Lagrange Multipliers: Pricing a Constraint in Rupees

A Lagrange multiplier, better known outside textbooks as a shadow price, says what one more unit of a limited thing would add to the answer, counted in the objective's own units. At the Amaltas workshop one more board is worth Rs 200/-, one more bench hour Rs 50/-, and one more roll of cloth Rs 0/-. Every one of those figures holds over a stated stretch and then stops.

The Amaltas workshop, an invented joinery, builds exactly two products, the plain crate and the lined crate, and its day runs into three ceilings. Boards arrive twenty to the day, and a plain crate swallows one while a lined crate swallows two. Bench hoursOne person working for one hour at one bench. Three benches across an eight hour day would give 24, and this workshop keeps two back for setting up. are capped at twenty two, and there the ratio reverses: two for a plain crate against one for a lined one. Cloth rollsLining cloth, cut so that one roll covers exactly one crate. Plain crates take no cloth, so this ceiling can only ever press on one of the two counts. arrive eight to the day and go only on lined crates. A plain crate is worth Rs 300/- of contributionWhat making one more of something adds to the day, after paying for whatever that one unit uses up. How the amount is arrived at belongs to another subject and is taken as given here. and a lined crate Rs 450/-, and under all of that the workshop's best day comes to 8 plain and 6 lined, worth Rs 5,100/-.

The search that produced that best plan is covered separately. Here the work picks up where that search put its pen down, and asks something the search never had to answer. Two of the three limits are used right up at that plan: twenty boards of twenty and twenty two hours of twenty two. Two rolls of cloth sit untouched. So what would a twenty first board actually be worth to the Amaltas workshop? Not what a board costs, and not what it feels like it ought to be worth. A shadow price is that question answered as a number, and all three of them come out of nothing more than two equations in two unknowns.

One note before the rupees start arriving. Each of the three prices below is derived twice over: once by algebra, and once by rebuilding the whole problem at a new setting of the limits. The two routes have to land on the same number, and either can be repeated with a pen.

What does a shadow price actually measure?

The recipe has four steps, and there is no calculus in any of them. The first is to take the best answer already in hand. The second is to loosen one limit by exactly one unit. The third is to work the whole problem out again. The fourth is to look at how much the answer improved. The improvement is the price of that limit, counted in the objective's own units rather than in the units of the limit that was loosened.

Run it once on the Amaltas workshop. The best plan under the three limits is 8 plain crates and 6 lined crates at Rs 5,100/- a day. Now let twenty one boards arrive instead of twenty and leave everything else alone. Work it out again and the day comes to Rs 5,300/- a day. The improvement is Rs 200/-. Rs 200/- is the price of a board at the Amaltas workshop: not two hundred boards, not two hundred anything else, but two hundred rupees of contribution a day. The objective is measured in rupees of contribution a day, so the price of a board is measured there too.

The units repay a moment's attention. Most of the confusion lives there. The limit that was loosened is counted in boards, and the answer came back counted in rupees. That is the whole trick. A shadow price is a rate of exchange between the scarce resource and the quantity being made large. The same question put to a household comes out the same way: a house with one bathroom and four people getting ready has a scarce bathroom, and the value of a second one is not measured in bathrooms. The value of a second bathroom is measured in minutes of nobody waiting. Minutes of nobody waiting is what the household was short of.

What are the Amaltas workshop's three prices?

Working each price out by loosening a limit and solving the whole problem again is honest, and it is also slow. There is a faster route, and it needs nothing beyond schoolroom algebra. Ask what has to be true of the two prices for the best plan to make sense at all, and the answer falls out in one step.

At the best plan the Amaltas workshop is making both kinds of crate, so both must be exactly worth making. A plain crate is worth Rs 300/- and it eats one board and two bench hours. A lined crate is worth Rs 450/- and it eats two boards and one bench hour. If a board is worth some amount and a bench hour is worth some amount, then what each crate consumes has to add up to what that crate is worth. Anything else and one of the two crates would be either a bargain worth making more of or a mistake worth making less of, and the plan would not have been the best one.

The two conditions just stated become two equations. One board plus two hours equals Rs 300/-. Two boards plus one hour equals Rs 450/-. Doubling the first gives two boards plus four hours equal to Rs 600/-; taking the second away from it makes the boards vanish, leaving three hours equal to Rs 150/-, so a bench hour is Rs 50/-. Putting the Rs 50/- back into the first makes a board Rs 300/- less two lots of Rs 50/-, or Rs 200/-. Two equations, two unknowns, four lines of arithmetic, and no calculus anywhere: the boards are worth Rs 200/- each and the bench hours Rs 50/- each.

