What an hour costs: prices and the ledger

Lesson 16 put every source in one unit, the number of your own hours that one hour of it replaces, and a rate does not choose a source: the twin arm and a 3 % simulator with hand labels each replace about one own hour. This lesson prices the hour. A motion hour is not a wage: it is a wall-clock hour divided by the share of it in which the robot moves and the demonstration is kept, and for a borrowed hour it is curation, filming and tracking, or compute and authoring spread over their use. A price divided by its rate is a cost per useful hour, and the ranking by that number is not the ranking by price. It is still the cost of the first hour, which lesson 18 takes apart.

1 · An own hour costs three times the wage

Lesson 16 left two sources with the same rate: an hour of the twin arm's demonstrations replaces 1.01 own hours, an hour of a 3 % simulator with hand-position labels 1.02. Which do you buy? The rates cannot say, because they count own hours and an own hour has no price. So price the unit first. An own-robot hour is a motion hour, 387.1 demonstrations of 9.3 s, and nobody is paid by the motion hour: a teleoperation session is paid by the wall-clock hour and the robot moves for part of it. Hire an operator and a robot cell for 1,000 hours:

p_own = (w + H / L) / (d (1 − x)) = ($40 + $4.5) / (0.4 × 0.9) = $123.6 per kept motion hour

That is 3.1 times the wage. Lesson 16's own curve counts kept demonstrations, so this is a price per kept hour. The hour is mostly labour (the cell is 10 % of a wall-clock hour): a cell twice as dear moves the price by a tenth, while the price is inversely proportional to the duty cycle. Every number is an assumption, because no source prints a dollar cost per collected hour. Published throughputs bound the shares:

Why a price per motion hour and not per something else? The price has to be in the unit of the rate, or dividing by the rate is not honest, and each alternative fails by a computation:

As in lesson 16, the Bench task is small: its whole own curve, 256 layouts, costs $81.7 at this price. Prices, ratios and the places where a ranking changes carry to larger tasks, totals do not, and every real value depends on the task, the policy and the year. Mobile ALOHA's seven tasks hold 2.9 hours of demonstration motion, DROID 350 hours, AgiBot World (2025) 2,976.

2 · The borrowed lines, and every discard paid once

Every other line is priced by the rule of the own hour: everything the line spends (labour, depreciation, review, tracking and GPU compute, authoring), divided by the hours that come out.

The simulator is 85 % authoring and 13 % compute, so what matters is how many hours a scene serves, not the GPU. An insertion hour costs $197.8 without the rig and $214.4 with it, 1.73 times an own hour: the sensor is $16.7 of the $90.8 extra, and the rest is that insertion records 25 % of the session, not 40.

Count a discard once. Where the Bench can measure the discards, it has, and they are inside the rate: lesson 16's rate is per attempted hour, and footage keeps only 41 of 64 attempted layouts, the older arm 50. Price footage per kept hour instead, $33.75 / 0.64 = $52.73, and divide by that rate, and the failed demonstrations are paid for twice: $322.5 per useful hour against $206.4, 56 % too high. For your own robot the Bench cannot measure them, since its scripted expert never fails (256 of 256 kept), so the operator's slips are an assumption and sit in the price. The own hour is a kept hour; every other source is an attempted hour.

3 · Divide by the rate

Lesson 16's rate ρ is the number of own hours that one attempted hour of a source replaces, read at an operating point: here 8 own layouts and 64 attempted layouts of the source, on one draw of layouts, the table's. A source that charges p for an attempted hour and replaces ρ own hours with it sells useful hours, own-equivalent hours, at

c = p / ρ dollars per useful hour, none when ρ ≤ 0

and the alternative is to record them yourself at p_own. Buying beats recording when p/ρ < p_own, that is, when ρ is above the break-even rate p/p_own. A simulator's gap is how far its arm model's link lengths are off; the labels are what the source records (joint angles, or hand positions as footage does).

Footage needs a rate of 0.27 and has 0.16: an attempted hour of it is worth $20.2 to you (ρ × p_own) and costs $33.75, so every hour bought loses $13.5, while an hour of the older arm is worth $46.8 and costs $15. The rate carries the sampling interval of the success it was read from (1,000 test layouts; the own curve's noise is not in it), and so does the cost: footage costs $175 to $250 per useful hour, above $123.6 throughout. A table is one draw of layouts, so repeat it: on twelve other draws footage's rate runs 0.09 to 0.21 and a useful hour of it costs $161 to $373, the older arm's $37 to $69, against $123.6 for an own hour on all of them. Who costs more than whom does not depend on the draw.

