

wip …

sources and further reading

Alfred Marshall, Principles of Economics (1890), for the cost curves and marginal analysis.
Carl Shapiro and Hal R. Varian, Information Rules (1998), for first-copy cost and information goods.
Jeremy Rifkin, The Zero Marginal Cost Society (2014), for the popular extrapolation, read critically.
Gary S. Becker, Human Capital (1964), for skills as investable capital.
T. P. Wright, “Factors Affecting the Cost of Airplanes” (1936), for the original learning curve.
Hal R. Arkes and Catherine Blumer, “The Psychology of Sunk Cost,” Organizational Behavior and Human Decision Processes (1985), for the sunk cost fallacy.

Kelly Criterion: How Much to Bet When You Think You Have an Edge

Fri, 12 Jun 2026 22:00:08 +0800

i ran into the kelly formula on social media, asked my AI to teach me, then had it dispatch a fleet of fact-checking agents to verify every number below against thorp’s papers and the original 1956 paper. one thing it taught me in chat turned out to be subtly wrong, which is exactly why this note exists.

🚧 reading status: not done. i’ve been taught this once and skimmed it once. future me: come back, work through the maths by hand, and tick the checklist at the bottom. until then, treat my own understanding here as unverified, even though the facts themselves are checked.

the question kelly actually answers

you have a bet with an edge. you get to repeat it many times, profits roll back into the bankroll. how much do you put in each time?

the naive answer is “maximise expected wealth”, and it is a trap. expected value is linear, so maximising $E[W]$ tells you to bet everything every round. do that repeatedly and you go broke with probability one (kelly says exactly this on p. 918 of the original paper). the expectation is propped up by an astronomically lucky path you will never live.

kelly’s move (bell labs, 1956) was to maximise $E[\log W]$ instead, which is the same as maximising the long-run compound growth rate. for a bet where you win $b$ per $1 staked with probability $p$ (lose the stake with probability $q = 1-p$), each round grows your log-wealth by

$$ g(f) = p\ln(1+bf) + q\ln(1-f) $$

take the derivative, set it to zero:

$$ f^{\ast} = \frac{bp-q}{b} = p - \frac{q}{b} $$

the second form is the one to remember: win rate, minus loss rate divided by the odds.

full kelly is more aggressive than you think

plug in numbers that sound conservative: 70% win rate, 1.5:1 payoff.

$$ f^{\ast} = 0.7 - \frac{0.3}{1.5} = 0.5 $$

half your bankroll. on one trade. the formula is not shy, and that is the first hint that nobody should run it raw.

the 2x kelly cliff

here is the property that makes kelly worth learning even if you never bet. the growth curve $g(f)$ is a hill with its peak at $f^{\ast}$, and in the continuous approximation (and with no risk-free rate) it is exactly parabolic. betting a fraction $c$ of full kelly gets you

$$ \text{growth share} = 2c - c^{2} $$

fraction of kelly	share of max growth
0.25x	44%
0.5x	75%
1x	100%
1.5x	75%
2x	0%
>2x	negative

at exactly twice kelly your excess growth rate is zero. you took on all that risk to compound at the risk-free rate. beyond 2x, the growth rate goes negative and your wealth shrinks towards zero almost surely, even though the strategy has a genuine edge and even though $E[W]$ is still going up. the average is rising while almost every actual path dies. concrete discrete example: a 60% coin at even odds has $f^{\ast} = 0.2$ and grows about 2.0% per round; bet 0.4 instead and you grind down at about minus 0.2% per round.

drawdowns: the part that converts people to half kelly

thorp’s survey has a remarkably clean set of results for full kelly (continuous approximation, no risk-free rate):

the probability your bankroll ever falls to a fraction $x$ of where it is now is just $x$. a 50% chance of halving at some point. a 10% chance of being down 90% at some point.
the probability you halve before you double is 1/3.

at a fraction $c$ of kelly, the ever-drop-to-$x$ probability becomes $x^{2/c - 1}$. at half kelly that is $x^{3}$: the chance of ever halving drops from 50% to 12.5%, and halve-before-double drops from 1/3 to 1/9.

so half kelly trades 25% of the growth for roughly a 4x improvement in the disaster odds. that trade is so lopsided that “half kelly” is basically the practitioner default.

where my AI taught me wrong (and the fact-check caught it)

in chat i was told the classic line: “the penalty for over-betting is steeper than for under-betting, that is why you go fractional.” the agents checked it against thorp and it is false in the continuous model: the growth parabola is symmetric, so 0.5x and 1.5x kelly both earn exactly 75% of max growth.