The algebra returns the same Rs 200/- that came out of solving the whole problem again above, and agreement between the two routes is exactly why both are worth doing. The slow method and the quick method have to agree, and when they do not, the plan they started from was not the best one.

Why is one more roll of cloth worth nothing?

The best plan makes six lined crates and each one takes a roll, so six rolls go out of the store and two stay in it. The leftover has a name, slackThe part of a ceiling nobody reached. Six rolls of the eight end up on crates and two never leave the store, so the cloth ceiling is carrying two rolls of it. The idea itself is covered on its own., and what slack means is settled elsewhere. The arithmetic that follows from the slack is what matters here.

Send a ninth roll of cloth to the Amaltas workshop tomorrow morning. Where does it go? The ninth roll goes on the shelf beside the two that are already there. A lined crate also needs two boards and an hour of bench, and there are no boards and no hours going spare, so the ninth roll cannot become a crate. So the day's contribution after the ninth roll arrives is Rs 5,100/-, exactly what it was before, and the improvement is Rs 0/-. A limit that has something left over has a price of zero, and that is arithmetic rather than an opinion about how important the thing feels.

The zero price catches people. Importance and price feel like the same idea, and they are not. Cloth is not unimportant to the Amaltas workshop. Take the cloth away entirely and there are no lined crates and the day collapses. Cloth is essential and its price is nevertheless zero. The price answers only one question: what would one more roll add? Think of a household that has bought a month of rice and has half a sack left at the end of it. Rice is not optional. Another kilo, this month, is worth nothing to that household, and the two statements sit together without any tension at all.

What is the check that has to close?

Three prices are now on the table and nothing so far has tested them as a set. There is a test, it takes one line, and it is the single most useful thing anybody handed a price list by somebody else can do with it. Multiply every limit by its own price, add the results, and the total has to come back to the value of the objective exactly.

Applied to the Amaltas workshop: twenty boards at Rs 200/- each is Rs 4,000/-. Twenty two bench hours at Rs 50/- each is Rs 1,100/-. Eight rolls of cloth at Rs 0/- each is Rs 0/-. The three added together come to Rs 5,100/-, the day's contribution to the last rupee. Not close to it, not within rounding of it, but the same number.

That is not luck and it is not a check somebody invented to be reassuring. The prices were built to make every crate exactly worth what it consumes, and the plan uses up the limits completely wherever the price is not zero, so the value of the limits and the value of the day are two ways of adding up the same thing. Read from one side, the day is worth Rs 5,100/- because of the crates it makes. Read from the other side, the day is worth Rs 5,100/- because of the boards and the hours that go into it. When a price list does not close, what has been found is not a rounding problem but a mistake, and the right response is to go back rather than to shrug.

What does the finished price list look like in one place?

Here is the whole thing in one place. Three prices, the check underneath them, and then the part almost everybody drops: how far each price reaches before it stops being true. The last column of the second table is the one that matters most.

The Rs 5,100/- in the corner of that table is the day's contribution, arrived at from the limits rather than from the crates. Now the ladder: the same problem worked out again once for every board supply from twenty upward.

Three boards at Rs 200/- each, and then nothing at all. One honest note about the two middle rows: at twenty one and twenty two boards the arithmetic runs through plans that are not whole cratesCrates go out of the door one by one, so eight and a third of them is not something the loading bay can ever hand across. Counting in whole units is treated on its own elsewhere.. The rows therefore describe a plan sliding rather than naming one. Both ends of the range are whole plans, 8 plain with 6 lined at twenty boards and 7 plain with 8 lined at twenty three, and the Rs 200/- a board runs cleanly between them.

The bench hours behave the same way and reach much further. Every extra hour is worth Rs 50/- right up to the fortieth hour, at which point the plan has become 20 plain crates and no lined crates at all, worth Rs 6,000/- a day, and the forty first hour is worth nothing. Check that against the price: Rs 5,100/- plus eighteen hours at Rs 50/- is Rs 6,000/-, and re-solving the whole problem at forty hours gives Rs 6,000/-. One table, three prices, two ranges, and a check that closes.

The green marker moves first. The marker climbs the bench hours edge while three boards go by and arrives at 7 plain and 8 lined. Then it will not budge again however far the control is pushed: the cloth has run out, and a further board has nothing left to turn into. Lower down, the strip of squares tells the same story. Exactly three of them fill and the remaining seven stay blank, and not one is ever half filled. The price does not taper off politely; it pays Rs 200/- three times, drops to Rs 0/-, and gives no warning at the join.