4 · Two rankings that disagree

Rank the five sources of the layouts task by what the invoice says and by what a useful hour costs (simulator at a gap of 1 %):

Footage and your own hour swap. Footage is the dearest of the borrowed hours to make and still 3.7 times cheaper than an own hour; its useful hour is the dearest of the five, 1.7 times an own hour's, because its rate, 0.16, is below the 0.27 that its price needs. A tie breaks. The twin and the older arm both charge $15, and a useful hour of the older arm costs 2.7 times as much.

The cheapest hour can sell nothing. The simulator is the cheapest hour by price at every gap, and by cost per useful hour at $0.20, $0.23 and $0.44 for gaps of 0.1, 0.3 and 1 %. At 3 % with joint-angle labels its rate is −0.12: an hour of it takes away 0.12 of an own hour, and no price makes p/ρ a price when ρ ≤ 0. The rate crosses zero between gaps of 1.5 % and 2 % (0.17 and −0.11, measured like the table's cells, the same signs on twelve other draws), a window half a point wide. The simulator's price is so low that the twin's useful hour is the cheaper only once the simulator's rate falls below 0.013, so the simulator does not slide down the ranking, it drops out of it. The same 3 % gap with hand-position labels costs $0.18 per useful hour (lessons 8 and 11: the label space decides).

Two more rankings fail by a computed counterexample:

The widget

PL.price = function (key, a) {
  var arm = a.arm_cost / a.arm_life_h, own = (a.wage_per_h + arm) / (a.duty * (1 - a.discard));
  switch (key) {
    case 'own': return own;
    case 'video': return (a.video_wage_per_h + a.track_per_h) / a.video_duty;
    case 'sim': return a.gpu_per_h / a.sim_speed + a.scene_weeks * a.week_cost / a.scene_uses_h + a.calib_h * own / a.scene_uses_h;
...
PL.cost = function (p, rho) { return rho > 0 ? p / rho : Infinity; };
...
var moved = function (key, d, g) { t = Object.assign({}, a); t[key] = d > 0 ? a[key] * g : a[key] / g; return t[key] <= (PL.LIMIT[key] || Infinity) && PL.rank(t, rho) !== ref; };
for (it = 1; it <= 150 && !hi; it++) { f = Math.pow(10, 3 * it / 150); if (moved(k, dir, f)) hi = f; else lo = f; }

What to try. Leave the defaults: duty 0.40, wage $40, 10 % discarded, GPU $2.5, 4 weeks of authoring, a simulator off by 1 %, 8 own and 64 attempted layouts. The own hour is $123.6; footage is $33.75 an hour and $206.4 per useful hour (interval $175 to $250), the dearest of the five; the older arm is $39.6, the twin $14.8, the simulator $0.44; counting footage's discard twice shows $322.5. Pick the 3 % simulator with joint labels: its bar is hatched and the cost reads none; with hand labels it reads $0.18. Slide the duty cycle down. The readout says the bars cross at 0.24; at 0.17, the bottom of ALOHA's range, the own hour costs $290.8 and is the dearest; at 0.67 it costs $73.8; at 0.06, one step from the left end and close to the 6.2 % of lesson 7's layouts task (§5), $824.1. Return to 0.4 and raise the GPU price tenfold, to $25, and authoring to 16 weeks: the simulator's useful hour goes from $0.44 to $2.11, still 7.0 times below the twin, and no bar changes place. Set the operating point to 8 own + 16 attempted and then 8 + 256: the older arm's useful hour goes from $19.7 to $88.7. At 32 own + 64 attempted footage's rate is 0.01 and a useful hour of it costs $3,342. The last readout names the assumption nearest to reversing a pair, and the factor.

5 · What the ranking depends on

Footage is the dearest useful hour only while your own hour is cheap enough. Your own hour costs what footage's useful hour costs at a duty cycle

d* = (w + H / L) / ((1 − x) · c_footage) = $44.5 / (0.9 × $206.4) = 0.24

Below it your own hour is the dearer and the two rankings agree on the pair. ALOHA's recorded fraction runs from 17 to 67 %, a factor 4, and the threshold, 24 %, is inside it. Which assumption moves the ranking most? For each one, scan its value up and down until a pair of the cost ranking reverses, the rates held where the table has them:

The duty cycle needs the smallest move, 1.67, and it is the only one of the three nearest assumptions with a published range, which spans the threshold; the two wages have none. The hardware is far from any change, and so is the simulator: each of its assumptions can move by a factor of 38 or more before it changes place, so its danger is its rate and not its price.

A session that rebuilds the scene for every demonstration records less than ALOHA's. Lesson 7 assumed 20 s to put the cup back and 120 s to rebuild the row: 9.3 s of motion in 149.3 s, a duty cycle of 6.2 %. At that duty cycle the own hour costs $793.8, footage is the cheaper useful hour, and the older arm's is 20 times cheaper.