the asymmetry is real but it lives in the consequences, not the slope:

only the over side has a cliff. every $f$ below $f^{\ast}$ still grows; past $2f^{\ast}$ you are in almost-sure-ruin territory. if you over-estimate your edge, only one direction can kill you.
variance rises monotonically with $f$. betting 1.5x kelly gives the same growth as 0.5x but strictly more pain, so over-betting is dominated. there is never a reason to be on the right side of the peak.
in the discrete setting (real bets, not the smooth approximation) the curve genuinely is steeper above $f^{\ast}$, because $\ln(1-f)$ blows up as you approach betting everything. thorp himself writes that over-betting is “much more severely penalized” in his sports betting section.

so the folk wisdom lands in the right place with the wrong proof. good reminder that plausible-sounding maths from an AI (or a screenshot) deserves a fact-check pass.

why you cannot run this in markets

for continuous returns (a stock, roughly geometric brownian motion) the kelly fraction becomes

$$ f^{\ast} = \frac{\mu - r}{\sigma^{2}} $$

excess return over variance, the same form as merton’s portfolio share under log utility. plug in long-run us equity numbers, about 5% excess return and 16% volatility: $0.05 / 0.0256 \approx 2$. raw kelly tells you to run 2x levered long equities. the formula is screaming its own weakness: $f^{\ast}$ is hypersensitive to $\mu$, and $\mu$ is the thing nobody can estimate.

in blackjack, $p$ and $b$ are counted from the deck. in markets:

there is no fixed $p$ and $b$. you estimate them from history, and the process is non-stationary.
backtested win rates are systematically inflated by overfitting and selection bias (see bailey and lópez de prado on the deflated sharpe ratio), so your estimated $p$ is biased exactly in the direction that pushes you over the cliff.
kelly assumes you know the distribution. fat tails plus leverage means one event you did not model can end you regardless of what the formula said.

the cautionary tale everyone reaches for is LTCM (1998). thorp’s framing, from his lectures: their approach was the “anti-kelly”, massive leverage to “pick up nickels in front of a bulldozer”, with little true edge underneath, which makes any sizeable position over-kelly by definition. honest caveat from the fact-check: thorp said this in talks, not in his written papers, and the mainstream post-mortems blame model risk and liquidity spirals rather than kelly arithmetic. “they died of over-betting” is the kelly community’s lens, not the consensus explanation. both can be true.

history, one paragraph

john l. kelly jr. published “a new interpretation of information rate” at bell labs in 1956. the original framing is a gambler with a private wire: under fair odds, the maximum growth rate of his bankroll equals the information rate of the channel, a direct bridge between shannon’s information theory and money (in general the channel’s information rate is the increase in growth it buys you). ed thorp took it to blackjack in beat the dealer, then to markets at princeton newport partners, and wrote the canonical practitioner treatment. poundstone’s fortune’s formula is the popular history of the whole cast.

what i actually take from this

no edge, no bet. if $f^{\ast} \le 0$, the formula says the correct position is zero. an unverified edge is not an edge, so for me, right now, kelly’s prescription for trading is exactly 0.
if i ever do have a verified edge, bet a quarter to a half of kelly. the 75%-growth-for-4x-safety trade is the whole lesson.
the deepest transferable idea: over-committing past your true edge does not just cost you, it can flip a winning game into a losing one. under-committing only costs you some upside. when the inputs are estimates, that one-sided cliff is the argument for leaving margin.
the framework still prices my non-market bets. the highest $p$, highest $b$ wager available to me is career capital, and it has no ruin branch. that is the one to size up.

references (all live-verified)

kelly (1956), a new interpretation of information rate: the original paper, 10 pages, surprisingly readable.
thorp (2006), the kelly criterion in blackjack, sports betting, and the stock market: the practitioner bible; section 7 has every drawdown formula quoted above.
maclean, thorp & ziemba (2010), good and bad properties of the kelly criterion: short and canonical on full vs fractional.
poundstone (2005), fortune’s formula: the popular history (shannon, kelly, thorp, and the mob).

reading checklist (for future me)

re-derive $f^{\ast} = p - q/b$ by hand from $g(f)$
verify the $2c - c^{2}$ growth-share claim by differentiating the continuous growth rate
read thorp (2006) section 7 properly, especially eq. 7.10 to 7.13 (the drawdown formulas)
read kelly (1956) end to end
read the good/bad properties paper
decide: do i ever want to measure a real edge (paper-trade journal), or is conclusion #1 final