How far does Rs 200/- a board hold before it stops?

The answer for the Amaltas workshop is three boards. The twenty first, the twenty second and the twenty third each add Rs 200/-, taking the day from Rs 5,100/- to Rs 5,300/- to Rs 5,500/- to Rs 5,700/-. At twenty three boards the best plan is 7 plain crates and 8 lined crates. Eight lined crates need eight rolls of cloth, and eight rolls is every roll the workshop has. The twenty fourth board is worth Rs 0/-.

Look at what actually happened there. The pattern is the general rule rather than a quirk of this workshop. The board price was Rs 200/- for as long as the boards and the bench hours were the two things holding the day back. Each new board let the plan trade a plain crate for a lined one, and that trade was worth Rs 200/- every time. Then the cloth ran out, a third limit joined the two that were already binding, and the trade the price described stopped being available. A shadow price is a rate over a range, not a standing offer, and the range ends exactly where the next limit starts to bind.

The range is the sentence to carry away. Everything else here follows from it. A price quoted with no range attached is not a small omission. The number on its own reads like a permanent property of a board, and it is nothing of the kind. The Rs 200/- is a property of the answer at one particular setting of the limits. The moment those limits move far enough, the answer reorganises itself around a different set of them and the old rate is simply gone.

Do two prices still work when both limits move at once?

Almost everything so far has moved one limit at a time. Real requests never arrive that neatly. Somebody asks the Amaltas workshop to take three more boards a day and put on another eighteen bench hours, both at once, and wants to know what the day would be worth. The two prices offer an obvious answer: Rs 5,100/- as it stands, plus three boards at Rs 200/-, plus eighteen hours at Rs 50/-. The sum is Rs 5,100/- plus Rs 600/- plus Rs 900/-, or Rs 6,600/- a day.

Now throw the prices away and solve the whole thing again from nothing. Twenty three boards, forty bench hours, eight rolls of cloth, and the same two contribution figures. The best plan that comes back is 19 plain crates and 2 lined crates. Count what it uses: nineteen boards plus two lots of two is twenty three boards exactly, and thirty eight hours plus two is forty hours exactly, with two rolls of cloth against eight available. The contribution is nineteen at Rs 300/- plus two at Rs 450/-, and that comes to Rs 5,700/- plus Rs 900/-, or Rs 6,600/- a day. Predicted Rs 6,600/- and re-solved Rs 6,600/-, and the two agreed to the rupee.

Do not read that as a promise that adding prices up always works. The reason it worked here is specific and worth naming. The prices hold while the same limits keep binding, and across that particular move the boards and the bench hours went on being the two things holding the day back the whole way. Notice how differently the plan itself behaved: it went from 8 plain and 6 lined to 19 plain and 2 lined, nearly a reversal, and the arithmetic still landed. A move large enough to change which limits are binding ends the arithmetic, and the prices give no signal when it happens.

Is a price the same as what the thing costs to buy?

A board is worth Rs 200/- to the Amaltas workshop's answer. The supplierWhoever the boards are bought from. Nothing about the supplier enters the workshop's own arithmetic except the rate it quotes. sells a board for Rs 120/-. The two numbers sit side by side, and it is very tempting to subtract one from the other and call the difference profit. Resist that for one paragraph. They are answering two different questions.

Rs 200/- is what the answer would gain. The Rs 200/- comes out of the workshop's own limits and its own contribution figures. Nothing about the supplier appears anywhere in the calculation that produced it, so the figure would be the same if boards were handed over free or cost a thousand rupees each. Rs 120/- is what the workshop would hand over. The Rs 120/- comes out of a conversation with somebody else entirely and has nothing to do with bench hours or cloth. One number describes the workshop's own arithmetic and the other describes a transaction, and a price list that has silently merged them is asserting something that is not in it.

Three things the Rs 200/- is not, and each of them has a range or a cost hiding behind it. The Rs 200/- is not what a board costs. A board costs Rs 120/-, and that is a separate matter. Nor is the Rs 200/- what a board will be worth after the plan has changed. As soon as the plan reorganises, the price is recomputed from the new plan. And it is not a licence to order boards without limit, as the manager below discovers.

The Amaltas workshop, the plain crate and the lined crate are invented.
Educational material. Not advice on any investment, tax, budget or market position.