6 · The columns the cheap hours do not carry

Force. In lesson 10's peg with a millimetre of clearance, demonstrations without force channels reach between 0.515 and 0.665 whatever their number (0.630 at 40); with force channels, 0.825 from one and 0.985 from ten. All the force-less hours together are worth less than one force-bearing demonstration, so however many are bought they supply almost no useful hour, and the cost per useful hour, p/ρ, grows without limit: none. Forty force-less insertions, at $197.8 an hour, cost $14.5 and reach 0.630; ten force-bearing ones cost $3.94 and reach 0.985. The column is cheaper from the dearer hour. A borrowed hour does not help: the sources of this lesson record images and positions, and DROID's listed features have no force or torque stream.

Recovery. A calm demonstration contains no displaced states (lesson 2), so calm hours do not supply recovery. A correction hour is a supervisor and the cell with the robot moving 50 % of the time: $89. Own calm demonstrations stay between 0.55 and 0.635 from 10 to 80 of them, and 80 cost $25.5; 20 calm demonstrations and three rounds of corrections (lesson 3), 15 supervised runs, reach 0.96 for $9.47 (the calm ones at $123.6 an hour, 0.035 hours of corrections at $89). AgiBot World (2025) reports that failure-recovery trajectories are about one percent of its data. The price of such a column is the price of an hour at the one place that has it, and before you can buy it you need an instrument, a force rig or a policy to run, which no price per hour includes; lesson 21 prices that.

7 · The cost of the first hour

The rate in the ledger is an average over the hours bought, (N_eq − n) / m (N_eq: the own layouts that give the same success, n the own layouts you hold, lesson 16), and it changes with the number m of attempted layouts. Set the operating point to 16, 64 and 256 attempted layouts: the older arm's rate is 0.76, 0.38 and 0.17, and its useful hour costs $19.7, $39.6 and $88.7. The twin's rate stays near one (1.28, 1.01, 0.93), but an own-equivalent hour is not a constant amount of success, because the own curve flattens. Twin arm, 8 own layouts, in blocks of attempted layouts:

The last 192 layouts cost 21 times as much per point as the first 16 (14 to 26 on twelve other draws), from a source that charges $15 an hour throughout and whose rate never fell below 0.93. A cost per useful hour is the cost of the first hours of a source at one operating point.

Common mistakes / failure modes

Checkpoint exercise

Where this points next

A price per motion hour, divided by the exchange rate, gives a cost per useful hour: $14.8 for the twin, $39.6 for the older arm, $0.20 to $0.44 for the simulator while its gap is 1 % or less, $123.6 for your own hour and $206.4 for footage, which costs less to make than an own hour and more to use. The 3 % simulator with joint labels sells nothing at any price, and calm and force-less hours cannot buy recovery or force. But each of these is the cost of the first hours at one operating point: the older arm's useful hour costs $19.7 after 16 attempted layouts and $88.7 after 256, and the twin's last 192 layouts cost 21 times as much per point of success as its first 16. The thousandth hour of a source teaches the policy less than the first. How fast does the value of one more hour fall?

Interview prompts

• Footage is cheaper per hour than your own demonstrations. Why can it be dearer per useful hour? (§3, §4 — its rate, 0.16, is below the break-even rate of 0.27, its price over your own.)
• A simulator costs $0.19 an hour. Why is it not always the right buy? (§4 — a cost per useful hour is price over rate, and the rate can be negative: the 3 % joint-label simulator removes 0.12 of an own hour per hour.)
• What does it mean to pay a discard once? (§2 — the rate is measured per attempted hour, so the failed attempts are already in it and the price must be per attempted hour; counting them in both overstates footage's cost by 56 %.)
• Which assumption would you check before trusting a ranking, and how would you find it? (§5 — move each alone until a pair reverses; the duty cycle needs the smallest change, ÷1.67, and its published range spans it.)
• What can no number of cheap hours buy? (§6 — a column the hours lack: calm demonstrations cannot supply recovery, force-less ones cannot supply force.)
• Why is a cost per useful hour not the price of the next hour? (§7 — it is an average over the hours bought: at 256 layouts the older arm's useful hour costs four and a half times what it costs at 16.)

Companion reads: Lesson 11 · The simulation gap (a wrong arm model turns a cheap hour into none), Lesson 10 · What cameras cannot see (the force column), Valuation · 09 Sensitivity and margin of safety (one assumption at a time; in Chinese) and Financials · 22 Cyclicals and heavy assets (utilisation and depreciation; in Chinese).